text:
@testset "4.1.2.1 (a+b sin)^m (c+d sin)^n" begin
    (A, B, a, b, c, d, e, f, m, n, p, x, ) = @variables A B a b c d e f m n p x

    #= ::Package:: =#

    #= ::Title::Closed:: =#
    #=Integrands*of*the*form*(d*sin(e+f*x))^n*(a+b*sin(e+f*x))^m=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(d*sin(e+f*x))^n*(a+a*sin(e+f*x))^m=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*sin(e+f*x)^n*(a+a*sin(e+f*x))^m=#

    #= ::Subsubsection::Closed:: =#
    #=n>0=#

    @test_int [sin(e + f*x)^3*(a + a*sin(e + f*x))^2, x, 9, (3*a^2*x)/4 - (2*a^2*cos(e + f*x))/f + (a^2*cos(e + f*x)^3)/f - (a^2*cos(e + f*x)^5)/(5*f) - (3*a^2*cos(e + f*x)*sin(e + f*x))/(4*f) - (a^2*cos(e + f*x)*sin(e + f*x)^3)/(2*f)]

    @test_int [sin(e + f*x)^3*(a + a*sin(e + f*x))^3, x, 13, (23*a^3*x)/16 - (4*a^3*cos(e + f*x))/f + (7*a^3*cos(e + f*x)^3)/(3*f) - (3*a^3*cos(e + f*x)^5)/(5*f) - (23*a^3*cos(e + f*x)*sin(e + f*x))/(16*f) - (23*a^3*cos(e + f*x)*sin(e + f*x)^3)/(24*f) - (a^3*cos(e + f*x)*sin(e + f*x)^5)/(6*f)]

    #= ::Subsubsection::Closed:: =#
    #=n<0=#

    @test_int [sin(x)^4/(a + a*sin(x)), x, 6, -((3*x)/(2*a)) - (4*cos(x))/a + (4*cos(x)^3)/(3*a) + (3*cos(x)*sin(x))/(2*a) + (cos(x)*sin(x)^3)/(a + a*sin(x))]
    @test_int [sin(x)^3/(a + a*sin(x)), x, 2, (3*x)/(2*a) + (2*cos(x))/a - (3*cos(x)*sin(x))/(2*a) + (cos(x)*sin(x)^2)/(a + a*sin(x))]
    @test_int [sin(x)^2/(a + a*sin(x)), x, 4, -(x/a) - cos(x)/a - cos(x)/(a*(1 + sin(x)))]
    @test_int [sin(x)^1/(a + a*sin(x)), x, 2, x/a + cos(x)/(a + a*sin(x))]
    @test_int [sin(x)^0/(a + a*sin(x)), x, 1, -(cos(x)/(a + a*sin(x)))]
    @test_int [csc(x)^1/(a + a*sin(x)), x, 3, -(atanh(cos(x))/a) + cos(x)/(a + a*sin(x))]
    @test_int [csc(x)^2/(a + a*sin(x)), x, 5, atanh(cos(x))/a - (2*cot(x))/a + cot(x)/(a + a*sin(x))]
    @test_int [csc(x)^3/(a + a*sin(x)), x, 6, -((3*atanh(cos(x)))/(2*a)) + (2*cot(x))/a - (3*cot(x)*csc(x))/(2*a) + (cot(x)*csc(x))/(a + a*sin(x))]
    @test_int [csc(x)^4/(a + a*sin(x)), x, 6, (3*atanh(cos(x)))/(2*a) - (4*cot(x))/a - (4*cot(x)^3)/(3*a) + (3*cot(x)*csc(x))/(2*a) + (cot(x)*csc(x)^2)/(a + a*sin(x))]

    @test_int [sin(x)^4/(a + a*sin(x))^2, x, 3, (7*x)/(2*a^2) + (16*cos(x))/(3*a^2) - (7*cos(x)*sin(x))/(2*a^2) + (8*cos(x)*sin(x)^2)/(3*a^2*(1 + sin(x))) + (cos(x)*sin(x)^3)/(3*(a + a*sin(x))^2)]
    @test_int [sin(x)^3/(a + a*sin(x))^2, x, 6, -((2*x)/a^2) - (4*cos(x))/(3*a^2) - (2*cos(x))/(a^2*(1 + sin(x))) + (cos(x)*sin(x)^2)/(3*(a + a*sin(x))^2)]
    @test_int [sin(x)^2/(a + a*sin(x))^2, x, 3, x/a^2 + (5*cos(x))/(3*a^2*(1 + sin(x))) - cos(x)/(3*(a + a*sin(x))^2)]
    @test_int [sin(x)^1/(a + a*sin(x))^2, x, 2, cos(x)/(3*(a + a*sin(x))^2) - (2*cos(x))/(3*(a^2 + a^2*sin(x)))]
    @test_int [sin(x)^0/(a + a*sin(x))^2, x, 2, -(cos(x)/(3*(a + a*sin(x))^2)) - cos(x)/(3*(a^2 + a^2*sin(x)))]
    @test_int [csc(x)^1/(a + a*sin(x))^2, x, 4, -(atanh(cos(x))/a^2) + (4*cos(x))/(3*a^2*(1 + sin(x))) + cos(x)/(3*(a + a*sin(x))^2)]
    @test_int [csc(x)^2/(a + a*sin(x))^2, x, 6, (2*atanh(cos(x)))/a^2 - (10*cot(x))/(3*a^2) + (2*cot(x))/(a^2*(1 + sin(x))) + cot(x)/(3*(a + a*sin(x))^2)]
    @test_int [csc(x)^3/(a + a*sin(x))^2, x, 7, -((7*atanh(cos(x)))/(2*a^2)) + (16*cot(x))/(3*a^2) - (7*cot(x)*csc(x))/(2*a^2) + (8*cot(x)*csc(x))/(3*a^2*(1 + sin(x))) + (cot(x)*csc(x))/(3*(a + a*sin(x))^2)]
    @test_int [csc(x)^4/(a + a*sin(x))^2, x, 7, (5*atanh(cos(x)))/a^2 - (4*cot(x))/a^2 - cot(x)^3/(3*a^2) + (cot(x)*csc(x))/a^2 - cos(x)/(3*a^2*(1 + sin(x))^2) - (13*cos(x))/(3*a^2*(1 + sin(x))), (5*atanh(cos(x)))/a^2 - (12*cot(x))/a^2 - (4*cot(x)^3)/a^2 + (5*cot(x)*csc(x))/a^2 + (10*cot(x)*csc(x)^2)/(3*a^2*(1 + sin(x))) + (cot(x)*csc(x)^2)/(3*(a + a*sin(x))^2)]

    @test_int [sin(x)^6/(a + a*sin(x))^3, x, 8, -((23*x)/(2*a^3)) - (136*cos(x))/(5*a^3) + (136*cos(x)^3)/(15*a^3) + (23*cos(x)*sin(x))/(2*a^3) + (cos(x)*sin(x)^5)/(5*(a + a*sin(x))^3) + (13*cos(x)*sin(x)^4)/(15*a*(a + a*sin(x))^2) + (23*cos(x)*sin(x)^3)/(3*(a^3 + a^3*sin(x)))]
    @test_int [sin(x)^5/(a + a*sin(x))^3, x, 4, (13*x)/(2*a^3) + (152*cos(x))/(15*a^3) - (13*cos(x)*sin(x))/(2*a^3) + (cos(x)*sin(x)^4)/(5*(a + a*sin(x))^3) + (11*cos(x)*sin(x)^3)/(15*a*(a + a*sin(x))^2) + (76*cos(x)*sin(x)^2)/(15*(a^3 + a^3*sin(x)))]
    @test_int [sin(x)^4/(a + a*sin(x))^3, x, 7, -((3*x)/a^3) - (9*cos(x))/(5*a^3) + (cos(x)*sin(x)^3)/(5*(a + a*sin(x))^3) + (3*cos(x)*sin(x)^2)/(5*a*(a + a*sin(x))^2) - (3*cos(x))/(a^3 + a^3*sin(x))]
    @test_int [sin(x)^3/(a + a*sin(x))^3, x, 5, x/a^3 + (cos(x)*sin(x)^2)/(5*(a + a*sin(x))^3) - (7*cos(x))/(15*a*(a + a*sin(x))^2) + (29*cos(x))/(15*(a^3 + a^3*sin(x)))]
    @test_int [sin(x)^2/(a + a*sin(x))^3, x, 3, -(cos(x)/(5*(a + a*sin(x))^3)) + (8*cos(x))/(15*a*(a + a*sin(x))^2) - (7*cos(x))/(15*(a^3 + a^3*sin(x)))]
    @test_int [sin(x)^1/(a + a*sin(x))^3, x, 3, cos(x)/(5*(a + a*sin(x))^3) - cos(x)/(5*a*(a + a*sin(x))^2) - cos(x)/(5*(a^3 + a^3*sin(x)))]
    @test_int [sin(x)^0/(a + a*sin(x))^3, x, 3, -(cos(x)/(5*(a + a*sin(x))^3)) - (2*cos(x))/(15*a*(a + a*sin(x))^2) - (2*cos(x))/(15*(a^3 + a^3*sin(x)))]
    @test_int [csc(x)^1/(a + a*sin(x))^3, x, 5, -(atanh(cos(x))/a^3) + cos(x)/(5*(a + a*sin(x))^3) + (7*cos(x))/(15*a*(a + a*sin(x))^2) + (22*cos(x))/(15*(a^3 + a^3*sin(x)))]
    @test_int [csc(x)^2/(a + a*sin(x))^3, x, 7, (3*atanh(cos(x)))/a^3 - (24*cot(x))/(5*a^3) + cot(x)/(5*(a + a*sin(x))^3) + (3*cot(x))/(5*a*(a + a*sin(x))^2) + (3*cot(x))/(a^3 + a^3*sin(x))]
    @test_int [csc(x)^3/(a + a*sin(x))^3, x, 8, -((13*atanh(cos(x)))/(2*a^3)) + (152*cot(x))/(15*a^3) - (13*cot(x)*csc(x))/(2*a^3) + (cot(x)*csc(x))/(5*(a + a*sin(x))^3) + (11*cot(x)*csc(x))/(15*a*(a + a*sin(x))^2) + (76*cot(x)*csc(x))/(15*(a^3 + a^3*sin(x)))]
    @test_int [csc(x)^4/(a + a*sin(x))^3, x, 8, (23*atanh(cos(x)))/(2*a^3) - (136*cot(x))/(5*a^3) - (136*cot(x)^3)/(15*a^3) + (23*cot(x)*csc(x))/(2*a^3) + (cot(x)*csc(x)^2)/(5*(a + a*sin(x))^3) + (13*cot(x)*csc(x)^2)/(15*a*(a + a*sin(x))^2) + (23*cot(x)*csc(x)^2)/(3*(a^3 + a^3*sin(x)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*sin(e+f*x)^n*(a+a*sin(e+f*x))^(m/2)=#

    #= ::Subsubsection::Closed:: =#
    #=n>0=#

    @test_int [sin(c + d*x)^4*sqrt(a + a*sin(c + d*x)), x, 5, -((32*a*cos(c + d*x))/(45*d*sqrt(a + a*sin(c + d*x)))) - (16*a*cos(c + d*x)*sin(c + d*x)^3)/(63*d*sqrt(a + a*sin(c + d*x))) - (2*a*cos(c + d*x)*sin(c + d*x)^4)/(9*d*sqrt(a + a*sin(c + d*x))) + (64*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(315*d) - (32*cos(c + d*x)*(a + a*sin(c + d*x))^(3/2))/(105*a*d)]
    @test_int [sin(c + d*x)^3*sqrt(a + a*sin(c + d*x)), x, 4, -((4*a*cos(c + d*x))/(5*d*sqrt(a + a*sin(c + d*x)))) - (2*a*cos(c + d*x)*sin(c + d*x)^3)/(7*d*sqrt(a + a*sin(c + d*x))) + (8*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(35*d) - (12*cos(c + d*x)*(a + a*sin(c + d*x))^(3/2))/(35*a*d)]
    @test_int [sin(c + d*x)^2*sqrt(a + a*sin(c + d*x)), x, 3, -((14*a*cos(c + d*x))/(15*d*sqrt(a + a*sin(c + d*x)))) + (4*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(15*d) - (2*cos(c + d*x)*(a + a*sin(c + d*x))^(3/2))/(5*a*d)]
    @test_int [sin(c + d*x)^1*sqrt(a + a*sin(c + d*x)), x, 2, -((2*a*cos(c + d*x))/(3*d*sqrt(a + a*sin(c + d*x)))) - (2*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(3*d)]
    @test_int [sin(c + d*x)^0*sqrt(a + a*sin(c + d*x)), x, 1, -((2*a*cos(c + d*x))/(d*sqrt(a + a*sin(c + d*x))))]
    @test_int [csc(c + d*x)^1*sqrt(a + a*sin(c + d*x)), x, 2, -((2*sqrt(a)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/d)]
    @test_int [csc(c + d*x)^2*sqrt(a + a*sin(c + d*x)), x, 3, -((sqrt(a)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/d) - (a*cot(c + d*x))/(d*sqrt(a + a*sin(c + d*x)))]
    @test_int [csc(c + d*x)^3*sqrt(a + a*sin(c + d*x)), x, 4, -((3*sqrt(a)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(4*d)) - (3*a*cot(c + d*x))/(4*d*sqrt(a + a*sin(c + d*x))) - (a*cot(c + d*x)*csc(c + d*x))/(2*d*sqrt(a + a*sin(c + d*x)))]
    @test_int [csc(c + d*x)^4*sqrt(a + a*sin(c + d*x)), x, 5, -((5*sqrt(a)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(8*d)) - (5*a*cot(c + d*x))/(8*d*sqrt(a + a*sin(c + d*x))) - (5*a*cot(c + d*x)*csc(c + d*x))/(12*d*sqrt(a + a*sin(c + d*x))) - (a*cot(c + d*x)*csc(c + d*x)^2)/(3*d*sqrt(a + a*sin(c + d*x)))]

    @test_int [csc(c + d*x)^1*sqrt(a - a*sin(c + d*x)), x, 2, -2*(sqrt(a)/d)*atanh(sqrt(a)*(cos(c + d*x)/sqrt(a - a*sin(c + d*x))))]

    @test_int [csc(c + d*x)^1*sqrt(-a + a*sin(c + d*x)), x, 2, 2*(sqrt(a)/d)*atan(sqrt(a)*(cos(c + d*x)/sqrt(-a + a*sin(c + d*x))))]

    @test_int [csc(c + d*x)^1*sqrt(-a - a*sin(c + d*x)), x, 2, 2*(sqrt(a)/d)*atan(sqrt(a)*(cos(c + d*x)/sqrt(-a - a*sin(c + d*x))))]

    @test_int [sin(c + d*x)^3*(a + a*sin(c + d*x))^(3/2), x, 6, -((68*a^2*cos(c + d*x))/(45*d*sqrt(a + a*sin(c + d*x)))) - (34*a^2*cos(c + d*x)*sin(c + d*x)^3)/(63*d*sqrt(a + a*sin(c + d*x))) - (2*a^2*cos(c + d*x)*sin(c + d*x)^4)/(9*d*sqrt(a + a*sin(c + d*x))) + (136*a*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(315*d) - (68*cos(c + d*x)*(a + a*sin(c + d*x))^(3/2))/(105*d)]
    @test_int [sin(c + d*x)^2*(a + a*sin(c + d*x))^(3/2), x, 4, -((152*a^2*cos(c + d*x))/(105*d*sqrt(a + a*sin(c + d*x)))) - (38*a*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(105*d) + (4*cos(c + d*x)*(a + a*sin(c + d*x))^(3/2))/(35*d) - (2*cos(c + d*x)*(a + a*sin(c + d*x))^(5/2))/(7*a*d)]
    @test_int [sin(c + d*x)^1*(a + a*sin(c + d*x))^(3/2), x, 3, -((8*a^2*cos(c + d*x))/(5*d*sqrt(a + a*sin(c + d*x)))) - (2*a*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(5*d) - (2*cos(c + d*x)*(a + a*sin(c + d*x))^(3/2))/(5*d)]
    @test_int [sin(c + d*x)^0*(a + a*sin(c + d*x))^(3/2), x, 2, -((8*a^2*cos(c + d*x))/(3*d*sqrt(a + a*sin(c + d*x)))) - (2*a*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(3*d)]
    @test_int [csc(c + d*x)^1*(a + a*sin(c + d*x))^(3/2), x, 4, -((2*a^(3/2)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/d) - (2*a^2*cos(c + d*x))/(d*sqrt(a + a*sin(c + d*x)))]
    @test_int [csc(c + d*x)^2*(a + a*sin(c + d*x))^(3/2), x, 4, -((3*a^(3/2)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/d) - (a^2*cot(c + d*x))/(d*sqrt(a + a*sin(c + d*x)))]
    @test_int [csc(c + d*x)^3*(a + a*sin(c + d*x))^(3/2), x, 5, -((7*a^(3/2)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(4*d)) - (7*a^2*cot(c + d*x))/(4*d*sqrt(a + a*sin(c + d*x))) - (a^2*cot(c + d*x)*csc(c + d*x))/(2*d*sqrt(a + a*sin(c + d*x)))]
    @test_int [csc(c + d*x)^4*(a + a*sin(c + d*x))^(3/2), x, 6, -((11*a^(3/2)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(8*d)) - (11*a^2*cot(c + d*x))/(8*d*sqrt(a + a*sin(c + d*x))) - (11*a^2*cot(c + d*x)*csc(c + d*x))/(12*d*sqrt(a + a*sin(c + d*x))) - (a^2*cot(c + d*x)*csc(c + d*x)^2)/(3*d*sqrt(a + a*sin(c + d*x)))]

    @test_int [sin(c + d*x)^3*(a + a*sin(c + d*x))^(5/2), x, 6, -((284*a^3*cos(c + d*x))/(99*d*sqrt(a + a*sin(c + d*x)))) - (710*a^3*cos(c + d*x)*sin(c + d*x)^3)/(693*d*sqrt(a + a*sin(c + d*x))) - (46*a^3*cos(c + d*x)*sin(c + d*x)^4)/(99*d*sqrt(a + a*sin(c + d*x))) + (568*a^2*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(693*d) - (2*a^2*cos(c + d*x)*sin(c + d*x)^4*sqrt(a + a*sin(c + d*x)))/(11*d) - (284*a*cos(c + d*x)*(a + a*sin(c + d*x))^(3/2))/(231*d)]
    @test_int [sin(c + d*x)^2*(a + a*sin(c + d*x))^(5/2), x, 5, -((832*a^3*cos(c + d*x))/(315*d*sqrt(a + a*sin(c + d*x)))) - (208*a^2*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(315*d) - (26*a*cos(c + d*x)*(a + a*sin(c + d*x))^(3/2))/(105*d) + (4*cos(c + d*x)*(a + a*sin(c + d*x))^(5/2))/(63*d) - (2*cos(c + d*x)*(a + a*sin(c + d*x))^(7/2))/(9*a*d)]
    @test_int [sin(c + d*x)^1*(a + a*sin(c + d*x))^(5/2), x, 4, -((64*a^3*cos(c + d*x))/(21*d*sqrt(a + a*sin(c + d*x)))) - (16*a^2*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(21*d) - (2*a*cos(c + d*x)*(a + a*sin(c + d*x))^(3/2))/(7*d) - (2*cos(c + d*x)*(a + a*sin(c + d*x))^(5/2))/(7*d)]
    @test_int [sin(c + d*x)^0*(a + a*sin(c + d*x))^(5/2), x, 3, -((64*a^3*cos(c + d*x))/(15*d*sqrt(a + a*sin(c + d*x)))) - (16*a^2*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(15*d) - (2*a*cos(c + d*x)*(a + a*sin(c + d*x))^(3/2))/(5*d)]
    @test_int [csc(c + d*x)^1*(a + a*sin(c + d*x))^(5/2), x, 4, -((2*a^(5/2)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/d) - (14*a^3*cos(c + d*x))/(3*d*sqrt(a + a*sin(c + d*x))) - (2*a^2*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(3*d)]
    @test_int [csc(c + d*x)^2*(a + a*sin(c + d*x))^(5/2), x, 4, -((5*a^(5/2)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/d) - (a^3*cos(c + d*x))/(d*sqrt(a + a*sin(c + d*x))) - (a^2*cot(c + d*x)*sqrt(a + a*sin(c + d*x)))/d]
    @test_int [csc(c + d*x)^3*(a + a*sin(c + d*x))^(5/2), x, 4, -((19*a^(5/2)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(4*d)) - (9*a^3*cot(c + d*x))/(4*d*sqrt(a + a*sin(c + d*x))) - (a^2*cot(c + d*x)*csc(c + d*x)*sqrt(a + a*sin(c + d*x)))/(2*d)]
    @test_int [csc(c + d*x)^4*(a + a*sin(c + d*x))^(5/2), x, 5, -((25*a^(5/2)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(8*d)) - (25*a^3*cot(c + d*x))/(8*d*sqrt(a + a*sin(c + d*x))) - (13*a^3*cot(c + d*x)*csc(c + d*x))/(12*d*sqrt(a + a*sin(c + d*x))) - (a^2*cot(c + d*x)*csc(c + d*x)^2*sqrt(a + a*sin(c + d*x)))/(3*d)]
    @test_int [csc(c + d*x)^5*(a + a*sin(c + d*x))^(5/2), x, 6, -((163*a^(5/2)*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(64*d)) - (163*a^3*cot(c + d*x))/(64*d*sqrt(a + a*sin(c + d*x))) - (163*a^3*cot(c + d*x)*csc(c + d*x))/(96*d*sqrt(a + a*sin(c + d*x))) - (17*a^3*cot(c + d*x)*csc(c + d*x)^2)/(24*d*sqrt(a + a*sin(c + d*x))) - (a^2*cot(c + d*x)*csc(c + d*x)^3*sqrt(a + a*sin(c + d*x)))/(4*d)]

    #= ::Subsubsection::Closed:: =#
    #=n<0=#

    @test_int [sin(c + d*x)^3/sqrt(a + a*sin(c + d*x)), x, 6, (sqrt(2)*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(sqrt(a)*d) - (28*cos(c + d*x))/(15*d*sqrt(a + a*sin(c + d*x))) - (2*cos(c + d*x)*sin(c + d*x)^2)/(5*d*sqrt(a + a*sin(c + d*x))) + (2*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(15*a*d)]
    @test_int [sin(c + d*x)^2/sqrt(a + a*sin(c + d*x)), x, 4, -((sqrt(2)*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(sqrt(a)*d)) + (4*cos(c + d*x))/(3*d*sqrt(a + a*sin(c + d*x))) - (2*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(3*a*d)]
    @test_int [sin(c + d*x)^1/sqrt(a + a*sin(c + d*x)), x, 3, (sqrt(2)*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(sqrt(a)*d) - (2*cos(c + d*x))/(d*sqrt(a + a*sin(c + d*x)))]
    @test_int [sin(c + d*x)^0/sqrt(a + a*sin(c + d*x)), x, 2, -((sqrt(2)*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(sqrt(a)*d))]
    @test_int [csc(c + d*x)^1/sqrt(a + a*sin(c + d*x)), x, 5, -((2*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(sqrt(a)*d)) + (sqrt(2)*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(sqrt(a)*d)]
    @test_int [csc(c + d*x)^2/sqrt(a + a*sin(c + d*x)), x, 6, atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x)))/(sqrt(a)*d) - (sqrt(2)*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(sqrt(a)*d) - cot(c + d*x)/(d*sqrt(a + a*sin(c + d*x)))]
    @test_int [csc(c + d*x)^3/sqrt(a + a*sin(c + d*x)), x, 7, -((7*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(4*sqrt(a)*d)) + (sqrt(2)*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(sqrt(a)*d) + cot(c + d*x)/(4*d*sqrt(a + a*sin(c + d*x))) - (cot(c + d*x)*csc(c + d*x))/(2*d*sqrt(a + a*sin(c + d*x)))]

    @test_int [sin(c + d*x)^4/(a + a*sin(c + d*x))^(3/2), x, 7, (15*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) + (cos(c + d*x)*sin(c + d*x)^3)/(2*d*(a + a*sin(c + d*x))^(3/2)) - (31*cos(c + d*x))/(5*a*d*sqrt(a + a*sin(c + d*x))) - (9*cos(c + d*x)*sin(c + d*x)^2)/(10*a*d*sqrt(a + a*sin(c + d*x))) + (13*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(10*a^2*d)]
    @test_int [sin(c + d*x)^3/(a + a*sin(c + d*x))^(3/2), x, 6, -((11*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d)) + (cos(c + d*x)*sin(c + d*x)^2)/(2*d*(a + a*sin(c + d*x))^(3/2)) + (13*cos(c + d*x))/(3*a*d*sqrt(a + a*sin(c + d*x))) - (7*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(6*a^2*d)]
    @test_int [sin(c + d*x)^2/(a + a*sin(c + d*x))^(3/2), x, 4, (7*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - cos(c + d*x)/(2*d*(a + a*sin(c + d*x))^(3/2)) - (2*cos(c + d*x))/(a*d*sqrt(a + a*sin(c + d*x)))]
    @test_int [sin(c + d*x)^1/(a + a*sin(c + d*x))^(3/2), x, 3, -((3*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d)) + cos(c + d*x)/(2*d*(a + a*sin(c + d*x))^(3/2))]
    @test_int [sin(c + d*x)^0/(a + a*sin(c + d*x))^(3/2), x, 3, -(atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x))))/(2*sqrt(2)*a^(3/2)*d)) - cos(c + d*x)/(2*d*(a + a*sin(c + d*x))^(3/2))]
    @test_int [csc(c + d*x)^1/(a + a*sin(c + d*x))^(3/2), x, 6, -((2*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(a^(3/2)*d)) + (5*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) + cos(c + d*x)/(2*d*(a + a*sin(c + d*x))^(3/2))]
    @test_int [csc(c + d*x)^2/(a + a*sin(c + d*x))^(3/2), x, 7, (3*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(a^(3/2)*d) - (9*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) + cot(c + d*x)/(2*d*(a + a*sin(c + d*x))^(3/2)) - (3*cot(c + d*x))/(2*a*d*sqrt(a + a*sin(c + d*x)))]
    @test_int [csc(c + d*x)^3/(a + a*sin(c + d*x))^(3/2), x, 8, -((19*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(4*a^(3/2)*d)) + (13*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) + (cot(c + d*x)*csc(c + d*x))/(2*d*(a + a*sin(c + d*x))^(3/2)) + (7*cot(c + d*x))/(4*a*d*sqrt(a + a*sin(c + d*x))) - (cot(c + d*x)*csc(c + d*x))/(a*d*sqrt(a + a*sin(c + d*x)))]

    @test_int [sin(c + d*x)^5/(a + a*sin(c + d*x))^(5/2), x, 8, (283*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) + (cos(c + d*x)*sin(c + d*x)^4)/(4*d*(a + a*sin(c + d*x))^(5/2)) + (21*cos(c + d*x)*sin(c + d*x)^3)/(16*a*d*(a + a*sin(c + d*x))^(3/2)) - (1729*cos(c + d*x))/(120*a^2*d*sqrt(a + a*sin(c + d*x))) - (157*cos(c + d*x)*sin(c + d*x)^2)/(80*a^2*d*sqrt(a + a*sin(c + d*x))) + (787*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(240*a^3*d)]
    @test_int [sin(c + d*x)^4/(a + a*sin(c + d*x))^(5/2), x, 7, -((163*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d)) + (cos(c + d*x)*sin(c + d*x)^3)/(4*d*(a + a*sin(c + d*x))^(5/2)) + (17*cos(c + d*x)*sin(c + d*x)^2)/(16*a*d*(a + a*sin(c + d*x))^(3/2)) + (197*cos(c + d*x))/(24*a^2*d*sqrt(a + a*sin(c + d*x))) - (95*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(48*a^3*d)]
    @test_int [sin(c + d*x)^3/(a + a*sin(c + d*x))^(5/2), x, 6, (75*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) + (cos(c + d*x)*sin(c + d*x)^2)/(4*d*(a + a*sin(c + d*x))^(5/2)) - (13*cos(c + d*x))/(16*a*d*(a + a*sin(c + d*x))^(3/2)) - (9*cos(c + d*x))/(4*a^2*d*sqrt(a + a*sin(c + d*x)))]
    @test_int [sin(c + d*x)^2/(a + a*sin(c + d*x))^(5/2), x, 4, -((19*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d)) - cos(c + d*x)/(4*d*(a + a*sin(c + d*x))^(5/2)) + (13*cos(c + d*x))/(16*a*d*(a + a*sin(c + d*x))^(3/2))]
    @test_int [sin(c + d*x)^1/(a + a*sin(c + d*x))^(5/2), x, 4, -((5*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d)) + cos(c + d*x)/(4*d*(a + a*sin(c + d*x))^(5/2)) - (5*cos(c + d*x))/(16*a*d*(a + a*sin(c + d*x))^(3/2))]
    @test_int [sin(c + d*x)^0/(a + a*sin(c + d*x))^(5/2), x, 4, -((3*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d)) - cos(c + d*x)/(4*d*(a + a*sin(c + d*x))^(5/2)) - (3*cos(c + d*x))/(16*a*d*(a + a*sin(c + d*x))^(3/2))]
    @test_int [csc(c + d*x)^1/(a + a*sin(c + d*x))^(5/2), x, 7, -((2*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(a^(5/2)*d)) + (43*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) + cos(c + d*x)/(4*d*(a + a*sin(c + d*x))^(5/2)) + (11*cos(c + d*x))/(16*a*d*(a + a*sin(c + d*x))^(3/2))]
    @test_int [csc(c + d*x)^2/(a + a*sin(c + d*x))^(5/2), x, 8, (5*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(a^(5/2)*d) - (115*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) + cot(c + d*x)/(4*d*(a + a*sin(c + d*x))^(5/2)) + (15*cot(c + d*x))/(16*a*d*(a + a*sin(c + d*x))^(3/2)) - (35*cot(c + d*x))/(16*a^2*d*sqrt(a + a*sin(c + d*x)))]
    @test_int [csc(c + d*x)^3/(a + a*sin(c + d*x))^(5/2), x, 9, -((39*atanh((sqrt(a)*cos(c + d*x))/sqrt(a + a*sin(c + d*x))))/(4*a^(5/2)*d)) + (219*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) + (cot(c + d*x)*csc(c + d*x))/(4*d*(a + a*sin(c + d*x))^(5/2)) + (19*cot(c + d*x)*csc(c + d*x))/(16*a*d*(a + a*sin(c + d*x))^(3/2)) + (63*cot(c + d*x))/(16*a^2*d*sqrt(a + a*sin(c + d*x))) - (31*cot(c + d*x)*csc(c + d*x))/(16*a^2*d*sqrt(a + a*sin(c + d*x)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d*sin(e+f*x))^(n/2)*(a+a*sin(e+f*x))^(m/2)=#

    #= ::Subsubsection::Closed:: =#
    #=n>0=#

    @test_int [sqrt(a + a*sin(e + f*x))/sqrt(sin(e + f*x)), x, 2, -((2*sqrt(a)*asin((sqrt(a)*cos(e + f*x))/sqrt(a + a*sin(e + f*x))))/f)]
    @test_int [sqrt(a - a*sin(e + f*x))/sqrt(-sin(e + f*x)), x, 2, (2*sqrt(a)*asin((sqrt(a)*cos(e + f*x))/sqrt(a - a*sin(e + f*x))))/f]

    #= ::Subsubsection::Closed:: =#
    #=n<0=#

    @test_int [1/(sqrt(1 + sin(x))*sqrt(sin(x))), x, 2, (-sqrt(2))*asin(cos(x)/(1 + sin(x)))]
    @test_int [1/(sqrt(a + a*sin(x))*sqrt(sin(x))), x, 2, -((sqrt(2)*atan((sqrt(a)*cos(x))/(sqrt(2)*sqrt(sin(x))*sqrt(a + a*sin(x)))))/sqrt(a))]

    @test_int [1/(sqrt(1 - sin(x))*sqrt(sin(x))), x, 2, sqrt(2)*atanh(cos(x)/(sqrt(2)*sqrt(1 - sin(x))*sqrt(sin(x))))]
    @test_int [1/(sqrt(a - a*sin(x))*sqrt(sin(x))), x, 2, (sqrt(2)*atanh((sqrt(a)*cos(x))/(sqrt(2)*sqrt(sin(x))*sqrt(a - a*sin(x)))))/sqrt(a)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d*sin(e+f*x))^(n/3)*(a+a*sin(e+f*x))^m=#

    @test_int [sin(c + d*x)^(1/3)/(a + a*sin(c + d*x))^2, x, 5, (4*cos(c + d*x)*HypergeometricFunctions._₂F₁(1/6, 1/2, 7/6, sin(c + d*x)^2)*sin(c + d*x)^(1/3))/(9*a^2*d*sqrt(cos(c + d*x)^2)) - (cos(c + d*x)*HypergeometricFunctions._₂F₁(1/2, 2/3, 5/3, sin(c + d*x)^2)*sin(c + d*x)^(4/3))/(36*a^2*d*sqrt(cos(c + d*x)^2)) - (cos(c + d*x)*sin(c + d*x)^(1/3))/(9*a^2*d*(1 + sin(c + d*x))) - (cos(c + d*x)*sin(c + d*x)^(1/3))/(3*d*(a + a*sin(c + d*x))^2)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d*sin(e+f*x))^n*(a+a*sin(e+f*x))^(m/3)=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [sin(c + d*x)^3*(a + a*sin(c + d*x))^(2/3), x, 6, -((63*cos(c + d*x)*(a + a*sin(c + d*x))^(2/3))/(220*d)) - (3*cos(c + d*x)*sin(c + d*x)^2*(a + a*sin(c + d*x))^(2/3))/(11*d) - (67*cos(c + d*x)*HypergeometricFunctions._₂F₁(-(1/6), 1/2, 3/2, (1/2)*(1 - sin(c + d*x)))*(a + a*sin(c + d*x))^(2/3))/(55*2^(5/6)*d*(1 + sin(c + d*x))^(7/6)) - (3*cos(c + d*x)*(a + a*sin(c + d*x))^(5/3))/(44*a*d)]
    @test_int [sin(c + d*x)^2*(a + a*sin(c + d*x))^(2/3), x, 4, (9*cos(c + d*x)*(a + a*sin(c + d*x))^(2/3))/(40*d) - (19*cos(c + d*x)*HypergeometricFunctions._₂F₁(-(1/6), 1/2, 3/2, (1/2)*(1 - sin(c + d*x)))*(a + a*sin(c + d*x))^(2/3))/(10*2^(5/6)*d*(1 + sin(c + d*x))^(7/6)) - (3*cos(c + d*x)*(a + a*sin(c + d*x))^(5/3))/(8*a*d)]
    @test_int [sin(c + d*x)^1*(a + a*sin(c + d*x))^(2/3), x, 3, -((3*cos(c + d*x)*(a + a*sin(c + d*x))^(2/3))/(5*d)) - (4*2^(1/6)*cos(c + d*x)*HypergeometricFunctions._₂F₁(-(1/6), 1/2, 3/2, (1/2)*(1 - sin(c + d*x)))*(a + a*sin(c + d*x))^(2/3))/(5*d*(1 + sin(c + d*x))^(7/6))]
    @test_int [sin(c + d*x)^0*(a + a*sin(c + d*x))^(2/3), x, 2, -((2*2^(1/6)*cos(c + d*x)*HypergeometricFunctions._₂F₁(-(1/6), 1/2, 3/2, (1/2)*(1 - sin(c + d*x)))*(a + a*sin(c + d*x))^(2/3))/(d*(1 + sin(c + d*x))^(7/6)))]
    @test_int [csc(c + d*x)^1*(a + a*sin(c + d*x))^(2/3), x, 4, -((2*2^(1/6)*AppellF1(1/2, 1, -(1/6), 3/2, 1 - sin(c + d*x), (1/2)*(1 - sin(c + d*x)))*cos(c + d*x)*(a + a*sin(c + d*x))^(2/3))/(d*(1 + sin(c + d*x))^(7/6)))]
    @test_int [csc(c + d*x)^2*(a + a*sin(c + d*x))^(2/3), x, 4, -((2*2^(1/6)*AppellF1(1/2, 2, -(1/6), 3/2, 1 - sin(c + d*x), (1/2)*(1 - sin(c + d*x)))*cos(c + d*x)*(a + a*sin(c + d*x))^(2/3))/(d*(1 + sin(c + d*x))^(7/6)))]

    @test_int [sin(c + d*x)^3*(a + a*sin(c + d*x))^(4/3), x, 6, -((388*2^(5/6)*a*cos(c + d*x)*HypergeometricFunctions._₂F₁(-(5/6), 1/2, 3/2, (1/2)*(1 - sin(c + d*x)))*(a + a*sin(c + d*x))^(1/3))/(455*d*(1 + sin(c + d*x))^(5/6))) - (72*cos(c + d*x)*(a + a*sin(c + d*x))^(4/3))/(455*d) - (3*cos(c + d*x)*sin(c + d*x)^2*(a + a*sin(c + d*x))^(4/3))/(13*d) - (6*cos(c + d*x)*(a + a*sin(c + d*x))^(7/3))/(65*a*d)]
    @test_int [sin(c + d*x)^2*(a + a*sin(c + d*x))^(4/3), x, 4, -((37*2^(5/6)*a*cos(c + d*x)*HypergeometricFunctions._₂F₁(-(5/6), 1/2, 3/2, (1/2)*(1 - sin(c + d*x)))*(a + a*sin(c + d*x))^(1/3))/(35*d*(1 + sin(c + d*x))^(5/6))) + (9*cos(c + d*x)*(a + a*sin(c + d*x))^(4/3))/(70*d) - (3*cos(c + d*x)*(a + a*sin(c + d*x))^(7/3))/(10*a*d)]
    @test_int [sin(c + d*x)^1*(a + a*sin(c + d*x))^(4/3), x, 3, -((8*2^(5/6)*a*cos(c + d*x)*HypergeometricFunctions._₂F₁(-(5/6), 1/2, 3/2, (1/2)*(1 - sin(c + d*x)))*(a + a*sin(c + d*x))^(1/3))/(7*d*(1 + sin(c + d*x))^(5/6))) - (3*cos(c + d*x)*(a + a*sin(c + d*x))^(4/3))/(7*d)]
    @test_int [sin(c + d*x)^0*(a + a*sin(c + d*x))^(4/3), x, 2, -((2*2^(5/6)*a*cos(c + d*x)*HypergeometricFunctions._₂F₁(-(5/6), 1/2, 3/2, (1/2)*(1 - sin(c + d*x)))*(a + a*sin(c + d*x))^(1/3))/(d*(1 + sin(c + d*x))^(5/6)))]
    @test_int [csc(c + d*x)^1*(a + a*sin(c + d*x))^(4/3), x, 4, -((2*2^(5/6)*a*AppellF1(1/2, 1, -(5/6), 3/2, 1 - sin(c + d*x), (1/2)*(1 - sin(c + d*x)))*cos(c + d*x)*(a + a*sin(c + d*x))^(1/3))/(d*(1 + sin(c + d*x))^(5/6)))]
    @test_int [csc(c + d*x)^2*(a + a*sin(c + d*x))^(4/3), x, 4, -((2*2^(5/6)*a*AppellF1(1/2, 2, -(5/6), 3/2, 1 - sin(c + d*x), (1/2)*(1 - sin(c + d*x)))*cos(c + d*x)*(a + a*sin(c + d*x))^(1/3))/(d*(1 + sin(c + d*x))^(5/6)))]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [sin(c + d*x)^3/(a + a*sin(c + d*x))^(1/3), x, 6, -((99*cos(c + d*x))/(80*d*(a + a*sin(c + d*x))^(1/3))) - (3*cos(c + d*x)*sin(c + d*x)^2)/(8*d*(a + a*sin(c + d*x))^(1/3)) + (37*cos(c + d*x)*HypergeometricFunctions._₂F₁(1/2, 5/6, 3/2, (1/2)*(1 - sin(c + d*x))))/(40*2^(5/6)*d*(1 + sin(c + d*x))^(1/6)*(a + a*sin(c + d*x))^(1/3)) + (3*cos(c + d*x)*(a + a*sin(c + d*x))^(2/3))/(40*a*d)]
    @test_int [sin(c + d*x)^2/(a + a*sin(c + d*x))^(1/3), x, 4, (9*cos(c + d*x))/(10*d*(a + a*sin(c + d*x))^(1/3)) - (7*cos(c + d*x)*HypergeometricFunctions._₂F₁(1/2, 5/6, 3/2, (1/2)*(1 - sin(c + d*x))))/(5*2^(5/6)*d*(1 + sin(c + d*x))^(1/6)*(a + a*sin(c + d*x))^(1/3)) - (3*cos(c + d*x)*(a + a*sin(c + d*x))^(2/3))/(5*a*d)]
    @test_int [sin(c + d*x)^1/(a + a*sin(c + d*x))^(1/3), x, 3, -((3*cos(c + d*x))/(2*d*(a + a*sin(c + d*x))^(1/3))) + (cos(c + d*x)*HypergeometricFunctions._₂F₁(1/2, 5/6, 3/2, (1/2)*(1 - sin(c + d*x))))/(2^(5/6)*d*(1 + sin(c + d*x))^(1/6)*(a + a*sin(c + d*x))^(1/3))]
    @test_int [sin(c + d*x)^0/(a + a*sin(c + d*x))^(1/3), x, 2, -((2^(1/6)*cos(c + d*x)*HypergeometricFunctions._₂F₁(1/2, 5/6, 3/2, (1/2)*(1 - sin(c + d*x))))/(d*(1 + sin(c + d*x))^(1/6)*(a + a*sin(c + d*x))^(1/3)))]
    @test_int [csc(c + d*x)^1/(a + a*sin(c + d*x))^(1/3), x, 4, -((2^(1/6)*AppellF1(1/2, 1, 5/6, 3/2, 1 - sin(c + d*x), (1/2)*(1 - sin(c + d*x)))*cos(c + d*x))/(d*(1 + sin(c + d*x))^(1/6)*(a + a*sin(c + d*x))^(1/3)))]
    @test_int [csc(c + d*x)^2/(a + a*sin(c + d*x))^(1/3), x, 4, -((2^(1/6)*AppellF1(1/2, 2, 5/6, 3/2, 1 - sin(c + d*x), (1/2)*(1 - sin(c + d*x)))*cos(c + d*x))/(d*(1 + sin(c + d*x))^(1/6)*(a + a*sin(c + d*x))^(1/3)))]

    @test_int [sin(c + d*x)^3/(a + a*sin(c + d*x))^(4/3), x, 6, (6*cos(c + d*x))/(5*d*(a + a*sin(c + d*x))^(4/3)) - (3*cos(c + d*x)*sin(c + d*x)^2)/(5*d*(a + a*sin(c + d*x))^(4/3)) + (6*cos(c + d*x))/(5*a*d*(a + a*sin(c + d*x))^(1/3)) - (2*2^(1/6)*cos(c + d*x)*HypergeometricFunctions._₂F₁(1/2, 5/6, 3/2, (1/2)*(1 - sin(c + d*x))))/(a*d*(1 + sin(c + d*x))^(1/6)*(a + a*sin(c + d*x))^(1/3))]
    @test_int [sin(c + d*x)^2/(a + a*sin(c + d*x))^(4/3), x, 4, -((3*cos(c + d*x))/(5*d*(a + a*sin(c + d*x))^(4/3))) - (3*cos(c + d*x))/(2*a*d*(a + a*sin(c + d*x))^(1/3)) + (13*cos(c + d*x)*HypergeometricFunctions._₂F₁(1/2, 5/6, 3/2, (1/2)*(1 - sin(c + d*x))))/(5*2^(5/6)*a*d*(1 + sin(c + d*x))^(1/6)*(a + a*sin(c + d*x))^(1/3))]
    @test_int [sin(c + d*x)^1/(a + a*sin(c + d*x))^(4/3), x, 3, (3*cos(c + d*x))/(5*d*(a + a*sin(c + d*x))^(4/3)) - (4*2^(1/6)*cos(c + d*x)*HypergeometricFunctions._₂F₁(1/2, 5/6, 3/2, (1/2)*(1 - sin(c + d*x))))/(5*a*d*(1 + sin(c + d*x))^(1/6)*(a + a*sin(c + d*x))^(1/3))]
    @test_int [sin(c + d*x)^0/(a + a*sin(c + d*x))^(4/3), x, 2, -((cos(c + d*x)*HypergeometricFunctions._₂F₁(1/2, 11/6, 3/2, (1/2)*(1 - sin(c + d*x))))/(2^(5/6)*a*d*(1 + sin(c + d*x))^(1/6)*(a + a*sin(c + d*x))^(1/3)))]
    @test_int [csc(c + d*x)^1/(a + a*sin(c + d*x))^(4/3), x, 4, -((AppellF1(1/2, 1, 11/6, 3/2, 1 - sin(c + d*x), (1/2)*(1 - sin(c + d*x)))*cos(c + d*x))/(2^(5/6)*a*d*(1 + sin(c + d*x))^(1/6)*(a + a*sin(c + d*x))^(1/3)))]
    @test_int [csc(c + d*x)^2/(a + a*sin(c + d*x))^(4/3), x, 4, -((AppellF1(1/2, 2, 11/6, 3/2, 1 - sin(c + d*x), (1/2)*(1 - sin(c + d*x)))*cos(c + d*x))/(2^(5/6)*a*d*(1 + sin(c + d*x))^(1/6)*(a + a*sin(c + d*x))^(1/3)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d*sin(e+f*x))^n*(a+a*sin(e+f*x))^m*with*n*symbolic=#

    @test_int [sin(e + f*x)^n*(1 + sin(e + f*x))^(3/2), x, 4, -((2*(5 + 4*n)*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, -n, 3/2, 1 - sin(e + f*x)))/(f*(3 + 2*n)*sqrt(1 + sin(e + f*x)))) - (2*cos(e + f*x)*sin(e + f*x)^(1 + n))/(f*(3 + 2*n)*sqrt(1 + sin(e + f*x)))]
    @test_int [sin(e + f*x)^n*(1 + sin(e + f*x))^(1/2), x, 2, -((2*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, -n, 3/2, 1 - sin(e + f*x)))/(f*sqrt(1 + sin(e + f*x))))]
    @test_int [sin(e + f*x)^n/(1 + sin(e + f*x))^(1/2), x, 3, -((AppellF1(1/2, -n, 1, 3/2, 1 - sin(e + f*x), (1/2)*(1 - sin(e + f*x)))*cos(e + f*x))/(f*sqrt(1 + sin(e + f*x))))]
    @test_int [sin(e + f*x)^n/(1 + sin(e + f*x))^(3/2), x, 3, -((AppellF1(1/2, -n, 2, 3/2, 1 - sin(e + f*x), (1/2)*(1 - sin(e + f*x)))*cos(e + f*x))/(2*f*sqrt(1 + sin(e + f*x))))]

    @test_int [sin(e + f*x)^n*(a + a*sin(e + f*x))^(3/2), x, 4, -((2*a^2*(5 + 4*n)*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, -n, 3/2, 1 - sin(e + f*x)))/(f*(3 + 2*n)*sqrt(a + a*sin(e + f*x)))) - (2*a^2*cos(e + f*x)*sin(e + f*x)^(1 + n))/(f*(3 + 2*n)*sqrt(a + a*sin(e + f*x)))]
    @test_int [sin(e + f*x)^n*(a + a*sin(e + f*x))^(1/2), x, 2, -((2*a*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, -n, 3/2, 1 - sin(e + f*x)))/(f*sqrt(a + a*sin(e + f*x))))]
    @test_int [sin(e + f*x)^n/(a + a*sin(e + f*x))^(1/2), x, 4, -((AppellF1(1/2, -n, 1, 3/2, 1 - sin(e + f*x), (1/2)*(1 - sin(e + f*x)))*cos(e + f*x))/(f*sqrt(a + a*sin(e + f*x))))]
    @test_int [sin(e + f*x)^n/(a + a*sin(e + f*x))^(3/2), x, 4, -((AppellF1(1/2, -n, 2, 3/2, 1 - sin(e + f*x), (1/2)*(1 - sin(e + f*x)))*cos(e + f*x))/(2*a*f*sqrt(a + a*sin(e + f*x))))]

    @test_int [(d*sin(e + f*x))^n*(1 + sin(e + f*x))^(3/2), x, 4, -((2*cos(e + f*x)*(d*sin(e + f*x))^(1 + n))/(d*f*(3 + 2*n)*sqrt(1 + sin(e + f*x)))) + ((5 + 4*n)*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, 1 + n, 2 + n, sin(e + f*x))*(d*sin(e + f*x))^(1 + n))/(d*f*(1 + n)*(3 + 2*n)*sqrt(1 - sin(e + f*x))*sqrt(1 + sin(e + f*x)))]
    @test_int [(d*sin(e + f*x))^n*(1 + sin(e + f*x))^(1/2), x, 2, (cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, 1 + n, 2 + n, sin(e + f*x))*(d*sin(e + f*x))^(1 + n))/(d*f*(1 + n)*sqrt(1 - sin(e + f*x))*sqrt(1 + sin(e + f*x)))]
    @test_int [(d*sin(e + f*x))^n/(1 + sin(e + f*x))^(1/2), x, 4, -((AppellF1(1/2, -n, 1, 3/2, 1 - sin(e + f*x), (1/2)*(1 - sin(e + f*x)))*cos(e + f*x)*(d*sin(e + f*x))^n)/(sin(e + f*x)^n*(f*sqrt(1 + sin(e + f*x)))))]
    @test_int [(d*sin(e + f*x))^n/(1 + sin(e + f*x))^(3/2), x, 4, -((AppellF1(1/2, -n, 2, 3/2, 1 - sin(e + f*x), (1/2)*(1 - sin(e + f*x)))*cos(e + f*x)*(d*sin(e + f*x))^n)/(sin(e + f*x)^n*(2*f*sqrt(1 + sin(e + f*x)))))]

    @test_int [(d*sin(e + f*x))^n*(a + a*sin(e + f*x))^(3/2), x, 5, -((2*a^2*(5 + 4*n)*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, -n, 3/2, 1 - sin(e + f*x))*(d*sin(e + f*x))^n)/(sin(e + f*x)^n*(f*(3 + 2*n)*sqrt(a + a*sin(e + f*x))))) - (2*a^2*cos(e + f*x)*(d*sin(e + f*x))^(1 + n))/(d*f*(3 + 2*n)*sqrt(a + a*sin(e + f*x)))]
    @test_int [(d*sin(e + f*x))^n*(a + a*sin(e + f*x))^(1/2), x, 3, -((2*a*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, -n, 3/2, 1 - sin(e + f*x))*(d*sin(e + f*x))^n)/(sin(e + f*x)^n*(f*sqrt(a + a*sin(e + f*x)))))]
    @test_int [(d*sin(e + f*x))^n/(a + a*sin(e + f*x))^(1/2), x, 5, -((AppellF1(1/2, -n, 1, 3/2, 1 - sin(e + f*x), (1/2)*(1 - sin(e + f*x)))*cos(e + f*x)*(d*sin(e + f*x))^n)/(sin(e + f*x)^n*(f*sqrt(a + a*sin(e + f*x)))))]
    @test_int [(d*sin(e + f*x))^n/(a + a*sin(e + f*x))^(3/2), x, 5, -((AppellF1(1/2, -n, 2, 3/2, 1 - sin(e + f*x), (1/2)*(1 - sin(e + f*x)))*cos(e + f*x)*(d*sin(e + f*x))^n)/(sin(e + f*x)^n*(2*a*f*sqrt(a + a*sin(e + f*x)))))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*sin(e+f*x)^n*(a+a*sin(e+f*x))^m*with*m*symbolic=#

    @test_int [sin(e + f*x)^n*(1 + sin(e + f*x))^m, x, 2, -((2^(1/2 + m)*AppellF1(1/2, -n, 1/2 - m, 3/2, 1 - sin(e + f*x), (1/2)*(1 - sin(e + f*x)))*cos(e + f*x))/(f*sqrt(1 + sin(e + f*x))))]
    @test_int [(-sin(e + f*x))^n*(1 - sin(e + f*x))^m, x, 2, (2^(1/2 + m)*AppellF1(1/2, -n, 1/2 - m, 3/2, 1 + sin(e + f*x), (1/2)*(1 + sin(e + f*x)))*cos(e + f*x))/(f*sqrt(1 - sin(e + f*x)))]

    @test_int [(d*sin(e + f*x))^n*(1 + sin(e + f*x))^m, x, 3, -((2^(1/2 + m)*AppellF1(1/2, -n, 1/2 - m, 3/2, 1 - sin(e + f*x), (1/2)*(1 - sin(e + f*x)))*cos(e + f*x)*(d*sin(e + f*x))^n)/(sin(e + f*x)^n*(f*sqrt(1 + sin(e + f*x)))))]
    @test_int [(d*sin(e + f*x))^n*(1 - sin(e + f*x))^m, x, 3, (2^(1/2 + m)*AppellF1(1/2, -n, 1/2 - m, 3/2, 1 + sin(e + f*x), (1/2)*(1 + sin(e + f*x)))*cos(e + f*x)*(d*sin(e + f*x))^n)/((-sin(e + f*x))^n*(f*sqrt(1 - sin(e + f*x))))]

    @test_int [sin(e + f*x)^n*(a + a*sin(e + f*x))^m, x, 3, -((2^(1/2 + m)*AppellF1(1/2, -n, 1/2 - m, 3/2, 1 - sin(e + f*x), (1/2)*(1 - sin(e + f*x)))*cos(e + f*x)*(1 + sin(e + f*x))^(-(1/2) - m)*(a + a*sin(e + f*x))^m)/f)]
    @test_int [(-sin(e + f*x))^n*(a - a*sin(e + f*x))^m, x, 3, (2^(1/2 + m)*AppellF1(1/2, -n, 1/2 - m, 3/2, 1 + sin(e + f*x), (1/2)*(1 + sin(e + f*x)))*cos(e + f*x)*(1 - sin(e + f*x))^(-(1/2) - m)*(a - a*sin(e + f*x))^m)/f]

    @test_int [(d*sin(e + f*x))^n*(a + a*sin(e + f*x))^m, x, 4, -((2^(1/2 + m)*AppellF1(1/2, -n, 1/2 - m, 3/2, 1 - sin(e + f*x), (1/2)*(1 - sin(e + f*x)))*cos(e + f*x)*(d*sin(e + f*x))^n*(1 + sin(e + f*x))^(-(1/2) - m)*(a + a*sin(e + f*x))^m)/(sin(e + f*x)^n*f))]
    @test_int [(d*sin(e + f*x))^n*(a - a*sin(e + f*x))^m, x, 4, (2^(1/2 + m)*AppellF1(1/2, -n, 1/2 - m, 3/2, 1 + sin(e + f*x), (1/2)*(1 + sin(e + f*x)))*cos(e + f*x)*(1 - sin(e + f*x))^(-(1/2) - m)*(d*sin(e + f*x))^n*(a - a*sin(e + f*x))^m)/((-sin(e + f*x))^n*f)]

    @test_int [sin(c + d*x)^4*(a + a*sin(c + d*x))^n, x, 7, If(13>=8, ((9 - n - n^2)*cos(c + d*x)*(a + a*sin(c + d*x))^n)/(d*(1 + n)*(2 + n)*(3 + n)*(4 + n)) - (n*cos(c + d*x)*sin(c + d*x)^2*(a + a*sin(c + d*x))^n)/(d*(3 + n)*(4 + n)) - (cos(c + d*x)*sin(c + d*x)^3*(a + a*sin(c + d*x))^n)/(d*(4 + n)) - (2^(1/2 + n)*(9 + 12*n + 17*n^2 + 6*n^3 + n^4)*cos(c + d*x)*HypergeometricFunctions._₂F₁(1/2, 1/2 - n, 3/2, (1/2)*(1 - sin(c + d*x)))*(1 + sin(c + d*x))^(-(1/2) - n)*(a + a*sin(c + d*x))^n)/(d*(1 + n)*(2 + n)*(3 + n)*(4 + n)) - ((9 + 3*n + n^2)*cos(c + d*x)*(a + a*sin(c + d*x))^(1 + n))/(a*d*(2 + n)*(3 + n)*(4 + n)), ((9 - n - n^2)*cos(c + d*x)*(a + a*sin(c + d*x))^n)/(d*(24 + 50*n + 35*n^2 + 10*n^3 + n^4)) - (n*cos(c + d*x)*sin(c + d*x)^2*(a + a*sin(c + d*x))^n)/(d*(3 + n)*(4 + n)) - (cos(c + d*x)*sin(c + d*x)^3*(a + a*sin(c + d*x))^n)/(d*(4 + n)) - (2^(1/2 + n)*(9 + 12*n + 17*n^2 + 6*n^3 + n^4)*cos(c + d*x)*HypergeometricFunctions._₂F₁(1/2, 1/2 - n, 3/2, (1/2)*(1 - sin(c + d*x)))*(1 + sin(c + d*x))^(-(1/2) - n)*(a + a*sin(c + d*x))^n)/(d*(4 + n)*(6 + 11*n + 6*n^2 + n^3)) - ((9 + 3*n + n^2)*cos(c + d*x)*(a + a*sin(c + d*x))^(1 + n))/(a*d*(4 + n)*(6 + 5*n + n^2)))]
    @test_int [sin(c + d*x)^3*(a + a*sin(c + d*x))^n, x, 6, If(13>=8, -(((4 + n)*cos(c + d*x)*(a + a*sin(c + d*x))^n)/(d*(1 + n)*(2 + n)*(3 + n))) - (cos(c + d*x)*sin(c + d*x)^2*(a + a*sin(c + d*x))^n)/(d*(3 + n)) - (2^(1/2 + n)*n*(5 + 3*n + n^2)*cos(c + d*x)*HypergeometricFunctions._₂F₁(1/2, 1/2 - n, 3/2, (1/2)*(1 - sin(c + d*x)))*(1 + sin(c + d*x))^(-(1/2) - n)*(a + a*sin(c + d*x))^n)/(d*(1 + n)*(2 + n)*(3 + n)) - (n*cos(c + d*x)*(a + a*sin(c + d*x))^(1 + n))/(a*d*(6 + 5*n + n^2)), -(((4 + n)*cos(c + d*x)*(a + a*sin(c + d*x))^n)/(d*(6 + 11*n + 6*n^2 + n^3))) - (cos(c + d*x)*sin(c + d*x)^2*(a + a*sin(c + d*x))^n)/(d*(3 + n)) - (2^(1/2 + n)*n*(5 + 3*n + n^2)*cos(c + d*x)*HypergeometricFunctions._₂F₁(1/2, 1/2 - n, 3/2, (1/2)*(1 - sin(c + d*x)))*(1 + sin(c + d*x))^(-(1/2) - n)*(a + a*sin(c + d*x))^n)/(d*(6 + 11*n + 6*n^2 + n^3)) - (n*cos(c + d*x)*(a + a*sin(c + d*x))^(1 + n))/(a*d*(6 + 5*n + n^2)))]
    @test_int [sin(c + d*x)^2*(a + a*sin(c + d*x))^n, x, 4, If(13>=8, (cos(c + d*x)*(a + a*sin(c + d*x))^n)/(d*(2 + 3*n + n^2)) - (2^(1/2 + n)*(1 + n + n^2)*cos(c + d*x)*HypergeometricFunctions._₂F₁(1/2, 1/2 - n, 3/2, (1/2)*(1 - sin(c + d*x)))*(1 + sin(c + d*x))^(-(1/2) - n)*(a + a*sin(c + d*x))^n)/(d*(1 + n)*(2 + n)) - (cos(c + d*x)*(a + a*sin(c + d*x))^(1 + n))/(a*d*(2 + n)), (cos(c + d*x)*(a + a*sin(c + d*x))^n)/(d*(2 + 3*n + n^2)) - (2^(1/2 + n)*(1 + n + n^2)*cos(c + d*x)*HypergeometricFunctions._₂F₁(1/2, 1/2 - n, 3/2, (1/2)*(1 - sin(c + d*x)))*(1 + sin(c + d*x))^(-(1/2) - n)*(a + a*sin(c + d*x))^n)/(d*(2 + 3*n + n^2)) - (cos(c + d*x)*(a + a*sin(c + d*x))^(1 + n))/(a*d*(2 + n)))]
    @test_int [sin(c + d*x)^1*(a + a*sin(c + d*x))^n, x, 3, -((cos(c + d*x)*(a + a*sin(c + d*x))^n)/(d*(1 + n))) - (2^(1/2 + n)*n*cos(c + d*x)*HypergeometricFunctions._₂F₁(1/2, 1/2 - n, 3/2, (1/2)*(1 - sin(c + d*x)))*(1 + sin(c + d*x))^(-(1/2) - n)*(a + a*sin(c + d*x))^n)/(d*(1 + n))]
    @test_int [sin(c + d*x)^0*(a + a*sin(c + d*x))^n, x, 2, -((2^(1/2 + n)*cos(c + d*x)*HypergeometricFunctions._₂F₁(1/2, 1/2 - n, 3/2, (1/2)*(1 - sin(c + d*x)))*(1 + sin(c + d*x))^(-(1/2) - n)*(a + a*sin(c + d*x))^n)/d)]
    @test_int [csc(c + d*x)^1*(a + a*sin(c + d*x))^n, x, 4, -((2^(1/2 + n)*AppellF1(1/2, 1, 1/2 - n, 3/2, 1 - sin(c + d*x), (1/2)*(1 - sin(c + d*x)))*cos(c + d*x)*(1 + sin(c + d*x))^(-(1/2) - n)*(a + a*sin(c + d*x))^n)/d)]
    @test_int [csc(c + d*x)^2*(a + a*sin(c + d*x))^n, x, 4, -((2^(1/2 + n)*AppellF1(1/2, 2, 1/2 - n, 3/2, 1 - sin(c + d*x), (1/2)*(1 - sin(c + d*x)))*cos(c + d*x)*(1 + sin(c + d*x))^(-(1/2) - n)*(a + a*sin(c + d*x))^n)/d)]

    @test_int [(1 + sin(c + d*x))^n, x, 1, -((2^(1/2 + n)*cos(c + d*x)*HypergeometricFunctions._₂F₁(1/2, 1/2 - n, 3/2, (1/2)*(1 - sin(c + d*x))))/(d*sqrt(1 + sin(c + d*x))))]
    @test_int [(1 - sin(c + d*x))^n, x, 1, (2^(1/2 + n)*cos(c + d*x)*HypergeometricFunctions._₂F₁(1/2, 1/2 - n, 3/2, (1/2)*(1 + sin(c + d*x))))/(d*sqrt(1 - sin(c + d*x)))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(d*sin(e+f*x))^n*(a+b*sin(e+f*x))^m=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*sin(e+f*x)^m*(a+b*sin(e+f*x))^n=#

    #= ::Subsubsection::Closed:: =#
    #=n>0=#

    @test_int [sin(e + f*x)^3*(a + b*sin(e + f*x)), x, 6, (3*b*x)/8 - (a*cos(e + f*x))/f + (a*cos(e + f*x)^3)/(3*f) - (3*b*cos(e + f*x)*sin(e + f*x))/(8*f) - (b*cos(e + f*x)*sin(e + f*x)^3)/(4*f)]
    @test_int [sin(e + f*x)^2*(a + b*sin(e + f*x)), x, 5, (a*x)/2 - (b*cos(e + f*x))/f + (b*cos(e + f*x)^3)/(3*f) - (a*cos(e + f*x)*sin(e + f*x))/(2*f)]
    @test_int [sin(e + f*x)^1*(a + b*sin(e + f*x)), x, 1, (b*x)/2 - (a*cos(e + f*x))/f - (b*cos(e + f*x)*sin(e + f*x))/(2*f)]
    @test_int [sin(e + f*x)^0*(a + b*sin(e + f*x)), x, 2, a*x - (b*cos(e + f*x))/f]
    @test_int [csc(e + f*x)^1*(a + b*sin(e + f*x)), x, 2, b*x - (a*atanh(cos(e + f*x)))/f]
    @test_int [csc(e + f*x)^2*(a + b*sin(e + f*x)), x, 4, -((b*atanh(cos(e + f*x)))/f) - (a*cot(e + f*x))/f]
    @test_int [csc(e + f*x)^3*(a + b*sin(e + f*x)), x, 5, -((a*atanh(cos(e + f*x)))/(2*f)) - (b*cot(e + f*x))/f - (a*cot(e + f*x)*csc(e + f*x))/(2*f)]
    @test_int [csc(e + f*x)^4*(a + b*sin(e + f*x)), x, 5, -((b*atanh(cos(e + f*x)))/(2*f)) - (a*cot(e + f*x))/f - (a*cot(e + f*x)^3)/(3*f) - (b*cot(e + f*x)*csc(e + f*x))/(2*f)]

    @test_int [sin(e + f*x)^3*(a + b*sin(e + f*x))^2, x, 7, (3*a*b*x)/4 - ((a^2 + b^2)*cos(e + f*x))/f + ((a^2 + 2*b^2)*cos(e + f*x)^3)/(3*f) - (b^2*cos(e + f*x)^5)/(5*f) - (3*a*b*cos(e + f*x)*sin(e + f*x))/(4*f) - (a*b*cos(e + f*x)*sin(e + f*x)^3)/(2*f)]
    @test_int [sin(e + f*x)^2*(a + b*sin(e + f*x))^2, x, 6, (1/8)*(4*a^2 + 3*b^2)*x - (2*a*b*cos(e + f*x))/f + (2*a*b*cos(e + f*x)^3)/(3*f) - ((4*a^2 + 3*b^2)*cos(e + f*x)*sin(e + f*x))/(8*f) - (b^2*cos(e + f*x)*sin(e + f*x)^3)/(4*f)]
    @test_int [sin(e + f*x)^1*(a + b*sin(e + f*x))^2, x, 2, a*b*x - (2*(a^2 + b^2)*cos(e + f*x))/(3*f) - (a*b*cos(e + f*x)*sin(e + f*x))/(3*f) - (cos(e + f*x)*(a + b*sin(e + f*x))^2)/(3*f)]
    @test_int [sin(e + f*x)^0*(a + b*sin(e + f*x))^2, x, 1, (1/2)*(2*a^2 + b^2)*x - (2*a*b*cos(e + f*x))/f - (b^2*cos(e + f*x)*sin(e + f*x))/(2*f)]
    @test_int [csc(e + f*x)^1*(a + b*sin(e + f*x))^2, x, 3, 2*a*b*x - (a^2*atanh(cos(e + f*x)))/f - (b^2*cos(e + f*x))/f]
    @test_int [csc(e + f*x)^2*(a + b*sin(e + f*x))^2, x, 4, b^2*x - (2*a*b*atanh(cos(e + f*x)))/f - (a^2*cot(e + f*x))/f]
    @test_int [csc(e + f*x)^3*(a + b*sin(e + f*x))^2, x, 5, -(((a^2 + 2*b^2)*atanh(cos(e + f*x)))/(2*f)) - (2*a*b*cot(e + f*x))/f - (a^2*cot(e + f*x)*csc(e + f*x))/(2*f)]
    @test_int [csc(e + f*x)^4*(a + b*sin(e + f*x))^2, x, 6, -((a*b*atanh(cos(e + f*x)))/f) - ((2*a^2 + 3*b^2)*cot(e + f*x))/(3*f) - (a*b*cot(e + f*x)*csc(e + f*x))/f - (a^2*cot(e + f*x)*csc(e + f*x)^2)/(3*f)]
    @test_int [csc(e + f*x)^5*(a + b*sin(e + f*x))^2, x, 6, -(((3*a^2 + 4*b^2)*atanh(cos(e + f*x)))/(8*f)) - (2*a*b*cot(e + f*x))/f - (2*a*b*cot(e + f*x)^3)/(3*f) - ((3*a^2 + 4*b^2)*cot(e + f*x)*csc(e + f*x))/(8*f) - (a^2*cot(e + f*x)*csc(e + f*x)^3)/(4*f)]

    @test_int [sin(e + f*x)^3*(a + b*sin(e + f*x))^3, x, 8, (1/16)*b*(18*a^2 + 5*b^2)*x - (a*(a^2 + 3*b^2)*cos(e + f*x))/f + (a*(a^2 + 6*b^2)*cos(e + f*x)^3)/(3*f) - (3*a*b^2*cos(e + f*x)^5)/(5*f) - (b*(18*a^2 + 5*b^2)*cos(e + f*x)*sin(e + f*x))/(16*f) - (b*(18*a^2 + 5*b^2)*cos(e + f*x)*sin(e + f*x)^3)/(24*f) - (b^3*cos(e + f*x)*sin(e + f*x)^5)/(6*f), (1/16)*b*(18*a^2 + 5*b^2)*x - (a*(5*a^2 + 12*b^2)*cos(e + f*x))/(5*f) + (a*(5*a^2 + 12*b^2)*cos(e + f*x)^3)/(15*f) - (b*(18*a^2 + 5*b^2)*cos(e + f*x)*sin(e + f*x))/(16*f) - (b*(18*a^2 + 5*b^2)*cos(e + f*x)*sin(e + f*x)^3)/(24*f) - (13*a*b^2*cos(e + f*x)*sin(e + f*x)^4)/(30*f) - (b^2*cos(e + f*x)*sin(e + f*x)^4*(a + b*sin(e + f*x)))/(6*f)]
    @test_int [sin(e + f*x)^2*(a + b*sin(e + f*x))^3, x, 4, (1/8)*a*(4*a^2 + 9*b^2)*x - (b*(15*a^2 + 4*b^2)*cos(e + f*x))/(5*f) + (b*(15*a^2 + 4*b^2)*cos(e + f*x)^3)/(15*f) - (a*(4*a^2 + 9*b^2)*cos(e + f*x)*sin(e + f*x))/(8*f) - (11*a*b^2*cos(e + f*x)*sin(e + f*x)^3)/(20*f) - (b^2*cos(e + f*x)*sin(e + f*x)^3*(a + b*sin(e + f*x)))/(5*f), (1/8)*a*(4*a^2 + 9*b^2)*x + ((3*a^4 - 52*a^2*b^2 - 16*b^4)*cos(e + f*x))/(30*b*f) + (a*(6*a^2 - 71*b^2)*cos(e + f*x)*sin(e + f*x))/(120*f) + ((3*a^2 - 16*b^2)*cos(e + f*x)*(a + b*sin(e + f*x))^2)/(60*b*f) + (a*cos(e + f*x)*(a + b*sin(e + f*x))^3)/(20*b*f) - (cos(e + f*x)*(a + b*sin(e + f*x))^4)/(5*b*f)]
    @test_int [sin(e + f*x)^1*(a + b*sin(e + f*x))^3, x, 3, (3/8)*b*(4*a^2 + b^2)*x - (a*(a^2 + 4*b^2)*cos(e + f*x))/(2*f) - (b*(2*a^2 + 3*b^2)*cos(e + f*x)*sin(e + f*x))/(8*f) - (a*cos(e + f*x)*(a + b*sin(e + f*x))^2)/(4*f) - (cos(e + f*x)*(a + b*sin(e + f*x))^3)/(4*f)]
    @test_int [sin(e + f*x)^0*(a + b*sin(e + f*x))^3, x, 2, (1/2)*a*(2*a^2 + 3*b^2)*x - (2*b*(4*a^2 + b^2)*cos(e + f*x))/(3*f) - (5*a*b^2*cos(e + f*x)*sin(e + f*x))/(6*f) - (b*cos(e + f*x)*(a + b*sin(e + f*x))^2)/(3*f)]
    @test_int [csc(e + f*x)^1*(a + b*sin(e + f*x))^3, x, 4, (1/2)*b*(6*a^2 + b^2)*x - (a^3*atanh(cos(e + f*x)))/f - (5*a*b^2*cos(e + f*x))/(2*f) - (b^2*cos(e + f*x)*(a + b*sin(e + f*x)))/(2*f)]
    @test_int [csc(e + f*x)^2*(a + b*sin(e + f*x))^3, x, 4, 3*a*b^2*x - (3*a^2*b*atanh(cos(e + f*x)))/f + (b*(a^2 - b^2)*cos(e + f*x))/f - (a^2*cot(e + f*x)*(a + b*sin(e + f*x)))/f]
    @test_int [csc(e + f*x)^3*(a + b*sin(e + f*x))^3, x, 4, b^3*x - (a*(a^2 + 6*b^2)*atanh(cos(e + f*x)))/(2*f) - (5*a^2*b*cot(e + f*x))/(2*f) - (a^2*cot(e + f*x)*csc(e + f*x)*(a + b*sin(e + f*x)))/(2*f)]
    @test_int [csc(e + f*x)^4*(a + b*sin(e + f*x))^3, x, 6, -((b*(3*a^2 + 2*b^2)*atanh(cos(e + f*x)))/(2*f)) - (a*(2*a^2 + 9*b^2)*cot(e + f*x))/(3*f) - (7*a^2*b*cot(e + f*x)*csc(e + f*x))/(6*f) - (a^2*cot(e + f*x)*csc(e + f*x)^2*(a + b*sin(e + f*x)))/(3*f)]
    @test_int [csc(e + f*x)^5*(a + b*sin(e + f*x))^3, x, 7, -((3*a*(a^2 + 4*b^2)*atanh(cos(e + f*x)))/(8*f)) - (b*(2*a^2 + b^2)*cot(e + f*x))/f - (3*a*(a^2 + 4*b^2)*cot(e + f*x)*csc(e + f*x))/(8*f) - (3*a^2*b*cot(e + f*x)*csc(e + f*x)^2)/(4*f) - (a^2*cot(e + f*x)*csc(e + f*x)^3*(a + b*sin(e + f*x)))/(4*f)]

    @test_int [sin(e + f*x)^0*(a + b*sin(e + f*x))^4, x, 3, (1/8)*(8*a^4 + 24*a^2*b^2 + 3*b^4)*x - (a*b*(19*a^2 + 16*b^2)*cos(e + f*x))/(6*f) - (b^2*(26*a^2 + 9*b^2)*cos(e + f*x)*sin(e + f*x))/(24*f) - (7*a*b*cos(e + f*x)*(a + b*sin(e + f*x))^2)/(12*f) - (b*cos(e + f*x)*(a + b*sin(e + f*x))^3)/(4*f)]

    #= ::Subsubsection::Closed:: =#
    #=n<0=#

    @test_int [sin(x)^4/(a + b*sin(x)), x, 7, -((a*(2*a^2 + b^2)*x)/(2*b^4)) + (2*a^4*atan((b + a*tan(x/2))/sqrt(a^2 - b^2)))/(b^4*sqrt(a^2 - b^2)) - ((3*a^2 + 2*b^2)*cos(x))/(3*b^3) + (a*cos(x)*sin(x))/(2*b^2) - (cos(x)*sin(x)^2)/(3*b)]
    @test_int [sin(x)^3/(a + b*sin(x)), x, 6, ((2*a^2 + b^2)*x)/(2*b^3) - (2*a^3*atan((b + a*tan(x/2))/sqrt(a^2 - b^2)))/(b^3*sqrt(a^2 - b^2)) + (a*cos(x))/b^2 - (cos(x)*sin(x))/(2*b)]
    @test_int [sin(x)^2/(a + b*sin(x)), x, 6, -((a*x)/b^2) + (2*a^2*atan((b + a*tan(x/2))/sqrt(a^2 - b^2)))/(b^2*sqrt(a^2 - b^2)) - cos(x)/b]
    @test_int [sin(x)^1/(a + b*sin(x)), x, 4, x/b - (2*a*atan((b + a*tan(x/2))/sqrt(a^2 - b^2)))/(b*sqrt(a^2 - b^2))]
    @test_int [sin(x)^0/(a + b*sin(x)), x, 3, (2*atan((b + a*tan(x/2))/sqrt(a^2 - b^2)))/sqrt(a^2 - b^2)]
    @test_int [csc(x)^1/(a + b*sin(x)), x, 5, -((2*b*atan((b + a*tan(x/2))/sqrt(a^2 - b^2)))/(a*sqrt(a^2 - b^2))) - atanh(cos(x))/a]
    @test_int [csc(x)^2/(a + b*sin(x)), x, 7, (2*b^2*atan((b + a*tan(x/2))/sqrt(a^2 - b^2)))/(a^2*sqrt(a^2 - b^2)) + (b*atanh(cos(x)))/a^2 - cot(x)/a]
    @test_int [csc(x)^3/(a + b*sin(x)), x, 7, -((2*b^3*atan((b + a*tan(x/2))/sqrt(a^2 - b^2)))/(a^3*sqrt(a^2 - b^2))) - ((a^2 + 2*b^2)*atanh(cos(x)))/(2*a^3) + (b*cot(x))/a^2 - (cot(x)*csc(x))/(2*a)]
    @test_int [csc(x)^4/(a + b*sin(x)), x, 8, (2*b^4*atan((b + a*tan(x/2))/sqrt(a^2 - b^2)))/(a^4*sqrt(a^2 - b^2)) + (b*(a^2 + 2*b^2)*atanh(cos(x)))/(2*a^4) - ((2*a^2 + 3*b^2)*cot(x))/(3*a^3) + (b*cot(x)*csc(x))/(2*a^2) - (cot(x)*csc(x)^2)/(3*a)]

    @test_int [sin(x)^4/(a + b*sin(x))^2, x, 7, ((6*a^2 + b^2)*x)/(2*b^4) - (2*a^3*(3*a^2 - 4*b^2)*atan((b + a*tan(x/2))/sqrt(a^2 - b^2)))/(b^4*(a^2 - b^2)^(3/2)) + (a*(3*a^2 - 2*b^2)*cos(x))/(b^3*(a^2 - b^2)) - ((3*a^2 - b^2)*cos(x)*sin(x))/(2*b^2*(a^2 - b^2)) + (a^2*cos(x)*sin(x)^2)/(b*(a^2 - b^2)*(a + b*sin(x)))]
    @test_int [sin(x)^3/(a + b*sin(x))^2, x, 6, -((2*a*x)/b^3) + (2*a^2*(2*a^2 - 3*b^2)*atan((b + a*tan(x/2))/sqrt(a^2 - b^2)))/(b^3*(a^2 - b^2)^(3/2)) - ((2*a^2 - b^2)*cos(x))/(b^2*(a^2 - b^2)) + (a^2*cos(x)*sin(x))/(b*(a^2 - b^2)*(a + b*sin(x)))]
    @test_int [sin(x)^2/(a + b*sin(x))^2, x, 5, x/b^2 - (2*a*(a^2 - 2*b^2)*atan((b + a*tan(x/2))/sqrt(a^2 - b^2)))/(b^2*(a^2 - b^2)^(3/2)) + (a^2*cos(x))/(b*(a^2 - b^2)*(a + b*sin(x)))]
    @test_int [sin(x)^1/(a + b*sin(x))^2, x, 5, -((2*b*atan((b + a*tan(x/2))/sqrt(a^2 - b^2)))/(a^2 - b^2)^(3/2)) - (a*cos(x))/((a^2 - b^2)*(a + b*sin(x)))]
    @test_int [sin(x)^0/(a + b*sin(x))^2, x, 5, (2*a*atan((b + a*tan(x/2))/sqrt(a^2 - b^2)))/(a^2 - b^2)^(3/2) + (b*cos(x))/((a^2 - b^2)*(a + b*sin(x)))]
    @test_int [csc(x)^1/(a + b*sin(x))^2, x, 6, -((2*b*(2*a^2 - b^2)*atan((b + a*tan(x/2))/sqrt(a^2 - b^2)))/(a^2*(a^2 - b^2)^(3/2))) - atanh(cos(x))/a^2 - (b^2*cos(x))/(a*(a^2 - b^2)*(a + b*sin(x)))]
    @test_int [csc(x)^2/(a + b*sin(x))^2, x, 7, (2*b^2*(3*a^2 - 2*b^2)*atan((b + a*tan(x/2))/sqrt(a^2 - b^2)))/(a^3*(a^2 - b^2)^(3/2)) + (2*b*atanh(cos(x)))/a^3 - ((a^2 - 2*b^2)*cot(x))/(a^2*(a^2 - b^2)) - (b^2*cot(x))/(a*(a^2 - b^2)*(a + b*sin(x)))]
    @test_int [csc(x)^3/(a + b*sin(x))^2, x, 8, -((2*b^3*(4*a^2 - 3*b^2)*atan((b + a*tan(x/2))/sqrt(a^2 - b^2)))/(a^4*(a^2 - b^2)^(3/2))) - ((a^2 + 6*b^2)*atanh(cos(x)))/(2*a^4) + (b*(2*a^2 - 3*b^2)*cot(x))/(a^3*(a^2 - b^2)) - ((a^2 - 3*b^2)*cot(x)*csc(x))/(2*a^2*(a^2 - b^2)) - (b^2*cot(x)*csc(x))/(a*(a^2 - b^2)*(a + b*sin(x)))]

    @test_int [sin(x)^5/(a + b*sin(x))^3, x, 8, ((12*a^2 + b^2)*x)/(2*b^5) - (a^3*(12*a^4 - 29*a^2*b^2 + 20*b^4)*atan((b + a*tan(x/2))/sqrt(a^2 - b^2)))/(b^5*(a^2 - b^2)^(5/2)) + (3*a*(4*a^4 - 7*a^2*b^2 + 2*b^4)*cos(x))/(2*b^4*(a^2 - b^2)^2) - ((6*a^4 - 10*a^2*b^2 + b^4)*cos(x)*sin(x))/(2*b^3*(a^2 - b^2)^2) + (a^2*cos(x)*sin(x)^3)/(2*b*(a^2 - b^2)*(a + b*sin(x))^2) + (a^2*(4*a^2 - 7*b^2)*cos(x)*sin(x)^2)/(2*b^2*(a^2 - b^2)^2*(a + b*sin(x)))]
    @test_int [sin(x)^4/(a + b*sin(x))^3, x, 7, -((3*a*x)/b^4) + (3*a^2*(2*a^4 - 5*a^2*b^2 + 4*b^4)*atan((b + a*tan(x/2))/sqrt(a^2 - b^2)))/(b^4*(a^2 - b^2)^(5/2)) - ((3*a^2 - 2*b^2)*cos(x))/(2*b^3*(a^2 - b^2)) + (a^2*cos(x)*sin(x)^2)/(2*b*(a^2 - b^2)*(a + b*sin(x))^2) - (3*a^3*(a^2 - 2*b^2)*cos(x))/(2*b^3*(a^2 - b^2)^2*(a + b*sin(x)))]
    @test_int [sin(x)^3/(a + b*sin(x))^3, x, 6, x/b^3 - (a*(2*a^4 - 5*a^2*b^2 + 6*b^4)*atan((b + a*tan(x/2))/sqrt(a^2 - b^2)))/(b^3*(a^2 - b^2)^(5/2)) + (a^2*cos(x)*sin(x))/(2*b*(a^2 - b^2)*(a + b*sin(x))^2) + (a^2*(2*a^2 - 5*b^2)*cos(x))/(2*b^2*(a^2 - b^2)^2*(a + b*sin(x)))]
    @test_int [sin(x)^2/(a + b*sin(x))^3, x, 6, ((a^2 + 2*b^2)*atan((b + a*tan(x/2))/sqrt(a^2 - b^2)))/(a^2 - b^2)^(5/2) + (a^2*cos(x))/(2*b*(a^2 - b^2)*(a + b*sin(x))^2) - (a*(a^2 - 4*b^2)*cos(x))/(2*b*(a^2 - b^2)^2*(a + b*sin(x)))]
    @test_int [sin(x)^1/(a + b*sin(x))^3, x, 6, -((3*a*b*atan((b + a*tan(x/2))/sqrt(a^2 - b^2)))/(a^2 - b^2)^(5/2)) - (a*cos(x))/(2*(a^2 - b^2)*(a + b*sin(x))^2) - ((a^2 + 2*b^2)*cos(x))/(2*(a^2 - b^2)^2*(a + b*sin(x)))]
    @test_int [sin(x)^0/(a + b*sin(x))^3, x, 6, ((2*a^2 + b^2)*atan((b + a*tan(x/2))/sqrt(a^2 - b^2)))/(a^2 - b^2)^(5/2) + (b*cos(x))/(2*(a^2 - b^2)*(a + b*sin(x))^2) + (3*a*b*cos(x))/(2*(a^2 - b^2)^2*(a + b*sin(x)))]
    @test_int [csc(x)^1/(a + b*sin(x))^3, x, 7, -((b*(6*a^4 - 5*a^2*b^2 + 2*b^4)*atan((b + a*tan(x/2))/sqrt(a^2 - b^2)))/(a^3*(a^2 - b^2)^(5/2))) - atanh(cos(x))/a^3 - (b^2*cos(x))/(2*a*(a^2 - b^2)*(a + b*sin(x))^2) - (b^2*(5*a^2 - 2*b^2)*cos(x))/(2*a^2*(a^2 - b^2)^2*(a + b*sin(x)))]
    @test_int [csc(x)^2/(a + b*sin(x))^3, x, 8, (3*b^2*(4*a^4 - 5*a^2*b^2 + 2*b^4)*atan((b + a*tan(x/2))/sqrt(a^2 - b^2)))/(a^4*(a^2 - b^2)^(5/2)) + (3*b*atanh(cos(x)))/a^4 - ((2*a^4 - 11*a^2*b^2 + 6*b^4)*cot(x))/(2*a^3*(a^2 - b^2)^2) - (b^2*cot(x))/(2*a*(a^2 - b^2)*(a + b*sin(x))^2) - (3*b^2*(2*a^2 - b^2)*cot(x))/(2*a^2*(a^2 - b^2)^2*(a + b*sin(x)))]
    @test_int [csc(x)^3/(a + b*sin(x))^3, x, 9, -((b^3*(20*a^4 - 29*a^2*b^2 + 12*b^4)*atan((b + a*tan(x/2))/sqrt(a^2 - b^2)))/(a^5*(a^2 - b^2)^(5/2))) - ((a^2 + 12*b^2)*atanh(cos(x)))/(2*a^5) + (3*b*(2*a^4 - 7*a^2*b^2 + 4*b^4)*cot(x))/(2*a^4*(a^2 - b^2)^2) - ((a^4 - 10*a^2*b^2 + 6*b^4)*cot(x)*csc(x))/(2*a^3*(a^2 - b^2)^2) - (b^2*cot(x)*csc(x))/(2*a*(a^2 - b^2)*(a + b*sin(x))^2) - (b^2*(7*a^2 - 4*b^2)*cot(x)*csc(x))/(2*a^2*(a^2 - b^2)^2*(a + b*sin(x)))]

    @test_int [sin(c + d*x)^0/(a + b*sin(c + d*x))^4, x, 7, (a*(2*a^2 + 3*b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(7/2)*d) + (b*cos(c + d*x))/(3*(a^2 - b^2)*d*(a + b*sin(c + d*x))^3) + (5*a*b*cos(c + d*x))/(6*(a^2 - b^2)^2*d*(a + b*sin(c + d*x))^2) + (b*(11*a^2 + 4*b^2)*cos(c + d*x))/(6*(a^2 - b^2)^3*d*(a + b*sin(c + d*x)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*sin(e+f*x)^m*(a+b*sin(e+f*x))^(n/2)=#

    #= ::Subsubsection::Closed:: =#
    #=n>0=#

    @test_int [sin(e + f*x)^1*sqrt(a + b*sin(e + f*x)), x, 6, -((2*cos(e + f*x)*sqrt(a + b*sin(e + f*x)))/(3*f)) + (2*a*Elliptic.E((1/2)*(e - pi/2 + f*x), (2*b)/(a + b))*sqrt(a + b*sin(e + f*x)))/(3*b*f*sqrt((a + b*sin(e + f*x))/(a + b))) - (2*(a^2 - b^2)*Elliptic.F((1/2)*(e - pi/2 + f*x), (2*b)/(a + b))*sqrt((a + b*sin(e + f*x))/(a + b)))/(3*b*f*sqrt(a + b*sin(e + f*x)))]
    @test_int [sin(e + f*x)^0*sqrt(a + b*sin(e + f*x)), x, 2, (2*Elliptic.E((1/2)*(e - pi/2 + f*x), (2*b)/(a + b))*sqrt(a + b*sin(e + f*x)))/(f*sqrt((a + b*sin(e + f*x))/(a + b)))]
    @test_int [csc(e + f*x)^1*sqrt(a + b*sin(e + f*x)), x, 5, (2*b*Elliptic.F((1/2)*(e - pi/2 + f*x), (2*b)/(a + b))*sqrt((a + b*sin(e + f*x))/(a + b)))/(f*sqrt(a + b*sin(e + f*x))) + (2*a*Elliptic.Pi(2, (1/2)*(e - pi/2 + f*x), (2*b)/(a + b))*sqrt((a + b*sin(e + f*x))/(a + b)))/(f*sqrt(a + b*sin(e + f*x)))]
    @test_int [csc(e + f*x)^2*sqrt(a + b*sin(e + f*x)), x, 9, -((cot(e + f*x)*sqrt(a + b*sin(e + f*x)))/f) - (Elliptic.E((1/2)*(e - pi/2 + f*x), (2*b)/(a + b))*sqrt(a + b*sin(e + f*x)))/(f*sqrt((a + b*sin(e + f*x))/(a + b))) + (a*Elliptic.F((1/2)*(e - pi/2 + f*x), (2*b)/(a + b))*sqrt((a + b*sin(e + f*x))/(a + b)))/(f*sqrt(a + b*sin(e + f*x))) + (b*Elliptic.Pi(2, (1/2)*(e - pi/2 + f*x), (2*b)/(a + b))*sqrt((a + b*sin(e + f*x))/(a + b)))/(f*sqrt(a + b*sin(e + f*x)))]

    #= ::Subsubsection::Closed:: =#
    #=n<0=#

    @test_int [sin(e + f*x)^1/sqrt(a + b*sin(e + f*x)), x, 5, (2*Elliptic.E((1/2)*(e - pi/2 + f*x), (2*b)/(a + b))*sqrt(a + b*sin(e + f*x)))/(b*f*sqrt((a + b*sin(e + f*x))/(a + b))) - (2*a*Elliptic.F((1/2)*(e - pi/2 + f*x), (2*b)/(a + b))*sqrt((a + b*sin(e + f*x))/(a + b)))/(b*f*sqrt(a + b*sin(e + f*x)))]
    @test_int [sin(e + f*x)^0/sqrt(a + b*sin(e + f*x)), x, 2, (2*Elliptic.F((1/2)*(e - pi/2 + f*x), (2*b)/(a + b))*sqrt((a + b*sin(e + f*x))/(a + b)))/(f*sqrt(a + b*sin(e + f*x)))]
    @test_int [csc(e + f*x)^1/sqrt(a + b*sin(e + f*x)), x, 2, (2*Elliptic.Pi(2, (1/2)*(e - pi/2 + f*x), (2*b)/(a + b))*sqrt((a + b*sin(e + f*x))/(a + b)))/(f*sqrt(a + b*sin(e + f*x)))]
    @test_int [csc(e + f*x)^2/sqrt(a + b*sin(e + f*x)), x, 9, -((cot(e + f*x)*sqrt(a + b*sin(e + f*x)))/(a*f)) - (Elliptic.E((1/2)*(e - pi/2 + f*x), (2*b)/(a + b))*sqrt(a + b*sin(e + f*x)))/(a*f*sqrt((a + b*sin(e + f*x))/(a + b))) + (Elliptic.F((1/2)*(e - pi/2 + f*x), (2*b)/(a + b))*sqrt((a + b*sin(e + f*x))/(a + b)))/(f*sqrt(a + b*sin(e + f*x))) - (b*Elliptic.Pi(2, (1/2)*(e - pi/2 + f*x), (2*b)/(a + b))*sqrt((a + b*sin(e + f*x))/(a + b)))/(a*f*sqrt(a + b*sin(e + f*x)))]

    #= ::Subsection:: =#
    #=Integrands*of*the*form*(d*sin(e+f*x))^(m/2)*(a+b*sin(e+f*x))^n=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d*sin(e+f*x))^(m/2)*(a+b*sin(e+f*x))^(n/2)=#

    @test_int [sqrt(sin(c + d*x))*sqrt(a + b*sin(c + d*x)), x, 7, -((cos(c + d*x)*sqrt(a + b*sin(c + d*x)))/(d*sqrt(sin(c + d*x)))) + ((a - b)*sqrt(a + b)*sqrt((a*(1 - csc(c + d*x)))/(a + b))*sqrt((a*(1 + csc(c + d*x)))/(a - b))*Elliptic.E(asin(sqrt(a + b*sin(c + d*x))/(sqrt(a + b)*sqrt(sin(c + d*x)))), -((a + b)/(a - b)))*tan(c + d*x))/(a*d) - (sqrt(a + b)*sqrt((a*(1 - csc(c + d*x)))/(a + b))*sqrt((a*(1 + csc(c + d*x)))/(a - b))*Elliptic.F(asin(sqrt(a + b*sin(c + d*x))/(sqrt(a + b)*sqrt(sin(c + d*x)))), -((a + b)/(a - b)))*tan(c + d*x))/d + (a*sqrt(a + b)*sqrt((a*(1 - csc(c + d*x)))/(a + b))*sqrt((a*(1 + csc(c + d*x)))/(a - b))*Elliptic.Pi((a + b)/b, asin(sqrt(a + b*sin(c + d*x))/(sqrt(a + b)*sqrt(sin(c + d*x)))), -((a + b)/(a - b)))*tan(c + d*x))/(b*d)]
    @test_int [1/(sqrt(sin(c + d*x))*sqrt(a + b*sin(c + d*x))), x, 1, -((2*sqrt(a + b)*sqrt((a*(1 - csc(c + d*x)))/(a + b))*sqrt((a*(1 + csc(c + d*x)))/(a - b))*Elliptic.F(asin(sqrt(a + b*sin(c + d*x))/(sqrt(a + b)*sqrt(sin(c + d*x)))), -((a + b)/(a - b)))*tan(c + d*x))/(a*d))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*sin(e+f*x)^m*(a+b*sin(e+f*x))^n*with*m*symbolic=#

    @test_int [(d*sin(e + f*x))^m*(a + b*sin(e + f*x))^3, x, 5, -((a*b^2*(7 + 2*m)*cos(e + f*x)*(d*sin(e + f*x))^(1 + m))/(d*f*(2 + m)*(3 + m))) + (a*(3*b^2*(1 + m) + a^2*(2 + m))*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, (1 + m)/2, (3 + m)/2, sin(e + f*x)^2)*(d*sin(e + f*x))^(1 + m))/(d*f*(1 + m)*(2 + m)*sqrt(cos(e + f*x)^2)) + (b*(b^2*(2 + m) + 3*a^2*(3 + m))*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, (2 + m)/2, (4 + m)/2, sin(e + f*x)^2)*(d*sin(e + f*x))^(2 + m))/(d^2*f*(2 + m)*(3 + m)*sqrt(cos(e + f*x)^2)) - (b^2*cos(e + f*x)*(d*sin(e + f*x))^(1 + m)*(a + b*sin(e + f*x)))/(d*f*(3 + m))]
    @test_int [(d*sin(e + f*x))^m*(a + b*sin(e + f*x))^2, x, 4, -((b^2*cos(e + f*x)*(d*sin(e + f*x))^(1 + m))/(d*f*(2 + m))) + ((b^2*(1 + m) + a^2*(2 + m))*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, (1 + m)/2, (3 + m)/2, sin(e + f*x)^2)*(d*sin(e + f*x))^(1 + m))/(d*f*(1 + m)*(2 + m)*sqrt(cos(e + f*x)^2)) + (2*a*b*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, (2 + m)/2, (4 + m)/2, sin(e + f*x)^2)*(d*sin(e + f*x))^(2 + m))/(d^2*f*(2 + m)*sqrt(cos(e + f*x)^2))]
    @test_int [(d*sin(e + f*x))^m*(a + b*sin(e + f*x))^1, x, 3, (a*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, (1 + m)/2, (3 + m)/2, sin(e + f*x)^2)*(d*sin(e + f*x))^(1 + m))/(d*f*(1 + m)*sqrt(cos(e + f*x)^2)) + (b*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, (2 + m)/2, (4 + m)/2, sin(e + f*x)^2)*(d*sin(e + f*x))^(2 + m))/(d^2*f*(2 + m)*sqrt(cos(e + f*x)^2))]
    @test_int [(d*sin(e + f*x))^m/(a + b*sin(e + f*x))^1, x, 5, -((a*d*AppellF1(1/2, (1 - m)/2, 1, 3/2, cos(e + f*x)^2, -((b^2*cos(e + f*x)^2)/(a^2 - b^2)))*cos(e + f*x)*(d*sin(e + f*x))^(-1 + m)*(sin(e + f*x)^2)^((1 - m)/2))/((a^2 - b^2)*f)) + (b*AppellF1(1/2, -(m/2), 1, 3/2, cos(e + f*x)^2, -((b^2*cos(e + f*x)^2)/(a^2 - b^2)))*cos(e + f*x)*(d*sin(e + f*x))^m)/((sin(e + f*x)^2)^(m/2)*((a^2 - b^2)*f))]
    @test_int [(d*sin(e + f*x))^m/(a + b*sin(e + f*x))^2, x, 10, -((b^2*AppellF1(1/2, (1/2)*(-1 - m), 2, 3/2, cos(e + f*x)^2, -((b^2*cos(e + f*x)^2)/(a^2 - b^2)))*cos(e + f*x)*(d*sin(e + f*x))^(1 + m)*(sin(e + f*x)^2)^((1/2)*(-1 - m)))/((a^2 - b^2)^2*d*f)) - (a^2*d*AppellF1(1/2, (1 - m)/2, 2, 3/2, cos(e + f*x)^2, -((b^2*cos(e + f*x)^2)/(a^2 - b^2)))*cos(e + f*x)*(d*sin(e + f*x))^(-1 + m)*(sin(e + f*x)^2)^((1 - m)/2))/((a^2 - b^2)^2*f) + (2*a*b*AppellF1(1/2, -(m/2), 2, 3/2, cos(e + f*x)^2, -((b^2*cos(e + f*x)^2)/(a^2 - b^2)))*cos(e + f*x)*(d*sin(e + f*x))^m)/((sin(e + f*x)^2)^(m/2)*((a^2 - b^2)^2*f))]
    @test_int [(d*sin(e + f*x))^m/(a + b*sin(e + f*x))^3, x, 13, -((3*a*b^2*AppellF1(1/2, (1/2)*(-1 - m), 3, 3/2, cos(e + f*x)^2, -((b^2*cos(e + f*x)^2)/(a^2 - b^2)))*cos(e + f*x)*(d*sin(e + f*x))^(1 + m)*(sin(e + f*x)^2)^((1/2)*(-1 - m)))/((a^2 - b^2)^3*d*f)) - (a^3*d*AppellF1(1/2, (1 - m)/2, 3, 3/2, cos(e + f*x)^2, -((b^2*cos(e + f*x)^2)/(a^2 - b^2)))*cos(e + f*x)*(d*sin(e + f*x))^(-1 + m)*(sin(e + f*x)^2)^((1 - m)/2))/((a^2 - b^2)^3*f) + (b^3*AppellF1(1/2, (1/2)*(-2 - m), 3, 3/2, cos(e + f*x)^2, -((b^2*cos(e + f*x)^2)/(a^2 - b^2)))*cos(e + f*x)*(d*sin(e + f*x))^m)/((sin(e + f*x)^2)^(m/2)*((a^2 - b^2)^3*f)) + (3*a^2*b*AppellF1(1/2, -(m/2), 3, 3/2, cos(e + f*x)^2, -((b^2*cos(e + f*x)^2)/(a^2 - b^2)))*cos(e + f*x)*(d*sin(e + f*x))^m)/((sin(e + f*x)^2)^(m/2)*((a^2 - b^2)^3*f))]

    @test_int [sin(c + d*x)^(-1 - a^2/(a^2 + b^2))*(a + b*sin(c + d*x))^2, x, 3, -(((a^2 + b^2)*cos(c + d*x))/(sin(c + d*x)^(a^2/(a^2 + b^2))*d)) + (2*a*(a^2 + b^2)*cos(c + d*x)*HypergeometricFunctions._₂F₁(1/2, b^2/(2*(a^2 + b^2)), (1/2)*(3 - a^2/(a^2 + b^2)), sin(c + d*x)^2)*sin(c + d*x)^(b^2/(a^2 + b^2)))/(b*d*sqrt(cos(c + d*x)^2))]

    @test_int [(1 + 2*sin(c + d*x))^2/sin(c + d*x)^(6/5), x, 3, -((5*cos(c + d*x))/(d*sin(c + d*x)^(1/5))) + (5*cos(c + d*x)*HypergeometricFunctions._₂F₁(2/5, 1/2, 7/5, sin(c + d*x)^2)*sin(c + d*x)^(4/5))/(d*sqrt(cos(c + d*x)^2))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*sin(e+f*x)^m*(a+b*sin(e+f*x))^n*with*n*symbolic=#

    @test_int [sin(c + d*x)^m*(a + b*sin(c + d*x))^n, x, 0, Unintegrable(sin(c + d*x)^m*(a + b*sin(c + d*x))^n, x)]

    @test_int [sin(c + d*x)^3*(a + b*sin(c + d*x))^n, x, 9, (2*a*cos(c + d*x)*(a + b*sin(c + d*x))^(1 + n))/(b^2*d*(2 + n)*(3 + n)) - (cos(c + d*x)*sin(c + d*x)*(a + b*sin(c + d*x))^(1 + n))/(b*d*(3 + n)) - (sqrt(2)*(a + b)*(2*a^2 + b^2*(2 + n)^2)*AppellF1(1/2, 1/2, -1 - n, 3/2, (1/2)*(1 - sin(c + d*x)), (b*(1 - sin(c + d*x)))/(a + b))*cos(c + d*x)*(a + b*sin(c + d*x))^n)/(((a + b*sin(c + d*x))/(a + b))^n*(b^3*d*(2 + n)*(3 + n)*sqrt(1 + sin(c + d*x)))) + (sqrt(2)*a*(2*a^2 + b^2*(4 + 5*n + n^2))*AppellF1(1/2, 1/2, -n, 3/2, (1/2)*(1 - sin(c + d*x)), (b*(1 - sin(c + d*x)))/(a + b))*cos(c + d*x)*(a + b*sin(c + d*x))^n)/(((a + b*sin(c + d*x))/(a + b))^n*(b^3*d*(2 + n)*(3 + n)*sqrt(1 + sin(c + d*x))))]
    @test_int [sin(c + d*x)^2*(a + b*sin(c + d*x))^n, x, 8, -((cos(c + d*x)*(a + b*sin(c + d*x))^(1 + n))/(b*d*(2 + n))) + (sqrt(2)*a*(a + b)*AppellF1(1/2, 1/2, -1 - n, 3/2, (1/2)*(1 - sin(c + d*x)), (b*(1 - sin(c + d*x)))/(a + b))*cos(c + d*x)*(a + b*sin(c + d*x))^n)/(((a + b*sin(c + d*x))/(a + b))^n*(b^2*d*(2 + n)*sqrt(1 + sin(c + d*x)))) - (sqrt(2)*(a^2 + b^2*(1 + n))*AppellF1(1/2, 1/2, -n, 3/2, (1/2)*(1 - sin(c + d*x)), (b*(1 - sin(c + d*x)))/(a + b))*cos(c + d*x)*(a + b*sin(c + d*x))^n)/(((a + b*sin(c + d*x))/(a + b))^n*(b^2*d*(2 + n)*sqrt(1 + sin(c + d*x))))]
    @test_int [sin(c + d*x)^1*(a + b*sin(c + d*x))^n, x, 7, -((sqrt(2)*(a + b)*AppellF1(1/2, 1/2, -1 - n, 3/2, (1/2)*(1 - sin(c + d*x)), (b*(1 - sin(c + d*x)))/(a + b))*cos(c + d*x)*(a + b*sin(c + d*x))^n)/(((a + b*sin(c + d*x))/(a + b))^n*(b*d*sqrt(1 + sin(c + d*x))))) + (sqrt(2)*a*AppellF1(1/2, 1/2, -n, 3/2, (1/2)*(1 - sin(c + d*x)), (b*(1 - sin(c + d*x)))/(a + b))*cos(c + d*x)*(a + b*sin(c + d*x))^n)/(((a + b*sin(c + d*x))/(a + b))^n*(b*d*sqrt(1 + sin(c + d*x))))]
    @test_int [sin(c + d*x)^0*(a + b*sin(c + d*x))^n, x, 3, -((sqrt(2)*AppellF1(1/2, 1/2, -n, 3/2, (1/2)*(1 - sin(c + d*x)), (b*(1 - sin(c + d*x)))/(a + b))*cos(c + d*x)*(a + b*sin(c + d*x))^n)/(((a + b*sin(c + d*x))/(a + b))^n*(d*sqrt(1 + sin(c + d*x)))))]
    @test_int [csc(c + d*x)^1*(a + b*sin(c + d*x))^n, x, 0, Unintegrable(csc(c + d*x)*(a + b*sin(c + d*x))^n, x)]

    #= ::Title::Closed:: =#
    #=Integrands*of*the*form*(a+b*sin(e+f*x))^m*(c+d*sin(e+f*x))^n=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^n=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^n=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [(a + a*sin(e + f*x))^1*(c - c*sin(e + f*x))^4, x, 6, (7/8)*a*c^4*x + (7*a*c^4*cos(e + f*x)^3)/(12*f) + (7*a*c^4*cos(e + f*x)*sin(e + f*x))/(8*f) + (a*cos(e + f*x)^3*(c^2 - c^2*sin(e + f*x))^2)/(5*f) + (7*a*cos(e + f*x)^3*(c^4 - c^4*sin(e + f*x)))/(20*f)]
    @test_int [(a + a*sin(e + f*x))^1*(c - c*sin(e + f*x))^3, x, 5, (5/8)*a*c^3*x + (5*a*c^3*cos(e + f*x)^3)/(12*f) + (5*a*c^3*cos(e + f*x)*sin(e + f*x))/(8*f) + (a*cos(e + f*x)^3*(c^3 - c^3*sin(e + f*x)))/(4*f)]
    @test_int [(a + a*sin(e + f*x))^1*(c - c*sin(e + f*x))^2, x, 4, (1/2)*a*c^2*x + (a*c^2*cos(e + f*x)^3)/(3*f) + (a*c^2*cos(e + f*x)*sin(e + f*x))/(2*f)]
    @test_int [(a + a*sin(e + f*x))^1*(c - c*sin(e + f*x))^1, x, 1, (a*c*x)/2 + (a*c*cos(e + f*x)*sin(e + f*x))/(2*f)]
    @test_int [(a + a*sin(e + f*x))^1/(c - c*sin(e + f*x))^1, x, 2, -((a*x)/c) + (2*a*cos(e + f*x))/(f*(c - c*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^1/(c - c*sin(e + f*x))^2, x, 2, (a*c*cos(e + f*x)^3)/(3*f*(c - c*sin(e + f*x))^3)]
    @test_int [(a + a*sin(e + f*x))^1/(c - c*sin(e + f*x))^3, x, 3, (a*c*cos(e + f*x)^3)/(5*f*(c - c*sin(e + f*x))^4) + (a*cos(e + f*x)^3)/(15*f*(c - c*sin(e + f*x))^3)]
    @test_int [(a + a*sin(e + f*x))^1/(c - c*sin(e + f*x))^4, x, 4, (a*c*cos(e + f*x)^3)/(7*f*(c - c*sin(e + f*x))^5) + (2*a*cos(e + f*x)^3)/(35*f*(c - c*sin(e + f*x))^4) + (2*a*cos(e + f*x)^3)/(105*c*f*(c - c*sin(e + f*x))^3)]
    @test_int [(a + a*sin(e + f*x))^1/(c - c*sin(e + f*x))^5, x, 5, (a*c*cos(e + f*x)^3)/(9*f*(c - c*sin(e + f*x))^6) + (a*cos(e + f*x)^3)/(21*f*(c - c*sin(e + f*x))^5) + (2*a*cos(e + f*x)^3)/(105*c*f*(c - c*sin(e + f*x))^4) + (2*a*c*cos(e + f*x)^3)/(315*f*(c^2 - c^2*sin(e + f*x))^3)]

    @test_int [(a + a*sin(e + f*x))^2*(c - c*sin(e + f*x))^5, x, 7, (9/16)*a^2*c^5*x + (3*a^2*c^5*cos(e + f*x)^5)/(10*f) + (9*a^2*c^5*cos(e + f*x)*sin(e + f*x))/(16*f) + (3*a^2*c^5*cos(e + f*x)^3*sin(e + f*x))/(8*f) + (a^2*c^3*cos(e + f*x)^5*(c - c*sin(e + f*x))^2)/(7*f) + (3*a^2*cos(e + f*x)^5*(c^5 - c^5*sin(e + f*x)))/(14*f)]
    @test_int [(a + a*sin(e + f*x))^2*(c - c*sin(e + f*x))^4, x, 6, (7/16)*a^2*c^4*x + (7*a^2*c^4*cos(e + f*x)^5)/(30*f) + (7*a^2*c^4*cos(e + f*x)*sin(e + f*x))/(16*f) + (7*a^2*c^4*cos(e + f*x)^3*sin(e + f*x))/(24*f) + (a^2*cos(e + f*x)^5*(c^4 - c^4*sin(e + f*x)))/(6*f)]
    @test_int [(a + a*sin(e + f*x))^2*(c - c*sin(e + f*x))^3, x, 5, (3/8)*a^2*c^3*x + (a^2*c^3*cos(e + f*x)^5)/(5*f) + (3*a^2*c^3*cos(e + f*x)*sin(e + f*x))/(8*f) + (a^2*c^3*cos(e + f*x)^3*sin(e + f*x))/(4*f)]
    @test_int [(a + a*sin(e + f*x))^2*(c - c*sin(e + f*x))^2, x, 4, (3/8)*a^2*c^2*x + (3*a^2*c^2*cos(e + f*x)*sin(e + f*x))/(8*f) + (a^2*c^2*cos(e + f*x)^3*sin(e + f*x))/(4*f)]
    @test_int [(a + a*sin(e + f*x))^2*(c - c*sin(e + f*x))^1, x, 4, (1/2)*a^2*c*x - (a^2*c*cos(e + f*x)^3)/(3*f) + (a^2*c*cos(e + f*x)*sin(e + f*x))/(2*f)]
    @test_int [(a + a*sin(e + f*x))^2/(c - c*sin(e + f*x))^1, x, 4, -((3*a^2*x)/c) + (3*a^2*cos(e + f*x))/(c*f) + (2*a^2*c*cos(e + f*x)^3)/(f*(c - c*sin(e + f*x))^2)]
    @test_int [(a + a*sin(e + f*x))^2/(c - c*sin(e + f*x))^2, x, 4, (a^2*x)/c^2 + (2*a^2*c*cos(e + f*x)^3)/(3*f*(c - c*sin(e + f*x))^3) - (2*a^2*cos(e + f*x))/(f*(c^2 - c^2*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^2/(c - c*sin(e + f*x))^3, x, 2, (a^2*c^2*cos(e + f*x)^5)/(5*f*(c - c*sin(e + f*x))^5)]
    @test_int [(a + a*sin(e + f*x))^2/(c - c*sin(e + f*x))^4, x, 3, (a^2*c^2*cos(e + f*x)^5)/(7*f*(c - c*sin(e + f*x))^6) + (a^2*c*cos(e + f*x)^5)/(35*f*(c - c*sin(e + f*x))^5)]
    @test_int [(a + a*sin(e + f*x))^2/(c - c*sin(e + f*x))^5, x, 4, (a^2*c^2*cos(e + f*x)^5)/(9*f*(c - c*sin(e + f*x))^7) + (2*a^2*c*cos(e + f*x)^5)/(63*f*(c - c*sin(e + f*x))^6) + (2*a^2*cos(e + f*x)^5)/(315*f*(c - c*sin(e + f*x))^5)]
    @test_int [(a + a*sin(e + f*x))^2/(c - c*sin(e + f*x))^6, x, 5, (a^2*c^2*cos(e + f*x)^5)/(11*f*(c - c*sin(e + f*x))^8) + (a^2*c*cos(e + f*x)^5)/(33*f*(c - c*sin(e + f*x))^7) + (2*a^2*cos(e + f*x)^5)/(231*f*(c - c*sin(e + f*x))^6) + (2*a^2*cos(e + f*x)^5)/(1155*c*f*(c - c*sin(e + f*x))^5)]

    @test_int [(a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^6, x, 8, (55/128)*a^3*c^6*x + (11*a^3*c^6*cos(e + f*x)^7)/(56*f) + (55*a^3*c^6*cos(e + f*x)*sin(e + f*x))/(128*f) + (55*a^3*c^6*cos(e + f*x)^3*sin(e + f*x))/(192*f) + (11*a^3*c^6*cos(e + f*x)^5*sin(e + f*x))/(48*f) + (a^3*cos(e + f*x)^7*(c^3 - c^3*sin(e + f*x))^2)/(9*f) + (11*a^3*cos(e + f*x)^7*(c^6 - c^6*sin(e + f*x)))/(72*f)]
    @test_int [(a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^5, x, 7, (45/128)*a^3*c^5*x + (9*a^3*c^5*cos(e + f*x)^7)/(56*f) + (45*a^3*c^5*cos(e + f*x)*sin(e + f*x))/(128*f) + (15*a^3*c^5*cos(e + f*x)^3*sin(e + f*x))/(64*f) + (3*a^3*c^5*cos(e + f*x)^5*sin(e + f*x))/(16*f) + (a^3*cos(e + f*x)^7*(c^5 - c^5*sin(e + f*x)))/(8*f)]
    @test_int [(a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^4, x, 6, (5/16)*a^3*c^4*x + (a^3*c^4*cos(e + f*x)^7)/(7*f) + (5*a^3*c^4*cos(e + f*x)*sin(e + f*x))/(16*f) + (5*a^3*c^4*cos(e + f*x)^3*sin(e + f*x))/(24*f) + (a^3*c^4*cos(e + f*x)^5*sin(e + f*x))/(6*f)]
    @test_int [(a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^3, x, 5, (5/16)*a^3*c^3*x + (5*a^3*c^3*cos(e + f*x)*sin(e + f*x))/(16*f) + (5*a^3*c^3*cos(e + f*x)^3*sin(e + f*x))/(24*f) + (a^3*c^3*cos(e + f*x)^5*sin(e + f*x))/(6*f)]
    @test_int [(a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^2, x, 5, (3/8)*a^3*c^2*x - (a^3*c^2*cos(e + f*x)^5)/(5*f) + (3*a^3*c^2*cos(e + f*x)*sin(e + f*x))/(8*f) + (a^3*c^2*cos(e + f*x)^3*sin(e + f*x))/(4*f)]
    @test_int [(a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^1, x, 5, (5/8)*a^3*c*x - (5*a^3*c*cos(e + f*x)^3)/(12*f) + (5*a^3*c*cos(e + f*x)*sin(e + f*x))/(8*f) - (c*cos(e + f*x)^3*(a^3 + a^3*sin(e + f*x)))/(4*f)]
    @test_int [(a + a*sin(e + f*x))^3/(c - c*sin(e + f*x))^1, x, 5, -((15*a^3*x)/(2*c)) + (15*a^3*cos(e + f*x))/(2*c*f) + (2*a^3*c^2*cos(e + f*x)^5)/(f*(c - c*sin(e + f*x))^3) + (5*a^3*cos(e + f*x)^3)/(2*f*(c - c*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^3/(c - c*sin(e + f*x))^2, x, 5, (5*a^3*x)/c^2 - (5*a^3*cos(e + f*x))/(c^2*f) + (2*a^3*c^2*cos(e + f*x)^5)/(3*f*(c - c*sin(e + f*x))^4) - (10*a^3*cos(e + f*x)^3)/(3*f*(c - c*sin(e + f*x))^2)]
    @test_int [(a + a*sin(e + f*x))^3/(c - c*sin(e + f*x))^3, x, 5, -((a^3*x)/c^3) + (2*a^3*c^2*cos(e + f*x)^5)/(5*f*(c - c*sin(e + f*x))^5) - (2*a^3*cos(e + f*x)^3)/(3*f*(c - c*sin(e + f*x))^3) + (2*a^3*cos(e + f*x))/(f*(c^3 - c^3*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^3/(c - c*sin(e + f*x))^4, x, 2, (a^3*c^3*cos(e + f*x)^7)/(7*f*(c - c*sin(e + f*x))^7)]
    @test_int [(a + a*sin(e + f*x))^3/(c - c*sin(e + f*x))^5, x, 3, (a^3*c^3*cos(e + f*x)^7)/(9*f*(c - c*sin(e + f*x))^8) + (a^3*c^2*cos(e + f*x)^7)/(63*f*(c - c*sin(e + f*x))^7)]
    @test_int [(a + a*sin(e + f*x))^3/(c - c*sin(e + f*x))^6, x, 4, (a^3*c^3*cos(e + f*x)^7)/(11*f*(c - c*sin(e + f*x))^9) + (2*a^3*c^2*cos(e + f*x)^7)/(99*f*(c - c*sin(e + f*x))^8) + (2*a^3*c*cos(e + f*x)^7)/(693*f*(c - c*sin(e + f*x))^7)]
    @test_int [(a + a*sin(e + f*x))^3/(c - c*sin(e + f*x))^7, x, 5, (a^3*c^3*cos(e + f*x)^7)/(13*f*(c - c*sin(e + f*x))^10) + (3*a^3*c^2*cos(e + f*x)^7)/(143*f*(c - c*sin(e + f*x))^9) + (2*a^3*c*cos(e + f*x)^7)/(429*f*(c - c*sin(e + f*x))^8) + (2*a^3*cos(e + f*x)^7)/(3003*f*(c - c*sin(e + f*x))^7)]
    @test_int [(a + a*sin(e + f*x))^3/(c - c*sin(e + f*x))^8, x, 6, (a^3*c^3*cos(e + f*x)^7)/(15*f*(c - c*sin(e + f*x))^11) + (4*a^3*c^2*cos(e + f*x)^7)/(195*f*(c - c*sin(e + f*x))^10) + (4*a^3*c*cos(e + f*x)^7)/(715*f*(c - c*sin(e + f*x))^9) + (8*a^3*cos(e + f*x)^7)/(6435*f*(c - c*sin(e + f*x))^8) + (8*a^3*cos(e + f*x)^7)/(45045*c*f*(c - c*sin(e + f*x))^7)]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [1/(a + a*sin(e + f*x))^1*(c - c*sin(e + f*x))^4, x, 6, -((35*c^4*x)/(2*a)) - (35*c^4*cos(e + f*x)^3)/(3*a*f) - (35*c^4*cos(e + f*x)*sin(e + f*x))/(2*a*f) - (2*a^3*c^4*cos(e + f*x)^7)/(f*(a + a*sin(e + f*x))^4) - (14*a*c^4*cos(e + f*x)^5)/(f*(a + a*sin(e + f*x))^2)]
    @test_int [1/(a + a*sin(e + f*x))^1*(c - c*sin(e + f*x))^3, x, 5, -((15*c^3*x)/(2*a)) - (15*c^3*cos(e + f*x))/(2*a*f) - (2*a^2*c^3*cos(e + f*x)^5)/(f*(a + a*sin(e + f*x))^3) - (5*c^3*cos(e + f*x)^3)/(2*f*(a + a*sin(e + f*x)))]
    @test_int [1/(a + a*sin(e + f*x))^1*(c - c*sin(e + f*x))^2, x, 4, -((3*c^2*x)/a) - (3*c^2*cos(e + f*x))/(a*f) - (2*a*c^2*cos(e + f*x)^3)/(f*(a + a*sin(e + f*x))^2)]
    @test_int [1/(a + a*sin(e + f*x))^1*(c - c*sin(e + f*x))^1, x, 2, -((c*x)/a) - (2*c*cos(e + f*x))/(f*(a + a*sin(e + f*x)))]
    @test_int [1/(a + a*sin(e + f*x))^1/(c - c*sin(e + f*x))^1, x, 3, tan(e + f*x)/(a*c*f)]
    @test_int [1/(a + a*sin(e + f*x))^1/(c - c*sin(e + f*x))^2, x, 4, sec(e + f*x)/(3*a*f*(c^2 - c^2*sin(e + f*x))) + (2*tan(e + f*x))/(3*a*c^2*f)]
    @test_int [1/(a + a*sin(e + f*x))^1/(c - c*sin(e + f*x))^3, x, 5, sec(e + f*x)/(5*a*c*f*(c - c*sin(e + f*x))^2) + sec(e + f*x)/(5*a*f*(c^3 - c^3*sin(e + f*x))) + (2*tan(e + f*x))/(5*a*c^3*f)]
    @test_int [1/(a + a*sin(e + f*x))^1/(c - c*sin(e + f*x))^4, x, 6, sec(e + f*x)/(7*a*c*f*(c - c*sin(e + f*x))^3) + (4*sec(e + f*x))/(35*a*f*(c^2 - c^2*sin(e + f*x))^2) + (4*sec(e + f*x))/(35*a*f*(c^4 - c^4*sin(e + f*x))) + (8*tan(e + f*x))/(35*a*c^4*f)]

    @test_int [1/(a + a*sin(e + f*x))^2*(c - c*sin(e + f*x))^5, x, 7, (105*c^5*x)/(2*a^2) + (35*c^5*cos(e + f*x)^3)/(a^2*f) + (105*c^5*cos(e + f*x)*sin(e + f*x))/(2*a^2*f) - (2*a^4*c^5*cos(e + f*x)^9)/(3*f*(a + a*sin(e + f*x))^6) + (6*a^2*c^5*cos(e + f*x)^7)/(f*(a + a*sin(e + f*x))^4) + (42*c^5*cos(e + f*x)^5)/(f*(a + a*sin(e + f*x))^2)]
    @test_int [1/(a + a*sin(e + f*x))^2*(c - c*sin(e + f*x))^4, x, 6, (35*c^4*x)/(2*a^2) + (35*c^4*cos(e + f*x))/(2*a^2*f) - (2*a^3*c^4*cos(e + f*x)^7)/(3*f*(a + a*sin(e + f*x))^5) + (14*a^4*c^4*cos(e + f*x)^5)/(3*f*(a^2 + a^2*sin(e + f*x))^3) + (35*c^4*cos(e + f*x)^3)/(6*f*(a^2 + a^2*sin(e + f*x)))]
    @test_int [1/(a + a*sin(e + f*x))^2*(c - c*sin(e + f*x))^3, x, 5, (5*c^3*x)/a^2 + (5*c^3*cos(e + f*x))/(a^2*f) - (2*a^2*c^3*cos(e + f*x)^5)/(3*f*(a + a*sin(e + f*x))^4) + (10*c^3*cos(e + f*x)^3)/(3*f*(a + a*sin(e + f*x))^2)]
    @test_int [1/(a + a*sin(e + f*x))^2*(c - c*sin(e + f*x))^2, x, 4, (c^2*x)/a^2 - (2*a*c^2*cos(e + f*x)^3)/(3*f*(a + a*sin(e + f*x))^3) + (2*c^2*cos(e + f*x))/(f*(a^2 + a^2*sin(e + f*x)))]
    @test_int [1/(a + a*sin(e + f*x))^2*(c - c*sin(e + f*x))^1, x, 2, -((a*c*cos(e + f*x)^3)/(3*f*(a + a*sin(e + f*x))^3))]
    @test_int [1/(a + a*sin(e + f*x))^2/(c - c*sin(e + f*x))^1, x, 4, -(sec(e + f*x)/(3*c*f*(a^2 + a^2*sin(e + f*x)))) + (2*tan(e + f*x))/(3*a^2*c*f)]
    @test_int [1/(a + a*sin(e + f*x))^2/(c - c*sin(e + f*x))^2, x, 3, tan(e + f*x)/(a^2*c^2*f) + tan(e + f*x)^3/(3*a^2*c^2*f)]
    @test_int [1/(a + a*sin(e + f*x))^2/(c - c*sin(e + f*x))^3, x, 4, sec(e + f*x)^3/(5*a^2*f*(c^3 - c^3*sin(e + f*x))) + (4*tan(e + f*x))/(5*a^2*c^3*f) + (4*tan(e + f*x)^3)/(15*a^2*c^3*f)]
    @test_int [1/(a + a*sin(e + f*x))^2/(c - c*sin(e + f*x))^4, x, 5, sec(e + f*x)^3/(7*a^2*f*(c^2 - c^2*sin(e + f*x))^2) + sec(e + f*x)^3/(7*a^2*f*(c^4 - c^4*sin(e + f*x))) + (4*tan(e + f*x))/(7*a^2*c^4*f) + (4*tan(e + f*x)^3)/(21*a^2*c^4*f)]
    @test_int [1/(a + a*sin(e + f*x))^2/(c - c*sin(e + f*x))^5, x, 6, sec(e + f*x)^3/(9*a^2*c^2*f*(c - c*sin(e + f*x))^3) + (2*sec(e + f*x)^3)/(21*a^2*c^3*f*(c - c*sin(e + f*x))^2) + (2*sec(e + f*x)^3)/(21*a^2*f*(c^5 - c^5*sin(e + f*x))) + (8*tan(e + f*x))/(21*a^2*c^5*f) + (8*tan(e + f*x)^3)/(63*a^2*c^5*f)]

    @test_int [1/(a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^5, x, 7, -((63*c^5*x)/(2*a^3)) - (63*c^5*cos(e + f*x))/(2*a^3*f) - (2*a^4*c^5*cos(e + f*x)^9)/(5*f*(a + a*sin(e + f*x))^7) + (6*a^2*c^5*cos(e + f*x)^7)/(5*f*(a + a*sin(e + f*x))^5) - (42*c^5*cos(e + f*x)^5)/(5*f*(a + a*sin(e + f*x))^3) - (21*c^5*cos(e + f*x)^3)/(2*f*(a^3 + a^3*sin(e + f*x)))]
    @test_int [1/(a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^4, x, 6, -((7*c^4*x)/a^3) - (7*c^4*cos(e + f*x))/(a^3*f) - (2*a^3*c^4*cos(e + f*x)^7)/(5*f*(a + a*sin(e + f*x))^6) + (14*a*c^4*cos(e + f*x)^5)/(15*f*(a + a*sin(e + f*x))^4) - (14*c^4*cos(e + f*x)^3)/(3*a*f*(a + a*sin(e + f*x))^2)]
    @test_int [1/(a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^3, x, 5, -((c^3*x)/a^3) - (2*a^2*c^3*cos(e + f*x)^5)/(5*f*(a + a*sin(e + f*x))^5) + (2*c^3*cos(e + f*x)^3)/(3*f*(a + a*sin(e + f*x))^3) - (2*c^3*cos(e + f*x))/(f*(a^3 + a^3*sin(e + f*x)))]
    @test_int [1/(a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^2, x, 2, -((a^2*c^2*cos(e + f*x)^5)/(5*f*(a + a*sin(e + f*x))^5))]
    @test_int [1/(a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^1, x, 3, -((a*c*cos(e + f*x)^3)/(5*f*(a + a*sin(e + f*x))^4)) - (c*cos(e + f*x)^3)/(15*f*(a + a*sin(e + f*x))^3)]
    @test_int [1/(a + a*sin(e + f*x))^3/(c - c*sin(e + f*x))^1, x, 5, -(sec(e + f*x)/(5*a*c*f*(a + a*sin(e + f*x))^2)) - sec(e + f*x)/(5*c*f*(a^3 + a^3*sin(e + f*x))) + (2*tan(e + f*x))/(5*a^3*c*f)]
    @test_int [1/(a + a*sin(e + f*x))^3/(c - c*sin(e + f*x))^2, x, 4, -(sec(e + f*x)^3/(5*c^2*f*(a^3 + a^3*sin(e + f*x)))) + (4*tan(e + f*x))/(5*a^3*c^2*f) + (4*tan(e + f*x)^3)/(15*a^3*c^2*f)]
    @test_int [1/(a + a*sin(e + f*x))^3/(c - c*sin(e + f*x))^3, x, 3, tan(e + f*x)/(a^3*c^3*f) + (2*tan(e + f*x)^3)/(3*a^3*c^3*f) + tan(e + f*x)^5/(5*a^3*c^3*f)]
    @test_int [1/(a + a*sin(e + f*x))^3/(c - c*sin(e + f*x))^4, x, 4, sec(e + f*x)^5/(7*a^3*f*(c^4 - c^4*sin(e + f*x))) + (6*tan(e + f*x))/(7*a^3*c^4*f) + (4*tan(e + f*x)^3)/(7*a^3*c^4*f) + (6*tan(e + f*x)^5)/(35*a^3*c^4*f)]
    @test_int [1/(a + a*sin(e + f*x))^3/(c - c*sin(e + f*x))^5, x, 5, sec(e + f*x)^5/(9*a^3*c^3*f*(c - c*sin(e + f*x))^2) + sec(e + f*x)^5/(9*a^3*f*(c^5 - c^5*sin(e + f*x))) + (2*tan(e + f*x))/(3*a^3*c^5*f) + (4*tan(e + f*x)^3)/(9*a^3*c^5*f) + (2*tan(e + f*x)^5)/(15*a^3*c^5*f)]
    @test_int [1/(a + a*sin(e + f*x))^3/(c - c*sin(e + f*x))^6, x, 6, sec(e + f*x)^5/(11*a^3*f*(c^2 - c^2*sin(e + f*x))^3) + (8*sec(e + f*x)^5)/(99*a^3*f*(c^3 - c^3*sin(e + f*x))^2) + (8*sec(e + f*x)^5)/(99*a^3*f*(c^6 - c^6*sin(e + f*x))) + (16*tan(e + f*x))/(33*a^3*c^6*f) + (32*tan(e + f*x)^3)/(99*a^3*c^6*f) + (16*tan(e + f*x)^5)/(165*a^3*c^6*f)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(n/2)=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [(a + a*sin(e + f*x))^1*(c - c*sin(e + f*x))^(7/2), x, 5, (256*a*c^5*cos(e + f*x)^3)/(315*f*(c - c*sin(e + f*x))^(3/2)) + (64*a*c^4*cos(e + f*x)^3)/(105*f*sqrt(c - c*sin(e + f*x))) + (8*a*c^3*cos(e + f*x)^3*sqrt(c - c*sin(e + f*x)))/(21*f) + (2*a*c^2*cos(e + f*x)^3*(c - c*sin(e + f*x))^(3/2))/(9*f)]
    @test_int [(a + a*sin(e + f*x))^1*(c - c*sin(e + f*x))^(5/2), x, 4, (64*a*c^4*cos(e + f*x)^3)/(105*f*(c - c*sin(e + f*x))^(3/2)) + (16*a*c^3*cos(e + f*x)^3)/(35*f*sqrt(c - c*sin(e + f*x))) + (2*a*c^2*cos(e + f*x)^3*sqrt(c - c*sin(e + f*x)))/(7*f)]
    @test_int [(a + a*sin(e + f*x))^1*(c - c*sin(e + f*x))^(3/2), x, 3, (8*a*c^3*cos(e + f*x)^3)/(15*f*(c - c*sin(e + f*x))^(3/2)) + (2*a*c^2*cos(e + f*x)^3)/(5*f*sqrt(c - c*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^1*(c - c*sin(e + f*x))^(1/2), x, 2, (2*a*c^2*cos(e + f*x)^3)/(3*f*(c - c*sin(e + f*x))^(3/2))]
    @test_int [(a + a*sin(e + f*x))^1/(c - c*sin(e + f*x))^(1/2), x, 4, (2*sqrt(2)*a*atanh((sqrt(c)*cos(e + f*x))/(sqrt(2)*sqrt(c - c*sin(e + f*x)))))/(sqrt(c)*f) - (2*a*cos(e + f*x))/(f*sqrt(c - c*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^1/(c - c*sin(e + f*x))^(3/2), x, 4, -((a*atanh((sqrt(c)*cos(e + f*x))/(sqrt(2)*sqrt(c - c*sin(e + f*x)))))/(sqrt(2)*c^(3/2)*f)) + (a*cos(e + f*x))/(f*(c - c*sin(e + f*x))^(3/2))]
    @test_int [(a + a*sin(e + f*x))^1/(c - c*sin(e + f*x))^(5/2), x, 5, -((a*atanh((sqrt(c)*cos(e + f*x))/(sqrt(2)*sqrt(c - c*sin(e + f*x)))))/(8*sqrt(2)*c^(5/2)*f)) + (a*cos(e + f*x))/(2*f*(c - c*sin(e + f*x))^(5/2)) - (a*cos(e + f*x))/(8*c*f*(c - c*sin(e + f*x))^(3/2))]
    @test_int [(a + a*sin(e + f*x))^1/(c - c*sin(e + f*x))^(7/2), x, 6, -((a*atanh((sqrt(c)*cos(e + f*x))/(sqrt(2)*sqrt(c - c*sin(e + f*x)))))/(32*sqrt(2)*c^(7/2)*f)) + (a*cos(e + f*x))/(3*f*(c - c*sin(e + f*x))^(7/2)) - (a*cos(e + f*x))/(24*c*f*(c - c*sin(e + f*x))^(5/2)) - (a*cos(e + f*x))/(32*c^2*f*(c - c*sin(e + f*x))^(3/2))]

    @test_int [(a + a*sin(e + f*x))^2*(c - c*sin(e + f*x))^(7/2), x, 5, (256*a^2*c^6*cos(e + f*x)^5)/(1155*f*(c - c*sin(e + f*x))^(5/2)) + (64*a^2*c^5*cos(e + f*x)^5)/(231*f*(c - c*sin(e + f*x))^(3/2)) + (8*a^2*c^4*cos(e + f*x)^5)/(33*f*sqrt(c - c*sin(e + f*x))) + (2*a^2*c^3*cos(e + f*x)^5*sqrt(c - c*sin(e + f*x)))/(11*f)]
    @test_int [(a + a*sin(e + f*x))^2*(c - c*sin(e + f*x))^(5/2), x, 4, (64*a^2*c^5*cos(e + f*x)^5)/(315*f*(c - c*sin(e + f*x))^(5/2)) + (16*a^2*c^4*cos(e + f*x)^5)/(63*f*(c - c*sin(e + f*x))^(3/2)) + (2*a^2*c^3*cos(e + f*x)^5)/(9*f*sqrt(c - c*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^2*(c - c*sin(e + f*x))^(3/2), x, 3, (8*a^2*c^4*cos(e + f*x)^5)/(35*f*(c - c*sin(e + f*x))^(5/2)) + (2*a^2*c^3*cos(e + f*x)^5)/(7*f*(c - c*sin(e + f*x))^(3/2))]
    @test_int [(a + a*sin(e + f*x))^2*(c - c*sin(e + f*x))^(1/2), x, 2, (2*a^2*c^3*cos(e + f*x)^5)/(5*f*(c - c*sin(e + f*x))^(5/2))]
    @test_int [(a + a*sin(e + f*x))^2/(c - c*sin(e + f*x))^(1/2), x, 5, (4*sqrt(2)*a^2*atanh((sqrt(c)*cos(e + f*x))/(sqrt(2)*sqrt(c - c*sin(e + f*x)))))/(sqrt(c)*f) - (2*a^2*c*cos(e + f*x)^3)/(3*f*(c - c*sin(e + f*x))^(3/2)) - (4*a^2*cos(e + f*x))/(f*sqrt(c - c*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^2/(c - c*sin(e + f*x))^(3/2), x, 5, -((3*sqrt(2)*a^2*atanh((sqrt(c)*cos(e + f*x))/(sqrt(2)*sqrt(c - c*sin(e + f*x)))))/(c^(3/2)*f)) + (a^2*c*cos(e + f*x)^3)/(f*(c - c*sin(e + f*x))^(5/2)) + (3*a^2*cos(e + f*x))/(c*f*sqrt(c - c*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^2/(c - c*sin(e + f*x))^(5/2), x, 5, (3*a^2*atanh((sqrt(c)*cos(e + f*x))/(sqrt(2)*sqrt(c - c*sin(e + f*x)))))/(4*sqrt(2)*c^(5/2)*f) + (a^2*c*cos(e + f*x)^3)/(2*f*(c - c*sin(e + f*x))^(7/2)) - (3*a^2*cos(e + f*x))/(4*c*f*(c - c*sin(e + f*x))^(3/2))]
    @test_int [(a + a*sin(e + f*x))^2/(c - c*sin(e + f*x))^(7/2), x, 6, (a^2*atanh((sqrt(c)*cos(e + f*x))/(sqrt(2)*sqrt(c - c*sin(e + f*x)))))/(16*sqrt(2)*c^(7/2)*f) + (a^2*c*cos(e + f*x)^3)/(3*f*(c - c*sin(e + f*x))^(9/2)) - (a^2*cos(e + f*x))/(4*c*f*(c - c*sin(e + f*x))^(5/2)) + (a^2*cos(e + f*x))/(16*c^2*f*(c - c*sin(e + f*x))^(3/2))]
    @test_int [(a + a*sin(e + f*x))^2/(c - c*sin(e + f*x))^(9/2), x, 7, (3*a^2*atanh((sqrt(c)*cos(e + f*x))/(sqrt(2)*sqrt(c - c*sin(e + f*x)))))/(256*sqrt(2)*c^(9/2)*f) + (a^2*c*cos(e + f*x)^3)/(4*f*(c - c*sin(e + f*x))^(11/2)) - (a^2*cos(e + f*x))/(8*c*f*(c - c*sin(e + f*x))^(7/2)) + (a^2*cos(e + f*x))/(64*c^2*f*(c - c*sin(e + f*x))^(5/2)) + (3*a^2*cos(e + f*x))/(256*c^3*f*(c - c*sin(e + f*x))^(3/2))]

    @test_int [(a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^(7/2), x, 5, (256*a^3*c^7*cos(e + f*x)^7)/(3003*f*(c - c*sin(e + f*x))^(7/2)) + (64*a^3*c^6*cos(e + f*x)^7)/(429*f*(c - c*sin(e + f*x))^(5/2)) + (24*a^3*c^5*cos(e + f*x)^7)/(143*f*(c - c*sin(e + f*x))^(3/2)) + (2*a^3*c^4*cos(e + f*x)^7)/(13*f*sqrt(c - c*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^(5/2), x, 4, (64*a^3*c^6*cos(e + f*x)^7)/(693*f*(c - c*sin(e + f*x))^(7/2)) + (16*a^3*c^5*cos(e + f*x)^7)/(99*f*(c - c*sin(e + f*x))^(5/2)) + (2*a^3*c^4*cos(e + f*x)^7)/(11*f*(c - c*sin(e + f*x))^(3/2))]
    @test_int [(a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^(3/2), x, 3, (8*a^3*c^5*cos(e + f*x)^7)/(63*f*(c - c*sin(e + f*x))^(7/2)) + (2*a^3*c^4*cos(e + f*x)^7)/(9*f*(c - c*sin(e + f*x))^(5/2))]
    @test_int [(a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^(1/2), x, 2, (2*a^3*c^4*cos(e + f*x)^7)/(7*f*(c - c*sin(e + f*x))^(7/2))]
    @test_int [(a + a*sin(e + f*x))^3/(c - c*sin(e + f*x))^(1/2), x, 6, (8*sqrt(2)*a^3*atanh((sqrt(c)*cos(e + f*x))/(sqrt(2)*sqrt(c - c*sin(e + f*x)))))/(sqrt(c)*f) - (2*a^3*c^2*cos(e + f*x)^5)/(5*f*(c - c*sin(e + f*x))^(5/2)) - (4*a^3*c*cos(e + f*x)^3)/(3*f*(c - c*sin(e + f*x))^(3/2)) - (8*a^3*cos(e + f*x))/(f*sqrt(c - c*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^3/(c - c*sin(e + f*x))^(3/2), x, 6, -((10*sqrt(2)*a^3*atanh((sqrt(c)*cos(e + f*x))/(sqrt(2)*sqrt(c - c*sin(e + f*x)))))/(c^(3/2)*f)) + (a^3*c^2*cos(e + f*x)^5)/(f*(c - c*sin(e + f*x))^(7/2)) + (5*a^3*cos(e + f*x)^3)/(3*f*(c - c*sin(e + f*x))^(3/2)) + (10*a^3*cos(e + f*x))/(c*f*sqrt(c - c*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^3/(c - c*sin(e + f*x))^(5/2), x, 6, (15*a^3*atanh((sqrt(c)*cos(e + f*x))/(sqrt(2)*sqrt(c - c*sin(e + f*x)))))/(2*sqrt(2)*c^(5/2)*f) + (a^3*c^2*cos(e + f*x)^5)/(2*f*(c - c*sin(e + f*x))^(9/2)) - (5*a^3*cos(e + f*x)^3)/(4*f*(c - c*sin(e + f*x))^(5/2)) - (15*a^3*cos(e + f*x))/(4*c^2*f*sqrt(c - c*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^3/(c - c*sin(e + f*x))^(7/2), x, 6, -((5*a^3*atanh((sqrt(c)*cos(e + f*x))/(sqrt(2)*sqrt(c - c*sin(e + f*x)))))/(8*sqrt(2)*c^(7/2)*f)) + (a^3*c^2*cos(e + f*x)^5)/(3*f*(c - c*sin(e + f*x))^(11/2)) - (5*a^3*cos(e + f*x)^3)/(12*f*(c - c*sin(e + f*x))^(7/2)) + (5*a^3*cos(e + f*x))/(8*c^2*f*(c - c*sin(e + f*x))^(3/2))]
    @test_int [(a + a*sin(e + f*x))^3/(c - c*sin(e + f*x))^(9/2), x, 7, -((5*a^3*atanh((sqrt(c)*cos(e + f*x))/(sqrt(2)*sqrt(c - c*sin(e + f*x)))))/(128*sqrt(2)*c^(9/2)*f)) + (a^3*c^2*cos(e + f*x)^5)/(4*f*(c - c*sin(e + f*x))^(13/2)) - (5*a^3*cos(e + f*x)^3)/(24*f*(c - c*sin(e + f*x))^(9/2)) + (5*a^3*cos(e + f*x))/(32*c^2*f*(c - c*sin(e + f*x))^(5/2)) - (5*a^3*cos(e + f*x))/(128*c^3*f*(c - c*sin(e + f*x))^(3/2))]
    @test_int [(a + a*sin(e + f*x))^3/(c - c*sin(e + f*x))^(11/2), x, 8, -((3*a^3*atanh((sqrt(c)*cos(e + f*x))/(sqrt(2)*sqrt(c - c*sin(e + f*x)))))/(512*sqrt(2)*c^(11/2)*f)) + (a^3*c^2*cos(e + f*x)^5)/(5*f*(c - c*sin(e + f*x))^(15/2)) - (a^3*cos(e + f*x)^3)/(8*f*(c - c*sin(e + f*x))^(11/2)) + (a^3*cos(e + f*x))/(16*c^2*f*(c - c*sin(e + f*x))^(7/2)) - (a^3*cos(e + f*x))/(128*c^3*f*(c - c*sin(e + f*x))^(5/2)) - (3*a^3*cos(e + f*x))/(512*c^4*f*(c - c*sin(e + f*x))^(3/2))]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [1/(a + a*sin(e + f*x))^1*(c - c*sin(e + f*x))^(7/2), x, 5, -((256*c^3*sec(e + f*x)*sqrt(c - c*sin(e + f*x)))/(5*a*f)) + (64*c^2*sec(e + f*x)*(c - c*sin(e + f*x))^(3/2))/(5*a*f) + (8*c*sec(e + f*x)*(c - c*sin(e + f*x))^(5/2))/(5*a*f) + (2*sec(e + f*x)*(c - c*sin(e + f*x))^(7/2))/(5*a*f)]
    @test_int [1/(a + a*sin(e + f*x))^1*(c - c*sin(e + f*x))^(5/2), x, 4, -((64*c^2*sec(e + f*x)*sqrt(c - c*sin(e + f*x)))/(3*a*f)) + (16*c*sec(e + f*x)*(c - c*sin(e + f*x))^(3/2))/(3*a*f) + (2*sec(e + f*x)*(c - c*sin(e + f*x))^(5/2))/(3*a*f)]
    @test_int [1/(a + a*sin(e + f*x))^1*(c - c*sin(e + f*x))^(3/2), x, 3, -((8*c*sec(e + f*x)*sqrt(c - c*sin(e + f*x)))/(a*f)) + (2*sec(e + f*x)*(c - c*sin(e + f*x))^(3/2))/(a*f)]
    @test_int [1/(a + a*sin(e + f*x))^1*(c - c*sin(e + f*x))^(1/2), x, 2, -((2*sec(e + f*x)*sqrt(c - c*sin(e + f*x)))/(a*f))]
    @test_int [1/(a + a*sin(e + f*x))^1/(c - c*sin(e + f*x))^(1/2), x, 4, atanh((sqrt(c)*cos(e + f*x))/(sqrt(2)*sqrt(c - c*sin(e + f*x))))/(sqrt(2)*a*sqrt(c)*f) - (sec(e + f*x)*sqrt(c - c*sin(e + f*x)))/(a*c*f)]
    @test_int [1/(a + a*sin(e + f*x))^1/(c - c*sin(e + f*x))^(3/2), x, 5, (3*atanh((sqrt(c)*cos(e + f*x))/(sqrt(2)*sqrt(c - c*sin(e + f*x)))))/(4*sqrt(2)*a*c^(3/2)*f) + (3*cos(e + f*x))/(4*a*f*(c - c*sin(e + f*x))^(3/2)) - sec(e + f*x)/(a*c*f*sqrt(c - c*sin(e + f*x)))]
    @test_int [1/(a + a*sin(e + f*x))^1/(c - c*sin(e + f*x))^(5/2), x, 6, (15*atanh((sqrt(c)*cos(e + f*x))/(sqrt(2)*sqrt(c - c*sin(e + f*x)))))/(32*sqrt(2)*a*c^(5/2)*f) + (15*cos(e + f*x))/(32*a*c*f*(c - c*sin(e + f*x))^(3/2)) + sec(e + f*x)/(4*a*c*f*(c - c*sin(e + f*x))^(3/2)) - (5*sec(e + f*x))/(8*a*c^2*f*sqrt(c - c*sin(e + f*x)))]

    @test_int [1/(a + a*sin(e + f*x))^2*(c - c*sin(e + f*x))^(9/2), x, 6, (4096*c^3*sec(e + f*x)^3*(c - c*sin(e + f*x))^(3/2))/(15*a^2*f) - (1024*c^2*sec(e + f*x)^3*(c - c*sin(e + f*x))^(5/2))/(5*a^2*f) + (128*c*sec(e + f*x)^3*(c - c*sin(e + f*x))^(7/2))/(5*a^2*f) + (32*sec(e + f*x)^3*(c - c*sin(e + f*x))^(9/2))/(15*a^2*f) + (2*sec(e + f*x)^3*(c - c*sin(e + f*x))^(11/2))/(5*a^2*c*f)]
    @test_int [1/(a + a*sin(e + f*x))^2*(c - c*sin(e + f*x))^(7/2), x, 5, (256*c^2*sec(e + f*x)^3*(c - c*sin(e + f*x))^(3/2))/(3*a^2*f) - (64*c*sec(e + f*x)^3*(c - c*sin(e + f*x))^(5/2))/(a^2*f) + (8*sec(e + f*x)^3*(c - c*sin(e + f*x))^(7/2))/(a^2*f) + (2*sec(e + f*x)^3*(c - c*sin(e + f*x))^(9/2))/(3*a^2*c*f)]
    @test_int [1/(a + a*sin(e + f*x))^2*(c - c*sin(e + f*x))^(5/2), x, 4, (64*c*sec(e + f*x)^3*(c - c*sin(e + f*x))^(3/2))/(3*a^2*f) - (16*sec(e + f*x)^3*(c - c*sin(e + f*x))^(5/2))/(a^2*f) + (2*sec(e + f*x)^3*(c - c*sin(e + f*x))^(7/2))/(a^2*c*f)]
    @test_int [1/(a + a*sin(e + f*x))^2*(c - c*sin(e + f*x))^(3/2), x, 3, (8*sec(e + f*x)^3*(c - c*sin(e + f*x))^(3/2))/(3*a^2*f) - (2*sec(e + f*x)^3*(c - c*sin(e + f*x))^(5/2))/(a^2*c*f)]
    @test_int [1/(a + a*sin(e + f*x))^2*(c - c*sin(e + f*x))^(1/2), x, 2, -((2*sec(e + f*x)^3*(c - c*sin(e + f*x))^(3/2))/(3*a^2*c*f))]
    @test_int [1/(a + a*sin(e + f*x))^2/(c - c*sin(e + f*x))^(1/2), x, 5, atanh((sqrt(c)*cos(e + f*x))/(sqrt(2)*sqrt(c - c*sin(e + f*x))))/(2*sqrt(2)*a^2*sqrt(c)*f) - (sec(e + f*x)*sqrt(c - c*sin(e + f*x)))/(2*a^2*c*f) - (sec(e + f*x)^3*(c - c*sin(e + f*x))^(3/2))/(3*a^2*c^2*f)]
    @test_int [1/(a + a*sin(e + f*x))^2/(c - c*sin(e + f*x))^(3/2), x, 6, (5*atanh((sqrt(c)*cos(e + f*x))/(sqrt(2)*sqrt(c - c*sin(e + f*x)))))/(8*sqrt(2)*a^2*c^(3/2)*f) + (5*cos(e + f*x))/(8*a^2*f*(c - c*sin(e + f*x))^(3/2)) - (5*sec(e + f*x))/(6*a^2*c*f*sqrt(c - c*sin(e + f*x))) - (sec(e + f*x)^3*sqrt(c - c*sin(e + f*x)))/(3*a^2*c^2*f)]
    @test_int [1/(a + a*sin(e + f*x))^2/(c - c*sin(e + f*x))^(5/2), x, 7, (35*atanh((sqrt(c)*cos(e + f*x))/(sqrt(2)*sqrt(c - c*sin(e + f*x)))))/(64*sqrt(2)*a^2*c^(5/2)*f) + (35*cos(e + f*x))/(64*a^2*c*f*(c - c*sin(e + f*x))^(3/2)) + (7*sec(e + f*x))/(24*a^2*c*f*(c - c*sin(e + f*x))^(3/2)) - (35*sec(e + f*x))/(48*a^2*c^2*f*sqrt(c - c*sin(e + f*x))) - sec(e + f*x)^3/(3*a^2*c^2*f*sqrt(c - c*sin(e + f*x)))]

    @test_int [1/(a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^(9/2), x, 6, -((4096*c^2*sec(e + f*x)^5*(c - c*sin(e + f*x))^(5/2))/(15*a^3*f)) + (1024*c*sec(e + f*x)^5*(c - c*sin(e + f*x))^(7/2))/(3*a^3*f) - (128*sec(e + f*x)^5*(c - c*sin(e + f*x))^(9/2))/(a^3*f) + (32*sec(e + f*x)^5*(c - c*sin(e + f*x))^(11/2))/(3*a^3*c*f) + (2*sec(e + f*x)^5*(c - c*sin(e + f*x))^(13/2))/(3*a^3*c^2*f)]
    @test_int [1/(a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^(7/2), x, 5, -((256*c*sec(e + f*x)^5*(c - c*sin(e + f*x))^(5/2))/(5*a^3*f)) + (64*sec(e + f*x)^5*(c - c*sin(e + f*x))^(7/2))/(a^3*f) - (24*sec(e + f*x)^5*(c - c*sin(e + f*x))^(9/2))/(a^3*c*f) + (2*sec(e + f*x)^5*(c - c*sin(e + f*x))^(11/2))/(a^3*c^2*f)]
    @test_int [1/(a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^(5/2), x, 4, -((64*sec(e + f*x)^5*(c - c*sin(e + f*x))^(5/2))/(15*a^3*f)) + (16*sec(e + f*x)^5*(c - c*sin(e + f*x))^(7/2))/(3*a^3*c*f) - (2*sec(e + f*x)^5*(c - c*sin(e + f*x))^(9/2))/(a^3*c^2*f)]
    @test_int [1/(a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^(3/2), x, 3, (8*sec(e + f*x)^5*(c - c*sin(e + f*x))^(5/2))/(15*a^3*c*f) - (2*sec(e + f*x)^5*(c - c*sin(e + f*x))^(7/2))/(3*a^3*c^2*f)]
    @test_int [1/(a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^(1/2), x, 2, -((2*sec(e + f*x)^5*(c - c*sin(e + f*x))^(5/2))/(5*a^3*c^2*f))]
    @test_int [1/(a + a*sin(e + f*x))^3/(c - c*sin(e + f*x))^(1/2), x, 6, atanh((sqrt(c)*cos(e + f*x))/(sqrt(2)*sqrt(c - c*sin(e + f*x))))/(4*sqrt(2)*a^3*sqrt(c)*f) - (sec(e + f*x)*sqrt(c - c*sin(e + f*x)))/(4*a^3*c*f) - (sec(e + f*x)^3*(c - c*sin(e + f*x))^(3/2))/(6*a^3*c^2*f) - (sec(e + f*x)^5*(c - c*sin(e + f*x))^(5/2))/(5*a^3*c^3*f)]
    @test_int [1/(a + a*sin(e + f*x))^3/(c - c*sin(e + f*x))^(3/2), x, 7, (7*atanh((sqrt(c)*cos(e + f*x))/(sqrt(2)*sqrt(c - c*sin(e + f*x)))))/(16*sqrt(2)*a^3*c^(3/2)*f) + (7*cos(e + f*x))/(16*a^3*f*(c - c*sin(e + f*x))^(3/2)) - (7*sec(e + f*x))/(12*a^3*c*f*sqrt(c - c*sin(e + f*x))) - (7*sec(e + f*x)^3*sqrt(c - c*sin(e + f*x)))/(30*a^3*c^2*f) - (sec(e + f*x)^5*(c - c*sin(e + f*x))^(3/2))/(5*a^3*c^3*f)]
    @test_int [1/(a + a*sin(e + f*x))^3/(c - c*sin(e + f*x))^(5/2), x, 8, (63*atanh((sqrt(c)*cos(e + f*x))/(sqrt(2)*sqrt(c - c*sin(e + f*x)))))/(128*sqrt(2)*a^3*c^(5/2)*f) + (63*cos(e + f*x))/(128*a^3*c*f*(c - c*sin(e + f*x))^(3/2)) + (21*sec(e + f*x))/(80*a^3*c*f*(c - c*sin(e + f*x))^(3/2)) - (21*sec(e + f*x))/(32*a^3*c^2*f*sqrt(c - c*sin(e + f*x))) - (3*sec(e + f*x)^3)/(10*a^3*c^2*f*sqrt(c - c*sin(e + f*x))) - (sec(e + f*x)^5*sqrt(c - c*sin(e + f*x)))/(5*a^3*c^3*f)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+a*sin(e+f*x))^(m/2)*(c-c*sin(e+f*x))^(n/2)=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [(a + a*sin(e + f*x))^(1/2)*(c - c*sin(e + f*x))^(7/2), x, 1, -((a*cos(e + f*x)*(c - c*sin(e + f*x))^(7/2))/(4*f*sqrt(a + a*sin(e + f*x))))]
    @test_int [(a + a*sin(e + f*x))^(1/2)*(c - c*sin(e + f*x))^(5/2), x, 1, -((a*cos(e + f*x)*(c - c*sin(e + f*x))^(5/2))/(3*f*sqrt(a + a*sin(e + f*x))))]
    @test_int [(a + a*sin(e + f*x))^(1/2)*(c - c*sin(e + f*x))^(3/2), x, 1, -((a*cos(e + f*x)*(c - c*sin(e + f*x))^(3/2))/(2*f*sqrt(a + a*sin(e + f*x))))]
    @test_int [(a + a*sin(e + f*x))^(1/2)*(c - c*sin(e + f*x))^(1/2), x, 1, -((a*cos(e + f*x)*sqrt(c - c*sin(e + f*x)))/(f*sqrt(a + a*sin(e + f*x))))]
    @test_int [(a + a*sin(e + f*x))^(1/2)/(c - c*sin(e + f*x))^(1/2), x, 3, -((a*cos(e + f*x)*log(1 - sin(e + f*x)))/(f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))))]
    @test_int [(a + a*sin(e + f*x))^(1/2)/(c - c*sin(e + f*x))^(3/2), x, 1, (a*cos(e + f*x))/(f*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2))]
    @test_int [(a + a*sin(e + f*x))^(1/2)/(c - c*sin(e + f*x))^(5/2), x, 1, (a*cos(e + f*x))/(2*f*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(5/2))]
    @test_int [(a + a*sin(e + f*x))^(1/2)/(c - c*sin(e + f*x))^(7/2), x, 1, (a*cos(e + f*x))/(3*f*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(7/2))]

    @test_int [(a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(7/2), x, 2, -((a^2*cos(e + f*x)*(c - c*sin(e + f*x))^(7/2))/(10*f*sqrt(a + a*sin(e + f*x)))) - (a*cos(e + f*x)*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(7/2))/(5*f)]
    @test_int [(a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(5/2), x, 2, -((a^2*cos(e + f*x)*(c - c*sin(e + f*x))^(5/2))/(6*f*sqrt(a + a*sin(e + f*x)))) - (a*cos(e + f*x)*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(5/2))/(4*f)]
    @test_int [(a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(3/2), x, 2, -((a^2*cos(e + f*x)*(c - c*sin(e + f*x))^(3/2))/(3*f*sqrt(a + a*sin(e + f*x)))) - (a*cos(e + f*x)*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2))/(3*f)]
    @test_int [(a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(1/2), x, 1, (c*cos(e + f*x)*(a + a*sin(e + f*x))^(3/2))/(2*f*sqrt(c - c*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^(3/2)/(c - c*sin(e + f*x))^(1/2), x, 4, -((2*a^2*cos(e + f*x)*log(1 - sin(e + f*x)))/(f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))) - (a*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(f*sqrt(c - c*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^(3/2)/(c - c*sin(e + f*x))^(3/2), x, 4, (a*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(f*(c - c*sin(e + f*x))^(3/2)) + (a^2*cos(e + f*x)*log(1 - sin(e + f*x)))/(c*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^(3/2)/(c - c*sin(e + f*x))^(5/2), x, 1, (cos(e + f*x)*(a + a*sin(e + f*x))^(3/2))/(4*f*(c - c*sin(e + f*x))^(5/2))]
    @test_int [(a + a*sin(e + f*x))^(3/2)/(c - c*sin(e + f*x))^(7/2), x, 2, (cos(e + f*x)*(a + a*sin(e + f*x))^(3/2))/(6*f*(c - c*sin(e + f*x))^(7/2)) + (cos(e + f*x)*(a + a*sin(e + f*x))^(3/2))/(24*c*f*(c - c*sin(e + f*x))^(5/2))]
    @test_int [(a + a*sin(e + f*x))^(3/2)/(c - c*sin(e + f*x))^(9/2), x, 2, (a*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(4*f*(c - c*sin(e + f*x))^(9/2)) - (a^2*cos(e + f*x))/(12*c*f*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(7/2))]
    @test_int [(a + a*sin(e + f*x))^(3/2)/(c - c*sin(e + f*x))^(11/2), x, 2, (a*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(5*f*(c - c*sin(e + f*x))^(11/2)) - (a^2*cos(e + f*x))/(20*c*f*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(9/2))]

    @test_int [(a + a*sin(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(7/2), x, 3, -((a^3*cos(e + f*x)*(c - c*sin(e + f*x))^(7/2))/(15*f*sqrt(a + a*sin(e + f*x)))) - (2*a^2*cos(e + f*x)*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(7/2))/(15*f) - (a*cos(e + f*x)*(a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(7/2))/(6*f)]
    @test_int [(a + a*sin(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(5/2), x, 3, -((2*a^3*cos(e + f*x)*(c - c*sin(e + f*x))^(5/2))/(15*f*sqrt(a + a*sin(e + f*x)))) - (a^2*cos(e + f*x)*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(5/2))/(5*f) - (a*cos(e + f*x)*(a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(5/2))/(5*f)]
    @test_int [(a + a*sin(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(3/2), x, 2, (c^2*cos(e + f*x)*(a + a*sin(e + f*x))^(5/2))/(6*f*sqrt(c - c*sin(e + f*x))) + (c*cos(e + f*x)*(a + a*sin(e + f*x))^(5/2)*sqrt(c - c*sin(e + f*x)))/(4*f)]
    @test_int [(a + a*sin(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(1/2), x, 1, (c*cos(e + f*x)*(a + a*sin(e + f*x))^(5/2))/(3*f*sqrt(c - c*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^(5/2)/(c - c*sin(e + f*x))^(1/2), x, 5, -((4*a^3*cos(e + f*x)*log(1 - sin(e + f*x)))/(f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))) - (2*a^2*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(f*sqrt(c - c*sin(e + f*x))) - (a*cos(e + f*x)*(a + a*sin(e + f*x))^(3/2))/(2*f*sqrt(c - c*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^(5/2)/(c - c*sin(e + f*x))^(3/2), x, 5, (a*cos(e + f*x)*(a + a*sin(e + f*x))^(3/2))/(f*(c - c*sin(e + f*x))^(3/2)) + (4*a^3*cos(e + f*x)*log(1 - sin(e + f*x)))/(c*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (2*a^2*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(c*f*sqrt(c - c*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^(5/2)/(c - c*sin(e + f*x))^(5/2), x, 5, (a*cos(e + f*x)*(a + a*sin(e + f*x))^(3/2))/(2*f*(c - c*sin(e + f*x))^(5/2)) - (a^2*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(c*f*(c - c*sin(e + f*x))^(3/2)) - (a^3*cos(e + f*x)*log(1 - sin(e + f*x)))/(c^2*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^(5/2)/(c - c*sin(e + f*x))^(7/2), x, 1, (cos(e + f*x)*(a + a*sin(e + f*x))^(5/2))/(6*f*(c - c*sin(e + f*x))^(7/2))]
    @test_int [(a + a*sin(e + f*x))^(5/2)/(c - c*sin(e + f*x))^(9/2), x, 2, (cos(e + f*x)*(a + a*sin(e + f*x))^(5/2))/(8*f*(c - c*sin(e + f*x))^(9/2)) + (cos(e + f*x)*(a + a*sin(e + f*x))^(5/2))/(48*c*f*(c - c*sin(e + f*x))^(7/2))]
    @test_int [(a + a*sin(e + f*x))^(5/2)/(c - c*sin(e + f*x))^(11/2), x, 3, (cos(e + f*x)*(a + a*sin(e + f*x))^(5/2))/(10*f*(c - c*sin(e + f*x))^(11/2)) + (cos(e + f*x)*(a + a*sin(e + f*x))^(5/2))/(40*c*f*(c - c*sin(e + f*x))^(9/2)) + (cos(e + f*x)*(a + a*sin(e + f*x))^(5/2))/(240*c^2*f*(c - c*sin(e + f*x))^(7/2))]
    @test_int [(a + a*sin(e + f*x))^(5/2)/(c - c*sin(e + f*x))^(13/2), x, 3, (a*cos(e + f*x)*(a + a*sin(e + f*x))^(3/2))/(6*f*(c - c*sin(e + f*x))^(13/2)) - (a^2*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(15*c*f*(c - c*sin(e + f*x))^(11/2)) + (a^3*cos(e + f*x))/(60*c^2*f*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(9/2))]

    @test_int [(a + a*sin(e + f*x))^(7/2)*(c - c*sin(e + f*x))^(9/2), x, 4, -((a^4*cos(e + f*x)*(c - c*sin(e + f*x))^(9/2))/(35*f*sqrt(a + a*sin(e + f*x)))) - (a^3*cos(e + f*x)*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(9/2))/(14*f) - (3*a^2*cos(e + f*x)*(a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(9/2))/(28*f) - (a*cos(e + f*x)*(a + a*sin(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(9/2))/(8*f)]
    @test_int [(a + a*sin(e + f*x))^(7/2)*(c - c*sin(e + f*x))^(7/2), x, 4, -((2*a^4*cos(e + f*x)*(c - c*sin(e + f*x))^(7/2))/(35*f*sqrt(a + a*sin(e + f*x)))) - (4*a^3*cos(e + f*x)*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(7/2))/(35*f) - (a^2*cos(e + f*x)*(a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(7/2))/(7*f) - (a*cos(e + f*x)*(a + a*sin(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(7/2))/(7*f)]
    @test_int [(a + a*sin(e + f*x))^(7/2)*(c - c*sin(e + f*x))^(5/2), x, 3, (c^3*cos(e + f*x)*(a + a*sin(e + f*x))^(7/2))/(15*f*sqrt(c - c*sin(e + f*x))) + (2*c^2*cos(e + f*x)*(a + a*sin(e + f*x))^(7/2)*sqrt(c - c*sin(e + f*x)))/(15*f) + (c*cos(e + f*x)*(a + a*sin(e + f*x))^(7/2)*(c - c*sin(e + f*x))^(3/2))/(6*f)]
    @test_int [(a + a*sin(e + f*x))^(7/2)*(c - c*sin(e + f*x))^(3/2), x, 2, (c^2*cos(e + f*x)*(a + a*sin(e + f*x))^(7/2))/(10*f*sqrt(c - c*sin(e + f*x))) + (c*cos(e + f*x)*(a + a*sin(e + f*x))^(7/2)*sqrt(c - c*sin(e + f*x)))/(5*f)]
    @test_int [(a + a*sin(e + f*x))^(7/2)*(c - c*sin(e + f*x))^(1/2), x, 1, (c*cos(e + f*x)*(a + a*sin(e + f*x))^(7/2))/(4*f*sqrt(c - c*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^(7/2)/(c - c*sin(e + f*x))^(1/2), x, 6, -((8*a^4*cos(e + f*x)*log(1 - sin(e + f*x)))/(f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))) - (4*a^3*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(f*sqrt(c - c*sin(e + f*x))) - (a^2*cos(e + f*x)*(a + a*sin(e + f*x))^(3/2))/(f*sqrt(c - c*sin(e + f*x))) - (a*cos(e + f*x)*(a + a*sin(e + f*x))^(5/2))/(3*f*sqrt(c - c*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^(7/2)/(c - c*sin(e + f*x))^(3/2), x, 6, (a*cos(e + f*x)*(a + a*sin(e + f*x))^(5/2))/(f*(c - c*sin(e + f*x))^(3/2)) + (12*a^4*cos(e + f*x)*log(1 - sin(e + f*x)))/(c*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (6*a^3*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(c*f*sqrt(c - c*sin(e + f*x))) + (3*a^2*cos(e + f*x)*(a + a*sin(e + f*x))^(3/2))/(2*c*f*sqrt(c - c*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^(7/2)/(c - c*sin(e + f*x))^(5/2), x, 6, (a*cos(e + f*x)*(a + a*sin(e + f*x))^(5/2))/(2*f*(c - c*sin(e + f*x))^(5/2)) - (3*a^2*cos(e + f*x)*(a + a*sin(e + f*x))^(3/2))/(2*c*f*(c - c*sin(e + f*x))^(3/2)) - (6*a^4*cos(e + f*x)*log(1 - sin(e + f*x)))/(c^2*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) - (3*a^3*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(c^2*f*sqrt(c - c*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^(7/2)/(c - c*sin(e + f*x))^(7/2), x, 6, (a*cos(e + f*x)*(a + a*sin(e + f*x))^(5/2))/(3*f*(c - c*sin(e + f*x))^(7/2)) - (a^2*cos(e + f*x)*(a + a*sin(e + f*x))^(3/2))/(2*c*f*(c - c*sin(e + f*x))^(5/2)) + (a^3*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(c^2*f*(c - c*sin(e + f*x))^(3/2)) + (a^4*cos(e + f*x)*log(1 - sin(e + f*x)))/(c^3*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^(7/2)/(c - c*sin(e + f*x))^(9/2), x, 1, (cos(e + f*x)*(a + a*sin(e + f*x))^(7/2))/(8*f*(c - c*sin(e + f*x))^(9/2))]
    @test_int [(a + a*sin(e + f*x))^(7/2)/(c - c*sin(e + f*x))^(11/2), x, 2, (cos(e + f*x)*(a + a*sin(e + f*x))^(7/2))/(10*f*(c - c*sin(e + f*x))^(11/2)) + (cos(e + f*x)*(a + a*sin(e + f*x))^(7/2))/(80*c*f*(c - c*sin(e + f*x))^(9/2))]
    @test_int [(a + a*sin(e + f*x))^(7/2)/(c - c*sin(e + f*x))^(13/2), x, 3, (cos(e + f*x)*(a + a*sin(e + f*x))^(7/2))/(12*f*(c - c*sin(e + f*x))^(13/2)) + (cos(e + f*x)*(a + a*sin(e + f*x))^(7/2))/(60*c*f*(c - c*sin(e + f*x))^(11/2)) + (cos(e + f*x)*(a + a*sin(e + f*x))^(7/2))/(480*c^2*f*(c - c*sin(e + f*x))^(9/2))]
    @test_int [(a + a*sin(e + f*x))^(7/2)/(c - c*sin(e + f*x))^(15/2), x, 4, (cos(e + f*x)*(a + a*sin(e + f*x))^(7/2))/(14*f*(c - c*sin(e + f*x))^(15/2)) + (cos(e + f*x)*(a + a*sin(e + f*x))^(7/2))/(56*c*f*(c - c*sin(e + f*x))^(13/2)) + (cos(e + f*x)*(a + a*sin(e + f*x))^(7/2))/(280*c^2*f*(c - c*sin(e + f*x))^(11/2)) + (cos(e + f*x)*(a + a*sin(e + f*x))^(7/2))/(2240*c^3*f*(c - c*sin(e + f*x))^(9/2))]
    @test_int [(a + a*sin(e + f*x))^(7/2)/(c - c*sin(e + f*x))^(17/2), x, 4, (a*cos(e + f*x)*(a + a*sin(e + f*x))^(5/2))/(8*f*(c - c*sin(e + f*x))^(17/2)) - (3*a^2*cos(e + f*x)*(a + a*sin(e + f*x))^(3/2))/(56*c*f*(c - c*sin(e + f*x))^(15/2)) + (a^3*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(56*c^2*f*(c - c*sin(e + f*x))^(13/2)) - (a^4*cos(e + f*x))/(280*c^3*f*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(11/2))]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [1/(a + a*sin(e + f*x))^(1/2)*(c - c*sin(e + f*x))^(5/2), x, 5, (4*c^3*cos(e + f*x)*log(1 + sin(e + f*x)))/(f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (2*c^2*cos(e + f*x)*sqrt(c - c*sin(e + f*x)))/(f*sqrt(a + a*sin(e + f*x))) + (c*cos(e + f*x)*(c - c*sin(e + f*x))^(3/2))/(2*f*sqrt(a + a*sin(e + f*x)))]
    @test_int [1/(a + a*sin(e + f*x))^(1/2)*(c - c*sin(e + f*x))^(3/2), x, 4, (2*c^2*cos(e + f*x)*log(1 + sin(e + f*x)))/(f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (c*cos(e + f*x)*sqrt(c - c*sin(e + f*x)))/(f*sqrt(a + a*sin(e + f*x)))]
    @test_int [1/(a + a*sin(e + f*x))^(1/2)*(c - c*sin(e + f*x))^(1/2), x, 3, (c*cos(e + f*x)*log(1 + sin(e + f*x)))/(f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [1/(a + a*sin(e + f*x))^(1/2)/(c - c*sin(e + f*x))^(1/2), x, 2, (atanh(sin(e + f*x))*cos(e + f*x))/(f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [1/(a + a*sin(e + f*x))^(1/2)/(c - c*sin(e + f*x))^(3/2), x, 3, cos(e + f*x)/(2*f*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2)) + (atanh(sin(e + f*x))*cos(e + f*x))/(2*c*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [1/(a + a*sin(e + f*x))^(1/2)/(c - c*sin(e + f*x))^(5/2), x, 4, cos(e + f*x)/(4*f*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(5/2)) + cos(e + f*x)/(4*c*f*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2)) + (atanh(sin(e + f*x))*cos(e + f*x))/(4*c^2*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]

    @test_int [1/(a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(7/2), x, 6, -((12*c^4*cos(e + f*x)*log(1 + sin(e + f*x)))/(a*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))) - (6*c^3*cos(e + f*x)*sqrt(c - c*sin(e + f*x)))/(a*f*sqrt(a + a*sin(e + f*x))) - (3*c^2*cos(e + f*x)*(c - c*sin(e + f*x))^(3/2))/(2*a*f*sqrt(a + a*sin(e + f*x))) - (c*cos(e + f*x)*(c - c*sin(e + f*x))^(5/2))/(f*(a + a*sin(e + f*x))^(3/2))]
    @test_int [1/(a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(5/2), x, 5, -((4*c^3*cos(e + f*x)*log(1 + sin(e + f*x)))/(a*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))) - (2*c^2*cos(e + f*x)*sqrt(c - c*sin(e + f*x)))/(a*f*sqrt(a + a*sin(e + f*x))) - (c*cos(e + f*x)*(c - c*sin(e + f*x))^(3/2))/(f*(a + a*sin(e + f*x))^(3/2))]
    @test_int [1/(a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(3/2), x, 4, -((c^2*cos(e + f*x)*log(1 + sin(e + f*x)))/(a*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))) - (c*cos(e + f*x)*sqrt(c - c*sin(e + f*x)))/(f*(a + a*sin(e + f*x))^(3/2))]
    @test_int [1/(a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(1/2), x, 1, -((c*cos(e + f*x))/(f*(a + a*sin(e + f*x))^(3/2)*sqrt(c - c*sin(e + f*x))))]
    @test_int [1/(a + a*sin(e + f*x))^(3/2)/(c - c*sin(e + f*x))^(1/2), x, 3, -(cos(e + f*x)/(2*f*(a + a*sin(e + f*x))^(3/2)*sqrt(c - c*sin(e + f*x)))) + (atanh(sin(e + f*x))*cos(e + f*x))/(2*a*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [1/(a + a*sin(e + f*x))^(3/2)/(c - c*sin(e + f*x))^(3/2), x, 4, -(cos(e + f*x)/(2*f*(a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(3/2))) + cos(e + f*x)/(2*a*f*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2)) + (atanh(sin(e + f*x))*cos(e + f*x))/(2*a*c*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [1/(a + a*sin(e + f*x))^(3/2)/(c - c*sin(e + f*x))^(5/2), x, 5, -(cos(e + f*x)/(2*f*(a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(5/2))) + (3*cos(e + f*x))/(8*a*f*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(5/2)) + (3*cos(e + f*x))/(8*a*c*f*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2)) + (3*atanh(sin(e + f*x))*cos(e + f*x))/(8*a*c^2*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]

    @test_int [1/(a + a*sin(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(9/2), x, 7, (24*c^5*cos(e + f*x)*log(1 + sin(e + f*x)))/(a^2*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (12*c^4*cos(e + f*x)*sqrt(c - c*sin(e + f*x)))/(a^2*f*sqrt(a + a*sin(e + f*x))) + (3*c^3*cos(e + f*x)*(c - c*sin(e + f*x))^(3/2))/(a^2*f*sqrt(a + a*sin(e + f*x))) + (2*c^2*cos(e + f*x)*(c - c*sin(e + f*x))^(5/2))/(a*f*(a + a*sin(e + f*x))^(3/2)) - (c*cos(e + f*x)*(c - c*sin(e + f*x))^(7/2))/(2*f*(a + a*sin(e + f*x))^(5/2))]
    @test_int [1/(a + a*sin(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(7/2), x, 6, (6*c^4*cos(e + f*x)*log(1 + sin(e + f*x)))/(a^2*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (3*c^3*cos(e + f*x)*sqrt(c - c*sin(e + f*x)))/(a^2*f*sqrt(a + a*sin(e + f*x))) + (3*c^2*cos(e + f*x)*(c - c*sin(e + f*x))^(3/2))/(2*a*f*(a + a*sin(e + f*x))^(3/2)) - (c*cos(e + f*x)*(c - c*sin(e + f*x))^(5/2))/(2*f*(a + a*sin(e + f*x))^(5/2))]
    @test_int [1/(a + a*sin(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(5/2), x, 5, (c^3*cos(e + f*x)*log(1 + sin(e + f*x)))/(a^2*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x))) + (c^2*cos(e + f*x)*sqrt(c - c*sin(e + f*x)))/(a*f*(a + a*sin(e + f*x))^(3/2)) - (c*cos(e + f*x)*(c - c*sin(e + f*x))^(3/2))/(2*f*(a + a*sin(e + f*x))^(5/2))]
    @test_int [1/(a + a*sin(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(3/2), x, 1, -((cos(e + f*x)*(c - c*sin(e + f*x))^(3/2))/(4*f*(a + a*sin(e + f*x))^(5/2)))]
    @test_int [1/(a + a*sin(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(1/2), x, 1, -((c*cos(e + f*x))/(2*f*(a + a*sin(e + f*x))^(5/2)*sqrt(c - c*sin(e + f*x))))]
    @test_int [1/(a + a*sin(e + f*x))^(5/2)/(c - c*sin(e + f*x))^(1/2), x, 4, -(cos(e + f*x)/(4*f*(a + a*sin(e + f*x))^(5/2)*sqrt(c - c*sin(e + f*x)))) - cos(e + f*x)/(4*a*f*(a + a*sin(e + f*x))^(3/2)*sqrt(c - c*sin(e + f*x))) + (atanh(sin(e + f*x))*cos(e + f*x))/(4*a^2*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [1/(a + a*sin(e + f*x))^(5/2)/(c - c*sin(e + f*x))^(3/2), x, 5, -(cos(e + f*x)/(4*f*(a + a*sin(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(3/2))) - (3*cos(e + f*x))/(8*a*f*(a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(3/2)) + (3*cos(e + f*x))/(8*a^2*f*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2)) + (3*atanh(sin(e + f*x))*cos(e + f*x))/(8*a^2*c*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]
    @test_int [1/(a + a*sin(e + f*x))^(5/2)/(c - c*sin(e + f*x))^(5/2), x, 6, -(cos(e + f*x)/(4*f*(a + a*sin(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(5/2))) - cos(e + f*x)/(2*a*f*(a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(5/2)) + (3*cos(e + f*x))/(8*a^2*f*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(5/2)) + (3*cos(e + f*x))/(8*a^2*c*f*sqrt(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2)) + (3*atanh(sin(e + f*x))*cos(e + f*x))/(8*a^2*c^2*f*sqrt(a + a*sin(e + f*x))*sqrt(c - c*sin(e + f*x)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^n*with*m*and/or*n*symbolic=#

    @test_int [(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^n, x, 4, (2^(1/2 + n)*c*cos(e + f*x)*HypergeometricFunctions._₂F₁((1/2)*(1 + 2*m), (1/2)*(1 - 2*n), (1/2)*(3 + 2*m), (1/2)*(1 + sin(e + f*x)))*(1 - sin(e + f*x))^(1/2 - n)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(-1 + n))/(f*(1 + 2*m))]

    @test_int [(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^3, x, 4, -((2^(1/2 + m)*a^4*c^3*cos(e + f*x)^7*HypergeometricFunctions._₂F₁(7/2, 1/2 - m, 9/2, (1/2)*(1 - sin(e + f*x)))*(1 + sin(e + f*x))^(1/2 - m)*(a + a*sin(e + f*x))^(-4 + m))/(7*f))]
    @test_int [(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^2, x, 4, -((2^(1/2 + m)*a^3*c^2*cos(e + f*x)^5*HypergeometricFunctions._₂F₁(5/2, 1/2 - m, 7/2, (1/2)*(1 - sin(e + f*x)))*(1 + sin(e + f*x))^(1/2 - m)*(a + a*sin(e + f*x))^(-3 + m))/(5*f))]
    @test_int [(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^1, x, 4, -((2^(1/2 + m)*a^2*c*cos(e + f*x)^3*HypergeometricFunctions._₂F₁(3/2, 1/2 - m, 5/2, (1/2)*(1 - sin(e + f*x)))*(1 + sin(e + f*x))^(1/2 - m)*(a + a*sin(e + f*x))^(-2 + m))/(3*f))]
    @test_int [(a + a*sin(e + f*x))^m/(c - c*sin(e + f*x))^1, x, 4, (2^(1/2 + m)*HypergeometricFunctions._₂F₁(-(1/2), 1/2 - m, 1/2, (1/2)*(1 - sin(e + f*x)))*sec(e + f*x)*(1 + sin(e + f*x))^(1/2 - m)*(a + a*sin(e + f*x))^m)/(c*f)]
    @test_int [(a + a*sin(e + f*x))^m/(c - c*sin(e + f*x))^2, x, 4, (2^(1/2 + m)*HypergeometricFunctions._₂F₁(-(3/2), 1/2 - m, -(1/2), (1/2)*(1 - sin(e + f*x)))*sec(e + f*x)^3*(1 + sin(e + f*x))^(1/2 - m)*(a + a*sin(e + f*x))^(1 + m))/(3*a*c^2*f)]
    @test_int [(a + a*sin(e + f*x))^m/(c - c*sin(e + f*x))^3, x, 4, (2^(1/2 + m)*HypergeometricFunctions._₂F₁(-(5/2), 1/2 - m, -(3/2), (1/2)*(1 - sin(e + f*x)))*sec(e + f*x)^5*(1 + sin(e + f*x))^(1/2 - m)*(a + a*sin(e + f*x))^(2 + m))/(5*a^2*c^3*f)]

    @test_int [(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(5/2), x, 3, (64*c^3*cos(e + f*x)*(a + a*sin(e + f*x))^m)/(f*(5 + 2*m)*(3 + 8*m + 4*m^2)*sqrt(c - c*sin(e + f*x))) + (16*c^2*cos(e + f*x)*(a + a*sin(e + f*x))^m*sqrt(c - c*sin(e + f*x)))/(f*(15 + 16*m + 4*m^2)) + (2*c*cos(e + f*x)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(3/2))/(f*(5 + 2*m))]
    @test_int [(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(3/2), x, 2, (8*c^2*cos(e + f*x)*(a + a*sin(e + f*x))^m)/(f*(3 + 8*m + 4*m^2)*sqrt(c - c*sin(e + f*x))) + (2*c*cos(e + f*x)*(a + a*sin(e + f*x))^m*sqrt(c - c*sin(e + f*x)))/(f*(3 + 2*m))]
    @test_int [(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(1/2), x, 1, (2*c*cos(e + f*x)*(a + a*sin(e + f*x))^m)/(f*(1 + 2*m)*sqrt(c - c*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^m/(c - c*sin(e + f*x))^(1/2), x, 3, (cos(e + f*x)*HypergeometricFunctions._₂F₁(1, 1/2 + m, 3/2 + m, (1/2)*(1 + sin(e + f*x)))*(a + a*sin(e + f*x))^m)/(f*(1 + 2*m)*sqrt(c - c*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^m/(c - c*sin(e + f*x))^(3/2), x, 3, (cos(e + f*x)*HypergeometricFunctions._₂F₁(2, 1/2 + m, 3/2 + m, (1/2)*(1 + sin(e + f*x)))*(a + a*sin(e + f*x))^m)/(2*c*f*(1 + 2*m)*sqrt(c - c*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^m/(c - c*sin(e + f*x))^(5/2), x, 3, (cos(e + f*x)*HypergeometricFunctions._₂F₁(3, 1/2 + m, 3/2 + m, (1/2)*(1 + sin(e + f*x)))*(a + a*sin(e + f*x))^m)/(4*c^2*f*(1 + 2*m)*sqrt(c - c*sin(e + f*x)))]

    #= The*same*rules*should*be*used*to*integrate*the*following*two*problems: =#
    @test_int [(a + a*sin(e + f*x))^m/(c - c*sin(e + f*x))^(1/2), x, 3, (cos(e + f*x)*HypergeometricFunctions._₂F₁(1, 1/2 + m, 3/2 + m, (1/2)*(1 + sin(e + f*x)))*(a + a*sin(e + f*x))^m)/(f*(1 + 2*m)*sqrt(c - c*sin(e + f*x)))]
    @test_int [(c + c*sin(e + f*x))^m/(a - a*sin(e + f*x))^(1/2), x, 3, (cos(e + f*x)*HypergeometricFunctions._₂F₁(1, 1/2 + m, 3/2 + m, (1/2)*(1 + sin(e + f*x)))*(c + c*sin(e + f*x))^m)/(f*(1 + 2*m)*sqrt(a - a*sin(e + f*x)))]

    @test_int [(a + a*sin(e + f*x))^m/(c - c*sin(e + f*x))^(m + 3), x, 3, If(13>=8, (cos(e + f*x)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(-3 - m))/(f*(5 + 2*m)) + (2*cos(e + f*x)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(-2 - m))/(c*f*(15 + 16*m + 4*m^2)) + (2*cos(e + f*x)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(-1 - m))/(c^2*f*(5 + 2*m)*(3 + 8*m + 4*m^2)), (cos(e + f*x)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(-3 - m))/(f*(5 + 2*m)) + (2*cos(e + f*x)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(-2 - m))/(c*f*(15 + 16*m + 4*m^2)) + (2*cos(e + f*x)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(-1 - m))/(c^2*f*(15 + 46*m + 36*m^2 + 8*m^3)))]
    @test_int [(a + a*sin(e + f*x))^m/(c - c*sin(e + f*x))^(m + 2), x, 2, (cos(e + f*x)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(-2 - m))/(f*(3 + 2*m)) + (cos(e + f*x)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(-1 - m))/(c*f*(3 + 8*m + 4*m^2))]
    @test_int [(a + a*sin(e + f*x))^m/(c - c*sin(e + f*x))^(m + 1), x, 1, (cos(e + f*x)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(-1 - m))/(f*(1 + 2*m))]
    @test_int [(a + a*sin(e + f*x))^m/(c - c*sin(e + f*x))^(m + 0), x, 4, (2^(1/2 - m)*c*cos(e + f*x)*HypergeometricFunctions._₂F₁((1/2)*(1 + 2*m), (1/2)*(1 + 2*m), (1/2)*(3 + 2*m), (1/2)*(1 + sin(e + f*x)))*(1 - sin(e + f*x))^(1/2 + m)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(-1 - m))/(f*(1 + 2*m))]
    @test_int [(a + a*sin(e + f*x))^m/(c - c*sin(e + f*x))^(m - 1), x, 4, (2^(3/2 - m)*c^2*cos(e + f*x)*HypergeometricFunctions._₂F₁((1/2)*(-1 + 2*m), (1/2)*(1 + 2*m), (1/2)*(3 + 2*m), (1/2)*(1 + sin(e + f*x)))*(1 - sin(e + f*x))^(1/2 + m)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(-1 - m))/(f*(1 + 2*m))]
    @test_int [(a + a*sin(e + f*x))^m/(c - c*sin(e + f*x))^(m - 2), x, 4, (2^(5/2 - m)*c^3*cos(e + f*x)*HypergeometricFunctions._₂F₁((1/2)*(-3 + 2*m), (1/2)*(1 + 2*m), (1/2)*(3 + 2*m), (1/2)*(1 + sin(e + f*x)))*(1 - sin(e + f*x))^(1/2 + m)*(a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(-1 - m))/(f*(1 + 2*m))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(a+a*sin(e+f*x))^m*(c+d*sin(e+f*x))^n=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+a*sin(e+f*x))^m*(c+d*sin(e+f*x))^n=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [(a + a*sin(e + f*x))^1*(c + d*sin(e + f*x))^4, x, 4, (1/8)*a*(8*c^4 + 16*c^3*d + 24*c^2*d^2 + 12*c*d^3 + 3*d^4)*x - (a*(12*c^4 + 95*c^3*d + 112*c^2*d^2 + 80*c*d^3 + 16*d^4)*cos(e + f*x))/(30*f) - (a*d*(24*c^3 + 130*c^2*d + 116*c*d^2 + 45*d^3)*cos(e + f*x)*sin(e + f*x))/(120*f) - (a*(12*c^2 + 35*c*d + 16*d^2)*cos(e + f*x)*(c + d*sin(e + f*x))^2)/(60*f) - (a*(4*c + 5*d)*cos(e + f*x)*(c + d*sin(e + f*x))^3)/(20*f) - (a*cos(e + f*x)*(c + d*sin(e + f*x))^4)/(5*f)]
    @test_int [(a + a*sin(e + f*x))^1*(c + d*sin(e + f*x))^3, x, 3, (1/8)*a*(8*c^3 + 12*c^2*d + 12*c*d^2 + 3*d^3)*x - (a*(3*c^3 + 16*c^2*d + 12*c*d^2 + 4*d^3)*cos(e + f*x))/(6*f) - (a*d*(6*c^2 + 20*c*d + 9*d^2)*cos(e + f*x)*sin(e + f*x))/(24*f) - (a*(3*c + 4*d)*cos(e + f*x)*(c + d*sin(e + f*x))^2)/(12*f) - (a*cos(e + f*x)*(c + d*sin(e + f*x))^3)/(4*f)]
    @test_int [(a + a*sin(e + f*x))^1*(c + d*sin(e + f*x))^2, x, 2, (1/2)*a*(2*c^2 + 2*c*d + d^2)*x - (2*a*(c^2 + 3*c*d + d^2)*cos(e + f*x))/(3*f) - (a*d*(2*c + 3*d)*cos(e + f*x)*sin(e + f*x))/(6*f) - (a*cos(e + f*x)*(c + d*sin(e + f*x))^2)/(3*f)]
    @test_int [(a + a*sin(e + f*x))^1*(c + d*sin(e + f*x))^1, x, 1, (1/2)*a*(2*c + d)*x - (a*(c + d)*cos(e + f*x))/f - (a*d*cos(e + f*x)*sin(e + f*x))/(2*f)]
    @test_int [(a + a*sin(e + f*x))^1*(c + d*sin(e + f*x))^0, x, 2, a*x - (a*cos(e + f*x))/f]
    @test_int [(a + a*sin(e + f*x))^1/(c + d*sin(e + f*x))^1, x, 4, (a*x)/d - (2*a*(c - d)*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/(d*sqrt(c^2 - d^2)*f)]
    @test_int [(a + a*sin(e + f*x))^1/(c + d*sin(e + f*x))^2, x, 5, (2*a*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/((c + d)*sqrt(c^2 - d^2)*f) - (a*cos(e + f*x))/((c + d)*f*(c + d*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^1/(c + d*sin(e + f*x))^3, x, 6, (a*(2*c - d)*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/((c + d)*(c^2 - d^2)^(3/2)*f) - (a*cos(e + f*x))/(2*(c + d)*f*(c + d*sin(e + f*x))^2) - (a*(c - 2*d)*cos(e + f*x))/(2*(c - d)*(c + d)^2*f*(c + d*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^1/(c + d*sin(e + f*x))^4, x, 7, (a*(2*c^2 - 2*c*d + d^2)*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/((c + d)*(c^2 - d^2)^(5/2)*f) - (a*cos(e + f*x))/(3*(c + d)*f*(c + d*sin(e + f*x))^3) - (a*(2*c - 3*d)*cos(e + f*x))/(6*(c - d)*(c + d)^2*f*(c + d*sin(e + f*x))^2) - (a*(c - 4*d)*(2*c - d)*cos(e + f*x))/(6*(c - d)^2*(c + d)^3*f*(c + d*sin(e + f*x)))]

    @test_int [(a + a*sin(e + f*x))^2*(c + d*sin(e + f*x))^4, x, 5, (1/16)*a^2*(24*c^4 + 64*c^3*d + 84*c^2*d^2 + 48*c*d^3 + 11*d^4)*x + (a^2*(4*c^5 - 48*c^4*d - 311*c^3*d^2 - 448*c^2*d^3 - 288*c*d^4 - 64*d^5)*cos(e + f*x))/(60*d*f) + (a^2*(8*c^4 - 96*c^3*d - 438*c^2*d^2 - 464*c*d^3 - 165*d^4)*cos(e + f*x)*sin(e + f*x))/(240*f) + (a^2*(4*c^3 - 48*c^2*d - 123*c*d^2 - 64*d^3)*cos(e + f*x)*(c + d*sin(e + f*x))^2)/(120*d*f) + (a^2*(4*c^2 - 48*c*d - 55*d^2)*cos(e + f*x)*(c + d*sin(e + f*x))^3)/(120*d*f) + (a^2*(c - 12*d)*cos(e + f*x)*(c + d*sin(e + f*x))^4)/(30*d*f) - (a^2*cos(e + f*x)*(c + d*sin(e + f*x))^5)/(6*d*f)]
    @test_int [(a + a*sin(e + f*x))^2*(c + d*sin(e + f*x))^3, x, 4, (3/8)*a^2*(2*c + d)*(2*c^2 + 3*c*d + 2*d^2)*x + (a^2*(c^4 - 10*c^3*d - 44*c^2*d^2 - 40*c*d^3 - 12*d^4)*cos(e + f*x))/(10*d*f) + (a^2*(2*c^3 - 20*c^2*d - 57*c*d^2 - 30*d^3)*cos(e + f*x)*sin(e + f*x))/(40*f) + (a^2*(c^2 - 10*c*d - 12*d^2)*cos(e + f*x)*(c + d*sin(e + f*x))^2)/(20*d*f) + (a^2*(c - 10*d)*cos(e + f*x)*(c + d*sin(e + f*x))^3)/(20*d*f) - (a^2*cos(e + f*x)*(c + d*sin(e + f*x))^4)/(5*d*f)]
    @test_int [(a + a*sin(e + f*x))^2*(c + d*sin(e + f*x))^2, x, 3, (1/8)*a^2*(12*c^2 + 16*c*d + 7*d^2)*x - (a^2*(12*c^2 + 16*c*d + 7*d^2)*cos(e + f*x))/(6*f) - (a^2*(12*c^2 + 16*c*d + 7*d^2)*cos(e + f*x)*sin(e + f*x))/(24*f) - ((8*c - d)*d*cos(e + f*x)*(a + a*sin(e + f*x))^2)/(12*f) - (d^2*cos(e + f*x)*(a + a*sin(e + f*x))^3)/(4*a*f)]
    @test_int [(a + a*sin(e + f*x))^2*(c + d*sin(e + f*x))^1, x, 2, (1/2)*a^2*(3*c + 2*d)*x - (2*a^2*(3*c + 2*d)*cos(e + f*x))/(3*f) - (a^2*(3*c + 2*d)*cos(e + f*x)*sin(e + f*x))/(6*f) - (d*cos(e + f*x)*(a + a*sin(e + f*x))^2)/(3*f)]
    @test_int [(a + a*sin(e + f*x))^2*(c + d*sin(e + f*x))^0, x, 1, (3*a^2*x)/2 - (2*a^2*cos(e + f*x))/f - (a^2*cos(e + f*x)*sin(e + f*x))/(2*f)]
    @test_int [(a + a*sin(e + f*x))^2/(c + d*sin(e + f*x))^1, x, 5, -((a^2*(c - 2*d)*x)/d^2) + (2*a^2*(c - d)^2*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/(d^2*sqrt(c^2 - d^2)*f) - (a^2*cos(e + f*x))/(d*f)]
    @test_int [(a + a*sin(e + f*x))^2/(c + d*sin(e + f*x))^2, x, 5, (a^2*x)/d^2 - (2*a^2*(c - d)^2*(c + 2*d)*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/(d^2*(c^2 - d^2)^(3/2)*f) + (a^2*(c - d)*cos(e + f*x))/(d*(c + d)*f*(c + d*sin(e + f*x))), (a^2*x)/d^2 - (2*a^2*(c - d)*(c + 2*d)*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/(d^2*(c + d)*sqrt(c^2 - d^2)*f) + (a^2*(c - d)*cos(e + f*x))/(d*(c + d)*f*(c + d*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^2/(c + d*sin(e + f*x))^3, x, 6, (3*a^2*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/((c + d)^2*sqrt(c^2 - d^2)*f) + (a^2*(c - d)*cos(e + f*x))/(2*d*(c + d)*f*(c + d*sin(e + f*x))^2) - (a^2*(c + 4*d)*cos(e + f*x))/(2*d*(c + d)^2*f*(c + d*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^2/(c + d*sin(e + f*x))^4, x, 7, (a^2*(3*c - 2*d)*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/((c - d)*(c + d)^3*sqrt(c^2 - d^2)*f) + (a^2*(c - d)*cos(e + f*x))/(3*d*(c + d)*f*(c + d*sin(e + f*x))^3) - (a^2*(c + 6*d)*cos(e + f*x))/(6*d*(c + d)^2*f*(c + d*sin(e + f*x))^2) - (a^2*(c^2 + 6*c*d - 10*d^2)*cos(e + f*x))/(6*(c - d)*d*(c + d)^3*f*(c + d*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^2/(c + d*sin(e + f*x))^5, x, 8, (a^2*(12*c^2 - 16*c*d + 7*d^2)*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/(4*(c - d)^2*(c + d)^4*sqrt(c^2 - d^2)*f) + (a^2*(c - d)*cos(e + f*x))/(4*d*(c + d)*f*(c + d*sin(e + f*x))^4) - (a^2*(c + 8*d)*cos(e + f*x))/(12*d*(c + d)^2*f*(c + d*sin(e + f*x))^3) - (a^2*(2*c^2 + 16*c*d - 21*d^2)*cos(e + f*x))/(24*(c - d)*d*(c + d)^3*f*(c + d*sin(e + f*x))^2) - (a^2*(2*c^3 + 16*c^2*d - 59*c*d^2 + 32*d^3)*cos(e + f*x))/(24*(c - d)^2*d*(c + d)^4*f*(c + d*sin(e + f*x)))]

    @test_int [(a + a*sin(e + f*x))^3*(c + d*sin(e + f*x))^3, x, 6, (1/16)*a^3*(40*c^3 + 90*c^2*d + 78*c*d^2 + 23*d^3)*x - (4*a^3*(c + d)^3*cos(e + f*x))/f + (a^3*(c + d)^2*(c + 7*d)*cos(e + f*x)^3)/(3*f) - (3*a^3*d^2*(c + d)*cos(e + f*x)^5)/(5*f) - (a^3*(24*c^3 + 90*c^2*d + 78*c*d^2 + 23*d^3)*cos(e + f*x)*sin(e + f*x))/(16*f) - (a^3*d*(18*c^2 + 54*c*d + 23*d^2)*cos(e + f*x)*sin(e + f*x)^3)/(24*f) - (a^3*d^3*cos(e + f*x)*sin(e + f*x)^5)/(6*f), (1/16)*a^3*(40*c^3 + 90*c^2*d + 78*c*d^2 + 23*d^3)*x - (a^3*(2*c^5 - 18*c^4*d + 107*c^3*d^2 + 472*c^2*d^3 + 456*c*d^4 + 136*d^5)*cos(e + f*x))/(60*d^2*f) - (a^3*(4*c^4 - 36*c^3*d + 216*c^2*d^2 + 626*c*d^3 + 345*d^4)*cos(e + f*x)*sin(e + f*x))/(240*d*f) - (a^3*(2*c^3 - 18*c^2*d + 111*c*d^2 + 136*d^3)*cos(e + f*x)*(c + d*sin(e + f*x))^2)/(120*d^2*f) - (a^3*(2*c^2 - 18*c*d + 115*d^2)*cos(e + f*x)*(c + d*sin(e + f*x))^3)/(120*d^2*f) + (a^3*(2*c - 13*d)*cos(e + f*x)*(c + d*sin(e + f*x))^4)/(30*d^2*f) - (cos(e + f*x)*(a^3 + a^3*sin(e + f*x))*(c + d*sin(e + f*x))^4)/(6*d*f)]
    @test_int [(a + a*sin(e + f*x))^3*(c + d*sin(e + f*x))^2, x, 9, (1/8)*a^3*(20*c^2 + 30*c*d + 13*d^2)*x - (4*a^3*(c + d)^2*cos(e + f*x))/f + (a^3*(c^2 + 6*c*d + 5*d^2)*cos(e + f*x)^3)/(3*f) - (a^3*d^2*cos(e + f*x)^5)/(5*f) - (a^3*(12*c^2 + 30*c*d + 13*d^2)*cos(e + f*x)*sin(e + f*x))/(8*f) - (a^3*d*(2*c + 3*d)*cos(e + f*x)*sin(e + f*x)^3)/(4*f), (1/8)*a^3*(20*c^2 + 30*c*d + 13*d^2)*x - (a^3*(20*c^2 + 30*c*d + 13*d^2)*cos(e + f*x))/(5*f) + (a^3*(20*c^2 + 30*c*d + 13*d^2)*cos(e + f*x)^3)/(60*f) - (3*a^3*(20*c^2 + 30*c*d + 13*d^2)*cos(e + f*x)*sin(e + f*x))/(40*f) - ((10*c - d)*d*cos(e + f*x)*(a + a*sin(e + f*x))^3)/(20*f) - (d^2*cos(e + f*x)*(a + a*sin(e + f*x))^4)/(5*a*f)]
    @test_int [(a + a*sin(e + f*x))^3*(c + d*sin(e + f*x))^1, x, 8, (5/8)*a^3*(4*c + 3*d)*x - (4*a^3*(c + d)*cos(e + f*x))/f + (a^3*(c + 3*d)*cos(e + f*x)^3)/(3*f) - (3*a^3*(4*c + 5*d)*cos(e + f*x)*sin(e + f*x))/(8*f) - (a^3*d*cos(e + f*x)*sin(e + f*x)^3)/(4*f), (5/8)*a^3*(4*c + 3*d)*x - (a^3*(4*c + 3*d)*cos(e + f*x))/f + (a^3*(4*c + 3*d)*cos(e + f*x)^3)/(12*f) - (3*a^3*(4*c + 3*d)*cos(e + f*x)*sin(e + f*x))/(8*f) - (d*cos(e + f*x)*(a + a*sin(e + f*x))^3)/(4*f)]
    @test_int [(a + a*sin(e + f*x))^3*(c + d*sin(e + f*x))^0, x, 7, (5*a^3*x)/2 - (4*a^3*cos(e + f*x))/f + (a^3*cos(e + f*x)^3)/(3*f) - (3*a^3*cos(e + f*x)*sin(e + f*x))/(2*f)]
    @test_int [(a + a*sin(e + f*x))^3/(c + d*sin(e + f*x))^1, x, 7, (a^3*(2*c^2 - 6*c*d + 7*d^2)*x)/(2*d^3) - (2*a^3*(c - d)^3*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/(d^3*sqrt(c^2 - d^2)*f) + (a^3*(2*c - 5*d)*cos(e + f*x))/(2*d^2*f) - (cos(e + f*x)*(a^3 + a^3*sin(e + f*x)))/(2*d*f)]
    @test_int [(a + a*sin(e + f*x))^3/(c + d*sin(e + f*x))^2, x, 7, -((a^3*(2*c - 3*d)*x)/d^3) + (2*a^3*(c - d)^2*(2*c + 3*d)*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/(d^3*(c + d)*sqrt(c^2 - d^2)*f) - (2*a^3*c*cos(e + f*x))/(d^2*(c + d)*f) + ((c - d)*cos(e + f*x)*(a^3 + a^3*sin(e + f*x)))/(d*(c + d)*f*(c + d*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^3/(c + d*sin(e + f*x))^3, x, 7, (a^3*x)/d^3 - (a^3*(c - d)*(2*c^2 + 6*c*d + 7*d^2)*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/(d^3*(c + d)^2*sqrt(c^2 - d^2)*f) + ((c - d)*cos(e + f*x)*(a^3 + a^3*sin(e + f*x)))/(2*d*(c + d)*f*(c + d*sin(e + f*x))^2) + (a^3*(c - d)*(2*c + 5*d)*cos(e + f*x))/(2*d^2*(c + d)^2*f*(c + d*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^3/(c + d*sin(e + f*x))^4, x, 8, (5*a^3*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/((c + d)^3*sqrt(c^2 - d^2)*f) + ((c - d)*cos(e + f*x)*(a^3 + a^3*sin(e + f*x)))/(3*d*(c + d)*f*(c + d*sin(e + f*x))^3) + (a^3*(c - d)*(2*c + 7*d)*cos(e + f*x))/(6*d^2*(c + d)^2*f*(c + d*sin(e + f*x))^2) - (a^3*(2*c^2 + 9*c*d + 22*d^2)*cos(e + f*x))/(6*d^2*(c + d)^3*f*(c + d*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^3/(c + d*sin(e + f*x))^5, x, 9, (5*a^3*(4*c - 3*d)*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/(4*(c - d)*(c + d)^4*sqrt(c^2 - d^2)*f) + ((c - d)*cos(e + f*x)*(a^3 + a^3*sin(e + f*x)))/(4*d*(c + d)*f*(c + d*sin(e + f*x))^4) + (a^3*(c - d)*(2*c + 9*d)*cos(e + f*x))/(12*d^2*(c + d)^2*f*(c + d*sin(e + f*x))^3) - (a^3*(2*c^2 + 12*c*d + 45*d^2)*cos(e + f*x))/(24*d^2*(c + d)^3*f*(c + d*sin(e + f*x))^2) - (a^3*(2*c^3 + 12*c^2*d + 43*c*d^2 - 72*d^3)*cos(e + f*x))/(24*(c - d)*d^2*(c + d)^4*f*(c + d*sin(e + f*x)))]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [1/(a + a*sin(e + f*x))^1*(c + d*sin(e + f*x))^4, x, 3, (d*(8*c^3 - 12*c^2*d + 12*c*d^2 - 3*d^3)*x)/(2*a) + (2*d*(3*c^3 - 16*c^2*d + 12*c*d^2 - 4*d^3)*cos(e + f*x))/(3*a*f) + (d^2*(6*c^2 - 20*c*d + 9*d^2)*cos(e + f*x)*sin(e + f*x))/(6*a*f) + ((3*c - 4*d)*d*cos(e + f*x)*(c + d*sin(e + f*x))^2)/(3*a*f) - ((c - d)*cos(e + f*x)*(c + d*sin(e + f*x))^3)/(f*(a + a*sin(e + f*x)))]
    @test_int [1/(a + a*sin(e + f*x))^1*(c + d*sin(e + f*x))^3, x, 2, (3*d*(2*c^2 - 2*c*d + d^2)*x)/(2*a) + (2*d*(c^2 - 3*c*d + d^2)*cos(e + f*x))/(a*f) + ((2*c - 3*d)*d^2*cos(e + f*x)*sin(e + f*x))/(2*a*f) - ((c - d)*cos(e + f*x)*(c + d*sin(e + f*x))^2)/(f*(a + a*sin(e + f*x)))]
    @test_int [1/(a + a*sin(e + f*x))^1*(c + d*sin(e + f*x))^2, x, 3, ((2*c - d)*d*x)/a - (d^2*cos(e + f*x))/(a*f) - ((c - d)^2*cos(e + f*x))/(a*f*(1 + sin(e + f*x)))]
    @test_int [1/(a + a*sin(e + f*x))^1*(c + d*sin(e + f*x))^1, x, 2, (d*x)/a - ((c - d)*cos(e + f*x))/(f*(a + a*sin(e + f*x)))]
    @test_int [1/(a + a*sin(e + f*x))^1*(c + d*sin(e + f*x))^0, x, 1, -(cos(e + f*x)/(f*(a + a*sin(e + f*x))))]
    @test_int [1/(a + a*sin(e + f*x))^1/(c + d*sin(e + f*x))^1, x, 5, -((2*d*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/(a*(c - d)*sqrt(c^2 - d^2)*f)) - cos(e + f*x)/((c - d)*f*(a + a*sin(e + f*x)))]
    @test_int [1/(a + a*sin(e + f*x))^1/(c + d*sin(e + f*x))^2, x, 6, -((2*d*(2*c + d)*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/(a*(c - d)*(c^2 - d^2)^(3/2)*f)) - (d*(c + 2*d)*cos(e + f*x))/(a*(c - d)^2*(c + d)*f*(c + d*sin(e + f*x))) - cos(e + f*x)/((c - d)*f*(a + a*sin(e + f*x))*(c + d*sin(e + f*x)))]
    @test_int [1/(a + a*sin(e + f*x))^1/(c + d*sin(e + f*x))^3, x, 7, -((3*d*(2*c^2 + 2*c*d + d^2)*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/(a*(c - d)*(c^2 - d^2)^(5/2)*f)) - (d*(2*c + 3*d)*cos(e + f*x))/(2*a*(c - d)^2*(c + d)*f*(c + d*sin(e + f*x))^2) - cos(e + f*x)/((c - d)*f*(a + a*sin(e + f*x))*(c + d*sin(e + f*x))^2) - (d*(2*c + d)*(c + 4*d)*cos(e + f*x))/(2*a*(c - d)^3*(c + d)^2*f*(c + d*sin(e + f*x)))]

    @test_int [1/(a + a*sin(e + f*x))^2*(c + d*sin(e + f*x))^5, x, 4, (5*(2*c - d)*d^2*(2*c^2 - 3*c*d + 2*d^2)*x)/(2*a^2) + (2*d*(c^4 + 10*c^3*d - 44*c^2*d^2 + 40*c*d^3 - 12*d^4)*cos(e + f*x))/(3*a^2*f) + (d^2*(2*c^3 + 20*c^2*d - 57*c*d^2 + 30*d^3)*cos(e + f*x)*sin(e + f*x))/(6*a^2*f) + (d*(c^2 + 10*c*d - 12*d^2)*cos(e + f*x)*(c + d*sin(e + f*x))^2)/(3*a^2*f) - ((c - d)*(c + 10*d)*cos(e + f*x)*(c + d*sin(e + f*x))^3)/(3*a^2*f*(1 + sin(e + f*x))) - ((c - d)*cos(e + f*x)*(c + d*sin(e + f*x))^4)/(3*f*(a + a*sin(e + f*x))^2)]
    @test_int [1/(a + a*sin(e + f*x))^2*(c + d*sin(e + f*x))^4, x, 3, (d^2*(12*c^2 - 16*c*d + 7*d^2)*x)/(2*a^2) + (2*d*(c^3 + 8*c^2*d - 20*c*d^2 + 8*d^3)*cos(e + f*x))/(3*a^2*f) + (d^2*(2*c^2 + 16*c*d - 21*d^2)*cos(e + f*x)*sin(e + f*x))/(6*a^2*f) - ((c - d)*(c + 8*d)*cos(e + f*x)*(c + d*sin(e + f*x))^2)/(3*a^2*f*(1 + sin(e + f*x))) - ((c - d)*cos(e + f*x)*(c + d*sin(e + f*x))^3)/(3*f*(a + a*sin(e + f*x))^2)]
    @test_int [1/(a + a*sin(e + f*x))^2*(c + d*sin(e + f*x))^3, x, 5, ((3*c - 2*d)*d^2*x)/a^2 + ((c - 4*d)*d^2*cos(e + f*x))/(3*a^2*f) - ((c - d)^2*(c + 6*d)*cos(e + f*x))/(3*a^2*f*(1 + sin(e + f*x))) - ((c - d)*cos(e + f*x)*(c + d*sin(e + f*x))^2)/(3*f*(a + a*sin(e + f*x))^2)]
    @test_int [1/(a + a*sin(e + f*x))^2*(c + d*sin(e + f*x))^2, x, 3, (d^2*x)/a^2 - ((c - d)*(c + 4*d)*cos(e + f*x))/(3*a^2*f*(1 + sin(e + f*x))) - ((c - d)*cos(e + f*x)*(c + d*sin(e + f*x)))/(3*f*(a + a*sin(e + f*x))^2)]
    @test_int [1/(a + a*sin(e + f*x))^2*(c + d*sin(e + f*x))^1, x, 2, -(((c - d)*cos(e + f*x))/(3*f*(a + a*sin(e + f*x))^2)) - ((c + 2*d)*cos(e + f*x))/(3*f*(a^2 + a^2*sin(e + f*x)))]
    @test_int [1/(a + a*sin(e + f*x))^2*(c + d*sin(e + f*x))^0, x, 2, -(cos(e + f*x)/(3*f*(a + a*sin(e + f*x))^2)) - cos(e + f*x)/(3*f*(a^2 + a^2*sin(e + f*x)))]
    @test_int [1/(a + a*sin(e + f*x))^2/(c + d*sin(e + f*x))^1, x, 6, (2*d^2*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/(a^2*(c - d)^2*sqrt(c^2 - d^2)*f) - ((c - 4*d)*cos(e + f*x))/(3*a^2*(c - d)^2*f*(1 + sin(e + f*x))) - cos(e + f*x)/(3*(c - d)*f*(a + a*sin(e + f*x))^2)]
    @test_int [1/(a + a*sin(e + f*x))^2/(c + d*sin(e + f*x))^2, x, 7, (2*d^2*(3*c + 2*d)*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/(a^2*(c - d)^3*(c + d)*sqrt(c^2 - d^2)*f) - (d*(c^2 - 6*c*d - 10*d^2)*cos(e + f*x))/(3*a^2*(c - d)^3*(c + d)*f*(c + d*sin(e + f*x))) - ((c - 6*d)*cos(e + f*x))/(3*a^2*(c - d)^2*f*(1 + sin(e + f*x))*(c + d*sin(e + f*x))) - cos(e + f*x)/(3*(c - d)*f*(a + a*sin(e + f*x))^2*(c + d*sin(e + f*x)))]
    @test_int [1/(a + a*sin(e + f*x))^2/(c + d*sin(e + f*x))^3, x, 8, (d^2*(12*c^2 + 16*c*d + 7*d^2)*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/(a^2*(c - d)^4*(c + d)^2*sqrt(c^2 - d^2)*f) - (d*(2*c^2 - 16*c*d - 21*d^2)*cos(e + f*x))/(6*a^2*(c - d)^3*(c + d)*f*(c + d*sin(e + f*x))^2) - ((c - 8*d)*cos(e + f*x))/(3*a^2*(c - d)^2*f*(1 + sin(e + f*x))*(c + d*sin(e + f*x))^2) - cos(e + f*x)/(3*(c - d)*f*(a + a*sin(e + f*x))^2*(c + d*sin(e + f*x))^2) - (d*(2*c^3 - 16*c^2*d - 59*c*d^2 - 32*d^3)*cos(e + f*x))/(6*a^2*(c - d)^4*(c + d)^2*f*(c + d*sin(e + f*x)))]

    @test_int [1/(a + a*sin(e + f*x))^3*(c + d*sin(e + f*x))^6, x, 5, (d^3*(40*c^3 - 90*c^2*d + 78*c*d^2 - 23*d^3)*x)/(2*a^3) + (2*d*(2*c^5 + 18*c^4*d + 107*c^3*d^2 - 472*c^2*d^3 + 456*c*d^4 - 136*d^5)*cos(e + f*x))/(15*a^3*f) + (d^2*(4*c^4 + 36*c^3*d + 216*c^2*d^2 - 626*c*d^3 + 345*d^4)*cos(e + f*x)*sin(e + f*x))/(30*a^3*f) + (d*(2*c^3 + 18*c^2*d + 111*c*d^2 - 136*d^3)*cos(e + f*x)*(c + d*sin(e + f*x))^2)/(15*a^3*f) - ((c - d)*(2*c^2 + 18*c*d + 115*d^2)*cos(e + f*x)*(c + d*sin(e + f*x))^3)/(15*f*(a^3 + a^3*sin(e + f*x))) - ((c - d)*(2*c + 13*d)*cos(e + f*x)*(c + d*sin(e + f*x))^4)/(15*a*f*(a + a*sin(e + f*x))^2) - ((c - d)*cos(e + f*x)*(c + d*sin(e + f*x))^5)/(5*f*(a + a*sin(e + f*x))^3)]
    @test_int [1/(a + a*sin(e + f*x))^3*(c + d*sin(e + f*x))^5, x, 4, (d^3*(20*c^2 - 30*c*d + 13*d^2)*x)/(2*a^3) + (2*d*(2*c^4 + 15*c^3*d + 72*c^2*d^2 - 180*c*d^3 + 76*d^4)*cos(e + f*x))/(15*a^3*f) + (d^2*(4*c^3 + 30*c^2*d + 146*c*d^2 - 195*d^3)*cos(e + f*x)*sin(e + f*x))/(30*a^3*f) - ((c - d)*(2*c^2 + 15*c*d + 76*d^2)*cos(e + f*x)*(c + d*sin(e + f*x))^2)/(15*f*(a^3 + a^3*sin(e + f*x))) - ((c - d)*(2*c + 11*d)*cos(e + f*x)*(c + d*sin(e + f*x))^3)/(15*a*f*(a + a*sin(e + f*x))^2) - ((c - d)*cos(e + f*x)*(c + d*sin(e + f*x))^4)/(5*f*(a + a*sin(e + f*x))^3)]
    @test_int [1/(a + a*sin(e + f*x))^3*(c + d*sin(e + f*x))^4, x, 6, ((4*c - 3*d)*d^3*x)/a^3 + (d^2*(2*c^2 + 10*c*d - 27*d^2)*cos(e + f*x))/(15*a^3*f) - ((c - d)^2*(2*c^2 + 12*c*d + 45*d^2)*cos(e + f*x))/(15*f*(a^3 + a^3*sin(e + f*x))) - ((c - d)*(2*c + 9*d)*cos(e + f*x)*(c + d*sin(e + f*x))^2)/(15*a*f*(a + a*sin(e + f*x))^2) - ((c - d)*cos(e + f*x)*(c + d*sin(e + f*x))^3)/(5*f*(a + a*sin(e + f*x))^3)]
    @test_int [1/(a + a*sin(e + f*x))^3*(c + d*sin(e + f*x))^3, x, 5, (d^3*x)/a^3 - ((c - d)^2*(2*c + 7*d)*cos(e + f*x))/(15*a*f*(a + a*sin(e + f*x))^2) - ((c - d)*(2*c^2 + 11*c*d + 29*d^2)*cos(e + f*x))/(15*f*(a^3 + a^3*sin(e + f*x))) - ((c - d)*cos(e + f*x)*(c + d*sin(e + f*x))^2)/(5*f*(a + a*sin(e + f*x))^3)]
    @test_int [1/(a + a*sin(e + f*x))^3*(c + d*sin(e + f*x))^2, x, 3, -(((c - d)*(2*c + 5*d)*cos(e + f*x))/(15*a*f*(a + a*sin(e + f*x))^2)) - ((2*c^2 + 6*c*d + 7*d^2)*cos(e + f*x))/(15*f*(a^3 + a^3*sin(e + f*x))) - ((c - d)*cos(e + f*x)*(c + d*sin(e + f*x)))/(5*f*(a + a*sin(e + f*x))^3)]
    @test_int [1/(a + a*sin(e + f*x))^3*(c + d*sin(e + f*x))^1, x, 3, -(((c - d)*cos(e + f*x))/(5*f*(a + a*sin(e + f*x))^3)) - ((2*c + 3*d)*cos(e + f*x))/(15*a*f*(a + a*sin(e + f*x))^2) - ((2*c + 3*d)*cos(e + f*x))/(15*f*(a^3 + a^3*sin(e + f*x)))]
    @test_int [1/(a + a*sin(e + f*x))^3*(c + d*sin(e + f*x))^0, x, 3, -(cos(e + f*x)/(5*f*(a + a*sin(e + f*x))^3)) - (2*cos(e + f*x))/(15*a*f*(a + a*sin(e + f*x))^2) - (2*cos(e + f*x))/(15*f*(a^3 + a^3*sin(e + f*x)))]
    @test_int [1/(a + a*sin(e + f*x))^3/(c + d*sin(e + f*x))^1, x, 7, -((2*d^3*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/(a^3*(c - d)^3*sqrt(c^2 - d^2)*f)) - cos(e + f*x)/(5*(c - d)*f*(a + a*sin(e + f*x))^3) - ((2*c - 7*d)*cos(e + f*x))/(15*a*(c - d)^2*f*(a + a*sin(e + f*x))^2) - ((2*c^2 - 9*c*d + 22*d^2)*cos(e + f*x))/(15*(c - d)^3*f*(a^3 + a^3*sin(e + f*x)))]
    @test_int [1/(a + a*sin(e + f*x))^3/(c + d*sin(e + f*x))^2, x, 8, -((2*d^3*(4*c + 3*d)*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/(a^3*(c - d)^4*(c + d)*sqrt(c^2 - d^2)*f)) - (d*(2*c^3 - 12*c^2*d + 43*c*d^2 + 72*d^3)*cos(e + f*x))/(15*a^3*(c - d)^4*(c + d)*f*(c + d*sin(e + f*x))) - cos(e + f*x)/(5*(c - d)*f*(a + a*sin(e + f*x))^3*(c + d*sin(e + f*x))) - ((2*c - 9*d)*cos(e + f*x))/(15*a*(c - d)^2*f*(a + a*sin(e + f*x))^2*(c + d*sin(e + f*x))) - ((2*c^2 - 12*c*d + 45*d^2)*cos(e + f*x))/(15*(c - d)^3*f*(a^3 + a^3*sin(e + f*x))*(c + d*sin(e + f*x)))]
    @test_int [1/(a + a*sin(e + f*x))^3/(c + d*sin(e + f*x))^3, x, 9, -((d^3*(20*c^2 + 30*c*d + 13*d^2)*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/(a^3*(c - d)^5*(c + d)^2*sqrt(c^2 - d^2)*f)) - (d*(4*c^3 - 30*c^2*d + 146*c*d^2 + 195*d^3)*cos(e + f*x))/(30*a^3*(c - d)^4*(c + d)*f*(c + d*sin(e + f*x))^2) - cos(e + f*x)/(5*(c - d)*f*(a + a*sin(e + f*x))^3*(c + d*sin(e + f*x))^2) - ((2*c - 11*d)*cos(e + f*x))/(15*a*(c - d)^2*f*(a + a*sin(e + f*x))^2*(c + d*sin(e + f*x))^2) - ((2*c^2 - 15*c*d + 76*d^2)*cos(e + f*x))/(15*(c - d)^3*f*(a^3 + a^3*sin(e + f*x))*(c + d*sin(e + f*x))^2) - (d*(4*c^4 - 30*c^3*d + 142*c^2*d^2 + 525*c*d^3 + 304*d^4)*cos(e + f*x))/(30*a^3*(c - d)^5*(c + d)^2*f*(c + d*sin(e + f*x)))]

    @test_int [(A + B*sin(x))/(1 + sin(x))^4, x, 4, -(((A - B)*cos(x))/(7*(1 + sin(x))^4)) - ((3*A + 4*B)*cos(x))/(35*(1 + sin(x))^3) - (2*(3*A + 4*B)*cos(x))/(105*(1 + sin(x))^2) - (2*(3*A + 4*B)*cos(x))/(105*(1 + sin(x)))]
    @test_int [(A + B*sin(x))/(1 - sin(x))^4, x, 4, ((A + B)*cos(x))/(7*(1 - sin(x))^4) + ((3*A - 4*B)*cos(x))/(35*(1 - sin(x))^3) + (2*(3*A - 4*B)*cos(x))/(105*(1 - sin(x))^2) + (2*(3*A - 4*B)*cos(x))/(105*(1 - sin(x)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+a*sin(e+f*x))^m*(c+d*sin(e+f*x))^(n/2)=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [(a + a*sin(e + f*x))*(c + d*sin(e + f*x))^(5/2), x, 8, (-2*a*(15*c^2 + 56*c*d + 25*d^2)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(105*f) - (2*a*(5*c + 7*d)*cos(e + f*x)*(c + d*sin(e + f*x))^(3/2))/(35*f) - (2*a*cos(e + f*x)*(c + d*sin(e + f*x))^(5/2))/(7*f) + (2*a*(15*c^3 + 161*c^2*d + 145*c*d^2 + 63*d^3)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(105*d*f*sqrt((c + d*sin(e + f*x))/(c + d))) - (2*a*(c^2 - d^2)*(15*c^2 + 56*c*d + 25*d^2)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(105*d*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))*(c + d*sin(e + f*x))^(3/2), x, 7, (-2*a*(3*c + 5*d)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(15*f) - (2*a*cos(e + f*x)*(c + d*sin(e + f*x))^(3/2))/(5*f) + (2*a*(3*c^2 + 20*c*d + 9*d^2)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(15*d*f*sqrt((c + d*sin(e + f*x))/(c + d))) - (2*a*(3*c + 5*d)*(c^2 - d^2)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(15*d*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))*(c + d*sin(e + f*x))^(1/2), x, 6, (-2*a*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(3*f) + (2*a*(c + 3*d)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(3*d*f*sqrt((c + d*sin(e + f*x))/(c + d))) - (2*a*(c^2 - d^2)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(3*d*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))/(c + d*sin(e + f*x))^(1/2), x, 5, (2*a*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(d*f*sqrt((c + d*sin(e + f*x))/(c + d))) - (2*a*(c - d)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(d*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))/(c + d*sin(e + f*x))^(3/2), x, 6, (-2*a*cos(e + f*x))/((c + d)*f*sqrt(c + d*sin(e + f*x))) - (2*a*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(d*(c + d)*f*sqrt((c + d*sin(e + f*x))/(c + d))) + (2*a*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(d*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))/(c + d*sin(e + f*x))^(5/2), x, 7, (-2*a*cos(e + f*x))/(3*(c + d)*f*(c + d*sin(e + f*x))^(3/2)) - (2*a*(c - 3*d)*cos(e + f*x))/(3*(c - d)*(c + d)^2*f*sqrt(c + d*sin(e + f*x))) - (2*a*(c - 3*d)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(3*(c - d)*d*(c + d)^2*f*sqrt((c + d*sin(e + f*x))/(c + d))) + (2*a*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(3*d*(c + d)*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))/(c + d*sin(e + f*x))^(7/2), x, 8, (-2*a*cos(e + f*x))/(5*(c + d)*f*(c + d*sin(e + f*x))^(5/2)) - (2*a*(3*c - 5*d)*cos(e + f*x))/(15*(c - d)*(c + d)^2*f*(c + d*sin(e + f*x))^(3/2)) - (2*a*(3*c^2 - 20*c*d + 9*d^2)*cos(e + f*x))/(15*(c - d)^2*(c + d)^3*f*sqrt(c + d*sin(e + f*x))) - (2*a*(3*c^2 - 20*c*d + 9*d^2)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(15*(c - d)^2*d*(c + d)^3*f*sqrt((c + d*sin(e + f*x))/(c + d))) + (2*a*(3*c - 5*d)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(15*(c - d)*d*(c + d)^2*f*sqrt(c + d*sin(e + f*x)))]

    @test_int [(a + a*sin(e + f*x))^2*(c + d*sin(e + f*x))^(5/2), x, 9, (4*a^2*(5*c^3 - 45*c^2*d - 141*c*d^2 - 75*d^3)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(315*d*f) + (4*a^2*(5*c*(c - 9*d) - 56*d^2)*cos(e + f*x)*(c + d*sin(e + f*x))^(3/2))/(315*d*f) + (4*a^2*(c - 9*d)*cos(e + f*x)*(c + d*sin(e + f*x))^(5/2))/(63*d*f) - (2*a^2*cos(e + f*x)*(c + d*sin(e + f*x))^(7/2))/(9*d*f) - (4*a^2*(5*c^4 - 45*c^3*d - 381*c^2*d^2 - 435*c*d^3 - 168*d^4)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(315*d^2*f*sqrt((c + d*sin(e + f*x))/(c + d))) + (4*a^2*(c^2 - d^2)*(5*c^3 - 45*c^2*d - 141*c*d^2 - 75*d^3)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(315*d^2*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^2*(c + d*sin(e + f*x))^(3/2), x, 8, (4*a^2*(c^2 - 7*c*d - 10*d^2)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(35*d*f) + (4*a^2*(c - 7*d)*cos(e + f*x)*(c + d*sin(e + f*x))^(3/2))/(35*d*f) - (2*a^2*cos(e + f*x)*(c + d*sin(e + f*x))^(5/2))/(7*d*f) - (4*a^2*(c + 3*d)*(c^2 - 10*c*d - 7*d^2)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(35*d^2*f*sqrt((c + d*sin(e + f*x))/(c + d))) + (4*a^2*(c^2 - 7*c*d - 10*d^2)*(c^2 - d^2)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(35*d^2*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^2*(c + d*sin(e + f*x))^(1/2), x, 7, (4*a^2*(c - 5*d)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(15*d*f) - (2*a^2*cos(e + f*x)*(c + d*sin(e + f*x))^(3/2))/(5*d*f) - (4*a^2*(c^2 - 5*c*d - 12*d^2)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(15*d^2*f*sqrt((c + d*sin(e + f*x))/(c + d))) + (4*a^2*(c - 5*d)*(c^2 - d^2)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(15*d^2*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^2/(c + d*sin(e + f*x))^(1/2), x, 6, (-2*a^2*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(3*d*f) - (4*a^2*(c - 3*d)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(3*d^2*f*sqrt((c + d*sin(e + f*x))/(c + d))) + (4*a^2*(c - 2*d)*(c - d)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(3*d^2*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^2/(c + d*sin(e + f*x))^(3/2), x, 6, (2*a^2*(c - d)*cos(e + f*x))/(d*(c + d)*f*sqrt(c + d*sin(e + f*x))) + (4*a^2*c*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(d^2*(c + d)*f*sqrt((c + d*sin(e + f*x))/(c + d))) - (4*a^2*(c - d)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(d^2*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^2/(c + d*sin(e + f*x))^(5/2), x, 7, (2*a^2*(c - d)*cos(e + f*x))/(3*d*(c + d)*f*(c + d*sin(e + f*x))^(3/2)) - (4*a^2*(c + 3*d)*cos(e + f*x))/(3*d*(c + d)^2*f*sqrt(c + d*sin(e + f*x))) - (4*a^2*(c + 3*d)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(3*d^2*(c + d)^2*f*sqrt((c + d*sin(e + f*x))/(c + d))) + (4*a^2*(c + 2*d)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(3*d^2*(c + d)*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^2/(c + d*sin(e + f*x))^(7/2), x, 8, (2*a^2*(c - d)*cos(e + f*x))/(5*d*(c + d)*f*(c + d*sin(e + f*x))^(5/2)) - (4*a^2*(c + 5*d)*cos(e + f*x))/(15*d*(c + d)^2*f*(c + d*sin(e + f*x))^(3/2)) - (4*a^2*(c^2 + 5*c*d - 12*d^2)*cos(e + f*x))/(15*(c - d)*d*(c + d)^3*f*sqrt(c + d*sin(e + f*x))) - (4*a^2*(c^2 + 5*c*d - 12*d^2)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(15*(c - d)*d^2*(c + d)^3*f*sqrt((c + d*sin(e + f*x))/(c + d))) + (4*a^2*(c + 5*d)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(15*d^2*(c + d)^2*f*sqrt(c + d*sin(e + f*x)))]

    @test_int [(a + a*sin(e + f*x))^3*(c + d*sin(e + f*x))^(5/2), x, 11, (-4*a^3*(4*c^4 - 33*c^3*d + 177*c^2*d^2 + 561*c*d^3 + 315*d^4)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(693*d^2*f) - (4*a^3*(4*c^3 - 33*c^2*d + 182*c*d^2 + 231*d^3)*cos(e + f*x)*(c + d*sin(e + f*x))^(3/2))/(693*d^2*f) - (4*a^3*(4*c^2 - 33*c*d + 189*d^2)*cos(e + f*x)*(c + d*sin(e + f*x))^(5/2))/(693*d^2*f) + (8*a^3*(c - 6*d)*cos(e + f*x)*(c + d*sin(e + f*x))^(7/2))/(99*d^2*f) - (2*cos(e + f*x)*(a^3 + a^3*sin(e + f*x))*(c + d*sin(e + f*x))^(7/2))/(11*d*f) + (4*a^3*(c + 3*d)*(4*c^4 - 45*c^3*d + 309*c^2*d^2 + 525*c*d^3 + 231*d^4)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(693*d^3*f*sqrt((c + d*sin(e + f*x))/(c + d))) - (4*a^3*(c^2 - d^2)*(4*c^4 - 33*c^3*d + 177*c^2*d^2 + 561*c*d^3 + 315*d^4)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(693*d^3*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^3*(c + d*sin(e + f*x))^(3/2), x, 10, (-4*a^3*(4*c^3 - 27*c^2*d + 114*c*d^2 + 165*d^3)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(315*d^2*f) - (4*a^3*(4*c^2 - 27*c*d + 119*d^2)*cos(e + f*x)*(c + d*sin(e + f*x))^(3/2))/(315*d^2*f) + (8*a^3*(c - 5*d)*cos(e + f*x)*(c + d*sin(e + f*x))^(5/2))/(63*d^2*f) - (2*cos(e + f*x)*(a^3 + a^3*sin(e + f*x))*(c + d*sin(e + f*x))^(5/2))/(9*d*f) + (4*a^3*(4*c^4 - 27*c^3*d + 111*c^2*d^2 + 579*c*d^3 + 357*d^4)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(315*d^3*f*sqrt((c + d*sin(e + f*x))/(c + d))) - (4*a^3*(c^2 - d^2)*(4*c^3 - 27*c^2*d + 114*c*d^2 + 165*d^3)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(315*d^3*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^3*(c + d*sin(e + f*x))^(1/2), x, 9, (-4*a^3*(4*c^2 - 21*c*d + 65*d^2)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(105*d^2*f) + (8*a^3*(c - 4*d)*cos(e + f*x)*(c + d*sin(e + f*x))^(3/2))/(35*d^2*f) - (2*cos(e + f*x)*(a^3 + a^3*sin(e + f*x))*(c + d*sin(e + f*x))^(3/2))/(7*d*f) + (4*a^3*(4*c^3 - 21*c^2*d + 62*c*d^2 + 147*d^3)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(105*d^3*f*sqrt((c + d*sin(e + f*x))/(c + d))) - (4*a^3*(c^2 - d^2)*(4*c^2 - 21*c*d + 65*d^2)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(105*d^3*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^3/(c + d*sin(e + f*x))^(1/2), x, 8, (8*a^3*(c - 3*d)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(15*d^2*f) - (2*cos(e + f*x)*(a^3 + a^3*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))/(5*d*f) + (4*a^3*(4*c^2 - 15*c*d + 27*d^2)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(15*d^3*f*sqrt((c + d*sin(e + f*x))/(c + d))) - (4*a^3*(c - d)*(4*c^2 - 11*c*d + 15*d^2)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(15*d^3*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^3/(c + d*sin(e + f*x))^(3/2), x, 8, (2*(c - d)*cos(e + f*x)*(a^3 + a^3*sin(e + f*x)))/(d*(c + d)*f*sqrt(c + d*sin(e + f*x))) - (4*a^3*(2*c - d)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(3*d^2*(c + d)*f) - (4*a^3*(4*c^2 - 5*c*d - 3*d^2)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(3*d^3*(c + d)*f*sqrt((c + d*sin(e + f*x))/(c + d))) + (4*a^3*(4*c - 5*d)*(c - d)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(3*d^3*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^3/(c + d*sin(e + f*x))^(5/2), x, 8, (2*(c - d)*cos(e + f*x)*(a^3 + a^3*sin(e + f*x)))/(3*d*(c + d)*f*(c + d*sin(e + f*x))^(3/2)) + (8*a^3*(c - d)*(c + 2*d)*cos(e + f*x))/(3*d^2*(c + d)^2*f*sqrt(c + d*sin(e + f*x))) + (4*a^3*(4*c^2 + 5*c*d - 3*d^2)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(3*d^3*(c + d)^2*f*sqrt((c + d*sin(e + f*x))/(c + d))) - (4*a^3*(c - d)*(4*c + 5*d)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(3*d^3*(c + d)*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^3/(c + d*sin(e + f*x))^(7/2), x, 9, (2*(c - d)*cos(e + f*x)*(a^3 + a^3*sin(e + f*x)))/(5*d*(c + d)*f*(c + d*sin(e + f*x))^(5/2)) + (8*a^3*(c - d)*(c + 3*d)*cos(e + f*x))/(15*d^2*(c + d)^2*f*(c + d*sin(e + f*x))^(3/2)) - (4*a^3*(4*c^2 + 15*c*d + 27*d^2)*cos(e + f*x))/(15*d^2*(c + d)^3*f*sqrt(c + d*sin(e + f*x))) - (4*a^3*(4*c^2 + 15*c*d + 27*d^2)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(15*d^3*(c + d)^3*f*sqrt((c + d*sin(e + f*x))/(c + d))) + (4*a^3*(4*c^2 + 11*c*d + 15*d^2)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(15*d^3*(c + d)^2*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^3/(c + d*sin(e + f*x))^(9/2), x, 10, (2*(c - d)*cos(e + f*x)*(a^3 + a^3*sin(e + f*x)))/(7*d*(c + d)*f*(c + d*sin(e + f*x))^(7/2)) + (8*a^3*(c - d)*(c + 4*d)*cos(e + f*x))/(35*d^2*(c + d)^2*f*(c + d*sin(e + f*x))^(5/2)) - (4*a^3*(4*c^2 + 21*c*d + 65*d^2)*cos(e + f*x))/(105*d^2*(c + d)^3*f*(c + d*sin(e + f*x))^(3/2)) - (4*a^3*(4*c^3 + 21*c^2*d + 62*c*d^2 - 147*d^3)*cos(e + f*x))/(105*(c - d)*d^2*(c + d)^4*f*sqrt(c + d*sin(e + f*x))) - (4*a^3*(4*c^3 + 21*c^2*d + 62*c*d^2 - 147*d^3)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(105*(c - d)*d^3*(c + d)^4*f*sqrt((c + d*sin(e + f*x))/(c + d))) + (4*a^3*(4*c^2 + 21*c*d + 65*d^2)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(105*d^3*(c + d)^3*f*sqrt(c + d*sin(e + f*x)))]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [(c + d*sin(e + f*x))^(5/2)/(a + a*sin(e + f*x)), x, 7, ((3*c - 5*d)*d*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(3*a*f) - ((c - d)*cos(e + f*x)*(c + d*sin(e + f*x))^(3/2))/(f*(a + a*sin(e + f*x))) - ((3*c^2 - 20*c*d + 9*d^2)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(3*a*f*sqrt((c + d*sin(e + f*x))/(c + d))) + ((3*c - 5*d)*(c^2 - d^2)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(3*a*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(c + d*sin(e + f*x))^(3/2)/(a + a*sin(e + f*x)), x, 6, -(((c - d)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(f*(a + a*sin(e + f*x)))) - ((c - 3*d)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(a*f*sqrt((c + d*sin(e + f*x))/(c + d))) + ((c^2 - d^2)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(a*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(c + d*sin(e + f*x))^(1/2)/(a + a*sin(e + f*x)), x, 6, -((cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(f*(a + a*sin(e + f*x)))) - (Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(a*f*sqrt((c + d*sin(e + f*x))/(c + d))) + ((c + d)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(a*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [1/((a + a*sin(e + f*x))*(c + d*sin(e + f*x))^(1/2)), x, 6, -((cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/((c - d)*f*(a + a*sin(e + f*x)))) - (Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(a*(c - d)*f*sqrt((c + d*sin(e + f*x))/(c + d))) + (Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(a*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [1/((a + a*sin(e + f*x))*(c + d*sin(e + f*x))^(3/2)), x, 7, -((d*(c + 3*d)*cos(e + f*x))/(a*(c - d)^2*(c + d)*f*sqrt(c + d*sin(e + f*x)))) - cos(e + f*x)/((c - d)*f*(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x))) - ((c + 3*d)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(a*(c - d)^2*(c + d)*f*sqrt((c + d*sin(e + f*x))/(c + d))) + (Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(a*(c - d)*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [1/((a + a*sin(e + f*x))*(c + d*sin(e + f*x))^(5/2)), x, 8, -(d*(3*c + 5*d)*cos(e + f*x))/(3*a*(c - d)^2*(c + d)*f*(c + d*sin(e + f*x))^(3/2)) - cos(e + f*x)/((c - d)*f*(a + a*sin(e + f*x))*(c + d*sin(e + f*x))^(3/2)) - (d*(3*c^2 + 20*c*d + 9*d^2)*cos(e + f*x))/(3*a*(c - d)^3*(c + d)^2*f*sqrt(c + d*sin(e + f*x))) - ((3*c^2 + 20*c*d + 9*d^2)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(3*a*(c - d)^3*(c + d)^2*f*sqrt((c + d*sin(e + f*x))/(c + d))) + ((3*c + 5*d)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(3*a*(c - d)^2*(c + d)*f*sqrt(c + d*sin(e + f*x)))]

    @test_int [(c + d*sin(e + f*x))^(5/2)/(a + a*sin(e + f*x))^2, x, 7, -((c - d)*(c + 5*d)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(3*a^2*f*(1 + sin(e + f*x))) - ((c - d)*cos(e + f*x)*(c + d*sin(e + f*x))^(3/2))/(3*f*(a + a*sin(e + f*x))^2) - ((c^2 + 5*c*d - 12*d^2)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(3*a^2*f*sqrt((c + d*sin(e + f*x))/(c + d))) + ((c + 5*d)*(c^2 - d^2)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(3*a^2*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(c + d*sin(e + f*x))^(3/2)/(a + a*sin(e + f*x))^2, x, 7, -((c + 3*d)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(3*a^2*f*(1 + sin(e + f*x))) - ((c - d)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(3*f*(a + a*sin(e + f*x))^2) - ((c + 3*d)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(3*a^2*f*sqrt((c + d*sin(e + f*x))/(c + d))) + ((c + d)*(c + 2*d)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(3*a^2*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(c + d*sin(e + f*x))^(1/2)/(a + a*sin(e + f*x))^2, x, 7, -(c*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(3*a^2*(c - d)*f*(1 + sin(e + f*x))) - (cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(3*f*(a + a*sin(e + f*x))^2) - (c*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(3*a^2*(c - d)*f*sqrt((c + d*sin(e + f*x))/(c + d))) + ((c + d)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(3*a^2*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [1/((a + a*sin(e + f*x))^2*(c + d*sin(e + f*x))^(1/2)), x, 7, -((c - 3*d)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(3*a^2*(c - d)^2*f*(1 + sin(e + f*x))) - (cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(3*(c - d)*f*(a + a*sin(e + f*x))^2) - ((c - 3*d)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(3*a^2*(c - d)^2*f*sqrt((c + d*sin(e + f*x))/(c + d))) + ((c - 2*d)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(3*a^2*(c - d)*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [1/((a + a*sin(e + f*x))^2*(c + d*sin(e + f*x))^(3/2)), x, 8, -(d*(c^2 - 5*c*d - 12*d^2)*cos(e + f*x))/(3*a^2*(c - d)^3*(c + d)*f*sqrt(c + d*sin(e + f*x))) - ((c - 5*d)*cos(e + f*x))/(3*a^2*(c - d)^2*f*(1 + sin(e + f*x))*sqrt(c + d*sin(e + f*x))) - cos(e + f*x)/(3*(c - d)*f*(a + a*sin(e + f*x))^2*sqrt(c + d*sin(e + f*x))) - ((c^2 - 5*c*d - 12*d^2)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(3*a^2*(c - d)^3*(c + d)*f*sqrt((c + d*sin(e + f*x))/(c + d))) + ((c - 5*d)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(3*a^2*(c - d)^2*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [1/((a + a*sin(e + f*x))^2*(c + d*sin(e + f*x))^(5/2)), x, 9, -(d*(c^2 - 7*c*d - 10*d^2)*cos(e + f*x))/(3*a^2*(c - d)^3*(c + d)*f*(c + d*sin(e + f*x))^(3/2)) - ((c - 7*d)*cos(e + f*x))/(3*a^2*(c - d)^2*f*(1 + sin(e + f*x))*(c + d*sin(e + f*x))^(3/2)) - cos(e + f*x)/(3*(c - d)*f*(a + a*sin(e + f*x))^2*(c + d*sin(e + f*x))^(3/2)) - (d*(c + 3*d)*(c^2 - 10*c*d - 7*d^2)*cos(e + f*x))/(3*a^2*(c - d)^4*(c + d)^2*f*sqrt(c + d*sin(e + f*x))) - ((c + 3*d)*(c^2 - 10*c*d - 7*d^2)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(3*a^2*(c - d)^4*(c + d)^2*f*sqrt((c + d*sin(e + f*x))/(c + d))) + ((c^2 - 7*c*d - 10*d^2)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(3*a^2*(c - d)^3*(c + d)*f*sqrt(c + d*sin(e + f*x)))]

    @test_int [(c + d*sin(e + f*x))^(5/2)/(a + a*sin(e + f*x))^3, x, 8, (-2*(c - d)*(c + 3*d)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(15*a*f*(a + a*sin(e + f*x))^2) - ((4*c^2 + 15*c*d + 27*d^2)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(30*f*(a^3 + a^3*sin(e + f*x))) - ((c - d)*cos(e + f*x)*(c + d*sin(e + f*x))^(3/2))/(5*f*(a + a*sin(e + f*x))^3) - ((4*c^2 + 15*c*d + 27*d^2)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(30*a^3*f*sqrt((c + d*sin(e + f*x))/(c + d))) + ((c + d)*(4*c^2 + 11*c*d + 15*d^2)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(30*a^3*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(c + d*sin(e + f*x))^(3/2)/(a + a*sin(e + f*x))^3, x, 8, -((c - d)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(5*f*(a + a*sin(e + f*x))^3) - (2*(c + 2*d)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(15*a*f*(a + a*sin(e + f*x))^2) - ((4*c^2 + 5*c*d - 3*d^2)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(30*(c - d)*f*(a^3 + a^3*sin(e + f*x))) - ((4*c^2 + 5*c*d - 3*d^2)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(30*a^3*(c - d)*f*sqrt((c + d*sin(e + f*x))/(c + d))) + ((c + d)*(4*c + 5*d)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(30*a^3*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(c + d*sin(e + f*x))^(1/2)/(a + a*sin(e + f*x))^3, x, 8, -(cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(5*f*(a + a*sin(e + f*x))^3) - ((2*c - d)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(15*a*(c - d)*f*(a + a*sin(e + f*x))^2) - ((4*c^2 - 5*c*d - 3*d^2)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(30*(c - d)^2*f*(a^3 + a^3*sin(e + f*x))) - ((4*c^2 - 5*c*d - 3*d^2)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(30*a^3*(c - d)^2*f*sqrt((c + d*sin(e + f*x))/(c + d))) + ((4*c - 5*d)*(c + d)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(30*a^3*(c - d)*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [1/((a + a*sin(e + f*x))^3*(c + d*sin(e + f*x))^(1/2)), x, 8, -(cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(5*(c - d)*f*(a + a*sin(e + f*x))^3) - (2*(c - 3*d)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(15*a*(c - d)^2*f*(a + a*sin(e + f*x))^2) - ((4*c^2 - 15*c*d + 27*d^2)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(30*(c - d)^3*f*(a^3 + a^3*sin(e + f*x))) - ((4*c^2 - 15*c*d + 27*d^2)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(30*a^3*(c - d)^3*f*sqrt((c + d*sin(e + f*x))/(c + d))) + ((4*c^2 - 11*c*d + 15*d^2)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(30*a^3*(c - d)^2*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [1/((a + a*sin(e + f*x))^3*(c + d*sin(e + f*x))^(3/2)), x, 9, -(d*(4*c^3 - 21*c^2*d + 62*c*d^2 + 147*d^3)*cos(e + f*x))/(30*a^3*(c - d)^4*(c + d)*f*sqrt(c + d*sin(e + f*x))) - cos(e + f*x)/(5*(c - d)*f*(a + a*sin(e + f*x))^3*sqrt(c + d*sin(e + f*x))) - (2*(c - 4*d)*cos(e + f*x))/(15*a*(c - d)^2*f*(a + a*sin(e + f*x))^2*sqrt(c + d*sin(e + f*x))) - ((4*c^2 - 21*c*d + 65*d^2)*cos(e + f*x))/(30*(c - d)^3*f*(a^3 + a^3*sin(e + f*x))*sqrt(c + d*sin(e + f*x))) - ((4*c^3 - 21*c^2*d + 62*c*d^2 + 147*d^3)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(30*a^3*(c - d)^4*(c + d)*f*sqrt((c + d*sin(e + f*x))/(c + d))) + ((4*c^2 - 21*c*d + 65*d^2)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(30*a^3*(c - d)^3*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [1/((a + a*sin(e + f*x))^3*(c + d*sin(e + f*x))^(5/2)), x, 10, -(d*(4*c^3 - 27*c^2*d + 114*c*d^2 + 165*d^3)*cos(e + f*x))/(30*a^3*(c - d)^4*(c + d)*f*(c + d*sin(e + f*x))^(3/2)) - cos(e + f*x)/(5*(c - d)*f*(a + a*sin(e + f*x))^3*(c + d*sin(e + f*x))^(3/2)) - (2*(c - 5*d)*cos(e + f*x))/(15*a*(c - d)^2*f*(a + a*sin(e + f*x))^2*(c + d*sin(e + f*x))^(3/2)) - ((4*c^2 - 27*c*d + 119*d^2)*cos(e + f*x))/(30*(c - d)^3*f*(a^3 + a^3*sin(e + f*x))*(c + d*sin(e + f*x))^(3/2)) - (d*(4*c^4 - 27*c^3*d + 111*c^2*d^2 + 579*c*d^3 + 357*d^4)*cos(e + f*x))/(30*a^3*(c - d)^5*(c + d)^2*f*sqrt(c + d*sin(e + f*x))) - ((4*c^4 - 27*c^3*d + 111*c^2*d^2 + 579*c*d^3 + 357*d^4)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(30*a^3*(c - d)^5*(c + d)^2*f*sqrt((c + d*sin(e + f*x))/(c + d))) + ((4*c^3 - 27*c^2*d + 114*c*d^2 + 165*d^3)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(30*a^3*(c - d)^4*(c + d)*f*sqrt(c + d*sin(e + f*x)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+a*sin(e+f*x))^(m/2)*(c+d*sin(e+f*x))^n=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [(a + a*sin(e + f*x))^(1/2)*(c + d*sin(e + f*x))^3, x, 4, -((4*a*(c + d)*(15*c^2 + 10*c*d + 7*d^2)*cos(e + f*x))/(35*f*sqrt(a + a*sin(e + f*x)))) - (8*(5*c - d)*d*(c + d)*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(35*f) - (12*d^2*(c + d)*cos(e + f*x)*(a + a*sin(e + f*x))^(3/2))/(35*a*f) - (2*a*cos(e + f*x)*(c + d*sin(e + f*x))^3)/(7*f*sqrt(a + a*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^(1/2)*(c + d*sin(e + f*x))^2, x, 3, -((2*a*(15*c^2 + 10*c*d + 7*d^2)*cos(e + f*x))/(15*f*sqrt(a + a*sin(e + f*x)))) - (4*(5*c - d)*d*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(15*f) - (2*d^2*cos(e + f*x)*(a + a*sin(e + f*x))^(3/2))/(5*a*f)]
    @test_int [(a + a*sin(e + f*x))^(1/2)*(c + d*sin(e + f*x))^1, x, 2, -((2*a*(3*c + d)*cos(e + f*x))/(3*f*sqrt(a + a*sin(e + f*x)))) - (2*d*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(3*f)]
    @test_int [(a + a*sin(e + f*x))^(1/2)*(c + d*sin(e + f*x))^0, x, 1, -((2*a*cos(e + f*x))/(f*sqrt(a + a*sin(e + f*x))))]
    @test_int [(a + a*sin(e + f*x))^(1/2)/(c + d*sin(e + f*x))^1, x, 2, -((2*sqrt(a)*atanh((sqrt(a)*sqrt(d)*cos(e + f*x))/(sqrt(c + d)*sqrt(a + a*sin(e + f*x)))))/(sqrt(d)*sqrt(c + d)*f))]
    @test_int [(a + a*sin(e + f*x))^(1/2)/(c + d*sin(e + f*x))^2, x, 3, -((sqrt(a)*atanh((sqrt(a)*sqrt(d)*cos(e + f*x))/(sqrt(c + d)*sqrt(a + a*sin(e + f*x)))))/(sqrt(d)*(c + d)^(3/2)*f)) - (a*cos(e + f*x))/((c + d)*f*sqrt(a + a*sin(e + f*x))*(c + d*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^(1/2)/(c + d*sin(e + f*x))^3, x, 4, -((3*sqrt(a)*atanh((sqrt(a)*sqrt(d)*cos(e + f*x))/(sqrt(c + d)*sqrt(a + a*sin(e + f*x)))))/(4*sqrt(d)*(c + d)^(5/2)*f)) - (a*cos(e + f*x))/(2*(c + d)*f*sqrt(a + a*sin(e + f*x))*(c + d*sin(e + f*x))^2) - (3*a*cos(e + f*x))/(4*(c + d)^2*f*sqrt(a + a*sin(e + f*x))*(c + d*sin(e + f*x)))]

    @test_int [(a + a*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))^3, x, 6, (4*a^2*(c - 17*d)*(c + d)*(15*c^2 + 10*c*d + 7*d^2)*cos(e + f*x))/(315*d*f*sqrt(a + a*sin(e + f*x))) + (8*a*(c - 17*d)*(5*c - d)*(c + d)*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(315*f) + (4*(c - 17*d)*d*(c + d)*cos(e + f*x)*(a + a*sin(e + f*x))^(3/2))/(105*f) + (2*a^2*(c - 17*d)*cos(e + f*x)*(c + d*sin(e + f*x))^3)/(63*d*f*sqrt(a + a*sin(e + f*x))) - (2*a^2*cos(e + f*x)*(c + d*sin(e + f*x))^4)/(9*d*f*sqrt(a + a*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))^2, x, 4, -((8*a^2*(35*c^2 + 42*c*d + 19*d^2)*cos(e + f*x))/(105*f*sqrt(a + a*sin(e + f*x)))) - (2*a*(35*c^2 + 42*c*d + 19*d^2)*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(105*f) - (4*(7*c - d)*d*cos(e + f*x)*(a + a*sin(e + f*x))^(3/2))/(35*f) - (2*d^2*cos(e + f*x)*(a + a*sin(e + f*x))^(5/2))/(7*a*f)]
    @test_int [(a + a*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))^1, x, 3, -((8*a^2*(5*c + 3*d)*cos(e + f*x))/(15*f*sqrt(a + a*sin(e + f*x)))) - (2*a*(5*c + 3*d)*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(15*f) - (2*d*cos(e + f*x)*(a + a*sin(e + f*x))^(3/2))/(5*f)]
    @test_int [(a + a*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))^0, x, 2, -((8*a^2*cos(e + f*x))/(3*f*sqrt(a + a*sin(e + f*x)))) - (2*a*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(3*f)]
    @test_int [(a + a*sin(e + f*x))^(3/2)/(c + d*sin(e + f*x))^1, x, 4, (2*a^(3/2)*(c - d)*atanh((sqrt(a)*sqrt(d)*cos(e + f*x))/(sqrt(c + d)*sqrt(a + a*sin(e + f*x)))))/(d^(3/2)*sqrt(c + d)*f) - (2*a^2*cos(e + f*x))/(d*f*sqrt(a + a*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^(3/2)/(c + d*sin(e + f*x))^2, x, 4, -((a^(3/2)*(c + 3*d)*atanh((sqrt(a)*sqrt(d)*cos(e + f*x))/(sqrt(c + d)*sqrt(a + a*sin(e + f*x)))))/(d^(3/2)*(c + d)^(3/2)*f)) + (a^2*(c - d)*cos(e + f*x))/(d*(c + d)*f*sqrt(a + a*sin(e + f*x))*(c + d*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^(3/2)/(c + d*sin(e + f*x))^3, x, 5, -((a^(3/2)*(c + 7*d)*atanh((sqrt(a)*sqrt(d)*cos(e + f*x))/(sqrt(c + d)*sqrt(a + a*sin(e + f*x)))))/(4*d^(3/2)*(c + d)^(5/2)*f)) + (a^2*(c - d)*cos(e + f*x))/(2*d*(c + d)*f*sqrt(a + a*sin(e + f*x))*(c + d*sin(e + f*x))^2) - (a^2*(c + 7*d)*cos(e + f*x))/(4*d*(c + d)^2*f*sqrt(a + a*sin(e + f*x))*(c + d*sin(e + f*x)))]

    @test_int [(a + a*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^3, x, 6, -((4*a^3*(c + d)*(15*c^2 + 10*c*d + 7*d^2)*(3*c^2 - 38*c*d + 355*d^2)*cos(e + f*x))/(3465*d^2*f*sqrt(a + a*sin(e + f*x)))) - (8*a^2*(5*c - d)*(c + d)*(3*c^2 - 38*c*d + 355*d^2)*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(3465*d*f) - (4*a*(c + d)*(3*c^2 - 38*c*d + 355*d^2)*cos(e + f*x)*(a + a*sin(e + f*x))^(3/2))/(1155*f) - (2*a^3*(3*c^2 - 38*c*d + 355*d^2)*cos(e + f*x)*(c + d*sin(e + f*x))^3)/(693*d^2*f*sqrt(a + a*sin(e + f*x))) + (2*a^3*(3*c - 23*d)*cos(e + f*x)*(c + d*sin(e + f*x))^4)/(99*d^2*f*sqrt(a + a*sin(e + f*x))) - (2*a^2*cos(e + f*x)*sqrt(a + a*sin(e + f*x))*(c + d*sin(e + f*x))^4)/(11*d*f)]
    @test_int [(a + a*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^2, x, 5, -((64*a^3*(21*c^2 + 30*c*d + 13*d^2)*cos(e + f*x))/(315*f*sqrt(a + a*sin(e + f*x)))) - (16*a^2*(21*c^2 + 30*c*d + 13*d^2)*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(315*f) - (2*a*(21*c^2 + 30*c*d + 13*d^2)*cos(e + f*x)*(a + a*sin(e + f*x))^(3/2))/(105*f) - (4*(9*c - d)*d*cos(e + f*x)*(a + a*sin(e + f*x))^(5/2))/(63*f) - (2*d^2*cos(e + f*x)*(a + a*sin(e + f*x))^(7/2))/(9*a*f)]
    @test_int [(a + a*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^1, x, 4, -((64*a^3*(7*c + 5*d)*cos(e + f*x))/(105*f*sqrt(a + a*sin(e + f*x)))) - (16*a^2*(7*c + 5*d)*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(105*f) - (2*a*(7*c + 5*d)*cos(e + f*x)*(a + a*sin(e + f*x))^(3/2))/(35*f) - (2*d*cos(e + f*x)*(a + a*sin(e + f*x))^(5/2))/(7*f)]
    @test_int [(a + a*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^0, x, 3, -((64*a^3*cos(e + f*x))/(15*f*sqrt(a + a*sin(e + f*x)))) - (16*a^2*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(15*f) - (2*a*cos(e + f*x)*(a + a*sin(e + f*x))^(3/2))/(5*f)]
    @test_int [(a + a*sin(e + f*x))^(5/2)/(c + d*sin(e + f*x))^1, x, 4, -((2*a^(5/2)*(c - d)^2*atanh((sqrt(a)*sqrt(d)*cos(e + f*x))/(sqrt(c + d)*sqrt(a + a*sin(e + f*x)))))/(d^(5/2)*sqrt(c + d)*f)) + (2*a^3*(3*c - 7*d)*cos(e + f*x))/(3*d^2*f*sqrt(a + a*sin(e + f*x))) - (2*a^2*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(3*d*f)]
    @test_int [(a + a*sin(e + f*x))^(5/2)/(c + d*sin(e + f*x))^2, x, 4, (a^(5/2)*(c - d)*(3*c + 5*d)*atanh((sqrt(a)*sqrt(d)*cos(e + f*x))/(sqrt(c + d)*sqrt(a + a*sin(e + f*x)))))/(d^(5/2)*(c + d)^(3/2)*f) - (a^3*(3*c + d)*cos(e + f*x))/(d^2*(c + d)*f*sqrt(a + a*sin(e + f*x))) + (a^2*(c - d)*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(d*(c + d)*f*(c + d*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^(5/2)/(c + d*sin(e + f*x))^3, x, 4, -((a^(5/2)*(3*c^2 + 10*c*d + 19*d^2)*atanh((sqrt(a)*sqrt(d)*cos(e + f*x))/(sqrt(c + d)*sqrt(a + a*sin(e + f*x)))))/(4*d^(5/2)*(c + d)^(5/2)*f)) + (a^2*(c - d)*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(2*d*(c + d)*f*(c + d*sin(e + f*x))^2) + (3*a^3*(c - d)*(c + 3*d)*cos(e + f*x))/(4*d^2*(c + d)^2*f*sqrt(a + a*sin(e + f*x))*(c + d*sin(e + f*x)))]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [1/(a + a*sin(e + f*x))^(1/2)*(c + d*sin(e + f*x))^3, x, 6, -((sqrt(2)*(c - d)^3*atanh((sqrt(a)*cos(e + f*x))/(sqrt(2)*sqrt(a + a*sin(e + f*x)))))/(sqrt(a)*f)) - (4*d*(21*c^2 - 12*c*d + 7*d^2)*cos(e + f*x))/(15*f*sqrt(a + a*sin(e + f*x))) - (2*(9*c - d)*d^2*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(15*a*f) - (2*d*cos(e + f*x)*(c + d*sin(e + f*x))^2)/(5*f*sqrt(a + a*sin(e + f*x)))]
    @test_int [1/(a + a*sin(e + f*x))^(1/2)*(c + d*sin(e + f*x))^2, x, 4, -((sqrt(2)*(c - d)^2*atanh((sqrt(a)*cos(e + f*x))/(sqrt(2)*sqrt(a + a*sin(e + f*x)))))/(sqrt(a)*f)) - (4*(3*c - d)*d*cos(e + f*x))/(3*f*sqrt(a + a*sin(e + f*x))) - (2*d^2*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(3*a*f)]
    @test_int [1/(a + a*sin(e + f*x))^(1/2)*(c + d*sin(e + f*x))^1, x, 3, -((sqrt(2)*(c - d)*atanh((sqrt(a)*cos(e + f*x))/(sqrt(2)*sqrt(a + a*sin(e + f*x)))))/(sqrt(a)*f)) - (2*d*cos(e + f*x))/(f*sqrt(a + a*sin(e + f*x)))]
    @test_int [1/(a + a*sin(e + f*x))^(1/2)*(c + d*sin(e + f*x))^0, x, 2, -((sqrt(2)*atanh((sqrt(a)*cos(e + f*x))/(sqrt(2)*sqrt(a + a*sin(e + f*x)))))/(sqrt(a)*f))]
    @test_int [1/(a + a*sin(e + f*x))^(1/2)/(c + d*sin(e + f*x))^1, x, 5, -((sqrt(2)*atanh((sqrt(a)*cos(e + f*x))/(sqrt(2)*sqrt(a + a*sin(e + f*x)))))/(sqrt(a)*(c - d)*f)) + (2*sqrt(d)*atanh((sqrt(a)*sqrt(d)*cos(e + f*x))/(sqrt(c + d)*sqrt(a + a*sin(e + f*x)))))/(sqrt(a)*(c - d)*sqrt(c + d)*f)]
    @test_int [1/(a + a*sin(e + f*x))^(1/2)/(c + d*sin(e + f*x))^2, x, 6, -((sqrt(2)*atanh((sqrt(a)*cos(e + f*x))/(sqrt(2)*sqrt(a + a*sin(e + f*x)))))/(sqrt(a)*(c - d)^2*f)) + (sqrt(d)*(3*c + d)*atanh((sqrt(a)*sqrt(d)*cos(e + f*x))/(sqrt(c + d)*sqrt(a + a*sin(e + f*x)))))/(sqrt(a)*(c - d)^2*(c + d)^(3/2)*f) + (d*cos(e + f*x))/((c^2 - d^2)*f*sqrt(a + a*sin(e + f*x))*(c + d*sin(e + f*x)))]
    @test_int [1/(a + a*sin(e + f*x))^(1/2)/(c + d*sin(e + f*x))^3, x, 7, -((sqrt(2)*atanh((sqrt(a)*cos(e + f*x))/(sqrt(2)*sqrt(a + a*sin(e + f*x)))))/(sqrt(a)*(c - d)^3*f)) + (sqrt(d)*(15*c^2 + 10*c*d + 7*d^2)*atanh((sqrt(a)*sqrt(d)*cos(e + f*x))/(sqrt(c + d)*sqrt(a + a*sin(e + f*x)))))/(4*sqrt(a)*(c - d)^3*(c + d)^(5/2)*f) + (d*cos(e + f*x))/(2*(c^2 - d^2)*f*sqrt(a + a*sin(e + f*x))*(c + d*sin(e + f*x))^2) + (d*(7*c + d)*cos(e + f*x))/(4*(c^2 - d^2)^2*f*sqrt(a + a*sin(e + f*x))*(c + d*sin(e + f*x)))]

    @test_int [1/(a + a*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))^3, x, 6, -(((c - d)^2*(c + 11*d)*atanh((sqrt(a)*cos(e + f*x))/(sqrt(2)*sqrt(a + a*sin(e + f*x)))))/(2*sqrt(2)*a^(3/2)*f)) + (d*(3*c^2 - 24*c*d + 13*d^2)*cos(e + f*x))/(3*a*f*sqrt(a + a*sin(e + f*x))) + ((3*c - 7*d)*d^2*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(6*a^2*f) - ((c - d)*cos(e + f*x)*(c + d*sin(e + f*x))^2)/(2*f*(a + a*sin(e + f*x))^(3/2))]
    @test_int [1/(a + a*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))^2, x, 4, -(((c - d)*(c + 7*d)*atanh((sqrt(a)*cos(e + f*x))/(sqrt(2)*sqrt(a + a*sin(e + f*x)))))/(2*sqrt(2)*a^(3/2)*f)) + ((c - 5*d)*d*cos(e + f*x))/(2*a*f*sqrt(a + a*sin(e + f*x))) - ((c - d)*cos(e + f*x)*(c + d*sin(e + f*x)))/(2*f*(a + a*sin(e + f*x))^(3/2))]
    @test_int [1/(a + a*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))^1, x, 3, -(((c + 3*d)*atanh((sqrt(a)*cos(e + f*x))/(sqrt(2)*sqrt(a + a*sin(e + f*x)))))/(2*sqrt(2)*a^(3/2)*f)) - ((c - d)*cos(e + f*x))/(2*f*(a + a*sin(e + f*x))^(3/2))]
    @test_int [1/(a + a*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))^0, x, 3, -(atanh((sqrt(a)*cos(e + f*x))/(sqrt(2)*sqrt(a + a*sin(e + f*x))))/(2*sqrt(2)*a^(3/2)*f)) - cos(e + f*x)/(2*f*(a + a*sin(e + f*x))^(3/2))]
    @test_int [1/(a + a*sin(e + f*x))^(3/2)/(c + d*sin(e + f*x))^1, x, 6, -(((c - 5*d)*atanh((sqrt(a)*cos(e + f*x))/(sqrt(2)*sqrt(a + a*sin(e + f*x)))))/(2*sqrt(2)*a^(3/2)*(c - d)^2*f)) - (2*d^(3/2)*atanh((sqrt(a)*sqrt(d)*cos(e + f*x))/(sqrt(c + d)*sqrt(a + a*sin(e + f*x)))))/(a^(3/2)*(c - d)^2*sqrt(c + d)*f) - cos(e + f*x)/(2*(c - d)*f*(a + a*sin(e + f*x))^(3/2))]
    @test_int [1/(a + a*sin(e + f*x))^(3/2)/(c + d*sin(e + f*x))^2, x, 7, -(((c - 9*d)*atanh((sqrt(a)*cos(e + f*x))/(sqrt(2)*sqrt(a + a*sin(e + f*x)))))/(2*sqrt(2)*a^(3/2)*(c - d)^3*f)) - (d^(3/2)*(5*c + 3*d)*atanh((sqrt(a)*sqrt(d)*cos(e + f*x))/(sqrt(c + d)*sqrt(a + a*sin(e + f*x)))))/(a^(3/2)*(c - d)^3*(c + d)^(3/2)*f) - cos(e + f*x)/(2*(c - d)*f*(a + a*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))) - (d*(c + 3*d)*cos(e + f*x))/(2*a*(c - d)^2*(c + d)*f*sqrt(a + a*sin(e + f*x))*(c + d*sin(e + f*x)))]
    @test_int [1/(a + a*sin(e + f*x))^(3/2)/(c + d*sin(e + f*x))^3, x, 8, -(((c - 13*d)*atanh((sqrt(a)*cos(e + f*x))/(sqrt(2)*sqrt(a + a*sin(e + f*x)))))/(2*sqrt(2)*a^(3/2)*(c - d)^4*f)) - (d^(3/2)*(35*c^2 + 42*c*d + 19*d^2)*atanh((sqrt(a)*sqrt(d)*cos(e + f*x))/(sqrt(c + d)*sqrt(a + a*sin(e + f*x)))))/(4*a^(3/2)*(c - d)^4*(c + d)^(5/2)*f) - cos(e + f*x)/(2*(c - d)*f*(a + a*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))^2) - (d*(c + 2*d)*cos(e + f*x))/(2*a*(c - d)^2*(c + d)*f*sqrt(a + a*sin(e + f*x))*(c + d*sin(e + f*x))^2) - (d*(2*c + d)*(c + 7*d)*cos(e + f*x))/(4*a*(c - d)^3*(c + d)^2*f*sqrt(a + a*sin(e + f*x))*(c + d*sin(e + f*x)))]

    @test_int [1/(a + a*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^3, x, 6, -((3*(c - d)*(c^2 + 6*c*d + 25*d^2)*atanh((sqrt(a)*cos(e + f*x))/(sqrt(2)*sqrt(a + a*sin(e + f*x)))))/(16*sqrt(2)*a^(5/2)*f)) - ((c - d)^2*(3*c + 13*d)*cos(e + f*x))/(16*a*f*(a + a*sin(e + f*x))^(3/2)) + ((c - 9*d)*d^2*cos(e + f*x))/(4*a^2*f*sqrt(a + a*sin(e + f*x))) - ((c - d)*cos(e + f*x)*(c + d*sin(e + f*x))^2)/(4*f*(a + a*sin(e + f*x))^(5/2))]
    @test_int [1/(a + a*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^2, x, 4, -(((3*c^2 + 10*c*d + 19*d^2)*atanh((sqrt(a)*cos(e + f*x))/(sqrt(2)*sqrt(a + a*sin(e + f*x)))))/(16*sqrt(2)*a^(5/2)*f)) - (3*(c - d)*(c + 3*d)*cos(e + f*x))/(16*a*f*(a + a*sin(e + f*x))^(3/2)) - ((c - d)*cos(e + f*x)*(c + d*sin(e + f*x)))/(4*f*(a + a*sin(e + f*x))^(5/2))]
    @test_int [1/(a + a*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^1, x, 4, -(((3*c + 5*d)*atanh((sqrt(a)*cos(e + f*x))/(sqrt(2)*sqrt(a + a*sin(e + f*x)))))/(16*sqrt(2)*a^(5/2)*f)) - ((c - d)*cos(e + f*x))/(4*f*(a + a*sin(e + f*x))^(5/2)) - ((3*c + 5*d)*cos(e + f*x))/(16*a*f*(a + a*sin(e + f*x))^(3/2))]
    @test_int [1/(a + a*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^0, x, 4, -((3*atanh((sqrt(a)*cos(e + f*x))/(sqrt(2)*sqrt(a + a*sin(e + f*x)))))/(16*sqrt(2)*a^(5/2)*f)) - cos(e + f*x)/(4*f*(a + a*sin(e + f*x))^(5/2)) - (3*cos(e + f*x))/(16*a*f*(a + a*sin(e + f*x))^(3/2))]
    @test_int [1/(a + a*sin(e + f*x))^(5/2)/(c + d*sin(e + f*x))^1, x, 7, -(((3*c^2 - 14*c*d + 43*d^2)*atanh((sqrt(a)*cos(e + f*x))/(sqrt(2)*sqrt(a + a*sin(e + f*x)))))/(16*sqrt(2)*a^(5/2)*(c - d)^3*f)) + (2*d^(5/2)*atanh((sqrt(a)*sqrt(d)*cos(e + f*x))/(sqrt(c + d)*sqrt(a + a*sin(e + f*x)))))/(a^(5/2)*(c - d)^3*sqrt(c + d)*f) - cos(e + f*x)/(4*(c - d)*f*(a + a*sin(e + f*x))^(5/2)) - ((3*c - 11*d)*cos(e + f*x))/(16*a*(c - d)^2*f*(a + a*sin(e + f*x))^(3/2))]
    @test_int [1/(a + a*sin(e + f*x))^(5/2)/(c + d*sin(e + f*x))^2, x, 8, -(((3*c^2 - 22*c*d + 115*d^2)*atanh((sqrt(a)*cos(e + f*x))/(sqrt(2)*sqrt(a + a*sin(e + f*x)))))/(16*sqrt(2)*a^(5/2)*(c - d)^4*f)) + (d^(5/2)*(7*c + 5*d)*atanh((sqrt(a)*sqrt(d)*cos(e + f*x))/(sqrt(c + d)*sqrt(a + a*sin(e + f*x)))))/(a^(5/2)*(c - d)^4*(c + d)^(3/2)*f) - cos(e + f*x)/(4*(c - d)*f*(a + a*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))) - (3*(c - 5*d)*cos(e + f*x))/(16*a*(c - d)^2*f*(a + a*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))) - ((c - 7*d)*d*(3*c + 5*d)*cos(e + f*x))/(16*a^2*(c - d)^3*(c + d)*f*sqrt(a + a*sin(e + f*x))*(c + d*sin(e + f*x)))]
    @test_int [1/(a + a*sin(e + f*x))^(5/2)/(c + d*sin(e + f*x))^3, x, 9, -((3*(c^2 - 10*c*d + 73*d^2)*atanh((sqrt(a)*cos(e + f*x))/(sqrt(2)*sqrt(a + a*sin(e + f*x)))))/(16*sqrt(2)*a^(5/2)*(c - d)^5*f)) + (3*d^(5/2)*(21*c^2 + 30*c*d + 13*d^2)*atanh((sqrt(a)*sqrt(d)*cos(e + f*x))/(sqrt(c + d)*sqrt(a + a*sin(e + f*x)))))/(4*a^(5/2)*(c - d)^5*(c + d)^(5/2)*f) - cos(e + f*x)/(4*(c - d)*f*(a + a*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^2) - ((3*c - 19*d)*cos(e + f*x))/(16*a*(c - d)^2*f*(a + a*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))^2) - (d*(3*c^2 - 20*c*d - 31*d^2)*cos(e + f*x))/(16*a^2*(c - d)^3*(c + d)*f*sqrt(a + a*sin(e + f*x))*(c + d*sin(e + f*x))^2) - (3*d*(c + 3*d)*(c^2 - 10*c*d - 7*d^2)*cos(e + f*x))/(16*a^2*(c - d)^4*(c + d)^2*f*sqrt(a + a*sin(e + f*x))*(c + d*sin(e + f*x)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+a*sin(e+f*x))^(m/2)*(c+d*sin(e+f*x))^(n/2)=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [(a + a*sin(e + f*x))^(1/2)*(c + d*sin(e + f*x))^(5/2), x, 5, -((5*sqrt(a)*(c + d)^3*atan((sqrt(a)*sqrt(d)*cos(e + f*x))/(sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))))/(8*sqrt(d)*f)) - (5*a*(c + d)^2*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(8*f*sqrt(a + a*sin(e + f*x))) - (5*a*(c + d)*cos(e + f*x)*(c + d*sin(e + f*x))^(3/2))/(12*f*sqrt(a + a*sin(e + f*x))) - (a*cos(e + f*x)*(c + d*sin(e + f*x))^(5/2))/(3*f*sqrt(a + a*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^(1/2)*(c + d*sin(e + f*x))^(3/2), x, 4, -((3*sqrt(a)*(c + d)^2*atan((sqrt(a)*sqrt(d)*cos(e + f*x))/(sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))))/(4*sqrt(d)*f)) - (3*a*(c + d)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(4*f*sqrt(a + a*sin(e + f*x))) - (a*cos(e + f*x)*(c + d*sin(e + f*x))^(3/2))/(2*f*sqrt(a + a*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^(1/2)*(c + d*sin(e + f*x))^(1/2), x, 3, -((sqrt(a)*(c + d)*atan((sqrt(a)*sqrt(d)*cos(e + f*x))/(sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))))/(sqrt(d)*f)) - (a*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(f*sqrt(a + a*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^(1/2)/(c + d*sin(e + f*x))^(1/2), x, 2, -((2*sqrt(a)*atan((sqrt(a)*sqrt(d)*cos(e + f*x))/(sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))))/(sqrt(d)*f))]
    @test_int [(a + a*sin(e + f*x))^(1/2)/(c + d*sin(e + f*x))^(3/2), x, 1, -((2*a*cos(e + f*x))/((c + d)*f*sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x))))]
    @test_int [(a + a*sin(e + f*x))^(1/2)/(c + d*sin(e + f*x))^(5/2), x, 2, -((2*a*cos(e + f*x))/(3*(c + d)*f*sqrt(a + a*sin(e + f*x))*(c + d*sin(e + f*x))^(3/2))) - (4*a*cos(e + f*x))/(3*(c + d)^2*f*sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^(1/2)/(c + d*sin(e + f*x))^(7/2), x, 3, -((2*a*cos(e + f*x))/(5*(c + d)*f*sqrt(a + a*sin(e + f*x))*(c + d*sin(e + f*x))^(5/2))) - (8*a*cos(e + f*x))/(15*(c + d)^2*f*sqrt(a + a*sin(e + f*x))*(c + d*sin(e + f*x))^(3/2)) - (16*a*cos(e + f*x))/(15*(c + d)^3*f*sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))]

    @test_int [(a + a*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))^(5/2), x, 7, (5*a^(3/2)*(c - 15*d)*(c + d)^3*atan((sqrt(a)*sqrt(d)*cos(e + f*x))/(sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))))/(64*d^(3/2)*f) + (5*a^2*(c - 15*d)*(c + d)^2*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(64*d*f*sqrt(a + a*sin(e + f*x))) + (5*a^2*(c - 15*d)*(c + d)*cos(e + f*x)*(c + d*sin(e + f*x))^(3/2))/(96*d*f*sqrt(a + a*sin(e + f*x))) + (a^2*(c - 15*d)*cos(e + f*x)*(c + d*sin(e + f*x))^(5/2))/(24*d*f*sqrt(a + a*sin(e + f*x))) - (a^2*cos(e + f*x)*(c + d*sin(e + f*x))^(7/2))/(4*d*f*sqrt(a + a*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))^(3/2), x, 6, (a^(3/2)*(c - 11*d)*(c + d)^2*atan((sqrt(a)*sqrt(d)*cos(e + f*x))/(sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))))/(8*d^(3/2)*f) + (a^2*(c - 11*d)*(c + d)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(8*d*f*sqrt(a + a*sin(e + f*x))) + (a^2*(c - 11*d)*cos(e + f*x)*(c + d*sin(e + f*x))^(3/2))/(12*d*f*sqrt(a + a*sin(e + f*x))) - (a^2*cos(e + f*x)*(c + d*sin(e + f*x))^(5/2))/(3*d*f*sqrt(a + a*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))^(1/2), x, 5, (a^(3/2)*(c - 7*d)*(c + d)*atan((sqrt(a)*sqrt(d)*cos(e + f*x))/(sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))))/(4*d^(3/2)*f) + (a^2*(c - 7*d)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(4*d*f*sqrt(a + a*sin(e + f*x))) - (a^2*cos(e + f*x)*(c + d*sin(e + f*x))^(3/2))/(2*d*f*sqrt(a + a*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^(3/2)/(c + d*sin(e + f*x))^(1/2), x, 4, (a^(3/2)*(c - 3*d)*atan((sqrt(a)*sqrt(d)*cos(e + f*x))/(sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))))/(d^(3/2)*f) - (a^2*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(d*f*sqrt(a + a*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^(3/2)/(c + d*sin(e + f*x))^(3/2), x, 4, -((2*a^(3/2)*atan((sqrt(a)*sqrt(d)*cos(e + f*x))/(sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))))/(d^(3/2)*f)) + (2*a^2*(c - d)*cos(e + f*x))/(d*(c + d)*f*sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^(3/2)/(c + d*sin(e + f*x))^(5/2), x, 3, (2*a^2*(c - d)*cos(e + f*x))/(3*d*(c + d)*f*sqrt(a + a*sin(e + f*x))*(c + d*sin(e + f*x))^(3/2)) - (2*a^2*(c + 5*d)*cos(e + f*x))/(3*d*(c + d)^2*f*sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^(3/2)/(c + d*sin(e + f*x))^(7/2), x, 4, (2*a^2*(c - d)*cos(e + f*x))/(5*d*(c + d)*f*sqrt(a + a*sin(e + f*x))*(c + d*sin(e + f*x))^(5/2)) - (2*a^2*(c + 9*d)*cos(e + f*x))/(15*d*(c + d)^2*f*sqrt(a + a*sin(e + f*x))*(c + d*sin(e + f*x))^(3/2)) - (4*a^2*(c + 9*d)*cos(e + f*x))/(15*d*(c + d)^3*f*sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^(3/2)/(c + d*sin(e + f*x))^(9/2), x, 5, (2*a^2*(c - d)*cos(e + f*x))/(7*d*(c + d)*f*sqrt(a + a*sin(e + f*x))*(c + d*sin(e + f*x))^(7/2)) - (2*a^2*(c + 13*d)*cos(e + f*x))/(35*d*(c + d)^2*f*sqrt(a + a*sin(e + f*x))*(c + d*sin(e + f*x))^(5/2)) - (8*a^2*(c + 13*d)*cos(e + f*x))/(105*d*(c + d)^3*f*sqrt(a + a*sin(e + f*x))*(c + d*sin(e + f*x))^(3/2)) - (16*a^2*(c + 13*d)*cos(e + f*x))/(105*d*(c + d)^4*f*sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))]

    @test_int [(a + a*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^(5/2), x, 7, -((a^(5/2)*(c + d)^3*(3*c^2 - 34*c*d + 283*d^2)*atan((sqrt(a)*sqrt(d)*cos(e + f*x))/(sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))))/(128*d^(5/2)*f)) - (a^3*(c + d)^2*(3*c^2 - 34*c*d + 283*d^2)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(128*d^2*f*sqrt(a + a*sin(e + f*x))) - (a^3*(c + d)*(3*c^2 - 34*c*d + 283*d^2)*cos(e + f*x)*(c + d*sin(e + f*x))^(3/2))/(192*d^2*f*sqrt(a + a*sin(e + f*x))) - (a^3*(3*c^2 - 34*c*d + 283*d^2)*cos(e + f*x)*(c + d*sin(e + f*x))^(5/2))/(240*d^2*f*sqrt(a + a*sin(e + f*x))) + (3*a^3*(c - 7*d)*cos(e + f*x)*(c + d*sin(e + f*x))^(7/2))/(40*d^2*f*sqrt(a + a*sin(e + f*x))) - (a^2*cos(e + f*x)*sqrt(a + a*sin(e + f*x))*(c + d*sin(e + f*x))^(7/2))/(5*d*f)]
    @test_int [(a + a*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^(3/2), x, 6, -((a^(5/2)*(c + d)^2*(3*c^2 - 26*c*d + 163*d^2)*atan((sqrt(a)*sqrt(d)*cos(e + f*x))/(sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))))/(64*d^(5/2)*f)) - (a^3*(c + d)*(3*c^2 - 26*c*d + 163*d^2)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(64*d^2*f*sqrt(a + a*sin(e + f*x))) - (a^3*(3*c^2 - 26*c*d + 163*d^2)*cos(e + f*x)*(c + d*sin(e + f*x))^(3/2))/(96*d^2*f*sqrt(a + a*sin(e + f*x))) + (a^3*(3*c - 17*d)*cos(e + f*x)*(c + d*sin(e + f*x))^(5/2))/(24*d^2*f*sqrt(a + a*sin(e + f*x))) - (a^2*cos(e + f*x)*sqrt(a + a*sin(e + f*x))*(c + d*sin(e + f*x))^(5/2))/(4*d*f)]
    @test_int [(a + a*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^(1/2), x, 5, -((a^(5/2)*(c + d)*(c^2 - 6*c*d + 25*d^2)*atan((sqrt(a)*sqrt(d)*cos(e + f*x))/(sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))))/(8*d^(5/2)*f)) - (a^3*(c^2 - 6*c*d + 25*d^2)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(8*d^2*f*sqrt(a + a*sin(e + f*x))) + (a^3*(3*c - 13*d)*cos(e + f*x)*(c + d*sin(e + f*x))^(3/2))/(12*d^2*f*sqrt(a + a*sin(e + f*x))) - (a^2*cos(e + f*x)*sqrt(a + a*sin(e + f*x))*(c + d*sin(e + f*x))^(3/2))/(3*d*f)]
    @test_int [(a + a*sin(e + f*x))^(5/2)/(c + d*sin(e + f*x))^(1/2), x, 4, -((a^(5/2)*(3*c^2 - 10*c*d + 19*d^2)*atan((sqrt(a)*sqrt(d)*cos(e + f*x))/(sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))))/(4*d^(5/2)*f)) + (3*a^3*(c - 3*d)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(4*d^2*f*sqrt(a + a*sin(e + f*x))) - (a^2*cos(e + f*x)*sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))/(2*d*f)]
    @test_int [(a + a*sin(e + f*x))^(5/2)/(c + d*sin(e + f*x))^(3/2), x, 4, (a^(5/2)*(3*c - 5*d)*atan((sqrt(a)*sqrt(d)*cos(e + f*x))/(sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))))/(d^(5/2)*f) + (2*a^2*(c - d)*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(d*(c + d)*f*sqrt(c + d*sin(e + f*x))) - (a^3*(3*c - d)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(d^2*(c + d)*f*sqrt(a + a*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^(5/2)/(c + d*sin(e + f*x))^(5/2), x, 4, -((2*a^(5/2)*atan((sqrt(a)*sqrt(d)*cos(e + f*x))/(sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))))/(d^(5/2)*f)) + (2*a^2*(c - d)*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(3*d*(c + d)*f*(c + d*sin(e + f*x))^(3/2)) + (2*a^3*(c - d)*(3*c + 7*d)*cos(e + f*x))/(3*d^2*(c + d)^2*f*sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^(5/2)/(c + d*sin(e + f*x))^(7/2), x, 3, (2*a^2*(c - d)*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(5*d*(c + d)*f*(c + d*sin(e + f*x))^(5/2)) + (2*a^3*(c - d)*(3*c + 11*d)*cos(e + f*x))/(15*d^2*(c + d)^2*f*sqrt(a + a*sin(e + f*x))*(c + d*sin(e + f*x))^(3/2)) - (2*a^3*(3*c^2 + 14*c*d + 43*d^2)*cos(e + f*x))/(15*d^2*(c + d)^3*f*sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^(5/2)/(c + d*sin(e + f*x))^(9/2), x, 4, (2*a^2*(c - d)*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(7*d*(c + d)*f*(c + d*sin(e + f*x))^(7/2)) + (6*a^3*(c - d)*(c + 5*d)*cos(e + f*x))/(35*d^2*(c + d)^2*f*sqrt(a + a*sin(e + f*x))*(c + d*sin(e + f*x))^(5/2)) - (2*a^3*(3*c^2 + 22*c*d + 115*d^2)*cos(e + f*x))/(105*d^2*(c + d)^3*f*sqrt(a + a*sin(e + f*x))*(c + d*sin(e + f*x))^(3/2)) - (4*a^3*(3*c^2 + 22*c*d + 115*d^2)*cos(e + f*x))/(105*d^2*(c + d)^4*f*sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^(5/2)/(c + d*sin(e + f*x))^(11/2), x, 5, (2*a^2*(c - d)*cos(e + f*x)*sqrt(a + a*sin(e + f*x)))/(9*d*(c + d)*f*(c + d*sin(e + f*x))^(9/2)) + (2*a^3*(c - d)*(3*c + 19*d)*cos(e + f*x))/(63*d^2*(c + d)^2*f*sqrt(a + a*sin(e + f*x))*(c + d*sin(e + f*x))^(7/2)) - (2*a^3*(c^2 + 10*c*d + 73*d^2)*cos(e + f*x))/(105*d^2*(c + d)^3*f*sqrt(a + a*sin(e + f*x))*(c + d*sin(e + f*x))^(5/2)) - (8*a^3*(c^2 + 10*c*d + 73*d^2)*cos(e + f*x))/(315*d^2*(c + d)^4*f*sqrt(a + a*sin(e + f*x))*(c + d*sin(e + f*x))^(3/2)) - (16*a^3*(c^2 + 10*c*d + 73*d^2)*cos(e + f*x))/(315*d^2*(c + d)^5*f*sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [1/(a + a*sin(e + f*x))^(1/2)*(c + d*sin(e + f*x))^(5/2), x, 7, -((sqrt(d)*(15*c^2 - 10*c*d + 7*d^2)*atan((sqrt(a)*sqrt(d)*cos(e + f*x))/(sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))))/(4*sqrt(a)*f)) - (sqrt(2)*(c - d)^(5/2)*atanh((sqrt(a)*sqrt(c - d)*cos(e + f*x))/(sqrt(2)*sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))))/(sqrt(a)*f) - ((7*c - d)*d*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(4*f*sqrt(a + a*sin(e + f*x))) - (d*cos(e + f*x)*(c + d*sin(e + f*x))^(3/2))/(2*f*sqrt(a + a*sin(e + f*x)))]
    @test_int [1/(a + a*sin(e + f*x))^(1/2)*(c + d*sin(e + f*x))^(3/2), x, 6, -(((3*c - d)*sqrt(d)*atan((sqrt(a)*sqrt(d)*cos(e + f*x))/(sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))))/(sqrt(a)*f)) - (sqrt(2)*(c - d)^(3/2)*atanh((sqrt(a)*sqrt(c - d)*cos(e + f*x))/(sqrt(2)*sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))))/(sqrt(a)*f) - (d*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(f*sqrt(a + a*sin(e + f*x)))]
    @test_int [1/(a + a*sin(e + f*x))^(1/2)*(c + d*sin(e + f*x))^(1/2), x, 5, -((2*sqrt(d)*atan((sqrt(a)*sqrt(d)*cos(e + f*x))/(sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))))/(sqrt(a)*f)) - (sqrt(2)*sqrt(c - d)*atanh((sqrt(a)*sqrt(c - d)*cos(e + f*x))/(sqrt(2)*sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))))/(sqrt(a)*f)]
    @test_int [1/(a + a*sin(e + f*x))^(1/2)/(c + d*sin(e + f*x))^(1/2), x, 2, -((sqrt(2)*atanh((sqrt(a)*sqrt(c - d)*cos(e + f*x))/(sqrt(2)*sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))))/(sqrt(a)*sqrt(c - d)*f))]
    @test_int [1/(a + a*sin(e + f*x))^(1/2)/(c + d*sin(e + f*x))^(3/2), x, 4, -((sqrt(2)*atanh((sqrt(a)*sqrt(c - d)*cos(e + f*x))/(sqrt(2)*sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))))/(sqrt(a)*(c - d)^(3/2)*f)) + (2*d*cos(e + f*x))/((c^2 - d^2)*f*sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))]
    @test_int [1/(a + a*sin(e + f*x))^(1/2)/(c + d*sin(e + f*x))^(5/2), x, 5, -((sqrt(2)*atanh((sqrt(a)*sqrt(c - d)*cos(e + f*x))/(sqrt(2)*sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))))/(sqrt(a)*(c - d)^(5/2)*f)) + (2*d*cos(e + f*x))/(3*(c^2 - d^2)*f*sqrt(a + a*sin(e + f*x))*(c + d*sin(e + f*x))^(3/2)) + (2*d*(5*c + d)*cos(e + f*x))/(3*(c^2 - d^2)^2*f*sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))]

    @test_int [1/(a + a*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))^(5/2), x, 7, -(((5*c - 3*d)*d^(3/2)*atan((sqrt(a)*sqrt(d)*cos(e + f*x))/(sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))))/(a^(3/2)*f)) - ((c - d)^(3/2)*(c + 9*d)*atanh((sqrt(a)*sqrt(c - d)*cos(e + f*x))/(sqrt(2)*sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))))/(2*sqrt(2)*a^(3/2)*f) + ((c - 3*d)*d*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(2*a*f*sqrt(a + a*sin(e + f*x))) - ((c - d)*cos(e + f*x)*(c + d*sin(e + f*x))^(3/2))/(2*f*(a + a*sin(e + f*x))^(3/2))]
    @test_int [1/(a + a*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))^(3/2), x, 6, -((2*d^(3/2)*atan((sqrt(a)*sqrt(d)*cos(e + f*x))/(sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))))/(a^(3/2)*f)) - (sqrt(c - d)*(c + 5*d)*atanh((sqrt(a)*sqrt(c - d)*cos(e + f*x))/(sqrt(2)*sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))))/(2*sqrt(2)*a^(3/2)*f) - ((c - d)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(2*f*(a + a*sin(e + f*x))^(3/2))]
    @test_int [1/(a + a*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))^(1/2), x, 4, -(((c + d)*atanh((sqrt(a)*sqrt(c - d)*cos(e + f*x))/(sqrt(2)*sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))))/(2*sqrt(2)*a^(3/2)*sqrt(c - d)*f)) - (cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(2*f*(a + a*sin(e + f*x))^(3/2))]
    @test_int [1/(a + a*sin(e + f*x))^(3/2)/(c + d*sin(e + f*x))^(1/2), x, 4, -(((c - 3*d)*atanh((sqrt(a)*sqrt(c - d)*cos(e + f*x))/(sqrt(2)*sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))))/(2*sqrt(2)*a^(3/2)*(c - d)^(3/2)*f)) - (cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(2*(c - d)*f*(a + a*sin(e + f*x))^(3/2))]
    @test_int [1/(a + a*sin(e + f*x))^(3/2)/(c + d*sin(e + f*x))^(3/2), x, 5, -(((c - 7*d)*atanh((sqrt(a)*sqrt(c - d)*cos(e + f*x))/(sqrt(2)*sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))))/(2*sqrt(2)*a^(3/2)*(c - d)^(5/2)*f)) - cos(e + f*x)/(2*(c - d)*f*(a + a*sin(e + f*x))^(3/2)*sqrt(c + d*sin(e + f*x))) - (d*(c + 5*d)*cos(e + f*x))/(2*a*(c - d)^2*(c + d)*f*sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))]
    @test_int [1/(a + a*sin(e + f*x))^(3/2)/(c + d*sin(e + f*x))^(5/2), x, 6, -(((c - 11*d)*atanh((sqrt(a)*sqrt(c - d)*cos(e + f*x))/(sqrt(2)*sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))))/(2*sqrt(2)*a^(3/2)*(c - d)^(7/2)*f)) - cos(e + f*x)/(2*(c - d)*f*(a + a*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))^(3/2)) - (d*(3*c + 7*d)*cos(e + f*x))/(6*a*(c - d)^2*(c + d)*f*sqrt(a + a*sin(e + f*x))*(c + d*sin(e + f*x))^(3/2)) - (d*(3*c^2 + 38*c*d + 19*d^2)*cos(e + f*x))/(6*a*(c - d)^3*(c + d)^2*f*sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))]

    @test_int [1/(a + a*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^(5/2), x, 7, -((2*d^(5/2)*atan((sqrt(a)*sqrt(d)*cos(e + f*x))/(sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))))/(a^(5/2)*f)) - (sqrt(c - d)*(3*c^2 + 14*c*d + 43*d^2)*atanh((sqrt(a)*sqrt(c - d)*cos(e + f*x))/(sqrt(2)*sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))))/(16*sqrt(2)*a^(5/2)*f) - ((c - d)*(3*c + 11*d)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(16*a*f*(a + a*sin(e + f*x))^(3/2)) - ((c - d)*cos(e + f*x)*(c + d*sin(e + f*x))^(3/2))/(4*f*(a + a*sin(e + f*x))^(5/2))]
    @test_int [1/(a + a*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^(3/2), x, 5, -((3*(c + d)^2*atanh((sqrt(a)*sqrt(c - d)*cos(e + f*x))/(sqrt(2)*sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))))/(16*sqrt(2)*a^(5/2)*sqrt(c - d)*f)) - ((c - d)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(4*f*(a + a*sin(e + f*x))^(5/2)) - ((3*c + 7*d)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(16*a*f*(a + a*sin(e + f*x))^(3/2))]
    @test_int [1/(a + a*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^(1/2), x, 5, -(((3*c - 5*d)*(c + d)*atanh((sqrt(a)*sqrt(c - d)*cos(e + f*x))/(sqrt(2)*sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))))/(16*sqrt(2)*a^(5/2)*(c - d)^(3/2)*f)) - (cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(4*f*(a + a*sin(e + f*x))^(5/2)) - ((3*c - d)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(16*a*(c - d)*f*(a + a*sin(e + f*x))^(3/2))]
    @test_int [1/(a + a*sin(e + f*x))^(5/2)/(c + d*sin(e + f*x))^(1/2), x, 5, -(((3*c^2 - 10*c*d + 19*d^2)*atanh((sqrt(a)*sqrt(c - d)*cos(e + f*x))/(sqrt(2)*sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))))/(16*sqrt(2)*a^(5/2)*(c - d)^(5/2)*f)) - (cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(4*(c - d)*f*(a + a*sin(e + f*x))^(5/2)) - (3*(c - 3*d)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(16*a*(c - d)^2*f*(a + a*sin(e + f*x))^(3/2))]
    @test_int [1/(a + a*sin(e + f*x))^(5/2)/(c + d*sin(e + f*x))^(3/2), x, 6, -((3*(c^2 - 6*c*d + 25*d^2)*atanh((sqrt(a)*sqrt(c - d)*cos(e + f*x))/(sqrt(2)*sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))))/(16*sqrt(2)*a^(5/2)*(c - d)^(7/2)*f)) - cos(e + f*x)/(4*(c - d)*f*(a + a*sin(e + f*x))^(5/2)*sqrt(c + d*sin(e + f*x))) - ((3*c - 13*d)*cos(e + f*x))/(16*a*(c - d)^2*f*(a + a*sin(e + f*x))^(3/2)*sqrt(c + d*sin(e + f*x))) - ((c - 7*d)*d*(3*c + 7*d)*cos(e + f*x))/(16*a^2*(c - d)^3*(c + d)*f*sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))]
    @test_int [1/(a + a*sin(e + f*x))^(5/2)/(c + d*sin(e + f*x))^(5/2), x, 7, -(((3*c^2 - 26*c*d + 163*d^2)*atanh((sqrt(a)*sqrt(c - d)*cos(e + f*x))/(sqrt(2)*sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))))/(16*sqrt(2)*a^(5/2)*(c - d)^(9/2)*f)) - cos(e + f*x)/(4*(c - d)*f*(a + a*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^(3/2)) - ((3*c - 17*d)*cos(e + f*x))/(16*a*(c - d)^2*f*(a + a*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))^(3/2)) - (d*(9*c^2 - 54*c*d - 95*d^2)*cos(e + f*x))/(48*a^2*(c - d)^3*(c + d)*f*sqrt(a + a*sin(e + f*x))*(c + d*sin(e + f*x))^(3/2)) - (d*(9*c^3 - 57*c^2*d - 493*c*d^2 - 299*d^3)*cos(e + f*x))/(48*a^2*(c - d)^4*(c + d)^2*f*sqrt(a + a*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+a*sin(e+f*x))^m*(c+d*sin(e+f*x))^n*with*m*and/or*n*symbolic=#

    @test_int [(a + a*sin(e + f*x))^m*(c + d*sin(e + f*x))^n, x, 4, (sqrt(2)*AppellF1(1/2 + m, 1/2, -n, 3/2 + m, (1/2)*(1 + sin(e + f*x)), -((d*(1 + sin(e + f*x)))/(c - d)))*cos(e + f*x)*(a + a*sin(e + f*x))^m*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c - d))^n*(f*(1 + 2*m)*sqrt(1 - sin(e + f*x))))]

    @test_int [(a + a*sin(e + f*x))^m*(c + d*sin(e + f*x))^3, x, 6, If(13>=8, -((d*(d^2*(4 + m) - c*d*(5 - 3*m - 2*m^2) + 2*c^2*(8 + 6*m + m^2))*cos(e + f*x)*(a + a*sin(e + f*x))^m)/(f*(1 + m)*(2 + m)*(3 + m))) - (2^(1/2 + m)*(d^3*m*(5 + 3*m + m^2) + 3*c^2*d*m*(6 + 5*m + m^2) + 3*c*d^2*(3 + 4*m + 4*m^2 + m^3) + c^3*(6 + 11*m + 6*m^2 + m^3))*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, 1/2 - m, 3/2, (1/2)*(1 - sin(e + f*x)))*(1 + sin(e + f*x))^(-(1/2) - m)*(a + a*sin(e + f*x))^m)/(f*(1 + m)*(2 + m)*(3 + m)) - (d^2*(d*m + c*(5 + m))*cos(e + f*x)*(a + a*sin(e + f*x))^(1 + m))/(a*f*(2 + m)*(3 + m)) - (d*cos(e + f*x)*(a + a*sin(e + f*x))^m*(c + d*sin(e + f*x))^2)/(f*(3 + m)), -((d*(d^2*(4 + m) - c*d*(5 - 3*m - 2*m^2) + 2*c^2*(8 + 6*m + m^2))*cos(e + f*x)*(a + a*sin(e + f*x))^m)/(f*(3 + m)*(2 + 3*m + m^2))) - (1/(f*(3 + m)*(2 + 3*m + m^2)))*(2^(1/2 + m)*(d^3*m*(5 + 3*m + m^2) + 3*c^2*d*m*(6 + 5*m + m^2) + 3*c*d^2*(3 + 4*m + 4*m^2 + m^3) + c^3*(6 + 11*m + 6*m^2 + m^3))*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, 1/2 - m, 3/2, (1/2)*(1 - sin(e + f*x)))*(1 + sin(e + f*x))^(-(1/2) - m)*(a + a*sin(e + f*x))^m) - (d^2*(d*m + c*(5 + m))*cos(e + f*x)*(a + a*sin(e + f*x))^(1 + m))/(a*f*(2 + m)*(3 + m)) - (d*cos(e + f*x)*(a + a*sin(e + f*x))^m*(c + d*sin(e + f*x))^2)/(f*(3 + m)))]
    @test_int [(a + a*sin(e + f*x))^m*(c + d*sin(e + f*x))^2, x, 4, If(13>=8, (d*(d - 2*c*(2 + m))*cos(e + f*x)*(a + a*sin(e + f*x))^m)/(f*(1 + m)*(2 + m)) - (2^(1/2 + m)*(2*c*d*m*(2 + m) + d^2*(1 + m + m^2) + c^2*(2 + 3*m + m^2))*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, 1/2 - m, 3/2, (1/2)*(1 - sin(e + f*x)))*(1 + sin(e + f*x))^(-(1/2) - m)*(a + a*sin(e + f*x))^m)/(f*(1 + m)*(2 + m)) - (d^2*cos(e + f*x)*(a + a*sin(e + f*x))^(1 + m))/(a*f*(2 + m)), (d*(d - 2*c*(2 + m))*cos(e + f*x)*(a + a*sin(e + f*x))^m)/(f*(1 + m)*(2 + m)) - (2^(1/2 + m)*(2*c*d*m*(2 + m) + d^2*(1 + m + m^2) + c^2*(2 + 3*m + m^2))*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, 1/2 - m, 3/2, (1/2)*(1 - sin(e + f*x)))*(1 + sin(e + f*x))^(-(1/2) - m)*(a + a*sin(e + f*x))^m)/(f*(2 + 3*m + m^2)) - (d^2*cos(e + f*x)*(a + a*sin(e + f*x))^(1 + m))/(a*f*(2 + m)))]
    @test_int [(a + a*sin(e + f*x))^m*(c + d*sin(e + f*x))^1, x, 3, -((d*cos(e + f*x)*(a + a*sin(e + f*x))^m)/(f*(1 + m))) - (2^(1/2 + m)*(c + c*m + d*m)*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, 1/2 - m, 3/2, (1/2)*(1 - sin(e + f*x)))*(1 + sin(e + f*x))^(-(1/2) - m)*(a + a*sin(e + f*x))^m)/(f*(1 + m))]
    @test_int [(a + a*sin(e + f*x))^m*(c + d*sin(e + f*x))^0, x, 2, -((2^(1/2 + m)*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, 1/2 - m, 3/2, (1/2)*(1 - sin(e + f*x)))*(1 + sin(e + f*x))^(-(1/2) - m)*(a + a*sin(e + f*x))^m)/f)]
    @test_int [(a + a*sin(e + f*x))^m/(c + d*sin(e + f*x))^1, x, 3, (sqrt(2)*AppellF1(1/2 + m, 1/2, 1, 3/2 + m, (1/2)*(1 + sin(e + f*x)), -((d*(1 + sin(e + f*x)))/(c - d)))*cos(e + f*x)*(a + a*sin(e + f*x))^m)/((c - d)*f*(1 + 2*m)*sqrt(1 - sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^m/(c + d*sin(e + f*x))^2, x, 3, (sqrt(2)*AppellF1(1/2 + m, 1/2, 2, 3/2 + m, (1/2)*(1 + sin(e + f*x)), -((d*(1 + sin(e + f*x)))/(c - d)))*cos(e + f*x)*(a + a*sin(e + f*x))^m)/((c - d)^2*f*(1 + 2*m)*sqrt(1 - sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^m/(c + d*sin(e + f*x))^3, x, 3, (sqrt(2)*AppellF1(1/2 + m, 1/2, 3, 3/2 + m, (1/2)*(1 + sin(e + f*x)), -((d*(1 + sin(e + f*x)))/(c - d)))*cos(e + f*x)*(a + a*sin(e + f*x))^m)/((c - d)^3*f*(1 + 2*m)*sqrt(1 - sin(e + f*x)))]

    @test_int [(a + a*sin(e + f*x))^m*(c + d*sin(e + f*x))^(5/2), x, 4, (sqrt(2)*(c - d)^2*AppellF1(1/2 + m, 1/2, -(5/2), 3/2 + m, (1/2)*(1 + sin(e + f*x)), -((d*(1 + sin(e + f*x)))/(c - d)))*cos(e + f*x)*(a + a*sin(e + f*x))^m*sqrt(c + d*sin(e + f*x)))/(f*(1 + 2*m)*sqrt(1 - sin(e + f*x))*sqrt((c + d*sin(e + f*x))/(c - d)))]
    @test_int [(a + a*sin(e + f*x))^m*(c + d*sin(e + f*x))^(3/2), x, 4, (sqrt(2)*(c - d)*AppellF1(1/2 + m, 1/2, -(3/2), 3/2 + m, (1/2)*(1 + sin(e + f*x)), -((d*(1 + sin(e + f*x)))/(c - d)))*cos(e + f*x)*(a + a*sin(e + f*x))^m*sqrt(c + d*sin(e + f*x)))/(f*(1 + 2*m)*sqrt(1 - sin(e + f*x))*sqrt((c + d*sin(e + f*x))/(c - d)))]
    @test_int [(a + a*sin(e + f*x))^m*(c + d*sin(e + f*x))^(1/2), x, 4, (sqrt(2)*AppellF1(1/2 + m, 1/2, -(1/2), 3/2 + m, (1/2)*(1 + sin(e + f*x)), -((d*(1 + sin(e + f*x)))/(c - d)))*cos(e + f*x)*(a + a*sin(e + f*x))^m*sqrt(c + d*sin(e + f*x)))/(f*(1 + 2*m)*sqrt(1 - sin(e + f*x))*sqrt((c + d*sin(e + f*x))/(c - d)))]
    @test_int [(a + a*sin(e + f*x))^m/(c + d*sin(e + f*x))^(1/2), x, 4, (sqrt(2)*AppellF1(1/2 + m, 1/2, 1/2, 3/2 + m, (1/2)*(1 + sin(e + f*x)), -((d*(1 + sin(e + f*x)))/(c - d)))*cos(e + f*x)*(a + a*sin(e + f*x))^m*sqrt((c + d*sin(e + f*x))/(c - d)))/(f*(1 + 2*m)*sqrt(1 - sin(e + f*x))*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^m/(c + d*sin(e + f*x))^(3/2), x, 4, (sqrt(2)*AppellF1(1/2 + m, 1/2, 3/2, 3/2 + m, (1/2)*(1 + sin(e + f*x)), -((d*(1 + sin(e + f*x)))/(c - d)))*cos(e + f*x)*(a + a*sin(e + f*x))^m*sqrt((c + d*sin(e + f*x))/(c - d)))/((c - d)*f*(1 + 2*m)*sqrt(1 - sin(e + f*x))*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + a*sin(e + f*x))^m/(c + d*sin(e + f*x))^(5/2), x, 4, (sqrt(2)*AppellF1(1/2 + m, 1/2, 5/2, 3/2 + m, (1/2)*(1 + sin(e + f*x)), -((d*(1 + sin(e + f*x)))/(c - d)))*cos(e + f*x)*(a + a*sin(e + f*x))^m*sqrt((c + d*sin(e + f*x))/(c - d)))/((c - d)^2*f*(1 + 2*m)*sqrt(1 - sin(e + f*x))*sqrt(c + d*sin(e + f*x)))]

    @test_int [(1 + sin(e + f*x))^m/(3 + 5*sin(e + f*x))^(m + 1), x, 2, -((4^(-1 - m)*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, 1 + m, 3/2, (1 - sin(e + f*x))/(4*(1 + sin(e + f*x)))))/(f*(1 + sin(e + f*x)))), -((2^(-1 - 2*m)*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, 1/2 - m, 3/2, -((1 - sin(e + f*x))/(3 + 5*sin(e + f*x))))*(1 + sin(e + f*x))^(-1 + m)*((1 + sin(e + f*x))/(3 + 5*sin(e + f*x)))^(1/2 - m))/((3 + 5*sin(e + f*x))^m*f))]
    @test_int [(1 + sin(e + f*x))^m/(3 + 4*sin(e + f*x))^(m + 1), x, 2, -(((7/2)^(-1 - m)*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, 1 + m, 3/2, (1 - sin(e + f*x))/(7*(1 + sin(e + f*x)))))/(f*(1 + sin(e + f*x)))), -((2^(1/2 + m)*7^(-(1/2) - m)*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, 1/2 - m, 3/2, -((1 - sin(e + f*x))/(2*(3 + 4*sin(e + f*x)))))*(1 + sin(e + f*x))^(-1 + m)*((1 + sin(e + f*x))/(3 + 4*sin(e + f*x)))^(1/2 - m))/((3 + 4*sin(e + f*x))^m*f))]
    @test_int [(1 + sin(e + f*x))^m/(3 + 3*sin(e + f*x))^(m + 1), x, 2, -((3^(-1 - m)*cos(e + f*x))/(f*(1 + sin(e + f*x))))]
    @test_int [(1 + sin(e + f*x))^m/(3 + 2*sin(e + f*x))^(m + 1), x, 2, -((2^(1/2 + m)*5^(-(1/2) - m)*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, 1/2 - m, 3/2, (1 - sin(e + f*x))/(2*(3 + 2*sin(e + f*x))))*(1 + sin(e + f*x))^(-1 + m)*((1 + sin(e + f*x))/(3 + 2*sin(e + f*x)))^(1/2 - m))/((3 + 2*sin(e + f*x))^m*f))]
    @test_int [(1 + sin(e + f*x))^m/(3 + 1*sin(e + f*x))^(m + 1), x, 2, -((2^(-(1/2) - m)*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, 1/2 - m, 3/2, (1 - sin(e + f*x))/(3 + sin(e + f*x)))*(1 + sin(e + f*x))^(-1 + m)*((1 + sin(e + f*x))/(3 + sin(e + f*x)))^(1/2 - m))/((3 + sin(e + f*x))^m*f))]
    @test_int [(1 + sin(e + f*x))^m/(3 + 0*sin(e + f*x))^(m + 1), x, 2, -((2^(1/2 + m)*3^(-1 - m)*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, 1/2 - m, 3/2, (1/2)*(1 - sin(e + f*x))))/(f*sqrt(1 + sin(e + f*x))))]
    @test_int [(1 + sin(e + f*x))^m/(3 - 1*sin(e + f*x))^(m + 1), x, 2, -((cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, 1 + m, 3/2, -((2*(1 - sin(e + f*x)))/(1 + sin(e + f*x))))*(3 - sin(e + f*x))^(-1 - m)*((3 - sin(e + f*x))/(1 + sin(e + f*x)))^(1 + m)*(1 + sin(e + f*x))^m)/f)]
    @test_int [(1 + sin(e + f*x))^m/(3 - 2*sin(e + f*x))^(m + 1), x, 2, (cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, -m, 1 - m, (2*(3 - 2*sin(e + f*x)))/(1 + sin(e + f*x)))*sqrt(-((1 - sin(e + f*x))/(1 + sin(e + f*x))))*(1 + sin(e + f*x))^m)/((3 - 2*sin(e + f*x))^m*(sqrt(5)*f*m*(1 - sin(e + f*x))))]
    @test_int [(1 + sin(e + f*x))^m/(3 - 3*sin(e + f*x))^(m + 1), x, 1, (cos(e + f*x)*(3 - 3*sin(e + f*x))^(-1 - m)*(1 + sin(e + f*x))^m)/(f*(1 + 2*m))]
    @test_int [(1 + sin(e + f*x))^m/(3 - 4*sin(e + f*x))^(m + 1), x, 2, (2^(1 + m)*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, 1 + m, 3/2, (7*(1 - sin(e + f*x)))/(1 + sin(e + f*x)))*(-3 + 4*sin(e + f*x))^m)/((3 - 4*sin(e + f*x))^m*(f*(1 + sin(e + f*x)))), (cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, -m, 1 - m, -((2*(3 - 4*sin(e + f*x)))/(1 + sin(e + f*x))))*sqrt((1 - sin(e + f*x))/(1 + sin(e + f*x)))*(1 + sin(e + f*x))^m)/((3 - 4*sin(e + f*x))^m*(sqrt(7)*f*m*(1 - sin(e + f*x))))]
    @test_int [(1 + sin(e + f*x))^m/(3 - 5*sin(e + f*x))^(m + 1), x, 2, (cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, 1 + m, 3/2, (4*(1 - sin(e + f*x)))/(1 + sin(e + f*x)))*(-3 + 5*sin(e + f*x))^m)/((3 - 5*sin(e + f*x))^m*(f*(1 + sin(e + f*x)))), (cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, -m, 1 - m, -((3 - 5*sin(e + f*x))/(1 + sin(e + f*x))))*sqrt((1 - sin(e + f*x))/(1 + sin(e + f*x)))*(1 + sin(e + f*x))^m)/((3 - 5*sin(e + f*x))^m*(4*f*m*(1 - sin(e + f*x))))]

    @test_int [(a + a*sin(e + f*x))^m/(3 + 5*sin(e + f*x))^(m + 1), x, 2, -((4^(-1 - m)*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, 1 + m, 3/2, (a - a*sin(e + f*x))/(4*(a + a*sin(e + f*x))))*(1 + sin(e + f*x))^(-1 - m)*(a + a*sin(e + f*x))^m)/f), -((cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, -m, 1 - m, (3 + 5*sin(e + f*x))/(4*(1 + sin(e + f*x))))*sqrt((1 - sin(e + f*x))/(1 + sin(e + f*x)))*(a + a*sin(e + f*x))^m)/((3 + 5*sin(e + f*x))^m*(4*f*m*(1 - sin(e + f*x)))))]
    @test_int [(a + a*sin(e + f*x))^m/(3 + 4*sin(e + f*x))^(m + 1), x, 2, -(((7/2)^(-1 - m)*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, 1 + m, 3/2, (a - a*sin(e + f*x))/(7*(a + a*sin(e + f*x))))*(1 + sin(e + f*x))^(-1 - m)*(a + a*sin(e + f*x))^m)/f), -((cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, -m, 1 - m, (2*(3 + 4*sin(e + f*x)))/(7*(1 + sin(e + f*x))))*sqrt((1 - sin(e + f*x))/(1 + sin(e + f*x)))*(a + a*sin(e + f*x))^m)/((3 + 4*sin(e + f*x))^m*(sqrt(7)*f*m*(1 - sin(e + f*x)))))]
    @test_int [(a + a*sin(e + f*x))^m/(3 + 3*sin(e + f*x))^(m + 1), x, 2, -((cos(e + f*x)*(3 + 3*sin(e + f*x))^(-1 - m)*(a + a*sin(e + f*x))^m)/f)]
    @test_int [(a + a*sin(e + f*x))^m/(3 + 2*sin(e + f*x))^(m + 1), x, 2, -(((5/2)^(-1 - m)*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, 1 + m, 3/2, -((a - a*sin(e + f*x))/(5*(a + a*sin(e + f*x)))))*(1 + sin(e + f*x))^(-1 - m)*(a + a*sin(e + f*x))^m)/f), (cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, -m, 1 - m, (2*(3 + 2*sin(e + f*x)))/(5*(1 + sin(e + f*x))))*sqrt(-((1 - sin(e + f*x))/(1 + sin(e + f*x))))*(a + a*sin(e + f*x))^m)/((3 + 2*sin(e + f*x))^m*(sqrt(5)*f*m*(1 - sin(e + f*x))))]
    @test_int [(a + a*sin(e + f*x))^m/(3 + 1*sin(e + f*x))^(m + 1), x, 2, -((2^(-1 - m)*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, 1 + m, 3/2, -((a - a*sin(e + f*x))/(2*(a + a*sin(e + f*x)))))*(1 + sin(e + f*x))^(-1 - m)*(a + a*sin(e + f*x))^m)/f), (cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, -m, 1 - m, (3 + sin(e + f*x))/(2*(1 + sin(e + f*x))))*sqrt(-((1 - sin(e + f*x))/(1 + sin(e + f*x))))*(a + a*sin(e + f*x))^m)/((3 + sin(e + f*x))^m*(2*sqrt(2)*f*m*(1 - sin(e + f*x))))]
    @test_int [(a + a*sin(e + f*x))^m/(3 + 0*sin(e + f*x))^(m + 1), x, 3, -((2^(1/2 + m)*3^(-1 - m)*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, 1/2 - m, 3/2, (1/2)*(1 - sin(e + f*x)))*(1 + sin(e + f*x))^(-(1/2) - m)*(a + a*sin(e + f*x))^m)/f)]
    @test_int [(a + a*sin(e + f*x))^m/(3 - 1*sin(e + f*x))^(m + 1), x, 2, -((cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, 1 + m, 3/2, -((2*(a - a*sin(e + f*x)))/(a + a*sin(e + f*x))))*(1 + sin(e + f*x))^(-1 - m)*(a + a*sin(e + f*x))^m)/f), (cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, -m, 1 - m, (3 - sin(e + f*x))/(1 + sin(e + f*x)))*sqrt(-((1 - sin(e + f*x))/(1 + sin(e + f*x))))*(a + a*sin(e + f*x))^m)/((3 - sin(e + f*x))^m*(2*sqrt(2)*f*m*(1 - sin(e + f*x))))]
    @test_int [(a + a*sin(e + f*x))^m/(3 - 2*sin(e + f*x))^(m + 1), x, 2, -((2^(1 + m)*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, 1 + m, 3/2, -((5*(a - a*sin(e + f*x)))/(a + a*sin(e + f*x))))*(1 + sin(e + f*x))^(-1 - m)*(a + a*sin(e + f*x))^m)/f), (cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, -m, 1 - m, (2*(3 - 2*sin(e + f*x)))/(1 + sin(e + f*x)))*sqrt(-((1 - sin(e + f*x))/(1 + sin(e + f*x))))*(a + a*sin(e + f*x))^m)/((3 - 2*sin(e + f*x))^m*(sqrt(5)*f*m*(1 - sin(e + f*x))))]
    @test_int [(a + a*sin(e + f*x))^m/(3 - 3*sin(e + f*x))^(m + 1), x, 1, (cos(e + f*x)*(3 - 3*sin(e + f*x))^(-1 - m)*(a + a*sin(e + f*x))^m)/(f*(1 + 2*m))]
    @test_int [(a + a*sin(e + f*x))^m/(3 - 4*sin(e + f*x))^(m + 1), x, 2, (cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, -m, 1 - m, -((2*(3 - 4*sin(e + f*x)))/(1 + sin(e + f*x))))*sqrt((1 - sin(e + f*x))/(1 + sin(e + f*x)))*(a + a*sin(e + f*x))^m)/((3 - 4*sin(e + f*x))^m*(sqrt(7)*f*m*(1 - sin(e + f*x))))]
    @test_int [(a + a*sin(e + f*x))^m/(3 - 5*sin(e + f*x))^(m + 1), x, 2, (cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, -m, 1 - m, -((3 - 5*sin(e + f*x))/(1 + sin(e + f*x))))*sqrt((1 - sin(e + f*x))/(1 + sin(e + f*x)))*(a + a*sin(e + f*x))^m)/((3 - 5*sin(e + f*x))^m*(4*f*m*(1 - sin(e + f*x))))]

    @test_int [(a + a*sin(e + f*x))^m/(-3 + 5*sin(e + f*x))^(m + 1), x, 2, -((cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, 1 + m, 3/2, (4*(a - a*sin(e + f*x)))/(a + a*sin(e + f*x)))*(1 + sin(e + f*x))^(-1 - m)*(a + a*sin(e + f*x))^m)/f), -((cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, -m, 1 - m, -((3 - 5*sin(e + f*x))/(1 + sin(e + f*x))))*sqrt((1 - sin(e + f*x))/(1 + sin(e + f*x)))*(a + a*sin(e + f*x))^m)/((-3 + 5*sin(e + f*x))^m*(4*f*m*(1 - sin(e + f*x)))))]
    @test_int [(a + a*sin(e + f*x))^m/(-3 + 4*sin(e + f*x))^(m + 1), x, 2, -((2^(1 + m)*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, 1 + m, 3/2, (7*(a - a*sin(e + f*x)))/(a + a*sin(e + f*x)))*(1 + sin(e + f*x))^(-1 - m)*(a + a*sin(e + f*x))^m)/f), -((cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, -m, 1 - m, -((2*(3 - 4*sin(e + f*x)))/(1 + sin(e + f*x))))*sqrt((1 - sin(e + f*x))/(1 + sin(e + f*x)))*(a + a*sin(e + f*x))^m)/((-3 + 4*sin(e + f*x))^m*(sqrt(7)*f*m*(1 - sin(e + f*x)))))]
    @test_int [(a + a*sin(e + f*x))^m/(-3 + 3*sin(e + f*x))^(m + 1), x, 1, (cos(e + f*x)*(-3 + 3*sin(e + f*x))^(-1 - m)*(a + a*sin(e + f*x))^m)/(f*(1 + 2*m))]
    @test_int [(a + a*sin(e + f*x))^m/(-3 + 2*sin(e + f*x))^(m + 1), x, 2, -((cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, -m, 1 - m, (2*(3 - 2*sin(e + f*x)))/(1 + sin(e + f*x)))*sqrt(-((1 - sin(e + f*x))/(1 + sin(e + f*x))))*(a + a*sin(e + f*x))^m)/((-3 + 2*sin(e + f*x))^m*(sqrt(5)*f*m*(1 - sin(e + f*x)))))]
    @test_int [(a + a*sin(e + f*x))^m/(-3 + 1*sin(e + f*x))^(m + 1), x, 2, -((cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, -m, 1 - m, (3 - sin(e + f*x))/(1 + sin(e + f*x)))*sqrt(-((1 - sin(e + f*x))/(1 + sin(e + f*x))))*(a + a*sin(e + f*x))^m)/((-3 + sin(e + f*x))^m*(2*sqrt(2)*f*m*(1 - sin(e + f*x)))))]
    @test_int [(a + a*sin(e + f*x))^m/(-3 + 0*sin(e + f*x))^(m + 1), x, 3, -(((-3)^(-1 - m)*2^(1/2 + m)*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, 1/2 - m, 3/2, (1/2)*(1 - sin(e + f*x)))*(1 + sin(e + f*x))^(-(1/2) - m)*(a + a*sin(e + f*x))^m)/f)]
    @test_int [(a + a*sin(e + f*x))^m/(-3 - 1*sin(e + f*x))^(m + 1), x, 2, -((cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, -m, 1 - m, (3 + sin(e + f*x))/(2*(1 + sin(e + f*x))))*sqrt(-((1 - sin(e + f*x))/(1 + sin(e + f*x))))*(a + a*sin(e + f*x))^m)/((-3 - sin(e + f*x))^m*(2*sqrt(2)*f*m*(1 - sin(e + f*x)))))]
    @test_int [(a + a*sin(e + f*x))^m/(-3 - 2*sin(e + f*x))^(m + 1), x, 2, -((cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, -m, 1 - m, (2*(3 + 2*sin(e + f*x)))/(5*(1 + sin(e + f*x))))*sqrt(-((1 - sin(e + f*x))/(1 + sin(e + f*x))))*(a + a*sin(e + f*x))^m)/((-3 - 2*sin(e + f*x))^m*(sqrt(5)*f*m*(1 - sin(e + f*x)))))]
    @test_int [(a + a*sin(e + f*x))^m/(-3 - 3*sin(e + f*x))^(m + 1), x, 2, -((cos(e + f*x)*(-3 - 3*sin(e + f*x))^(-1 - m)*(a + a*sin(e + f*x))^m)/f)]
    @test_int [(a + a*sin(e + f*x))^m/(-3 - 4*sin(e + f*x))^(m + 1), x, 2, (cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, -m, 1 - m, (2*(3 + 4*sin(e + f*x)))/(7*(1 + sin(e + f*x))))*sqrt((1 - sin(e + f*x))/(1 + sin(e + f*x)))*(a + a*sin(e + f*x))^m)/((-3 - 4*sin(e + f*x))^m*(sqrt(7)*f*m*(1 - sin(e + f*x))))]
    @test_int [(a + a*sin(e + f*x))^m/(-3 - 5*sin(e + f*x))^(m + 1), x, 2, (cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, -m, 1 - m, (3 + 5*sin(e + f*x))/(4*(1 + sin(e + f*x))))*sqrt((1 - sin(e + f*x))/(1 + sin(e + f*x)))*(a + a*sin(e + f*x))^m)/((-3 - 5*sin(e + f*x))^m*(4*f*m*(1 - sin(e + f*x))))]

    @test_int [(a + a*sin(e + f*x))^m/(d*sin(e + f*x))^(m + 1), x, 4, -((cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2 - m, -m, 1 - m, -((2*sin(e + f*x))/(1 - sin(e + f*x))))*((1 + sin(e + f*x))/(1 - sin(e + f*x)))^(1/2 - m)*(a + a*sin(e + f*x))^m)/((d*sin(e + f*x))^m*(d*f*m*(1 + sin(e + f*x)))))]
    @test_int [(a + a*sin(e + f*x))^m/(c + d*sin(e + f*x))^(m + 1), x, 2, -((1/((c + d)*f))*((2^(1/2 + m)*a*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, 1/2 - m, 3/2, ((c - d)*(1 - sin(e + f*x)))/(2*(c + d*sin(e + f*x))))*(a + a*sin(e + f*x))^(-1 + m)*(((c + d)*(1 + sin(e + f*x)))/(c + d*sin(e + f*x)))^(1/2 - m))/(c + d*sin(e + f*x))^m))]

    @test_int [(a + a*sin(e + f*x))^3*(c + d*sin(e + f*x))^n, x, 3, -((8*sqrt(2)*a^3*AppellF1(1/2, -(5/2), -n, 3/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*cos(e + f*x)*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(f*sqrt(1 + sin(e + f*x)))))]
    @test_int [(a + a*sin(e + f*x))^2*(c + d*sin(e + f*x))^n, x, 3, -((4*sqrt(2)*a^2*AppellF1(1/2, -(3/2), -n, 3/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*cos(e + f*x)*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(f*sqrt(1 + sin(e + f*x)))))]
    @test_int [(a + a*sin(e + f*x))^1*(c + d*sin(e + f*x))^n, x, 3, -((2*sqrt(2)*a*AppellF1(1/2, -(1/2), -n, 3/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*cos(e + f*x)*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(f*sqrt(1 + sin(e + f*x)))))]
    @test_int [(a + a*sin(e + f*x))^0*(c + d*sin(e + f*x))^n, x, 3, -((sqrt(2)*AppellF1(1/2, 1/2, -n, 3/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*cos(e + f*x)*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(f*sqrt(1 + sin(e + f*x)))))]
    @test_int [1/(a + a*sin(e + f*x))^1*(c + d*sin(e + f*x))^n, x, 3, -((AppellF1(1/2, 3/2, -n, 3/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*cos(e + f*x)*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(sqrt(2)*a*f*sqrt(1 + sin(e + f*x)))))]
    @test_int [1/(a + a*sin(e + f*x))^2*(c + d*sin(e + f*x))^n, x, 3, -((AppellF1(1/2, 5/2, -n, 3/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*cos(e + f*x)*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(2*sqrt(2)*a^2*f*sqrt(1 + sin(e + f*x)))))]
    @test_int [1/(a + a*sin(e + f*x))^3*(c + d*sin(e + f*x))^n, x, 3, -((AppellF1(1/2, 7/2, -n, 3/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*cos(e + f*x)*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(4*sqrt(2)*a^3*f*sqrt(1 + sin(e + f*x)))))]

    @test_int [(a + a*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^n, x, 5, (2*a^3*(3*c - d*(11 + 4*n))*cos(e + f*x)*(c + d*sin(e + f*x))^(1 + n))/(d^2*f*(3 + 2*n)*(5 + 2*n)*sqrt(a + a*sin(e + f*x))) - (2*a^2*cos(e + f*x)*sqrt(a + a*sin(e + f*x))*(c + d*sin(e + f*x))^(1 + n))/(d*f*(5 + 2*n)) - (2*a^3*(3*c^2 - 2*c*d*(7 + 4*n) + d^2*(43 + 56*n + 16*n^2))*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, -n, 3/2, (d*(1 - sin(e + f*x)))/(c + d))*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(d^2*f*(3 + 2*n)*(5 + 2*n)*sqrt(a + a*sin(e + f*x))))]
    @test_int [(a + a*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))^n, x, 5, -((2*a^2*cos(e + f*x)*(c + d*sin(e + f*x))^(1 + n))/(d*f*(3 + 2*n)*sqrt(a + a*sin(e + f*x)))) + (2*a^2*(c - d*(5 + 4*n))*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, -n, 3/2, (d*(1 - sin(e + f*x)))/(c + d))*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(d*f*(3 + 2*n)*sqrt(a + a*sin(e + f*x))))]
    @test_int [(a + a*sin(e + f*x))^(1/2)*(c + d*sin(e + f*x))^n, x, 3, -((2*a*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, -n, 3/2, (d*(1 - sin(e + f*x)))/(c + d))*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(f*sqrt(a + a*sin(e + f*x)))))]
    @test_int [1/(a + a*sin(e + f*x))^(1/2)*(c + d*sin(e + f*x))^n, x, 3, -((AppellF1(1/2, -n, 1, 3/2, (d*(1 - sin(e + f*x)))/(c + d), (1/2)*(1 - sin(e + f*x)))*cos(e + f*x)*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(f*sqrt(a + a*sin(e + f*x))))), -((AppellF1(1 + n, 1/2, 1, 2 + n, (c + d*sin(e + f*x))/(c + d), (c + d*sin(e + f*x))/(c - d))*cos(e + f*x)*sqrt((d*(1 - sin(e + f*x)))/(c + d))*(c + d*sin(e + f*x))^(1 + n))/((c - d)*f*(1 + n)*(1 - sin(e + f*x))*sqrt(a + a*sin(e + f*x))))]
    @test_int [1/(a + a*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))^n, x, 3, -((AppellF1(1/2, -n, 2, 3/2, (d*(1 - sin(e + f*x)))/(c + d), (1/2)*(1 - sin(e + f*x)))*cos(e + f*x)*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(2*a*f*sqrt(a + a*sin(e + f*x))))), (d*AppellF1(1 + n, 1/2, 2, 2 + n, (c + d*sin(e + f*x))/(c + d), (c + d*sin(e + f*x))/(c - d))*cos(e + f*x)*sqrt((d*(1 - sin(e + f*x)))/(c + d))*(c + d*sin(e + f*x))^(1 + n))/((c - d)^2*f*(1 + n)*(a - a*sin(e + f*x))*sqrt(a + a*sin(e + f*x)))]
    @test_int [1/(a + a*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^n, x, 3, -((AppellF1(1/2, -n, 3, 3/2, (d*(1 - sin(e + f*x)))/(c + d), (1/2)*(1 - sin(e + f*x)))*cos(e + f*x)*(c + d*sin(e + f*x))^n)/(((c + d*sin(e + f*x))/(c + d))^n*(4*a^2*f*sqrt(a + a*sin(e + f*x))))), -((d^2*AppellF1(1 + n, 1/2, 3, 2 + n, (c + d*sin(e + f*x))/(c + d), (c + d*sin(e + f*x))/(c - d))*cos(e + f*x)*sqrt((d*(1 - sin(e + f*x)))/(c + d))*(c + d*sin(e + f*x))^(1 + n))/((c - d)^3*f*(1 + n)*sqrt(a + a*sin(e + f*x))*(a^2 - a^2*sin(e + f*x))))]

    @test_int [(a + a*sin(e + f*x))*(c + d*sin(e + f*x))^(1/3), x, 3, -((2*sqrt(2)*a*AppellF1(1/2, -(1/2), -(1/3), 3/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*cos(e + f*x)*(c + d*sin(e + f*x))^(1/3))/(f*sqrt(1 + sin(e + f*x))*((c + d*sin(e + f*x))/(c + d))^(1/3)))]
    @test_int [(a + a*sin(e + f*x))/(c + d*sin(e + f*x))^(1/3), x, 3, -((2*sqrt(2)*a*AppellF1(1/2, -(1/2), 1/3, 3/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*cos(e + f*x)*((c + d*sin(e + f*x))/(c + d))^(1/3))/(f*sqrt(1 + sin(e + f*x))*(c + d*sin(e + f*x))^(1/3)))]
    @test_int [(a + a*sin(e + f*x))/(c + d*sin(e + f*x))^(4/3), x, 3, -((2*sqrt(2)*a*AppellF1(1/2, -(1/2), 4/3, 3/2, (1/2)*(1 - sin(e + f*x)), (d*(1 - sin(e + f*x)))/(c + d))*cos(e + f*x)*((c + d*sin(e + f*x))/(c + d))^(1/3))/((c + d)*f*sqrt(1 + sin(e + f*x))*(c + d*sin(e + f*x))^(1/3)))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(a+b*sin(e+f*x))^m*(c+d*sin(e+f*x))^n=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+b*sin(e+f*x))^m*(c+d*sin(e+f*x))^n=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [(a + b*sin(e + f*x))^1*(c + d*sin(e + f*x))^3, x, 3, (1/8)*(8*a*c^3 + 12*b*c^2*d + 12*a*c*d^2 + 3*b*d^3)*x - ((4*a*d*(4*c^2 + d^2) + 3*b*(c^3 + 4*c*d^2))*cos(e + f*x))/(6*f) - (d*(6*b*c^2 + 20*a*c*d + 9*b*d^2)*cos(e + f*x)*sin(e + f*x))/(24*f) - ((3*b*c + 4*a*d)*cos(e + f*x)*(c + d*sin(e + f*x))^2)/(12*f) - (b*cos(e + f*x)*(c + d*sin(e + f*x))^3)/(4*f)]
    @test_int [(a + b*sin(e + f*x))^1*(c + d*sin(e + f*x))^2, x, 2, (1/2)*(2*b*c*d + a*(2*c^2 + d^2))*x - (2*(3*a*c*d + b*(c^2 + d^2))*cos(e + f*x))/(3*f) - (d*(2*b*c + 3*a*d)*cos(e + f*x)*sin(e + f*x))/(6*f) - (b*cos(e + f*x)*(c + d*sin(e + f*x))^2)/(3*f)]
    @test_int [(a + b*sin(e + f*x))^1*(c + d*sin(e + f*x))^1, x, 1, (1/2)*(2*a*c + b*d)*x - ((b*c + a*d)*cos(e + f*x))/f - (b*d*cos(e + f*x)*sin(e + f*x))/(2*f)]
    @test_int [(a + b*sin(e + f*x))^1*(c + d*sin(e + f*x))^0, x, 2, a*x - (b*cos(e + f*x))/f]
    @test_int [(a + b*sin(e + f*x))^1/(c + d*sin(e + f*x))^1, x, 4, (b*x)/d - (2*(b*c - a*d)*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/(d*sqrt(c^2 - d^2)*f)]
    @test_int [(a + b*sin(e + f*x))^1/(c + d*sin(e + f*x))^2, x, 5, (2*(a*c - b*d)*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/((c^2 - d^2)^(3/2)*f) - ((b*c - a*d)*cos(e + f*x))/((c^2 - d^2)*f*(c + d*sin(e + f*x)))]
    @test_int [(a + b*sin(e + f*x))^1/(c + d*sin(e + f*x))^3, x, 6, -(((3*b*c*d - a*(2*c^2 + d^2))*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/((c^2 - d^2)^(5/2)*f)) - ((b*c - a*d)*cos(e + f*x))/(2*(c^2 - d^2)*f*(c + d*sin(e + f*x))^2) + ((3*a*c*d - b*(c^2 + 2*d^2))*cos(e + f*x))/(2*(c^2 - d^2)^2*f*(c + d*sin(e + f*x)))]

    @test_int [(a + b*sin(e + f*x))^2*(c + d*sin(e + f*x))^3, x, 4, (1/8)*(6*a*b*d*(4*c^2 + d^2) + b^2*c*(4*c^2 + 9*d^2) + 4*a^2*(2*c^3 + 3*c*d^2))*x - ((20*a^2*d^2*(4*c^2 + d^2) + 30*a*b*c*d*(c^2 + 4*d^2) - b^2*(3*c^4 - 52*c^2*d^2 - 16*d^4))*cos(e + f*x))/(30*d*f) - ((100*a^2*c*d^2 + 30*a*b*d*(2*c^2 + 3*d^2) - b^2*(6*c^3 - 71*c*d^2))*cos(e + f*x)*sin(e + f*x))/(120*f) - ((4*(5*a^2 + 4*b^2)*d^2 - 3*b*c*(b*c - 10*a*d))*cos(e + f*x)*(c + d*sin(e + f*x))^2)/(60*d*f) + (b*(b*c - 10*a*d)*cos(e + f*x)*(c + d*sin(e + f*x))^3)/(20*d*f) - (b^2*cos(e + f*x)*(c + d*sin(e + f*x))^4)/(5*d*f)]
    @test_int [(a + b*sin(e + f*x))^2*(c + d*sin(e + f*x))^2, x, 3, (1/8)*(16*a*b*c*d + 4*a^2*(2*c^2 + d^2) + b^2*(4*c^2 + 3*d^2))*x - ((8*a^2*b*c*d + 8*b^3*c*d - a^3*d^2 + 4*a*b^2*(3*c^2 + 2*d^2))*cos(e + f*x))/(6*b*f) - ((2*a*d*(8*b*c - a*d) + 3*b^2*(4*c^2 + 3*d^2))*cos(e + f*x)*sin(e + f*x))/(24*f) - (d*(8*b*c - a*d)*cos(e + f*x)*(a + b*sin(e + f*x))^2)/(12*b*f) - (d^2*cos(e + f*x)*(a + b*sin(e + f*x))^3)/(4*b*f)]
    @test_int [(a + b*sin(e + f*x))^2*(c + d*sin(e + f*x))^1, x, 2, (1/2)*(2*a^2*c + b^2*c + 2*a*b*d)*x - (2*(3*a*b*c + a^2*d + b^2*d)*cos(e + f*x))/(3*f) - (b*(3*b*c + 2*a*d)*cos(e + f*x)*sin(e + f*x))/(6*f) - (d*cos(e + f*x)*(a + b*sin(e + f*x))^2)/(3*f)]
    @test_int [(a + b*sin(e + f*x))^2*(c + d*sin(e + f*x))^0, x, 1, (1/2)*(2*a^2 + b^2)*x - (2*a*b*cos(e + f*x))/f - (b^2*cos(e + f*x)*sin(e + f*x))/(2*f)]
    @test_int [(a + b*sin(e + f*x))^2/(c + d*sin(e + f*x))^1, x, 5, -((b*(b*c - 2*a*d)*x)/d^2) + (2*(b*c - a*d)^2*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/(d^2*sqrt(c^2 - d^2)*f) - (b^2*cos(e + f*x))/(d*f)]
    @test_int [(a + b*sin(e + f*x))^2/(c + d*sin(e + f*x))^2, x, 5, (b^2*x)/d^2 - (2*(b*c - a*d)*(a*c*d + b*(c^2 - 2*d^2))*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/(d^2*(c^2 - d^2)^(3/2)*f) + ((b*c - a*d)^2*cos(e + f*x))/(d*(c^2 - d^2)*f*(c + d*sin(e + f*x)))]
    @test_int [(a + b*sin(e + f*x))^2/(c + d*sin(e + f*x))^3, x, 6, -(((6*a*b*c*d - a^2*(2*c^2 + d^2) - b^2*(c^2 + 2*d^2))*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/((c^2 - d^2)^(5/2)*f)) + ((b*c - a*d)^2*cos(e + f*x))/(2*d*(c^2 - d^2)*f*(c + d*sin(e + f*x))^2) - ((b*c - a*d)*(3*a*c*d + b*(c^2 - 4*d^2))*cos(e + f*x))/(2*d*(c^2 - d^2)^2*f*(c + d*sin(e + f*x)))]
    @test_int [(a + b*sin(e + f*x))^2/(c + d*sin(e + f*x))^4, x, 7, -(((2*a*b*d*(4*c^2 + d^2) - b^2*c*(c^2 + 4*d^2) - a^2*(2*c^3 + 3*c*d^2))*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/((c^2 - d^2)^(7/2)*f)) + ((b*c - a*d)^2*cos(e + f*x))/(3*d*(c^2 - d^2)*f*(c + d*sin(e + f*x))^3) - ((b*c - a*d)*(5*a*c*d + b*(c^2 - 6*d^2))*cos(e + f*x))/(6*d*(c^2 - d^2)^2*f*(c + d*sin(e + f*x))^2) + ((a^2*d^2*(11*c^2 + 4*d^2) - a*b*(4*c^3*d + 26*c*d^3) - b^2*(c^4 - 10*c^2*d^2 - 6*d^4))*cos(e + f*x))/(6*d*(c^2 - d^2)^3*f*(c + d*sin(e + f*x)))]

    @test_int [(a + b*sin(e + f*x))^3*(c + d*sin(e + f*x))^3, x, 5, (1/16)*(18*a^2*b*d*(4*c^2 + d^2) + b^3*d*(18*c^2 + 5*d^2) + 6*a*b^2*c*(4*c^2 + 9*d^2) + 8*a^3*(2*c^3 + 3*c*d^2))*x - ((3*a*b^2*d*(3*c^2 + d^2) + 3*a^2*b*c*(c^2 + 3*d^2) + b^3*c*(c^2 + 3*d^2) + a^3*(3*c^2*d + d^3))*cos(e + f*x))/f + ((b*c + a*d)*(8*a*b*c*d + a^2*d^2 + b^2*(c^2 + 6*d^2))*cos(e + f*x)^3)/(3*f) - (3*b^2*d^2*(b*c + a*d)*cos(e + f*x)^5)/(5*f) - ((24*a^3*c*d^2 + 18*a^2*b*d*(4*c^2 + d^2) + b^3*d*(18*c^2 + 5*d^2) + 6*a*b^2*c*(4*c^2 + 9*d^2))*cos(e + f*x)*sin(e + f*x))/(16*f) - (5*b^3*d^3*cos(e + f*x)*sin(e + f*x)^3)/(24*f) - (3*b*d*(b^2*c^2 + 3*a*b*c*d + a^2*d^2)*cos(e + f*x)*sin(e + f*x)^3)/(4*f) - (b^3*d^3*cos(e + f*x)*sin(e + f*x)^5)/(6*f), (1/16)*(18*a^2*b*d*(4*c^2 + d^2) + b^3*d*(18*c^2 + 5*d^2) + 6*a*b^2*c*(4*c^2 + 9*d^2) + 8*a^3*(2*c^3 + 3*c*d^2))*x - ((40*a^3*d^3*(4*c^2 + d^2) + 90*a^2*b*c*d^2*(c^2 + 4*d^2) - 6*a*b^2*d*(3*c^4 - 52*c^2*d^2 - 16*d^4) + b^3*(2*c^5 + 17*c^3*d^2 + 96*c*d^4))*cos(e + f*x))/(60*d^2*f) - ((200*a^3*c*d^3 + 90*a^2*b*d^2*(2*c^2 + 3*d^2) - 6*a*b^2*d*(6*c^3 - 71*c*d^2) + b^3*(4*c^4 + 36*c^2*d^2 + 75*d^4))*cos(e + f*x)*sin(e + f*x))/(240*d*f) - ((90*a^2*b*c*d^2 + 40*a^3*d^3 + b^3*(2*c^3 + 21*c*d^2) - a*b^2*(18*c^2*d - 96*d^3))*cos(e + f*x)*(c + d*sin(e + f*x))^2)/(120*d^2*f) + (b*(18*a*b*c*d - 90*a^2*d^2 - b^2*(2*c^2 + 25*d^2))*cos(e + f*x)*(c + d*sin(e + f*x))^3)/(120*d^2*f) + (b^2*(2*b*c - 13*a*d)*cos(e + f*x)*(c + d*sin(e + f*x))^4)/(30*d^2*f) - (b^2*cos(e + f*x)*(a + b*sin(e + f*x))*(c + d*sin(e + f*x))^4)/(6*d*f)]
    @test_int [(a + b*sin(e + f*x))^3*(c + d*sin(e + f*x))^2, x, 4, (1/8)*(24*a^2*b*c*d + 6*b^3*c*d + 4*a^3*(2*c^2 + d^2) + 3*a*b^2*(4*c^2 + 3*d^2))*x - ((30*a^3*b*c*d + 120*a*b^3*c*d - 3*a^4*d^2 + 4*b^4*(5*c^2 + 4*d^2) + 4*a^2*b^2*(20*c^2 + 13*d^2))*cos(e + f*x))/(30*b*f) - ((60*a^2*b*c*d + 90*b^3*c*d - 6*a^3*d^2 + a*b^2*(100*c^2 + 71*d^2))*cos(e + f*x)*sin(e + f*x))/(120*f) - ((3*a*d*(10*b*c - a*d) + 4*b^2*(5*c^2 + 4*d^2))*cos(e + f*x)*(a + b*sin(e + f*x))^2)/(60*b*f) - (d*(10*b*c - a*d)*cos(e + f*x)*(a + b*sin(e + f*x))^3)/(20*b*f) - (d^2*cos(e + f*x)*(a + b*sin(e + f*x))^4)/(5*b*f)]
    @test_int [(a + b*sin(e + f*x))^3*(c + d*sin(e + f*x))^1, x, 3, (1/8)*(8*a^3*c + 12*a*b^2*c + 12*a^2*b*d + 3*b^3*d)*x - ((16*a^2*b*c + 4*b^3*c + 3*a^3*d + 12*a*b^2*d)*cos(e + f*x))/(6*f) - (b*(20*a*b*c + 6*a^2*d + 9*b^2*d)*cos(e + f*x)*sin(e + f*x))/(24*f) - ((4*b*c + 3*a*d)*cos(e + f*x)*(a + b*sin(e + f*x))^2)/(12*f) - (d*cos(e + f*x)*(a + b*sin(e + f*x))^3)/(4*f)]
    @test_int [(a + b*sin(e + f*x))^3*(c + d*sin(e + f*x))^0, x, 2, (1/2)*a*(2*a^2 + 3*b^2)*x - (2*b*(4*a^2 + b^2)*cos(e + f*x))/(3*f) - (5*a*b^2*cos(e + f*x)*sin(e + f*x))/(6*f) - (b*cos(e + f*x)*(a + b*sin(e + f*x))^2)/(3*f)]
    @test_int [(a + b*sin(e + f*x))^3/(c + d*sin(e + f*x))^1, x, 6, -((b*(6*a*b*c*d - 6*a^2*d^2 - b^2*(2*c^2 + d^2))*x)/(2*d^3)) - (2*(b*c - a*d)^3*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/(d^3*sqrt(c^2 - d^2)*f) + (b^2*(2*b*c - 5*a*d)*cos(e + f*x))/(2*d^2*f) - (b^2*cos(e + f*x)*(a + b*sin(e + f*x)))/(2*d*f)]
    @test_int [(a + b*sin(e + f*x))^3/(c + d*sin(e + f*x))^2, x, 6, -((b^2*(2*b*c - 3*a*d)*x)/d^3) + (2*(b*c - a*d)^2*(2*b*c^2 + a*c*d - 3*b*d^2)*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/(d^3*(c^2 - d^2)^(3/2)*f) + (b*(2*a*b*c*d - a^2*d^2 - b^2*(2*c^2 - d^2))*cos(e + f*x))/(d^2*(c^2 - d^2)*f) + ((b*c - a*d)^2*cos(e + f*x)*(a + b*sin(e + f*x)))/(d*(c^2 - d^2)*f*(c + d*sin(e + f*x)))]
    @test_int [(a + b*sin(e + f*x))^3/(c + d*sin(e + f*x))^3, x, 6, (b^3*x)/d^3 - ((9*a^2*b*c*d^4 - a^3*d^3*(2*c^2 + d^2) - 3*a*b^2*d^3*(c^2 + 2*d^2) + b^3*(2*c^5 - 5*c^3*d^2 + 6*c*d^4))*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/(d^3*(c^2 - d^2)^(5/2)*f) + ((b*c - a*d)^2*cos(e + f*x)*(a + b*sin(e + f*x)))/(2*d*(c^2 - d^2)*f*(c + d*sin(e + f*x))^2) + ((b*c - a*d)^2*(2*b*c^2 + 3*a*c*d - 5*b*d^2)*cos(e + f*x))/(2*d^2*(c^2 - d^2)^2*f*(c + d*sin(e + f*x)))]
    @test_int [(a + b*sin(e + f*x))^3/(c + d*sin(e + f*x))^4, x, 7, -(((a*c - b*d)*(10*a*b*c*d - b^2*(3*c^2 + 2*d^2) - a^2*(2*c^2 + 3*d^2))*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/((c^2 - d^2)^(7/2)*f)) + ((b*c - a*d)^2*cos(e + f*x)*(a + b*sin(e + f*x)))/(3*d*(c^2 - d^2)*f*(c + d*sin(e + f*x))^3) + ((b*c - a*d)^2*(2*b*c^2 + 5*a*c*d - 7*b*d^2)*cos(e + f*x))/(6*d^2*(c^2 - d^2)^2*f*(c + d*sin(e + f*x))^2) - ((b*c - a*d)*(5*a*b*c*d*(c^2 - 7*d^2) + a^2*d^2*(11*c^2 + 4*d^2) + b^2*(2*c^4 - 5*c^2*d^2 + 18*d^4))*cos(e + f*x))/(6*d^2*(c^2 - d^2)^3*f*(c + d*sin(e + f*x)))]

    @test_int [(b*B/a + B*sin(x))/(a + b*sin(x)), x, 4, (B*x)/b - (2*sqrt(a^2 - b^2)*B*atan((b + a*tan(x/2))/sqrt(a^2 - b^2)))/(a*b)]
    @test_int [(a*B/b + B*sin(x))/(a + b*sin(x)), x, 2, (B*x)/b]

    @test_int [(a + b*sin(x))/(b + a*sin(x))^2, x, 2, -(cos(x)/(b + a*sin(x)))]
    @test_int [(2 - sin(x))/(2 + sin(x)), x, 2, -x + (4*x)/sqrt(3) + (8*atan(cos(x)/(2 + sqrt(3) + sin(x))))/sqrt(3)]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [1/(a + b*sin(e + f*x))^1*(c + d*sin(e + f*x))^4, x, 7, (d*(8*a^2*b*c*d^2 - 2*a^3*d^3 + 4*b^3*c*(2*c^2 + d^2) - a*b^2*d*(12*c^2 + d^2))*x)/(2*b^4) + (2*(b*c - a*d)^4*atan((b + a*tan((1/2)*(e + f*x)))/sqrt(a^2 - b^2)))/(b^4*sqrt(a^2 - b^2)*f) + (d^2*(12*a*b*c*d - 3*a^2*d^2 - b^2*(17*c^2 + 2*d^2))*cos(e + f*x))/(3*b^3*f) - (d^3*(8*b*c - 3*a*d)*cos(e + f*x)*sin(e + f*x))/(6*b^2*f) - (d^2*cos(e + f*x)*(c + d*sin(e + f*x))^2)/(3*b*f)]
    @test_int [1/(a + b*sin(e + f*x))^1*(c + d*sin(e + f*x))^3, x, 6, -((d*(6*a*b*c*d - 2*a^2*d^2 - b^2*(6*c^2 + d^2))*x)/(2*b^3)) + (2*(b*c - a*d)^3*atan((b + a*tan((1/2)*(e + f*x)))/sqrt(a^2 - b^2)))/(b^3*sqrt(a^2 - b^2)*f) - (d^2*(5*b*c - 2*a*d)*cos(e + f*x))/(2*b^2*f) - (d^2*cos(e + f*x)*(c + d*sin(e + f*x)))/(2*b*f)]
    @test_int [1/(a + b*sin(e + f*x))^1*(c + d*sin(e + f*x))^2, x, 5, (d*(2*b*c - a*d)*x)/b^2 + (2*(b*c - a*d)^2*atan((b + a*tan((1/2)*(e + f*x)))/sqrt(a^2 - b^2)))/(b^2*sqrt(a^2 - b^2)*f) - (d^2*cos(e + f*x))/(b*f)]
    @test_int [1/(a + b*sin(e + f*x))^1*(c + d*sin(e + f*x))^1, x, 4, (d*x)/b + (2*(b*c - a*d)*atan((b + a*tan((1/2)*(e + f*x)))/sqrt(a^2 - b^2)))/(b*sqrt(a^2 - b^2)*f)]
    @test_int [1/(a + b*sin(e + f*x))^1*(c + d*sin(e + f*x))^0, x, 3, (2*atan((b + a*tan((1/2)*(e + f*x)))/sqrt(a^2 - b^2)))/(sqrt(a^2 - b^2)*f)]
    @test_int [1/(a + b*sin(e + f*x))^1/(c + d*sin(e + f*x))^1, x, 7, (2*b*atan((b + a*tan((1/2)*(e + f*x)))/sqrt(a^2 - b^2)))/(sqrt(a^2 - b^2)*(b*c - a*d)*f) - (2*d*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/((b*c - a*d)*sqrt(c^2 - d^2)*f)]
    @test_int [1/(a + b*sin(e + f*x))^1/(c + d*sin(e + f*x))^2, x, 8, (2*b^2*atan((b + a*tan((1/2)*(e + f*x)))/sqrt(a^2 - b^2)))/(sqrt(a^2 - b^2)*(b*c - a*d)^2*f) + (2*d*(a*c*d - b*(2*c^2 - d^2))*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/((b*c - a*d)^2*(c^2 - d^2)^(3/2)*f) - (d^2*cos(e + f*x))/((b*c - a*d)*(c^2 - d^2)*f*(c + d*sin(e + f*x)))]
    @test_int [1/(a + b*sin(e + f*x))^1/(c + d*sin(e + f*x))^3, x, 9, (2*b^3*atan((b + a*tan((1/2)*(e + f*x)))/sqrt(a^2 - b^2)))/(sqrt(a^2 - b^2)*(b*c - a*d)^3*f) + (d*(6*a*b*c^3*d - a^2*d^2*(2*c^2 + d^2) - b^2*(6*c^4 - 5*c^2*d^2 + 2*d^4))*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/((b*c - a*d)^3*(c^2 - d^2)^(5/2)*f) - (d^2*cos(e + f*x))/(2*(b*c - a*d)*(c^2 - d^2)*f*(c + d*sin(e + f*x))^2) - (d^2*(5*b*c^2 - 3*a*c*d - 2*b*d^2)*cos(e + f*x))/(2*(b*c - a*d)^2*(c^2 - d^2)^2*f*(c + d*sin(e + f*x)))]

    @test_int [1/(a + b*sin(e + f*x))^2*(c + d*sin(e + f*x))^4, x, 7, -((d^2*(16*a*b*c*d - 6*a^2*d^2 - b^2*(12*c^2 + d^2))*x)/(2*b^4)) + (2*(b*c - a*d)^3*(a*b*c + 3*a^2*d - 4*b^2*d)*atan((b + a*tan((1/2)*(e + f*x)))/sqrt(a^2 - b^2)))/(b^4*(a^2 - b^2)^(3/2)*f) + (d*(2*b*c - a*d)*(2*a*b*c*d - 3*a^2*d^2 - b^2*(c^2 - 2*d^2))*cos(e + f*x))/(b^3*(a^2 - b^2)*f) + (d^2*(4*a*b*c*d - 3*a^2*d^2 - b^2*(2*c^2 - d^2))*cos(e + f*x)*sin(e + f*x))/(2*b^2*(a^2 - b^2)*f) + ((b*c - a*d)^2*cos(e + f*x)*(c + d*sin(e + f*x))^2)/(b*(a^2 - b^2)*f*(a + b*sin(e + f*x)))]
    @test_int [1/(a + b*sin(e + f*x))^2*(c + d*sin(e + f*x))^3, x, 6, (d^2*(3*b*c - 2*a*d)*x)/b^3 + (2*(b*c - a*d)^2*(a*b*c + 2*a^2*d - 3*b^2*d)*atan((b + a*tan((1/2)*(e + f*x)))/sqrt(a^2 - b^2)))/(b^3*(a^2 - b^2)^(3/2)*f) + (d*(2*a*b*c*d - 2*a^2*d^2 - b^2*(c^2 - d^2))*cos(e + f*x))/(b^2*(a^2 - b^2)*f) + ((b*c - a*d)^2*cos(e + f*x)*(c + d*sin(e + f*x)))/(b*(a^2 - b^2)*f*(a + b*sin(e + f*x)))]
    @test_int [1/(a + b*sin(e + f*x))^2*(c + d*sin(e + f*x))^2, x, 5, (d^2*x)/b^2 + (2*(b*c - a*d)*(a*b*c + a^2*d - 2*b^2*d)*atan((b + a*tan((1/2)*(e + f*x)))/sqrt(a^2 - b^2)))/(b^2*(a^2 - b^2)^(3/2)*f) + ((b*c - a*d)^2*cos(e + f*x))/(b*(a^2 - b^2)*f*(a + b*sin(e + f*x)))]
    @test_int [1/(a + b*sin(e + f*x))^2*(c + d*sin(e + f*x))^1, x, 5, (2*(a*c - b*d)*atan((b + a*tan((1/2)*(e + f*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(3/2)*f) + ((b*c - a*d)*cos(e + f*x))/((a^2 - b^2)*f*(a + b*sin(e + f*x)))]
    @test_int [1/(a + b*sin(e + f*x))^2*(c + d*sin(e + f*x))^0, x, 5, (2*a*atan((b + a*tan((1/2)*(e + f*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(3/2)*f) + (b*cos(e + f*x))/((a^2 - b^2)*f*(a + b*sin(e + f*x)))]
    @test_int [1/(a + b*sin(e + f*x))^2/(c + d*sin(e + f*x))^1, x, 8, (2*b*(a*b*c - 2*a^2*d + b^2*d)*atan((b + a*tan((1/2)*(e + f*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(3/2)*(b*c - a*d)^2*f) + (2*d^2*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/((b*c - a*d)^2*sqrt(c^2 - d^2)*f) + (b^2*cos(e + f*x))/((a^2 - b^2)*(b*c - a*d)*f*(a + b*sin(e + f*x)))]
    @test_int [1/(a + b*sin(e + f*x))^2/(c + d*sin(e + f*x))^2, x, 9, (2*b^2*(a*b*c - 3*a^2*d + 2*b^2*d)*atan((b + a*tan((1/2)*(e + f*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(3/2)*(b*c - a*d)^3*f) + (2*d^2*(3*b*c^2 - a*c*d - 2*b*d^2)*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/((b*c - a*d)^3*(c^2 - d^2)^(3/2)*f) + (d*(a^2*d^2 + b^2*(c^2 - 2*d^2))*cos(e + f*x))/((a^2 - b^2)*(b*c - a*d)^2*(c^2 - d^2)*f*(c + d*sin(e + f*x))) + (b^2*cos(e + f*x))/((a^2 - b^2)*(b*c - a*d)*f*(a + b*sin(e + f*x))*(c + d*sin(e + f*x)))]
    @test_int [1/(a + b*sin(e + f*x))^2/(c + d*sin(e + f*x))^3, x, 10, (2*b^3*(a*b*c - 4*a^2*d + 3*b^2*d)*atan((b + a*tan((1/2)*(e + f*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(3/2)*(b*c - a*d)^4*f) - (d^2*(2*a*b*c*d*(4*c^2 - d^2) - a^2*d^2*(2*c^2 + d^2) - 3*b^2*(4*c^4 - 5*c^2*d^2 + 2*d^4))*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/((b*c - a*d)^4*(c^2 - d^2)^(5/2)*f) + (d*(a^2*d^2 + b^2*(2*c^2 - 3*d^2))*cos(e + f*x))/(2*(a^2 - b^2)*(b*c - a*d)^2*(c^2 - d^2)*f*(c + d*sin(e + f*x))^2) + (b^2*cos(e + f*x))/((a^2 - b^2)*(b*c - a*d)*f*(a + b*sin(e + f*x))*(c + d*sin(e + f*x))^2) - ((3*a^3*c*d^4 - 3*a*b^2*c*d^4 - a^2*b*d^3*(7*c^2 - 4*d^2) - b^3*(2*c^4*d - 11*c^2*d^3 + 6*d^5))*cos(e + f*x))/(2*(a^2 - b^2)*(b*c - a*d)^3*(c^2 - d^2)^2*f*(c + d*sin(e + f*x)))]

    @test_int [1/(a + b*sin(e + f*x))^3*(c + d*sin(e + f*x))^5, x, 8, -((d^3*(30*a*b*c*d - 12*a^2*d^2 - b^2*(20*c^2 + d^2))*x)/(2*b^5)) + ((b*c - a*d)^3*(6*a^3*b*c*d - 12*a*b^3*c*d + 12*a^4*d^2 + a^2*b^2*(2*c^2 - 29*d^2) + b^4*(c^2 + 20*d^2))*atan((b + a*tan((1/2)*(e + f*x)))/sqrt(a^2 - b^2)))/(b^5*(a^2 - b^2)^(5/2)*f) - (d*(30*a^4*b*c*d^3 - 12*a^5*d^4 - a^3*b^2*d^2*(16*c^2 - 21*d^2) - b^5*c*d*(17*c^2 - 10*d^2) - a^2*b^3*c*d*(4*c^2 + 55*d^2) + a*b^4*(6*c^4 + 43*c^2*d^2 - 6*d^4))*cos(e + f*x))/(2*b^4*(a^2 - b^2)^2*f) + (d^2*(7*a^3*b*c*d^2 - 6*a^4*d^3 + b^4*d*(8*c^2 - d^2) + a^2*b^2*d*(c^2 + 10*d^2) - a*b^3*c*(3*c^2 + 16*d^2))*cos(e + f*x)*sin(e + f*x))/(2*b^3*(a^2 - b^2)^2*f) + ((b*c - a*d)^2*(3*a*b*c + 4*a^2*d - 7*b^2*d)*cos(e + f*x)*(c + d*sin(e + f*x))^2)/(2*b^2*(a^2 - b^2)^2*f*(a + b*sin(e + f*x))) + ((b*c - a*d)^2*cos(e + f*x)*(c + d*sin(e + f*x))^3)/(2*b*(a^2 - b^2)*f*(a + b*sin(e + f*x))^2)]
    @test_int [1/(a + b*sin(e + f*x))^3*(c + d*sin(e + f*x))^4, x, 7, (d^3*(4*b*c - 3*a*d)*x)/b^4 + ((b*c - a*d)^2*(4*a^3*b*c*d - 10*a*b^3*c*d + 6*a^4*d^2 + a^2*b^2*(2*c^2 - 15*d^2) + b^4*(c^2 + 12*d^2))*atan((b + a*tan((1/2)*(e + f*x)))/sqrt(a^2 - b^2)))/(b^4*(a^2 - b^2)^(5/2)*f) + (d^2*(2*a*b*c*d - 3*a^2*d^2 - b^2*(c^2 - 2*d^2))*cos(e + f*x))/(2*b^3*(a^2 - b^2)*f) + (3*(b*c - a*d)^3*(a*b*c + a^2*d - 2*b^2*d)*cos(e + f*x))/(2*b^3*(a^2 - b^2)^2*f*(a + b*sin(e + f*x))) + ((b*c - a*d)^2*cos(e + f*x)*(c + d*sin(e + f*x))^2)/(2*b*(a^2 - b^2)*f*(a + b*sin(e + f*x))^2)]
    @test_int [1/(a + b*sin(e + f*x))^3*(c + d*sin(e + f*x))^3, x, 6, (d^3*x)/b^3 + ((b*c - a*d)*(2*a^3*b*c*d - 8*a*b^3*c*d + 2*a^4*d^2 + a^2*b^2*(2*c^2 - 5*d^2) + b^4*(c^2 + 6*d^2))*atan((b + a*tan((1/2)*(e + f*x)))/sqrt(a^2 - b^2)))/(b^3*(a^2 - b^2)^(5/2)*f) + ((b*c - a*d)^2*(3*a*b*c + 2*a^2*d - 5*b^2*d)*cos(e + f*x))/(2*b^2*(a^2 - b^2)^2*f*(a + b*sin(e + f*x))) + ((b*c - a*d)^2*cos(e + f*x)*(c + d*sin(e + f*x)))/(2*b*(a^2 - b^2)*f*(a + b*sin(e + f*x))^2)]
    @test_int [1/(a + b*sin(e + f*x))^3*(c + d*sin(e + f*x))^2, x, 6, -(((6*a*b*c*d - a^2*(2*c^2 + d^2) - b^2*(c^2 + 2*d^2))*atan((b + a*tan((1/2)*(e + f*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(5/2)*f)) + ((b*c - a*d)^2*cos(e + f*x))/(2*b*(a^2 - b^2)*f*(a + b*sin(e + f*x))^2) + ((b*c - a*d)*(3*a*b*c + a^2*d - 4*b^2*d)*cos(e + f*x))/(2*b*(a^2 - b^2)^2*f*(a + b*sin(e + f*x)))]
    @test_int [1/(a + b*sin(e + f*x))^3*(c + d*sin(e + f*x))^1, x, 6, ((2*a^2*c + b^2*c - 3*a*b*d)*atan((b + a*tan((1/2)*(e + f*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(5/2)*f) + ((b*c - a*d)*cos(e + f*x))/(2*(a^2 - b^2)*f*(a + b*sin(e + f*x))^2) + ((3*a*b*c - a^2*d - 2*b^2*d)*cos(e + f*x))/(2*(a^2 - b^2)^2*f*(a + b*sin(e + f*x)))]
    @test_int [1/(a + b*sin(e + f*x))^3*(c + d*sin(e + f*x))^0, x, 6, ((2*a^2 + b^2)*atan((b + a*tan((1/2)*(e + f*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(5/2)*f) + (b*cos(e + f*x))/(2*(a^2 - b^2)*f*(a + b*sin(e + f*x))^2) + (3*a*b*cos(e + f*x))/(2*(a^2 - b^2)^2*f*(a + b*sin(e + f*x)))]
    @test_int [1/(a + b*sin(e + f*x))^3/(c + d*sin(e + f*x))^1, x, 9, -((b*(6*a^3*b*c*d - 6*a^4*d^2 - a^2*b^2*(2*c^2 - 5*d^2) - b^4*(c^2 + 2*d^2))*atan((b + a*tan((1/2)*(e + f*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(5/2)*(b*c - a*d)^3*f)) - (2*d^3*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/((b*c - a*d)^3*sqrt(c^2 - d^2)*f) + (b^2*cos(e + f*x))/(2*(a^2 - b^2)*(b*c - a*d)*f*(a + b*sin(e + f*x))^2) + (b^2*(3*a*b*c - 5*a^2*d + 2*b^2*d)*cos(e + f*x))/(2*(a^2 - b^2)^2*(b*c - a*d)^2*f*(a + b*sin(e + f*x)))]
    @test_int [1/(a + b*sin(e + f*x))^3/(c + d*sin(e + f*x))^2, x, 10, -((b^2*(8*a^3*b*c*d - 2*a*b^3*c*d - 12*a^4*d^2 - a^2*b^2*(2*c^2 - 15*d^2) - b^4*(c^2 + 6*d^2))*atan((b + a*tan((1/2)*(e + f*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(5/2)*(b*c - a*d)^4*f)) - (2*d^3*(4*b*c^2 - a*c*d - 3*b*d^2)*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/((b*c - a*d)^4*(c^2 - d^2)^(3/2)*f) - (d*(2*a^4*d^3 + a^2*b^2*d*(7*c^2 - 11*d^2) - 2*b^4*d*(2*c^2 - 3*d^2) - 3*a*b^3*c*(c^2 - d^2))*cos(e + f*x))/(2*(a^2 - b^2)^2*(b*c - a*d)^3*(c^2 - d^2)*f*(c + d*sin(e + f*x))) + (b^2*cos(e + f*x))/(2*(a^2 - b^2)*(b*c - a*d)*f*(a + b*sin(e + f*x))^2*(c + d*sin(e + f*x))) + (3*b^2*(a*b*c - 2*a^2*d + b^2*d)*cos(e + f*x))/(2*(a^2 - b^2)^2*(b*c - a*d)^2*f*(a + b*sin(e + f*x))*(c + d*sin(e + f*x)))]
    @test_int [1/(a + b*sin(e + f*x))^3/(c + d*sin(e + f*x))^3, x, 11, -((b^3*(10*a^3*b*c*d - 4*a*b^3*c*d - 20*a^4*d^2 - a^2*b^2*(2*c^2 - 29*d^2) - b^4*(c^2 + 12*d^2))*atan((b + a*tan((1/2)*(e + f*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(5/2)*(b*c - a*d)^5*f)) - (d^3*(a^2*d^2*(2*c^2 + d^2) - a*b*(10*c^3*d - 4*c*d^3) + b^2*(20*c^4 - 29*c^2*d^2 + 12*d^4))*atan((d + c*tan((1/2)*(e + f*x)))/sqrt(c^2 - d^2)))/((b*c - a*d)^5*(c^2 - d^2)^(5/2)*f) - (d*(a^4*d^3 - b^4*d*(5*c^2 - 6*d^2) + 2*a^2*b^2*d*(4*c^2 - 5*d^2) - 3*a*b^3*c*(c^2 - d^2))*cos(e + f*x))/(2*(a^2 - b^2)^2*(b*c - a*d)^3*(c^2 - d^2)*f*(c + d*sin(e + f*x))^2) + (b^2*cos(e + f*x))/(2*(a^2 - b^2)*(b*c - a*d)*f*(a + b*sin(e + f*x))^2*(c + d*sin(e + f*x))^2) + (b^2*(3*a*b*c - 7*a^2*d + 4*b^2*d)*cos(e + f*x))/(2*(a^2 - b^2)^2*(b*c - a*d)^2*f*(a + b*sin(e + f*x))*(c + d*sin(e + f*x))^2) + (3*d*(a^5*c*d^4 - 2*a^3*b^2*c*d^4 + a*b^4*c*(c^4 - 2*c^2*d^2 + 2*d^4) + b^5*d*(2*c^4 - 7*c^2*d^2 + 4*d^4) - a^2*b^3*d*(3*c^4 - 12*c^2*d^2 + 7*d^4) - a^4*b*(3*c^2*d^3 - 2*d^5))*cos(e + f*x))/(2*(a^2 - b^2)^2*(b*c - a*d)^4*(c^2 - d^2)^2*f*(c + d*sin(e + f*x)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+b*sin(e+f*x))^m*(c+d*sin(e+f*x))^(n/2)=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [(a + b*sin(e + f*x))*(c + d*sin(e + f*x))^(5/2), x, 8, (-2*(15*b*c^2 + 56*a*c*d + 25*b*d^2)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(105*f) - (2*(5*b*c + 7*a*d)*cos(e + f*x)*(c + d*sin(e + f*x))^(3/2))/(35*f) - (2*b*cos(e + f*x)*(c + d*sin(e + f*x))^(5/2))/(7*f) + (2*(15*b*c^3 + 161*a*c^2*d + 145*b*c*d^2 + 63*a*d^3)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(105*d*f*sqrt((c + d*sin(e + f*x))/(c + d))) - (2*(c^2 - d^2)*(15*b*c^2 + 56*a*c*d + 25*b*d^2)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(105*d*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + b*sin(e + f*x))*(c + d*sin(e + f*x))^(3/2), x, 7, (-2*(3*b*c + 5*a*d)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(15*f) - (2*b*cos(e + f*x)*(c + d*sin(e + f*x))^(3/2))/(5*f) + (2*(20*a*c*d + 3*b*(c^2 + 3*d^2))*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(15*d*f*sqrt((c + d*sin(e + f*x))/(c + d))) - (2*(3*b*c + 5*a*d)*(c^2 - d^2)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(15*d*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + b*sin(e + f*x))*(c + d*sin(e + f*x))^(1/2), x, 6, (-2*b*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(3*f) + (2*(b*c + 3*a*d)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(3*d*f*sqrt((c + d*sin(e + f*x))/(c + d))) - (2*b*(c^2 - d^2)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(3*d*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + b*sin(e + f*x))/(c + d*sin(e + f*x))^(1/2), x, 5, (2*b*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(d*f*sqrt((c + d*sin(e + f*x))/(c + d))) - (2*(b*c - a*d)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(d*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + b*sin(e + f*x))/(c + d*sin(e + f*x))^(3/2), x, 6, (-2*(b*c - a*d)*cos(e + f*x))/((c^2 - d^2)*f*sqrt(c + d*sin(e + f*x))) - (2*(b*c - a*d)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(d*(c^2 - d^2)*f*sqrt((c + d*sin(e + f*x))/(c + d))) + (2*b*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(d*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + b*sin(e + f*x))/(c + d*sin(e + f*x))^(5/2), x, 7, (-2*(b*c - a*d)*cos(e + f*x))/(3*(c^2 - d^2)*f*(c + d*sin(e + f*x))^(3/2)) + (2*(4*a*c*d - b*(c^2 + 3*d^2))*cos(e + f*x))/(3*(c^2 - d^2)^2*f*sqrt(c + d*sin(e + f*x))) + (2*(4*a*c*d - b*(c^2 + 3*d^2))*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(3*d*(c^2 - d^2)^2*f*sqrt((c + d*sin(e + f*x))/(c + d))) + (2*(b*c - a*d)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(3*d*(c^2 - d^2)*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + b*sin(e + f*x))/(c + d*sin(e + f*x))^(7/2), x, 8, (-2*(b*c - a*d)*cos(e + f*x))/(5*(c^2 - d^2)*f*(c + d*sin(e + f*x))^(5/2)) - (2*(3*b*c^2 - 8*a*c*d + 5*b*d^2)*cos(e + f*x))/(15*(c^2 - d^2)^2*f*(c + d*sin(e + f*x))^(3/2)) - (2*(3*b*c^3 - 23*a*c^2*d + 29*b*c*d^2 - 9*a*d^3)*cos(e + f*x))/(15*(c^2 - d^2)^3*f*sqrt(c + d*sin(e + f*x))) - (2*(3*b*c^3 - 23*a*c^2*d + 29*b*c*d^2 - 9*a*d^3)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(15*d*(c^2 - d^2)^3*f*sqrt((c + d*sin(e + f*x))/(c + d))) + (2*(3*b*c^2 - 8*a*c*d + 5*b*d^2)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(15*d*(c^2 - d^2)^2*f*sqrt(c + d*sin(e + f*x)))]

    @test_int [(a + b*sin(e + f*x))^2*(c + d*sin(e + f*x))^(5/2), x, 9, (-4*(84*a^2*c*d^2 + 15*a*b*d*(3*c^2 + 5*d^2) - b^2*(5*c^3 - 57*c*d^2))*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(315*d*f) - (2*(7*(9*a^2 + 7*b^2)*d^2 - 10*b*c*(b*c - 9*a*d))*cos(e + f*x)*(c + d*sin(e + f*x))^(3/2))/(315*d*f) + (4*b*(b*c - 9*a*d)*cos(e + f*x)*(c + d*sin(e + f*x))^(5/2))/(63*d*f) - (2*b^2*cos(e + f*x)*(c + d*sin(e + f*x))^(7/2))/(9*d*f) + (2*(21*a^2*d^2*(23*c^2 + 9*d^2) + 30*a*b*d*(3*c^3 + 29*c*d^2) - b^2*(10*c^4 - 279*c^2*d^2 - 147*d^4))*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(315*d^2*f*sqrt((c + d*sin(e + f*x))/(c + d))) + (4*(c^2 - d^2)*(5*b^2*c^3 - 45*a*b*c^2*d - 84*a^2*c*d^2 - 57*b^2*c*d^2 - 75*a*b*d^3)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(315*d^2*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + b*sin(e + f*x))^2*(c + d*sin(e + f*x))^(3/2), x, 8, (-2*(5*(7*a^2 + 5*b^2)*d^2 - 6*b*c*(b*c - 7*a*d))*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(105*d*f) + (4*b*(b*c - 7*a*d)*cos(e + f*x)*(c + d*sin(e + f*x))^(3/2))/(35*d*f) - (2*b^2*cos(e + f*x)*(c + d*sin(e + f*x))^(5/2))/(7*d*f) + (4*(70*a^2*c*d^2 + 21*a*b*d*(c^2 + 3*d^2) - b^2*(3*c^3 - 41*c*d^2))*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(105*d^2*f*sqrt((c + d*sin(e + f*x))/(c + d))) - (2*(c^2 - d^2)*(42*a*b*c*d + 35*a^2*d^2 - b^2*(6*c^2 - 25*d^2))*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(105*d^2*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + b*sin(e + f*x))^2*(c + d*sin(e + f*x))^(1/2), x, 7, (4*b*(b*c - 5*a*d)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(15*d*f) - (2*b^2*cos(e + f*x)*(c + d*sin(e + f*x))^(3/2))/(5*d*f) + (2*(3*(5*a^2 + 3*b^2)*d^2 - 2*b*c*(b*c - 5*a*d))*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(15*d^2*f*sqrt((c + d*sin(e + f*x))/(c + d))) + (4*b*(b*c - 5*a*d)*(c^2 - d^2)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(15*d^2*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + b*sin(e + f*x))^2/(c + d*sin(e + f*x))^(1/2), x, 6, (-2*b^2*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(3*d*f) - (4*b*(b*c - 3*a*d)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(3*d^2*f*sqrt((c + d*sin(e + f*x))/(c + d))) + (2*((3*a^2 + b^2)*d^2 + 2*b*c*(b*c - 3*a*d))*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(3*d^2*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + b*sin(e + f*x))^2/(c + d*sin(e + f*x))^(3/2), x, 6, (2*(b*c - a*d)^2*cos(e + f*x))/(d*(c^2 - d^2)*f*sqrt(c + d*sin(e + f*x))) + (2*(2*b^2*c^2 - 2*a*b*c*d + (a^2 - b^2)*d^2)*Elliptic.E((1/2)*(e - pi/2 + f*x), (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(d^2*(c^2 - d^2)*f*sqrt((c + d*sin(e + f*x))/(c + d))) - (4*b*(b*c - a*d)*Elliptic.F((1/2)*(e - pi/2 + f*x), (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(d^2*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + b*sin(e + f*x))^2/(c + d*sin(e + f*x))^(5/2), x, 7, (2*(b*c - a*d)^2*cos(e + f*x))/(3*d*(c^2 - d^2)*f*(c + d*sin(e + f*x))^(3/2)) - (4*(b*c - a*d)*(2*a*c*d + b*(c^2 - 3*d^2))*cos(e + f*x))/(3*d*(c^2 - d^2)^2*f*sqrt(c + d*sin(e + f*x))) - (4*(b*c - a*d)*(2*a*c*d + b*(c^2 - 3*d^2))*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(3*d^2*(c^2 - d^2)^2*f*sqrt((c + d*sin(e + f*x))/(c + d))) + (2*(2*a*b*c*d - a^2*d^2 + b^2*(2*c^2 - 3*d^2))*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(3*d^2*(c^2 - d^2)*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + b*sin(e + f*x))^2/(c + d*sin(e + f*x))^(7/2), x, 8, (2*(b*c - a*d)^2*cos(e + f*x))/(5*d*(c^2 - d^2)*f*(c + d*sin(e + f*x))^(5/2)) - (4*(b*c - a*d)*(4*a*c*d + b*(c^2 - 5*d^2))*cos(e + f*x))/(15*d*(c^2 - d^2)^2*f*(c + d*sin(e + f*x))^(3/2)) + (2*(a^2*d^2*(23*c^2 + 9*d^2) - a*b*(6*c^3*d + 58*c*d^3) - b^2*(2*c^4 - 19*c^2*d^2 - 15*d^4))*cos(e + f*x))/(15*d*(c^2 - d^2)^3*f*sqrt(c + d*sin(e + f*x))) + (2*(a^2*d^2*(23*c^2 + 9*d^2) - a*b*(6*c^3*d + 58*c*d^3) - b^2*(2*c^4 - 19*c^2*d^2 - 15*d^4))*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(15*d^2*(c^2 - d^2)^3*f*sqrt((c + d*sin(e + f*x))/(c + d))) + (4*(b*c - a*d)*(4*a*c*d + b*(c^2 - 5*d^2))*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(15*d^2*(c^2 - d^2)^2*f*sqrt(c + d*sin(e + f*x)))]

    @test_int [(a + b*sin(e + f*x))^3*(c + d*sin(e + f*x))^(5/2), x, 10, (-2*(1848*a^3*c*d^3 + 495*a^2*b*d^2*(3*c^2 + 5*d^2) - 66*a*b^2*d*(5*c^3 - 57*c*d^2) + 5*b^3*(8*c^4 + 57*c^2*d^2 + 135*d^4))*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(3465*d^2*f) - (2*(1485*a^2*b*c*d^2 + 693*a^3*d^3 - 33*a*b^2*d*(10*c^2 - 49*d^2) + 5*b^3*(8*c^3 + 67*c*d^2))*cos(e + f*x)*(c + d*sin(e + f*x))^(3/2))/(3465*d^2*f) + (2*b*(66*a*b*c*d - 297*a^2*d^2 - b^2*(8*c^2 + 81*d^2))*cos(e + f*x)*(c + d*sin(e + f*x))^(5/2))/(693*d^2*f) + (8*b^2*(b*c - 6*a*d)*cos(e + f*x)*(c + d*sin(e + f*x))^(7/2))/(99*d^2*f) - (2*b^2*cos(e + f*x)*(a + b*sin(e + f*x))*(c + d*sin(e + f*x))^(7/2))/(11*d*f) + (2*(231*a^3*d^3*(23*c^2 + 9*d^2) + 495*a^2*b*c*d^2*(3*c^2 + 29*d^2) - 33*a*b^2*d*(10*c^4 - 279*c^2*d^2 - 147*d^4) + 5*b^3*(8*c^5 + 51*c^3*d^2 + 741*c*d^4))*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(3465*d^3*f*sqrt((c + d*sin(e + f*x))/(c + d))) - (2*(c^2 - d^2)*(1848*a^3*c*d^3 + 495*a^2*b*d^2*(3*c^2 + 5*d^2) - 66*a*b^2*d*(5*c^3 - 57*c*d^2) + 5*b^3*(8*c^4 + 57*c^2*d^2 + 135*d^4))*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(3465*d^3*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + b*sin(e + f*x))^3*(c + d*sin(e + f*x))^(3/2), x, 9, (-2*(189*a^2*b*c*d^2 + 105*a^3*d^3 - 9*a*b^2*d*(6*c^2 - 25*d^2) + b^3*(8*c^3 + 39*c*d^2))*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(315*d^2*f) + (2*b*(54*a*b*c*d - 189*a^2*d^2 - b^2*(8*c^2 + 49*d^2))*cos(e + f*x)*(c + d*sin(e + f*x))^(3/2))/(315*d^2*f) + (8*b^2*(b*c - 5*a*d)*cos(e + f*x)*(c + d*sin(e + f*x))^(5/2))/(63*d^2*f) - (2*b^2*cos(e + f*x)*(a + b*sin(e + f*x))*(c + d*sin(e + f*x))^(5/2))/(9*d*f) + (2*(420*a^3*c*d^3 + 189*a^2*b*d^2*(c^2 + 3*d^2) - a*b^2*(54*c^3*d - 738*c*d^3) + b^3*(8*c^4 + 33*c^2*d^2 + 147*d^4))*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(315*d^3*f*sqrt((c + d*sin(e + f*x))/(c + d))) - (2*(c^2 - d^2)*(189*a^2*b*c*d^2 + 105*a^3*d^3 - 9*a*b^2*d*(6*c^2 - 25*d^2) + b^3*(8*c^3 + 39*c*d^2))*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(315*d^3*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + b*sin(e + f*x))^3*(c + d*sin(e + f*x))^(1/2), x, 8, (2*b*(42*a*b*c*d - 105*a^2*d^2 - b^2*(8*c^2 + 25*d^2))*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(105*d^2*f) + (8*b^2*(b*c - 4*a*d)*cos(e + f*x)*(c + d*sin(e + f*x))^(3/2))/(35*d^2*f) - (2*b^2*cos(e + f*x)*(a + b*sin(e + f*x))*(c + d*sin(e + f*x))^(3/2))/(7*d*f) + (2*(105*a^2*b*c*d^2 + 105*a^3*d^3 - 21*a*b^2*d*(2*c^2 - 9*d^2) + b^3*(8*c^3 + 19*c*d^2))*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(105*d^3*f*sqrt((c + d*sin(e + f*x))/(c + d))) + (2*b*(c^2 - d^2)*(42*a*b*c*d - 105*a^2*d^2 - b^2*(8*c^2 + 25*d^2))*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(105*d^3*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + b*sin(e + f*x))^3/(c + d*sin(e + f*x))^(1/2), x, 7, (8*b^2*(b*c - 3*a*d)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(15*d^2*f) - (2*b^2*cos(e + f*x)*(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))/(5*d*f) - (2*b*(30*a*b*c*d - 45*a^2*d^2 - b^2*(8*c^2 + 9*d^2))*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(15*d^3*f*sqrt((c + d*sin(e + f*x))/(c + d))) - (2*(45*a^2*b*c*d^2 - 15*a^3*d^3 - 15*a*b^2*d*(2*c^2 + d^2) + b^3*(8*c^3 + 7*c*d^2))*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(15*d^3*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + b*sin(e + f*x))^3/(c + d*sin(e + f*x))^(3/2), x, 7, (2*(b*c - a*d)^2*cos(e + f*x)*(a + b*sin(e + f*x)))/(d*(c^2 - d^2)*f*sqrt(c + d*sin(e + f*x))) + (2*b*(6*a*b*c*d - 3*a^2*d^2 - b^2*(4*c^2 - d^2))*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(3*d^2*(c^2 - d^2)*f) - (2*(9*a^2*b*c*d^2 - 3*a^3*d^3 - 9*a*b^2*d*(2*c^2 - d^2) + b^3*(8*c^3 - 5*c*d^2))*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(3*d^3*(c^2 - d^2)*f*sqrt((c + d*sin(e + f*x))/(c + d))) - (2*b*(18*a*b*c*d - 9*a^2*d^2 - b^2*(8*c^2 + d^2))*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(3*d^3*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + b*sin(e + f*x))^3/(c + d*sin(e + f*x))^(5/2), x, 7, (2*(b*c - a*d)^2*cos(e + f*x)*(a + b*sin(e + f*x)))/(3*d*(c^2 - d^2)*f*(c + d*sin(e + f*x))^(3/2)) + (8*(b*c - a*d)^2*(a*c*d + b*(c^2 - 2*d^2))*cos(e + f*x))/(3*d^2*(c^2 - d^2)^2*f*sqrt(c + d*sin(e + f*x))) + (2*(4*a^3*c*d^3 - 6*a*b^2*c*d*(c^2 - 3*d^2) - 3*a^2*b*d^2*(c^2 + 3*d^2) + b^3*(8*c^4 - 15*c^2*d^2 + 3*d^4))*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(3*d^3*(c^2 - d^2)^2*f*sqrt((c + d*sin(e + f*x))/(c + d))) - (2*(b*c - a*d)*(2*a*b*c*d - a^2*d^2 + b^2*(8*c^2 - 9*d^2))*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(3*d^3*(c^2 - d^2)*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + b*sin(e + f*x))^3/(c + d*sin(e + f*x))^(7/2), x, 8, (2*(b*c - a*d)^2*cos(e + f*x)*(a + b*sin(e + f*x)))/(5*d*(c^2 - d^2)*f*(c + d*sin(e + f*x))^(5/2)) + (8*(b*c - a*d)^2*(2*a*c*d + b*(c^2 - 3*d^2))*cos(e + f*x))/(15*d^2*(c^2 - d^2)^2*f*(c + d*sin(e + f*x))^(3/2)) - (2*(b*c - a*d)*(a^2*d^2*(23*c^2 + 9*d^2) + 2*a*b*d*(7*c^3 - 39*c*d^2) + b^2*(8*c^4 - 21*c^2*d^2 + 45*d^4))*cos(e + f*x))/(15*d^2*(c^2 - d^2)^3*f*sqrt(c + d*sin(e + f*x))) - (2*(b*c - a*d)*(a^2*d^2*(23*c^2 + 9*d^2) + 2*a*b*d*(7*c^3 - 39*c*d^2) + b^2*(8*c^4 - 21*c^2*d^2 + 45*d^4))*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(15*d^3*(c^2 - d^2)^3*f*sqrt((c + d*sin(e + f*x))/(c + d))) - (2*(8*a^3*c*d^3 - 6*a*b^2*c*d*(c^2 - 5*d^2) - 3*a^2*b*d^2*(3*c^2 + 5*d^2) - b^3*(8*c^4 - 15*c^2*d^2 + 15*d^4))*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(15*d^3*(c^2 - d^2)^2*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(a + b*sin(e + f*x))^3/(c + d*sin(e + f*x))^(9/2), x, 9, (2*(b*c - a*d)^2*cos(e + f*x)*(a + b*sin(e + f*x)))/(7*d*(c^2 - d^2)*f*(c + d*sin(e + f*x))^(7/2)) + (8*(b*c - a*d)^2*(3*a*c*d + b*(c^2 - 4*d^2))*cos(e + f*x))/(35*d^2*(c^2 - d^2)^2*f*(c + d*sin(e + f*x))^(5/2)) - (2*(b*c - a*d)*(a^2*d^2*(71*c^2 + 25*d^2) + a*b*(26*c^3*d - 218*c*d^3) + b^2*(8*c^4 - 17*c^2*d^2 + 105*d^4))*cos(e + f*x))/(105*d^2*(c^2 - d^2)^3*f*(c + d*sin(e + f*x))^(3/2)) + (2*(16*a^3*c*d^3*(11*c^2 + 13*d^2) - 6*a*b^2*c*d*(3*c^4 - 62*c^2*d^2 - 133*d^4) - 9*a^2*b*d^2*(5*c^4 + 102*c^2*d^2 + 21*d^4) - b^3*(8*c^6 - 23*c^4*d^2 + 294*c^2*d^4 + 105*d^6))*cos(e + f*x))/(105*d^2*(c^2 - d^2)^4*f*sqrt(c + d*sin(e + f*x))) + (2*(16*a^3*c*d^3*(11*c^2 + 13*d^2) - 6*a*b^2*c*d*(3*c^4 - 62*c^2*d^2 - 133*d^4) - 9*a^2*b*d^2*(5*c^4 + 102*c^2*d^2 + 21*d^4) - b^3*(8*c^6 - 23*c^4*d^2 + 294*c^2*d^4 + 105*d^6))*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(105*d^3*(c^2 - d^2)^4*f*sqrt((c + d*sin(e + f*x))/(c + d))) + (2*(b*c - a*d)*(a^2*d^2*(71*c^2 + 25*d^2) + a*b*(26*c^3*d - 218*c*d^3) + b^2*(8*c^4 - 17*c^2*d^2 + 105*d^4))*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(105*d^3*(c^2 - d^2)^3*f*sqrt(c + d*sin(e + f*x)))]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [(c + d*sin(e + f*x))^(5/2)/(a + b*sin(e + f*x)), x, 9, (-2*d^2*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(3*b*f) + (2*d*(7*b*c - 3*a*d)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(3*b^2*f*sqrt((c + d*sin(e + f*x))/(c + d))) - (2*d*(6*a*b*c*d - 3*a^2*d^2 - b^2*(2*c^2 + d^2))*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(3*b^3*f*sqrt(c + d*sin(e + f*x))) + (2*(b*c - a*d)^3*Elliptic.Pi((2*b)/(a + b), (e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(b^3*(a + b)*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(c + d*sin(e + f*x))^(3/2)/(a + b*sin(e + f*x)), x, 8, (2*d*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(b*f*sqrt((c + d*sin(e + f*x))/(c + d))) + (2*d*(b*c - a*d)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(b^2*f*sqrt(c + d*sin(e + f*x))) + (2*(b*c - a*d)^2*Elliptic.Pi((2*b)/(a + b), (e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(b^2*(a + b)*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(c + d*sin(e + f*x))^(1/2)/(a + b*sin(e + f*x)), x, 5, (2*d*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(b*f*sqrt(c + d*sin(e + f*x))) + (2*(b*c - a*d)*Elliptic.Pi((2*b)/(a + b), (e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(b*(a + b)*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [1/((a + b*sin(e + f*x))*(c + d*sin(e + f*x))^(1/2)), x, 2, (2*Elliptic.Pi((2*b)/(a + b), (e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/((a + b)*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [1/((a + b*sin(e + f*x))*(c + d*sin(e + f*x))^(3/2)), x, 7, (-2*d^2*cos(e + f*x))/((b*c - a*d)*(c^2 - d^2)*f*sqrt(c + d*sin(e + f*x))) - (2*d*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/((b*c - a*d)*(c^2 - d^2)*f*sqrt((c + d*sin(e + f*x))/(c + d))) + (2*b*Elliptic.Pi((2*b)/(a + b), (e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/((a + b)*(b*c - a*d)*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [1/((a + b*sin(e + f*x))*(c + d*sin(e + f*x))^(5/2)), x, 10, (-2*d^2*cos(e + f*x))/(3*(b*c - a*d)*(c^2 - d^2)*f*(c + d*sin(e + f*x))^(3/2)) - (2*d^2*(7*b*c^2 - 4*a*c*d - 3*b*d^2)*cos(e + f*x))/(3*(b*c - a*d)^2*(c^2 - d^2)^2*f*sqrt(c + d*sin(e + f*x))) - (2*d*(7*b*c^2 - 4*a*c*d - 3*b*d^2)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(3*(b*c - a*d)^2*(c^2 - d^2)^2*f*sqrt((c + d*sin(e + f*x))/(c + d))) + (2*d*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(3*(b*c - a*d)*(c^2 - d^2)*f*sqrt(c + d*sin(e + f*x))) + (2*b^2*Elliptic.Pi((2*b)/(a + b), (e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/((a + b)*(b*c - a*d)^2*f*sqrt(c + d*sin(e + f*x)))]

    @test_int [(c + d*sin(e + f*x))^(7/2)/(a + b*sin(e + f*x))^2, x, 10, (d*(6*a*b*c*d - 5*a^2*d^2 - b^2*(3*c^2 - 2*d^2))*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(3*b^2*(a^2 - b^2)*f) + ((b*c - a*d)^2*cos(e + f*x)*(c + d*sin(e + f*x))^(3/2))/(b*(a^2 - b^2)*f*(a + b*sin(e + f*x))) + ((29*a^2*b*c*d^2 - 15*a^3*d^3 + b^3*(3*c^3 - 20*c*d^2) - a*b^2*(9*c^2*d - 12*d^3))*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(3*b^3*(a^2 - b^2)*f*sqrt((c + d*sin(e + f*x))/(c + d))) - ((24*a^3*b*c*d^3 - 15*a^4*d^4 - 12*a*b^3*c*d*(c^2 + 3*d^2) + 2*a^2*b^2*d^2*(c^2 + 8*d^2) + b^4*(3*c^4 + 16*c^2*d^2 + 2*d^4))*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(3*b^4*(a^2 - b^2)*f*sqrt(c + d*sin(e + f*x))) + ((b*c - a*d)^3*(2*a*b*c + 5*a^2*d - 7*b^2*d)*Elliptic.Pi((2*b)/(a + b), (e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/((a - b)*b^4*(a + b)^2*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(c + d*sin(e + f*x))^(5/2)/(a + b*sin(e + f*x))^2, x, 9, ((b*c - a*d)^2*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(b*(a^2 - b^2)*f*(a + b*sin(e + f*x))) - ((2*a*b*c*d - 3*a^2*d^2 - b^2*(c^2 - 2*d^2))*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(b^2*(a^2 - b^2)*f*sqrt((c + d*sin(e + f*x))/(c + d))) + ((b*c - a*d)*(2*a*b*c*d + 3*a^2*d^2 - b^2*(c^2 + 4*d^2))*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(b^3*(a^2 - b^2)*f*sqrt(c + d*sin(e + f*x))) + ((b*c - a*d)^2*(2*a*b*c + 3*a^2*d - 5*b^2*d)*Elliptic.Pi((2*b)/(a + b), (e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/((a - b)*b^3*(a + b)^2*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(c + d*sin(e + f*x))^(3/2)/(a + b*sin(e + f*x))^2, x, 9, ((b*c - a*d)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/((a^2 - b^2)*f*(a + b*sin(e + f*x))) + ((b*c - a*d)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(b*(a^2 - b^2)*f*sqrt((c + d*sin(e + f*x))/(c + d))) + ((2*a*b*c*d + a^2*d^2 - b^2*(c^2 + 2*d^2))*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(b^2*(a^2 - b^2)*f*sqrt(c + d*sin(e + f*x))) + ((b*c - a*d)*(2*a*b*c + a^2*d - 3*b^2*d)*Elliptic.Pi((2*b)/(a + b), (e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/((a - b)*b^2*(a + b)^2*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(c + d*sin(e + f*x))^(1/2)/(a + b*sin(e + f*x))^2, x, 9, (b*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/((a^2 - b^2)*f*(a + b*sin(e + f*x))) + (Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/((a^2 - b^2)*f*sqrt((c + d*sin(e + f*x))/(c + d))) - ((b*c - a*d)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(b*(a^2 - b^2)*f*sqrt(c + d*sin(e + f*x))) + ((2*a*b*c - a^2*d - b^2*d)*Elliptic.Pi((2*b)/(a + b), (e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/((a - b)*b*(a + b)^2*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [1/((a + b*sin(e + f*x))^2*(c + d*sin(e + f*x))^(1/2)), x, 9, (b^2*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/((a^2 - b^2)*(b*c - a*d)*f*(a + b*sin(e + f*x))) + (b*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/((a^2 - b^2)*(b*c - a*d)*f*sqrt((c + d*sin(e + f*x))/(c + d))) - (Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/((a^2 - b^2)*f*sqrt(c + d*sin(e + f*x))) + ((2*a*b*c - 3*a^2*d + b^2*d)*Elliptic.Pi((2*b)/(a + b), (e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/((a - b)*(a + b)^2*(b*c - a*d)*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [1/((a + b*sin(e + f*x))^2*(c + d*sin(e + f*x))^(3/2)), x, 10, (d*(2*a^2*d^2 + b^2*(c^2 - 3*d^2))*cos(e + f*x))/((a^2 - b^2)*(b*c - a*d)^2*(c^2 - d^2)*f*sqrt(c + d*sin(e + f*x))) + (b^2*cos(e + f*x))/((a^2 - b^2)*(b*c - a*d)*f*(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))) + ((2*a^2*d^2 + b^2*(c^2 - 3*d^2))*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/((a^2 - b^2)*(b*c - a*d)^2*(c^2 - d^2)*f*sqrt((c + d*sin(e + f*x))/(c + d))) - (b*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/((a^2 - b^2)*(b*c - a*d)*f*sqrt(c + d*sin(e + f*x))) + (b*(2*a*b*c - 5*a^2*d + 3*b^2*d)*Elliptic.Pi((2*b)/(a + b), (e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/((a - b)*(a + b)^2*(b*c - a*d)^2*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [1/((a + b*sin(e + f*x))^2*(c + d*sin(e + f*x))^(5/2)), x, 11, (d*(2*a^2*d^2 + b^2*(3*c^2 - 5*d^2))*cos(e + f*x))/(3*(a^2 - b^2)*(b*c - a*d)^2*(c^2 - d^2)*f*(c + d*sin(e + f*x))^(3/2)) + (b^2*cos(e + f*x))/((a^2 - b^2)*(b*c - a*d)*f*(a + b*sin(e + f*x))*(c + d*sin(e + f*x))^(3/2)) - ((8*a^3*c*d^4 - 8*a*b^2*c*d^4 - 4*a^2*b*d^3*(5*c^2 - 3*d^2) - b^3*(3*c^4*d - 26*c^2*d^3 + 15*d^5))*cos(e + f*x))/(3*(a^2 - b^2)*(b*c - a*d)^3*(c^2 - d^2)^2*f*sqrt(c + d*sin(e + f*x))) - ((8*a^3*c*d^3 - 8*a*b^2*c*d^3 - 4*a^2*b*d^2*(5*c^2 - 3*d^2) - b^3*(3*c^4 - 26*c^2*d^2 + 15*d^4))*Elliptic.E((1/2)*(e - pi/2 + f*x), (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(3*(a^2 - b^2)*(b*c - a*d)^3*(c^2 - d^2)^2*f*sqrt((c + d*sin(e + f*x))/(c + d))) - ((2*a^2*d^2 + b^2*(3*c^2 - 5*d^2))*Elliptic.F((1/2)*(e - pi/2 + f*x), (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(3*(a^2 - b^2)*(b*c - a*d)^2*(c^2 - d^2)*f*sqrt(c + d*sin(e + f*x))) + (b^2*(2*a*b*c - 7*a^2*d + 5*b^2*d)*Elliptic.Pi((2*b)/(a + b), (1/2)*(e - pi/2 + f*x), (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/((a - b)*(a + b)^2*(b*c - a*d)^3*f*sqrt(c + d*sin(e + f*x)))]

    @test_int [(c + d*sin(e + f*x))^(9/2)/(a + b*sin(e + f*x))^3, x, 11, (d*(36*a^3*b*c*d^2 - 35*a^4*d^3 + b^4*d*(45*c^2 - 8*d^2) - 18*a*b^3*c*(c^2 + 5*d^2) + a^2*b^2*d*(9*c^2 + 61*d^2))*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(12*b^3*(a^2 - b^2)^2*f) + ((b*c - a*d)^2*(6*a*b*c + 7*a^2*d - 13*b^2*d)*cos(e + f*x)*(c + d*sin(e + f*x))^(3/2))/(4*b^2*(a^2 - b^2)^2*f*(a + b*sin(e + f*x))) + ((b*c - a*d)^2*cos(e + f*x)*(c + d*sin(e + f*x))^(5/2))/(2*b*(a^2 - b^2)*f*(a + b*sin(e + f*x))^2) + ((185*a^4*b*c*d^3 - 105*a^5*d^4 - b^5*c*d*(51*c^2 - 104*d^2) - 15*a^3*b^2*d^2*(3*c^2 - 13*d^2) - a^2*b^3*c*d*(21*c^2 + 361*d^2) + 9*a*b^4*(2*c^4 + 17*c^2*d^2 - 8*d^4))*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(12*b^4*(a^2 - b^2)^2*f*sqrt((c + d*sin(e + f*x))/(c + d))) - ((150*a^5*b*c*d^4 - 105*a^6*d^5 - 12*a^3*b^3*c*d^2*(4*c^2 + 29*d^2) + a^4*b^2*d^3*(26*c^2 + 223*d^2) - b^6*d*(57*c^4 + 136*c^2*d^2 + 8*d^4) + 6*a*b^5*c*(3*c^4 + 38*c^2*d^2 + 48*d^4) - a^2*b^4*d*(33*c^4 + 70*c^2*d^2 + 128*d^4))*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(12*b^5*(a^2 - b^2)^2*f*sqrt(c + d*sin(e + f*x))) + ((b*c - a*d)^3*(20*a^3*b*c*d - 44*a*b^3*c*d + 35*a^4*d^2 + 2*a^2*b^2*(4*c^2 - 43*d^2) + b^4*(4*c^2 + 63*d^2))*Elliptic.Pi((2*b)/(a + b), (e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(4*(a - b)^2*b^5*(a + b)^3*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(c + d*sin(e + f*x))^(7/2)/(a + b*sin(e + f*x))^3, x, 10, ((b*c - a*d)^2*(6*a*b*c + 5*a^2*d - 11*b^2*d)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(4*b^2*(a^2 - b^2)^2*f*(a + b*sin(e + f*x))) + ((b*c - a*d)^2*cos(e + f*x)*(c + d*sin(e + f*x))^(3/2))/(2*b*(a^2 - b^2)*f*(a + b*sin(e + f*x))^2) - ((8*a^3*b*c*d^2 - 15*a^4*d^3 + b^4*d*(13*c^2 - 8*d^2) - 2*a*b^3*c*(3*c^2 + 13*d^2) + a^2*b^2*d*(5*c^2 + 29*d^2))*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(4*b^3*(a^2 - b^2)^2*f*sqrt((c + d*sin(e + f*x))/(c + d))) + (3*(b*c - a*d)*(4*a^3*b*c*d^2 + 5*a^4*d^3 + a^2*b^2*d*(c^2 - 11*d^2) - 2*a*b^3*c*(c^2 + 5*d^2) + b^4*d*(5*c^2 + 8*d^2))*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(4*b^4*(a^2 - b^2)^2*f*sqrt(c + d*sin(e + f*x))) + ((b*c - a*d)^2*(12*a^3*b*c*d - 36*a*b^3*c*d + 15*a^4*d^2 + 2*a^2*b^2*(4*c^2 - 19*d^2) + b^4*(4*c^2 + 35*d^2))*Elliptic.Pi((2*b)/(a + b), (e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(4*(a - b)^2*b^4*(a + b)^3*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(c + d*sin(e + f*x))^(5/2)/(a + b*sin(e + f*x))^3, x, 10, ((b*c - a*d)^2*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(2*b*(a^2 - b^2)*f*(a + b*sin(e + f*x))^2) + (3*(b*c - a*d)*(2*a*b*c + a^2*d - 3*b^2*d)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(4*b*(a^2 - b^2)^2*f*(a + b*sin(e + f*x))) + (3*(b*c - a*d)*(2*a*b*c + a^2*d - 3*b^2*d)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(4*b^2*(a^2 - b^2)^2*f*sqrt((c + d*sin(e + f*x))/(c + d))) + ((4*a^3*b*c*d^2 + 3*a^4*d^3 + a^2*b^2*d*(7*c^2 - 5*d^2) + b^4*d*(11*c^2 + 8*d^2) - 2*a*b^3*c*(3*c^2 + 11*d^2))*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(4*b^3*(a^2 - b^2)^2*f*sqrt(c + d*sin(e + f*x))) + ((b*c - a*d)*(4*a^3*b*c*d - 28*a*b^3*c*d + 3*a^4*d^2 + 2*a^2*b^2*(4*c^2 - 3*d^2) + b^4*(4*c^2 + 15*d^2))*Elliptic.Pi((2*b)/(a + b), (e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(4*(a - b)^2*b^3*(a + b)^3*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(c + d*sin(e + f*x))^(3/2)/(a + b*sin(e + f*x))^3, x, 10, ((b*c - a*d)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(2*(a^2 - b^2)*f*(a + b*sin(e + f*x))^2) + ((6*a*b*c - a^2*d - 5*b^2*d)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(4*(a^2 - b^2)^2*f*(a + b*sin(e + f*x))) + ((6*a*b*c - a^2*d - 5*b^2*d)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(4*b*(a^2 - b^2)^2*f*sqrt((c + d*sin(e + f*x))/(c + d))) - ((b*c - a*d)*(6*a*b*c + a^2*d - 7*b^2*d)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(4*b^2*(a^2 - b^2)^2*f*sqrt(c + d*sin(e + f*x))) - ((4*a^3*b*c*d + 20*a*b^3*c*d + a^4*d^2 - b^4*(4*c^2 + 3*d^2) - 2*a^2*b^2*(4*c^2 + 5*d^2))*Elliptic.Pi((2*b)/(a + b), (e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(4*(a - b)^2*b^2*(a + b)^3*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [(c + d*sin(e + f*x))^(1/2)/(a + b*sin(e + f*x))^3, x, 10, (b*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(2*(a^2 - b^2)*f*(a + b*sin(e + f*x))^2) + (b*(6*a*b*c - 5*a^2*d - b^2*d)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(4*(a^2 - b^2)^2*(b*c - a*d)*f*(a + b*sin(e + f*x))) + ((6*a*b*c - 5*a^2*d - b^2*d)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(4*(a^2 - b^2)^2*(b*c - a*d)*f*sqrt((c + d*sin(e + f*x))/(c + d))) - (3*(2*a*b*c - a^2*d - b^2*d)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(4*b*(a^2 - b^2)^2*f*sqrt(c + d*sin(e + f*x))) - ((12*a^3*b*c*d + 12*a*b^3*c*d - 3*a^4*d^2 - b^4*(4*c^2 - d^2) - 2*a^2*b^2*(4*c^2 + 5*d^2))*Elliptic.Pi((2*b)/(a + b), (e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(4*(a - b)^2*b*(a + b)^3*(b*c - a*d)*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [1/((a + b*sin(e + f*x))^3*(c + d*sin(e + f*x))^(1/2)), x, 10, (b^2*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(2*(a^2 - b^2)*(b*c - a*d)*f*(a + b*sin(e + f*x))^2) + (3*b^2*(2*a*b*c - 3*a^2*d + b^2*d)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(4*(a^2 - b^2)^2*(b*c - a*d)^2*f*(a + b*sin(e + f*x))) + (3*b*(2*a*b*c - 3*a^2*d + b^2*d)*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(4*(a^2 - b^2)^2*(b*c - a*d)^2*f*sqrt((c + d*sin(e + f*x))/(c + d))) - ((6*a*b*c - 7*a^2*d + b^2*d)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(4*(a^2 - b^2)^2*(b*c - a*d)*f*sqrt(c + d*sin(e + f*x))) - ((20*a^3*b*c*d + 4*a*b^3*c*d - 15*a^4*d^2 - 2*a^2*b^2*(4*c^2 - 3*d^2) - b^4*(4*c^2 + 3*d^2))*Elliptic.Pi((2*b)/(a + b), (e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(4*(a - b)^2*(a + b)^3*(b*c - a*d)^2*f*sqrt(c + d*sin(e + f*x)))]
    @test_int [1/((a + b*sin(e + f*x))^3*(c + d*sin(e + f*x))^(3/2)), x, 11, -(d*(8*a^4*d^3 + a^2*b^2*d*(13*c^2 - 29*d^2) - b^4*d*(7*c^2 - 15*d^2) - 6*a*b^3*c*(c^2 - d^2))*cos(e + f*x))/(4*(a^2 - b^2)^2*(b*c - a*d)^3*(c^2 - d^2)*f*sqrt(c + d*sin(e + f*x))) + (b^2*cos(e + f*x))/(2*(a^2 - b^2)*(b*c - a*d)*f*(a + b*sin(e + f*x))^2*sqrt(c + d*sin(e + f*x))) + (b^2*(6*a*b*c - 11*a^2*d + 5*b^2*d)*cos(e + f*x))/(4*(a^2 - b^2)^2*(b*c - a*d)^2*f*(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))) - ((8*a^4*d^3 + a^2*b^2*d*(13*c^2 - 29*d^2) - b^4*d*(7*c^2 - 15*d^2) - 6*a*b^3*c*(c^2 - d^2))*Elliptic.E((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(4*(a^2 - b^2)^2*(b*c - a*d)^3*(c^2 - d^2)*f*sqrt((c + d*sin(e + f*x))/(c + d))) - (b*(6*a*b*c - 11*a^2*d + 5*b^2*d)*Elliptic.F((e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(4*(a^2 - b^2)^2*(b*c - a*d)^2*f*sqrt(c + d*sin(e + f*x))) - (b*(28*a^3*b*c*d - 4*a*b^3*c*d - 35*a^4*d^2 - 2*a^2*b^2*(4*c^2 - 19*d^2) - b^4*(4*c^2 + 15*d^2))*Elliptic.Pi((2*b)/(a + b), (e - pi/2 + f*x)/2, (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(4*(a - b)^2*(a + b)^3*(b*c - a*d)^3*f*sqrt(c + d*sin(e + f*x)))]
    #= [1/(a + b*sin(e + f*x))^3/(c + d*sin(e + f*x))^(5/2), x, 12, -((d*(8*a^4*d^3 + a^2*b^2*d*(45*c^2 - 61*d^2) - b^4*d*(27*c^2 - 35*d^2) - 18*a*b^3*c*(c^2 - d^2))*cos(e + f*x))/(12*(a^2 - b^2)^2*(b*c - a*d)^3*(c^2 - d^2)*f*(c + d*sin(e + f*x))^(3/2))) + (b^2*cos(e + f*x))/(2*(a^2 - b^2)*(b*c - a*d)*f*(a + b*sin(e + f*x))^2*(c + d*sin(e + f*x))^(3/2)) + (b^2*(6*a*b*c - 13*a^2*d + 7*b^2*d)*cos(e + f*x))/(4*(a^2 - b^2)^2*(b*c - a*d)^2*f*(a + b*sin(e + f*x))*(c + d*sin(e + f*x))^(3/2)) + (d*(32*a^5*c*d^4 - 64*a^3*b^2*c*d^4 - 8*a^4*b*d^3*(13*c^2 - 9*d^2) + 2*a*b^4*c*(9*c^4 - 18*c^2*d^2 + 25*d^4) + b^5*d*(33*c^4 - 170*c^2*d^2 + 105*d^4) - a^2*b^3*d*(51*c^4 - 310*c^2*d^2 + 195*d^4))*cos(e + f*x))/(12*(a^2 - b^2)^2*(b*c - a*d)^4*(c^2 - d^2)^2*f*sqrt(c + d*sin(e + f*x))) - ((32*a^5*c*d^4 - 64*a^3*b^2*c*d^4 - 8*a^4*b*d^3*(13*c^2 - 9*d^2) + 2*a*b^4*c*(9*c^4 - 18*c^2*d^2 + 25*d^4) + b^5*d*(33*c^4 - 170*c^2*d^2 + 105*d^4) - a^2*b^3*d*(51*c^4 - 310*c^2*d^2 + 195*d^4))*Elliptic.E(pi/4 + (1/2)*(-e - f*x), (2*d)/(c + d))*sqrt(c + d*sin(e + f*x)))/(12*(a^2 - b^2)^2*(b*c - a*d)^4*(c^2 - d^2)^2*f*sqrt((c + d*sin(e + f*x))/(c + d))) - ((8*a^4*d^3 + a^2*b^2*d*(45*c^2 - 61*d^2) - b^4*d*(27*c^2 - 35*d^2) - 18*a*b^3*c*(c^2 - d^2))*Elliptic.F(pi/4 + (1/2)*(-e - f*x), (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(12*(a^2 - b^2)^2*(b*c - a*d)^3*(c^2 - d^2)*f*sqrt(c + d*sin(e + f*x))) - (b^2*(36*a^3*b*c*d - 12*a*b^3*c*d - 63*a^4*d^2 - 2*a^2*b^2*(4*c^2 - 43*d^2) - b^4*(4*c^2 + 35*d^2))*Elliptic.Pi((2*b)/(a + b), -(pi/4) + (1/2)*(e + f*x), (2*d)/(c + d))*sqrt((c + d*sin(e + f*x))/(c + d)))/(4*(a - b)^2*(a + b)^3*(b*c - a*d)^4*f*sqrt(c + d*sin(e + f*x)))] =#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+b*sin(e+f*x))^(m/2)*(c+d*sin(e+f*x))^(n/2)=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [(a + b*sin(e + f*x))^(1/2)*(c + d*sin(e + f*x))^(5/2), x, 8, (sqrt(a + b)*(c - d)*sqrt(c + d)*(14*a*b*c*d - 3*a^2*d^2 + b^2*(33*c^2 + 16*d^2))*Elliptic.E(asin((sqrt(a + b)*sqrt(c + d*sin(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sin(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sec(e + f*x)*sqrt(-(((b*c - a*d)*(1 - sin(e + f*x)))/((c + d)*(a + b*sin(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sin(e + f*x)))/((c - d)*(a + b*sin(e + f*x))))*(a + b*sin(e + f*x)))/(24*b^2*(b*c - a*d)*f) - (1/(8*b^3*sqrt(a + b)*d*f))*(sqrt(c + d)*(5*a^2*b*c*d^2 - a^3*d^3 - a*b^2*d*(15*c^2 + 4*d^2) - 5*b^3*(c^3 + 4*c*d^2))*Elliptic.Pi((b*(c + d))/((a + b)*d), asin((sqrt(a + b)*sqrt(c + d*sin(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sin(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sec(e + f*x)*sqrt(-(((b*c - a*d)*(1 - sin(e + f*x)))/((c + d)*(a + b*sin(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sin(e + f*x)))/((c - d)*(a + b*sin(e + f*x))))*(a + b*sin(e + f*x))) - ((14*a*b*c*d - 3*a^2*d^2 + b^2*(33*c^2 + 16*d^2))*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(24*b*f*sqrt(a + b*sin(e + f*x))) - (d*(13*b*c - 3*a*d)*cos(e + f*x)*sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))/(12*b*f) - (d^2*cos(e + f*x)*(a + b*sin(e + f*x))^(3/2)*sqrt(c + d*sin(e + f*x)))/(3*b*f) + ((a + b)^(3/2)*(3*a^2*d^2 - 6*a*b*d*(2*c + d) + b^2*(33*c^2 + 26*c*d + 16*d^2))*Elliptic.F(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/(24*b^3*sqrt(c + d)*f)]
    @test_int [(a + b*sin(e + f*x))^(1/2)*(c + d*sin(e + f*x))^(3/2), x, 8, (sqrt(a + b)*(c - d)*sqrt(c + d)*(5*b*c + a*d)*Elliptic.E(asin((sqrt(a + b)*sqrt(c + d*sin(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sin(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sec(e + f*x)*sqrt(-(((b*c - a*d)*(1 - sin(e + f*x)))/((c + d)*(a + b*sin(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sin(e + f*x)))/((c - d)*(a + b*sin(e + f*x))))*(a + b*sin(e + f*x)))/(4*b*(b*c - a*d)*f) + (sqrt(c + d)*(6*a*b*c*d - a^2*d^2 + b^2*(3*c^2 + 4*d^2))*Elliptic.Pi((b*(c + d))/((a + b)*d), asin((sqrt(a + b)*sqrt(c + d*sin(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sin(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sec(e + f*x)*sqrt(-(((b*c - a*d)*(1 - sin(e + f*x)))/((c + d)*(a + b*sin(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sin(e + f*x)))/((c - d)*(a + b*sin(e + f*x))))*(a + b*sin(e + f*x)))/(4*b^2*sqrt(a + b)*d*f) + ((b*c - a*d)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(2*f*sqrt(a + b*sin(e + f*x))) - ((5*b*c + a*d)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(4*f*sqrt(a + b*sin(e + f*x))) + ((a + b)^(3/2)*(5*b*c - a*d + 2*b*d)*Elliptic.F(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/(4*b^2*sqrt(c + d)*f) - (b*cos(e + f*x)*(c + d*sin(e + f*x))^(3/2))/(2*f*sqrt(a + b*sin(e + f*x)))]
    @test_int [(a + b*sin(e + f*x))^(1/2)*(c + d*sin(e + f*x))^(1/2), x, 7, (sqrt(a + b)*(c - d)*sqrt(c + d)*Elliptic.E(asin((sqrt(a + b)*sqrt(c + d*sin(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sin(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sec(e + f*x)*sqrt(-(((b*c - a*d)*(1 - sin(e + f*x)))/((c + d)*(a + b*sin(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sin(e + f*x)))/((c - d)*(a + b*sin(e + f*x))))*(a + b*sin(e + f*x)))/((b*c - a*d)*f) + (sqrt(c + d)*(b*c + a*d)*Elliptic.Pi((b*(c + d))/((a + b)*d), asin((sqrt(a + b)*sqrt(c + d*sin(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sin(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sec(e + f*x)*sqrt(-(((b*c - a*d)*(1 - sin(e + f*x)))/((c + d)*(a + b*sin(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sin(e + f*x)))/((c - d)*(a + b*sin(e + f*x))))*(a + b*sin(e + f*x)))/(b*sqrt(a + b)*d*f) - (b*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(f*sqrt(a + b*sin(e + f*x))) + ((a + b)^(3/2)*Elliptic.F(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/(b*sqrt(c + d)*f)]
    @test_int [(a + b*sin(e + f*x))^(1/2)/(c + d*sin(e + f*x))^(1/2), x, 1, (2*sqrt(c + d)*Elliptic.Pi((b*(c + d))/((a + b)*d), asin((sqrt(a + b)*sqrt(c + d*sin(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sin(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sec(e + f*x)*sqrt(-(((b*c - a*d)*(1 - sin(e + f*x)))/((c + d)*(a + b*sin(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sin(e + f*x)))/((c - d)*(a + b*sin(e + f*x))))*(a + b*sin(e + f*x)))/(sqrt(a + b)*d*f)]
    @test_int [(a + b*sin(e + f*x))^(1/2)/(c + d*sin(e + f*x))^(3/2), x, 3, -((2*(a - b)*sqrt(a + b)*Elliptic.E(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/((c - d)*sqrt(c + d)*(b*c - a*d)*f)) + (2*(a - b)*sqrt(a + b)*Elliptic.F(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/((c - d)*sqrt(c + d)*(b*c - a*d)*f)]
    @test_int [(a + b*sin(e + f*x))^(1/2)/(c + d*sin(e + f*x))^(5/2), x, 4, (2*d*cos(e + f*x)*sqrt(a + b*sin(e + f*x)))/(3*(c^2 - d^2)*f*(c + d*sin(e + f*x))^(3/2)) + (2*(a - b)*sqrt(a + b)*(4*a*c*d - b*(3*c^2 + d^2))*Elliptic.E(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/(3*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)^2*f) + (2*(a - b)*sqrt(a + b)*(3*c + d)*Elliptic.F(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/(3*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)*f)]

    @test_int [(a + b*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))^(5/2), x, 9, (1/(192*b^2*d*(b*c - a*d)*f))*(sqrt(a + b)*(c - d)*sqrt(c + d)*(57*a^2*b*c*d^2 - 9*a^3*d^3 + a*b^2*d*(337*c^2 + 156*d^2) + b^3*(15*c^3 + 284*c*d^2))*Elliptic.E(asin((sqrt(a + b)*sqrt(c + d*sin(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sin(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sec(e + f*x)*sqrt(-(((b*c - a*d)*(1 - sin(e + f*x)))/((c + d)*(a + b*sin(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sin(e + f*x)))/((c - d)*(a + b*sin(e + f*x))))*(a + b*sin(e + f*x))) - (1/(64*b^3*sqrt(a + b)*d^2*f))*(sqrt(c + d)*(20*a^3*b*c*d^3 - 3*a^4*d^4 - 60*a*b^3*c*d*(c^2 + 4*d^2) - 6*a^2*b^2*d^2*(15*c^2 + 4*d^2) + b^4*(5*c^4 - 120*c^2*d^2 - 48*d^4))*Elliptic.Pi((b*(c + d))/((a + b)*d), asin((sqrt(a + b)*sqrt(c + d*sin(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sin(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sec(e + f*x)*sqrt(-(((b*c - a*d)*(1 - sin(e + f*x)))/((c + d)*(a + b*sin(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sin(e + f*x)))/((c - d)*(a + b*sin(e + f*x))))*(a + b*sin(e + f*x))) - ((57*a^2*b*c*d^2 - 9*a^3*d^3 + a*b^2*d*(337*c^2 + 156*d^2) + b^3*(15*c^3 + 284*c*d^2))*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(192*b*d*f*sqrt(a + b*sin(e + f*x))) - ((54*a*b*c*d - 9*a^2*d^2 + b^2*(59*c^2 + 36*d^2))*cos(e + f*x)*sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))/(96*b*f) - (d*(17*b*c - 3*a*d)*cos(e + f*x)*(a + b*sin(e + f*x))^(3/2)*sqrt(c + d*sin(e + f*x)))/(24*b*f) - (d^2*cos(e + f*x)*(a + b*sin(e + f*x))^(5/2)*sqrt(c + d*sin(e + f*x)))/(4*b*f) + (1/(192*b^3*d*sqrt(c + d)*f))*((a + b)^(3/2)*(9*a^3*d^3 - 3*a^2*b*d^2*(17*c + 6*d) + 3*a*b^2*d*(73*c^2 + 36*c*d + 28*d^2) + b^3*(15*c^3 + 118*c^2*d + 284*c*d^2 + 72*d^3))*Elliptic.F(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))]
    @test_int [(a + b*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))^(3/2), x, 8, (sqrt(a + b)*(c - d)*sqrt(c + d)*(38*a*b*c*d + 3*a^2*d^2 + b^2*(3*c^2 + 16*d^2))*Elliptic.E(asin((sqrt(a + b)*sqrt(c + d*sin(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sin(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sec(e + f*x)*sqrt(-(((b*c - a*d)*(1 - sin(e + f*x)))/((c + d)*(a + b*sin(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sin(e + f*x)))/((c - d)*(a + b*sin(e + f*x))))*(a + b*sin(e + f*x)))/(24*b*d*(b*c - a*d)*f) + (sqrt(c + d)*(b*c + a*d)*(10*a*b*c*d - a^2*d^2 - b^2*(c^2 - 12*d^2))*Elliptic.Pi((b*(c + d))/((a + b)*d), asin((sqrt(a + b)*sqrt(c + d*sin(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sin(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sec(e + f*x)*sqrt(-(((b*c - a*d)*(1 - sin(e + f*x)))/((c + d)*(a + b*sin(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sin(e + f*x)))/((c - d)*(a + b*sin(e + f*x))))*(a + b*sin(e + f*x)))/(8*b^2*sqrt(a + b)*d^2*f) - ((38*a*b*c*d + 3*a^2*d^2 + b^2*(3*c^2 + 16*d^2))*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(24*d*f*sqrt(a + b*sin(e + f*x))) - ((3*b*c + 7*a*d)*cos(e + f*x)*sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))/(12*f) - ((a + b)^(3/2)*(3*a^2*d^2 - 6*a*b*d*(4*c + d) - b^2*(3*c^2 + 14*c*d + 16*d^2))*Elliptic.F(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/(24*b^2*d*sqrt(c + d)*f) - (b*cos(e + f*x)*sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))^(3/2))/(3*f)]
    @test_int [(a + b*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))^(1/2), x, 7, (sqrt(a + b)*(c - d)*sqrt(c + d)*(b*c + 5*a*d)*Elliptic.E(asin((sqrt(a + b)*sqrt(c + d*sin(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sin(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sec(e + f*x)*sqrt(-(((b*c - a*d)*(1 - sin(e + f*x)))/((c + d)*(a + b*sin(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sin(e + f*x)))/((c - d)*(a + b*sin(e + f*x))))*(a + b*sin(e + f*x)))/(4*d*(b*c - a*d)*f) + (sqrt(c + d)*(6*a*b*c*d + 3*a^2*d^2 - b^2*(c^2 - 4*d^2))*Elliptic.Pi((b*(c + d))/((a + b)*d), asin((sqrt(a + b)*sqrt(c + d*sin(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sin(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sec(e + f*x)*sqrt(-(((b*c - a*d)*(1 - sin(e + f*x)))/((c + d)*(a + b*sin(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sin(e + f*x)))/((c - d)*(a + b*sin(e + f*x))))*(a + b*sin(e + f*x)))/(4*b*sqrt(a + b)*d^2*f) - (b*(b*c + 5*a*d)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(4*d*f*sqrt(a + b*sin(e + f*x))) - (b*cos(e + f*x)*sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))/(2*f) + ((a + b)^(3/2)*(3*a*d + b*(c + 2*d))*Elliptic.F(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/(4*b*d*sqrt(c + d)*f)]
    @test_int [(a + b*sin(e + f*x))^(3/2)/(c + d*sin(e + f*x))^(1/2), x, 6, -((b*cos(e + f*x)*sqrt(a + b*sin(e + f*x)))/(f*sqrt(c + d*sin(e + f*x)))) - ((a - b)*b*sqrt(a + b)*sqrt(c + d)*Elliptic.E(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/(d*(b*c - a*d)*f) + (sqrt(a + b)*(b*(c - d) - 2*a*d)*Elliptic.F(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/(d^2*sqrt(c + d)*f) - (sqrt(a + b)*(b*c - 3*a*d)*Elliptic.Pi(((a + b)*d)/(b*(c + d)), asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/(d^2*sqrt(c + d)*f)]
    @test_int [(a + b*sin(e + f*x))^(3/2)/(c + d*sin(e + f*x))^(3/2), x, 5, (2*(a - b)*sqrt(a + b)*Elliptic.E(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/((c - d)*d*sqrt(c + d)*f) - (1/((c - d)*d^2*sqrt(c + d)*f))*2*sqrt(a + b)*(b*(c - 2*d) + a*d)*Elliptic.F(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)) + (2*b*sqrt(a + b)*Elliptic.Pi(((a + b)*d)/(b*(c + d)), asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/(d^2*sqrt(c + d)*f)]
    @test_int [(a + b*sin(e + f*x))^(3/2)/(c + d*sin(e + f*x))^(5/2), x, 4, -((2*(b*c - a*d)*cos(e + f*x)*sqrt(a + b*sin(e + f*x)))/(3*(c^2 - d^2)*f*(c + d*sin(e + f*x))^(3/2))) - (8*(a - b)*sqrt(a + b)*(a*c - b*d)*Elliptic.E(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/(3*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)*f) + (2*(a - b)*sqrt(a + b)*(a*(3*c + d) - b*(c + 3*d))*Elliptic.F(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/(3*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)*f)]

    @test_int [(a + b*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^(5/2), x, 10, (1/(1920*b^2*d^2*(b*c - a*d)*f))*(sqrt(a + b)*(c - d)*sqrt(c + d)*(360*a^3*b*c*d^3 - 45*a^4*d^4 + 2*a^2*b^2*d^2*(1877*c^2 + 846*d^2) + 8*a*b^3*d*(45*c^3 + 791*c*d^2) - b^4*(45*c^4 - 1692*c^2*d^2 - 1024*d^4))*Elliptic.E(asin((sqrt(a + b)*sqrt(c + d*sin(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sin(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sec(e + f*x)*sqrt(-(((b*c - a*d)*(1 - sin(e + f*x)))/((c + d)*(a + b*sin(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sin(e + f*x)))/((c - d)*(a + b*sin(e + f*x))))*(a + b*sin(e + f*x))) - (1/(128*b^3*sqrt(a + b)*d^3*f))*(sqrt(c + d)*(b*c + a*d)*(28*a^3*b*c*d^3 - 3*a^4*d^4 + 28*a*b^3*c*d*(c^2 - 20*d^2) - 2*a^2*b^2*d^2*(89*c^2 + 20*d^2) - b^4*(3*c^4 + 40*c^2*d^2 + 240*d^4))*Elliptic.Pi((b*(c + d))/((a + b)*d), asin((sqrt(a + b)*sqrt(c + d*sin(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sin(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sec(e + f*x)*sqrt(-(((b*c - a*d)*(1 - sin(e + f*x)))/((c + d)*(a + b*sin(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sin(e + f*x)))/((c - d)*(a + b*sin(e + f*x))))*(a + b*sin(e + f*x))) - ((360*a^3*b*c*d^3 - 45*a^4*d^4 + 2*a^2*b^2*d^2*(1877*c^2 + 846*d^2) + 8*a*b^3*d*(45*c^3 + 791*c*d^2) - b^4*(45*c^4 - 1692*c^2*d^2 - 1024*d^4))*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(1920*b*d^2*f*sqrt(a + b*sin(e + f*x))) - ((917*a^2*b*c*d^2 + 15*a^3*d^3 + a*b^2*d*(345*c^2 + 772*d^2) - b^3*(45*c^3 - 516*c*d^2))*cos(e + f*x)*sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))/(960*b*d*f) + (1/(1920*b^3*d^2*sqrt(c + d)*f))*((a + b)^(3/2)*(45*a^4*d^4 - 30*a^3*b*d^3*(11*c + 3*d) + 30*a^2*b^2*d^2*(64*c^2 + 23*c*d + 22*d^2) + 2*a*b^3*d*(165*c^3 + 917*c^2*d + 2392*c*d^2 + 516*d^3) - b^4*(45*c^4 - 30*c^3*d - 1692*c^2*d^2 - 1544*c*d^3 - 1024*d^4))*Elliptic.F(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x))) - ((110*a*b*c*d + 93*a^2*d^2 - b^2*(15*c^2 - 64*d^2))*cos(e + f*x)*sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))^(3/2))/(240*d*f) + (3*b*(b*c - 7*a*d)*cos(e + f*x)*sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))^(5/2))/(40*d*f) - (b^2*cos(e + f*x)*sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))^(7/2))/(5*d*f)]
    @test_int [(a + b*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^(3/2), x, 9, (1/(192*b*d^2*(b*c - a*d)*f))*(sqrt(a + b)*(c - d)*sqrt(c + d)*(337*a^2*b*c*d^2 + 15*a^3*d^3 + a*b^2*d*(57*c^2 + 284*d^2) - b^3*(9*c^3 - 156*c*d^2))*Elliptic.E(asin((sqrt(a + b)*sqrt(c + d*sin(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sin(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sec(e + f*x)*sqrt(-(((b*c - a*d)*(1 - sin(e + f*x)))/((c + d)*(a + b*sin(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sin(e + f*x)))/((c - d)*(a + b*sin(e + f*x))))*(a + b*sin(e + f*x))) + (1/(64*b^2*sqrt(a + b)*d^3*f))*(sqrt(c + d)*(60*a^3*b*c*d^3 - 5*a^4*d^4 - 20*a*b^3*c*d*(c^2 - 12*d^2) + 3*b^4*(c^2 + 4*d^2)^2 + 30*a^2*b^2*d^2*(3*c^2 + 4*d^2))*Elliptic.Pi((b*(c + d))/((a + b)*d), asin((sqrt(a + b)*sqrt(c + d*sin(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sin(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sec(e + f*x)*sqrt(-(((b*c - a*d)*(1 - sin(e + f*x)))/((c + d)*(a + b*sin(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sin(e + f*x)))/((c - d)*(a + b*sin(e + f*x))))*(a + b*sin(e + f*x))) - ((337*a^2*b*c*d^2 + 15*a^3*d^3 + a*b^2*d*(57*c^2 + 284*d^2) - b^3*(9*c^3 - 156*c*d^2))*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(192*d^2*f*sqrt(a + b*sin(e + f*x))) - ((54*a*b*c*d + 59*a^2*d^2 - 9*b^2*(c^2 - 4*d^2))*cos(e + f*x)*sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))/(96*d*f) - (1/(192*b^2*d^2*sqrt(c + d)*f))*((a + b)^(3/2)*(15*a^3*d^3 - 15*a^2*b*d^2*(11*c + 2*d) - a*b^2*d*(51*c^2 + 172*c*d + 212*d^2) + b^3*(9*c^3 - 6*c^2*d - 156*c*d^2 - 72*d^3))*Elliptic.F(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x))) + (b*(3*b*c - 17*a*d)*cos(e + f*x)*sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))^(3/2))/(24*d*f) - (b^2*cos(e + f*x)*sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))^(5/2))/(4*d*f)]
    @test_int [(a + b*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^(1/2), x, 8, (sqrt(a + b)*(c - d)*sqrt(c + d)*(14*a*b*c*d + 33*a^2*d^2 - b^2*(3*c^2 - 16*d^2))*Elliptic.E(asin((sqrt(a + b)*sqrt(c + d*sin(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sin(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sec(e + f*x)*sqrt(-(((b*c - a*d)*(1 - sin(e + f*x)))/((c + d)*(a + b*sin(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sin(e + f*x)))/((c - d)*(a + b*sin(e + f*x))))*(a + b*sin(e + f*x)))/(24*d^2*(b*c - a*d)*f) + (sqrt(c + d)*(15*a^2*b*c*d^2 + 5*a^3*d^3 - 5*a*b^2*d*(c^2 - 4*d^2) + b^3*(c^3 + 4*c*d^2))*Elliptic.Pi((b*(c + d))/((a + b)*d), asin((sqrt(a + b)*sqrt(c + d*sin(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sin(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sec(e + f*x)*sqrt(-(((b*c - a*d)*(1 - sin(e + f*x)))/((c + d)*(a + b*sin(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sin(e + f*x)))/((c - d)*(a + b*sin(e + f*x))))*(a + b*sin(e + f*x)))/(8*b*sqrt(a + b)*d^3*f) - (b*(14*a*b*c*d + 33*a^2*d^2 - b^2*(3*c^2 - 16*d^2))*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(24*d^2*f*sqrt(a + b*sin(e + f*x))) + (b*(3*b*c - 13*a*d)*cos(e + f*x)*sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))/(12*d*f) + ((a + b)^(3/2)*(15*a^2*d^2 + 6*a*b*d*(2*c + 3*d) - b^2*(3*c^2 - 2*c*d - 16*d^2))*Elliptic.F(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/(24*b*d^2*sqrt(c + d)*f) - (b^2*cos(e + f*x)*sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))^(3/2))/(3*d*f)]
    @test_int [(a + b*sin(e + f*x))^(5/2)/(c + d*sin(e + f*x))^(1/2), x, 7, -((3*b*sqrt(a + b)*(c - d)*sqrt(c + d)*(b*c - 3*a*d)*Elliptic.E(asin((sqrt(a + b)*sqrt(c + d*sin(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sin(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sec(e + f*x)*sqrt(-(((b*c - a*d)*(1 - sin(e + f*x)))/((c + d)*(a + b*sin(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sin(e + f*x)))/((c - d)*(a + b*sin(e + f*x))))*(a + b*sin(e + f*x)))/(4*d^2*(b*c - a*d)*f)) - (sqrt(c + d)*(10*a*b*c*d - 15*a^2*d^2 - b^2*(3*c^2 + 4*d^2))*Elliptic.Pi((b*(c + d))/((a + b)*d), asin((sqrt(a + b)*sqrt(c + d*sin(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sin(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sec(e + f*x)*sqrt(-(((b*c - a*d)*(1 - sin(e + f*x)))/((c + d)*(a + b*sin(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sin(e + f*x)))/((c - d)*(a + b*sin(e + f*x))))*(a + b*sin(e + f*x)))/(4*sqrt(a + b)*d^3*f) + (3*b^2*(b*c - 3*a*d)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(4*d^2*f*sqrt(a + b*sin(e + f*x))) - (b^2*cos(e + f*x)*sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))/(2*d*f) - ((a + b)^(3/2)*(3*b*c - 7*a*d - 2*b*d)*Elliptic.F(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/(4*d^2*sqrt(c + d)*f)]
    @test_int [(a + b*sin(e + f*x))^(5/2)/(c + d*sin(e + f*x))^(3/2), x, 7, -((sqrt(a + b)*(4*a*b*c*d - 2*a^2*d^2 - b^2*(3*c^2 - d^2))*Elliptic.E(asin((sqrt(a + b)*sqrt(c + d*sin(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sin(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sec(e + f*x)*sqrt(-(((b*c - a*d)*(1 - sin(e + f*x)))/((c + d)*(a + b*sin(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sin(e + f*x)))/((c - d)*(a + b*sin(e + f*x))))*(a + b*sin(e + f*x)))/(d^2*sqrt(c + d)*(b*c - a*d)*f)) - (b*sqrt(c + d)*(3*b*c - 5*a*d)*Elliptic.Pi((b*(c + d))/((a + b)*d), asin((sqrt(a + b)*sqrt(c + d*sin(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sin(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sec(e + f*x)*sqrt(-(((b*c - a*d)*(1 - sin(e + f*x)))/((c + d)*(a + b*sin(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sin(e + f*x)))/((c - d)*(a + b*sin(e + f*x))))*(a + b*sin(e + f*x)))/(sqrt(a + b)*d^3*f) + (2*(b*c - a*d)^2*cos(e + f*x)*sqrt(a + b*sin(e + f*x)))/(d*(c^2 - d^2)*f*sqrt(c + d*sin(e + f*x))) + (b*(4*a*b*c*d - 2*a^2*d^2 - b^2*(3*c^2 - d^2))*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(d^2*(c^2 - d^2)*f*sqrt(a + b*sin(e + f*x))) - ((a + b)^(3/2)*(2*a*d - b*(3*c + d))*Elliptic.F(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/(d^2*(c + d)^(3/2)*f)]
    @test_int [(a + b*sin(e + f*x))^(5/2)/(c + d*sin(e + f*x))^(5/2), x, 6, (2*(b*c - a*d)^2*cos(e + f*x)*sqrt(a + b*sin(e + f*x)))/(3*d*(c^2 - d^2)*f*(c + d*sin(e + f*x))^(3/2)) + (2*(a - b)*sqrt(a + b)*(3*b*c^2 + 4*a*c*d - 7*b*d^2)*Elliptic.E(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/(3*(c - d)^2*d^2*(c + d)^(3/2)*f) - (1/(3*(c - d)^2*d^3*(c + d)^(3/2)*f))*(2*sqrt(a + b)*(a^2*d^2*(3*c + d) + a*b*d*(3*c^2 - 4*c*d - 7*d^2) + b^2*(3*c^3 - 6*c^2*d - 2*c*d^2 + 9*d^3))*Elliptic.F(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x))) + (2*b^2*sqrt(a + b)*Elliptic.Pi(((a + b)*d)/(b*(c + d)), asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/(d^3*sqrt(c + d)*f)]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [1/(a + b*sin(e + f*x))^(1/2)*(c + d*sin(e + f*x))^(5/2), x, 7, (3*sqrt(a + b)*(c - d)*d*sqrt(c + d)*(3*b*c - a*d)*Elliptic.E(asin((sqrt(a + b)*sqrt(c + d*sin(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sin(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sec(e + f*x)*sqrt(-(((b*c - a*d)*(1 - sin(e + f*x)))/((c + d)*(a + b*sin(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sin(e + f*x)))/((c - d)*(a + b*sin(e + f*x))))*(a + b*sin(e + f*x)))/(4*b^2*(b*c - a*d)*f) - (sqrt(c + d)*(10*a*b*c*d - 3*a^2*d^2 - b^2*(15*c^2 + 4*d^2))*Elliptic.Pi((b*(c + d))/((a + b)*d), asin((sqrt(a + b)*sqrt(c + d*sin(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sin(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sec(e + f*x)*sqrt(-(((b*c - a*d)*(1 - sin(e + f*x)))/((c + d)*(a + b*sin(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sin(e + f*x)))/((c - d)*(a + b*sin(e + f*x))))*(a + b*sin(e + f*x)))/(4*b^3*sqrt(a + b)*f) - (3*d*(3*b*c - a*d)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(4*b*f*sqrt(a + b*sin(e + f*x))) - (d^2*cos(e + f*x)*sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))/(2*b*f) + (sqrt(a + b)*(3*a^2*d^2 - a*b*d*(7*c + 3*d) + b^2*(8*c^2 + 9*c*d + 2*d^2))*Elliptic.F(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/(4*b^3*sqrt(c + d)*f)]
    @test_int [1/(a + b*sin(e + f*x))^(1/2)*(c + d*sin(e + f*x))^(3/2), x, 6, (sqrt(a + b)*(c - d)*d*sqrt(c + d)*Elliptic.E(asin((sqrt(a + b)*sqrt(c + d*sin(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sin(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sec(e + f*x)*sqrt(-(((b*c - a*d)*(1 - sin(e + f*x)))/((c + d)*(a + b*sin(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sin(e + f*x)))/((c - d)*(a + b*sin(e + f*x))))*(a + b*sin(e + f*x)))/(b*(b*c - a*d)*f) + (sqrt(c + d)*(3*b*c - a*d)*Elliptic.Pi((b*(c + d))/((a + b)*d), asin((sqrt(a + b)*sqrt(c + d*sin(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sin(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sec(e + f*x)*sqrt(-(((b*c - a*d)*(1 - sin(e + f*x)))/((c + d)*(a + b*sin(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sin(e + f*x)))/((c - d)*(a + b*sin(e + f*x))))*(a + b*sin(e + f*x)))/(b^2*sqrt(a + b)*f) - (d*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(f*sqrt(a + b*sin(e + f*x))) - (sqrt(a + b)*(a*d - b*(2*c + d))*Elliptic.F(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/(b^2*sqrt(c + d)*f)]
    @test_int [1/(a + b*sin(e + f*x))^(1/2)*(c + d*sin(e + f*x))^(1/2), x, 1, (2*sqrt(a + b)*Elliptic.Pi(((a + b)*d)/(b*(c + d)), asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/(b*sqrt(c + d)*f)]
    @test_int [1/(a + b*sin(e + f*x))^(1/2)/(c + d*sin(e + f*x))^(1/2), x, 1, (2*sqrt(a + b)*Elliptic.F(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/(sqrt(c + d)*(b*c - a*d)*f)]
    @test_int [1/(a + b*sin(e + f*x))^(1/2)/(c + d*sin(e + f*x))^(3/2), x, 3, (2*(a - b)*sqrt(a + b)*d*Elliptic.E(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/((c - d)*sqrt(c + d)*(b*c - a*d)^2*f) + (2*sqrt(a + b)*Elliptic.F(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/((c - d)*sqrt(c + d)*(b*c - a*d)*f)]
    @test_int [1/(a + b*sin(e + f*x))^(1/2)/(c + d*sin(e + f*x))^(5/2), x, 4, -((2*d^2*cos(e + f*x)*sqrt(a + b*sin(e + f*x)))/(3*(b*c - a*d)*(c^2 - d^2)*f*(c + d*sin(e + f*x))^(3/2))) - (4*(a - b)*sqrt(a + b)*d*(2*a*c*d - b*(3*c^2 - d^2))*Elliptic.E(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/(3*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)^3*f) - (2*sqrt(a + b)*(a*d*(3*c + d) - b*(3*c^2 + 3*c*d - 2*d^2))*Elliptic.F(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/(3*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)^2*f)]

    @test_int [1/(a + b*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))^(5/2), x, 7, ((c - d)*sqrt(c + d)*(2*b^2*c^2 - 4*a*b*c*d + 3*a^2*d^2 - b^2*d^2)*Elliptic.E(asin((sqrt(a + b)*sqrt(c + d*sin(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sin(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sec(e + f*x)*sqrt(-(((b*c - a*d)*(1 - sin(e + f*x)))/((c + d)*(a + b*sin(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sin(e + f*x)))/((c - d)*(a + b*sin(e + f*x))))*(a + b*sin(e + f*x)))/((a - b)*b^2*sqrt(a + b)*(b*c - a*d)*f) + (d*sqrt(c + d)*(5*b*c - 3*a*d)*Elliptic.Pi((b*(c + d))/((a + b)*d), asin((sqrt(a + b)*sqrt(c + d*sin(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sin(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sec(e + f*x)*sqrt(-(((b*c - a*d)*(1 - sin(e + f*x)))/((c + d)*(a + b*sin(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sin(e + f*x)))/((c - d)*(a + b*sin(e + f*x))))*(a + b*sin(e + f*x)))/(b^3*sqrt(a + b)*f) + (2*(b*c - a*d)^2*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(b*(a^2 - b^2)*f*sqrt(a + b*sin(e + f*x))) + ((4*a*b*c*d - 3*a^2*d^2 - b^2*(2*c^2 - d^2))*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(b*(a^2 - b^2)*f*sqrt(a + b*sin(e + f*x))) - (sqrt(a + b)*(3*a^2*d^2 - 2*a*b*d*(c + 3*d) - b^2*(2*c^2 - 6*c*d - d^2))*Elliptic.F(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/((a - b)*b^3*sqrt(c + d)*f)]
    @test_int [1/(a + b*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))^(3/2), x, 5, (1/((a - b)*b*sqrt(a + b)*f))*2*(c - d)*sqrt(c + d)*Elliptic.E(asin((sqrt(a + b)*sqrt(c + d*sin(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sin(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sec(e + f*x)*sqrt(-(((b*c - a*d)*(1 - sin(e + f*x)))/((c + d)*(a + b*sin(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sin(e + f*x)))/((c - d)*(a + b*sin(e + f*x))))*(a + b*sin(e + f*x)) + (1/(b^2*sqrt(a + b)*f))*2*d*sqrt(c + d)*Elliptic.Pi((b*(c + d))/((a + b)*d), asin((sqrt(a + b)*sqrt(c + d*sin(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sin(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sec(e + f*x)*sqrt(-(((b*c - a*d)*(1 - sin(e + f*x)))/((c + d)*(a + b*sin(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sin(e + f*x)))/((c - d)*(a + b*sin(e + f*x))))*(a + b*sin(e + f*x)) + (1/((a - b)*b^2*sqrt(c + d)*f))*2*sqrt(a + b)*(b*(c - 2*d) + a*d)*Elliptic.F(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x))]
    @test_int [1/(a + b*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))^(1/2), x, 3, (2*(c - d)*sqrt(c + d)*Elliptic.E(asin((sqrt(a + b)*sqrt(c + d*sin(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sin(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sec(e + f*x)*sqrt(-(((b*c - a*d)*(1 - sin(e + f*x)))/((c + d)*(a + b*sin(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sin(e + f*x)))/((c - d)*(a + b*sin(e + f*x))))*(a + b*sin(e + f*x)))/((a - b)*sqrt(a + b)*(b*c - a*d)*f) + (2*sqrt(a + b)*(c - d)*Elliptic.F(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/((a - b)*sqrt(c + d)*(b*c - a*d)*f)]
    @test_int [1/(a + b*sin(e + f*x))^(3/2)/(c + d*sin(e + f*x))^(1/2), x, 3, (2*b*(c - d)*sqrt(c + d)*Elliptic.E(asin((sqrt(a + b)*sqrt(c + d*sin(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sin(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sec(e + f*x)*sqrt(-(((b*c - a*d)*(1 - sin(e + f*x)))/((c + d)*(a + b*sin(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sin(e + f*x)))/((c - d)*(a + b*sin(e + f*x))))*(a + b*sin(e + f*x)))/((a - b)*sqrt(a + b)*(b*c - a*d)^2*f) + (2*sqrt(a + b)*Elliptic.F(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/((a - b)*sqrt(c + d)*(b*c - a*d)*f)]
    @test_int [1/(a + b*sin(e + f*x))^(3/2)/(c + d*sin(e + f*x))^(3/2), x, 4, (2*b^2*cos(e + f*x))/((a^2 - b^2)*(b*c - a*d)*f*sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))) - (2*(a^2*d^2 + b^2*(c^2 - 2*d^2))*Elliptic.E(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/(sqrt(a + b)*(c - d)*sqrt(c + d)*(b*c - a*d)^3*f) + (2*(b*(c - 2*d) - a*d)*Elliptic.F(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/(sqrt(a + b)*(c - d)*sqrt(c + d)*(b*c - a*d)^2*f)]
    @test_int [1/(a + b*sin(e + f*x))^(3/2)/(c + d*sin(e + f*x))^(5/2), x, 5, (2*b^2*cos(e + f*x))/((a^2 - b^2)*(b*c - a*d)*f*sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))^(3/2)) + (2*d*(a^2*d^2 + b^2*(3*c^2 - 4*d^2))*cos(e + f*x)*sqrt(a + b*sin(e + f*x)))/(3*(a^2 - b^2)*(b*c - a*d)^2*(c^2 - d^2)*f*(c + d*sin(e + f*x))^(3/2)) + (2*(4*a^3*c*d^3 - 4*a*b^2*c*d^3 - a^2*b*d^2*(9*c^2 - 5*d^2) - b^3*(3*c^4 - 15*c^2*d^2 + 8*d^4))*Elliptic.E(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/(3*sqrt(a + b)*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)^4*f) + (2*(a^2*d^2*(3*c + d) - 6*a*b*d*(c^2 - d^2) + b^2*(3*c^3 - 9*c^2*d - 6*c*d^2 + 8*d^3))*Elliptic.F(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/(3*sqrt(a + b)*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)^3*f)]

    @test_int [1/(a + b*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^(5/2), x, 6, (2*(c - d)*sqrt(c + d)*(4*a*b*c + 3*a^2*d - 7*b^2*d)*Elliptic.E(asin((sqrt(a + b)*sqrt(c + d*sin(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sin(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sec(e + f*x)*sqrt(-(((b*c - a*d)*(1 - sin(e + f*x)))/((c + d)*(a + b*sin(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sin(e + f*x)))/((c - d)*(a + b*sin(e + f*x))))*(a + b*sin(e + f*x)))/(3*(a - b)^2*b^2*(a + b)^(3/2)*f) + (2*d^2*sqrt(c + d)*Elliptic.Pi((b*(c + d))/((a + b)*d), asin((sqrt(a + b)*sqrt(c + d*sin(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sin(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sec(e + f*x)*sqrt(-(((b*c - a*d)*(1 - sin(e + f*x)))/((c + d)*(a + b*sin(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sin(e + f*x)))/((c - d)*(a + b*sin(e + f*x))))*(a + b*sin(e + f*x)))/(b^3*sqrt(a + b)*f) + (2*(b*c - a*d)^2*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(3*b*(a^2 - b^2)*f*(a + b*sin(e + f*x))^(3/2)) + (2*(3*a^2*b*(c - 2*d)*d + 3*a^3*d^2 + a*b^2*(3*c^2 - 4*c*d - 2*d^2) + b^3*(c^2 - 7*c*d + 9*d^2))*Elliptic.F(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/(3*(a - b)^2*b^3*sqrt(a + b)*sqrt(c + d)*f)]
    @test_int [1/(a + b*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^(3/2), x, 4, (8*(c - d)*sqrt(c + d)*(a*c - b*d)*Elliptic.E(asin((sqrt(a + b)*sqrt(c + d*sin(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sin(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sec(e + f*x)*sqrt(-(((b*c - a*d)*(1 - sin(e + f*x)))/((c + d)*(a + b*sin(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sin(e + f*x)))/((c - d)*(a + b*sin(e + f*x))))*(a + b*sin(e + f*x)))/(3*(a - b)^2*(a + b)^(3/2)*(b*c - a*d)*f) + (2*(b*c - a*d)*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(3*(a^2 - b^2)*f*(a + b*sin(e + f*x))^(3/2)) + (2*(c - d)*(3*a*c + b*c - a*d - 3*b*d)*Elliptic.F(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/(3*(a - b)^2*sqrt(a + b)*sqrt(c + d)*(b*c - a*d)*f)]
    @test_int [1/(a + b*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^(1/2), x, 4, (2*(c - d)*sqrt(c + d)*(4*a*b*c - 3*a^2*d - b^2*d)*Elliptic.E(asin((sqrt(a + b)*sqrt(c + d*sin(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sin(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sec(e + f*x)*sqrt(-(((b*c - a*d)*(1 - sin(e + f*x)))/((c + d)*(a + b*sin(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sin(e + f*x)))/((c - d)*(a + b*sin(e + f*x))))*(a + b*sin(e + f*x)))/(3*(a - b)^2*(a + b)^(3/2)*(b*c - a*d)^2*f) + (2*b*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(3*(a^2 - b^2)*f*(a + b*sin(e + f*x))^(3/2)) + (2*(3*a + b)*(c - d)*Elliptic.F(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/(3*(a - b)^2*sqrt(a + b)*sqrt(c + d)*(b*c - a*d)*f)]
    @test_int [1/(a + b*sin(e + f*x))^(5/2)/(c + d*sin(e + f*x))^(1/2), x, 4, (4*b*(c - d)*sqrt(c + d)*(2*a*b*c - 3*a^2*d + b^2*d)*Elliptic.E(asin((sqrt(a + b)*sqrt(c + d*sin(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sin(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sec(e + f*x)*sqrt(-(((b*c - a*d)*(1 - sin(e + f*x)))/((c + d)*(a + b*sin(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sin(e + f*x)))/((c - d)*(a + b*sin(e + f*x))))*(a + b*sin(e + f*x)))/(3*(a - b)^2*(a + b)^(3/2)*(b*c - a*d)^3*f) + (2*b^2*cos(e + f*x)*sqrt(c + d*sin(e + f*x)))/(3*(a^2 - b^2)*(b*c - a*d)*f*(a + b*sin(e + f*x))^(3/2)) + (2*(3*a*b*(c - d) - 3*a^2*d + b^2*(c + 2*d))*Elliptic.F(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/(3*(a - b)^2*sqrt(a + b)*sqrt(c + d)*(b*c - a*d)^2*f)]
    @test_int [1/(a + b*sin(e + f*x))^(5/2)/(c + d*sin(e + f*x))^(3/2), x, 5, (2*b^2*cos(e + f*x))/(3*(a^2 - b^2)*(b*c - a*d)*f*(a + b*sin(e + f*x))^(3/2)*sqrt(c + d*sin(e + f*x))) + (8*b^2*(a*b*c - 2*a^2*d + b^2*d)*cos(e + f*x))/(3*(a^2 - b^2)^2*(b*c - a*d)^2*f*sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))) + (2*(3*a^4*d^3 - b^4*d*(5*c^2 - 8*d^2) + 3*a^2*b^2*d*(3*c^2 - 5*d^2) - 4*a*b^3*c*(c^2 - d^2))*Elliptic.E(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/(3*sqrt(a + b)*(a^2 - b^2)*(c - d)*sqrt(c + d)*(b*c - a*d)^4*f) - (2*(3*a^2*b*(2*c - 3*d)*d - 3*a^3*d^2 - 3*a*b^2*(c^2 - 2*d^2) + b^3*(c^2 - 6*c*d + 8*d^2))*Elliptic.F(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))/(3*sqrt(a + b)*(a^2 - b^2)*(c - d)*sqrt(c + d)*(b*c - a*d)^3*f)]
    @test_int [1/(a + b*sin(e + f*x))^(5/2)/(c + d*sin(e + f*x))^(5/2), x, 6, (2*b^2*cos(e + f*x))/(3*(a^2 - b^2)*(b*c - a*d)*f*(a + b*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))^(3/2)) + (4*b^2*(2*a*b*c - 5*a^2*d + 3*b^2*d)*cos(e + f*x))/(3*(a^2 - b^2)^2*(b*c - a*d)^2*f*sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))^(3/2)) - (2*d*(a^4*d^3 + a^2*b^2*d*(11*c^2 - 13*d^2) - b^4*d*(7*c^2 - 8*d^2) - 4*a*b^3*c*(c^2 - d^2))*cos(e + f*x)*sqrt(a + b*sin(e + f*x)))/(3*(a^2 - b^2)^2*(b*c - a*d)^3*(c^2 - d^2)*f*(c + d*sin(e + f*x))^(3/2)) - (1/(3*sqrt(a + b)*(a^2 - b^2)*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)^5*f))*(8*(a^5*c*d^4 - 2*a^3*b^2*c*d^4 + a*b^4*c*(c^4 - 2*c^2*d^2 + 2*d^4) + b^5*d*(2*c^4 - 7*c^2*d^2 + 4*d^4) - a^2*b^3*d*(3*c^4 - 12*c^2*d^2 + 7*d^4) - a^4*b*(3*c^2*d^3 - 2*d^5))*Elliptic.E(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x))) - (1/(3*sqrt(a + b)*(a^2 - b^2)*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)^4*f))*(2*(a^4*d^3*(3*c + d) - 9*a^3*b*d^2*(c^2 - d^2) + a^2*b^2*d*(9*c^3 - 18*c^2*d - 15*c*d^2 + 16*d^3) + b^4*(c^4 - 9*c^3*d + 16*c^2*d^2 + 12*c*d^3 - 16*d^4) - 3*a*b^3*(c^4 - 5*c^2*d^2 + 4*d^4))*Elliptic.F(asin((sqrt(c + d)*sqrt(a + b*sin(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sin(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sec(e + f*x)*sqrt(((b*c - a*d)*(1 - sin(e + f*x)))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sin(e + f*x)))/((a - b)*(c + d*sin(e + f*x)))))*(c + d*sin(e + f*x)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+b*sin(e+f*x))^m*(c+d*sin(e+f*x))^n*with*m*and/or*n*symbolic=#

    @test_int [(a + b*sin(e + f*x))^m*(c + d*sin(e + f*x))^n, x, 0, Unintegrable((a + b*sin(e + f*x))^m*(c + d*sin(e + f*x))^n, x)]

    @test_int [(a + b*sin(e + f*x))^m*(c + d*sin(e + f*x))^2, x, 8, -((d^2*cos(e + f*x)*(a + b*sin(e + f*x))^(1 + m))/(b*f*(2 + m))) + (sqrt(2)*(a + b)*d*(a*d - 2*b*c*(2 + m))*AppellF1(1/2, 1/2, -1 - m, 3/2, (1/2)*(1 - sin(e + f*x)), (b*(1 - sin(e + f*x)))/(a + b))*cos(e + f*x)*(a + b*sin(e + f*x))^m)/(((a + b*sin(e + f*x))/(a + b))^m*(b^2*f*(2 + m)*sqrt(1 + sin(e + f*x)))) - (sqrt(2)*(a*d*(a*d - 2*b*c*(2 + m)) + b^2*(d^2*(1 + m) + c^2*(2 + m)))*AppellF1(1/2, 1/2, -m, 3/2, (1/2)*(1 - sin(e + f*x)), (b*(1 - sin(e + f*x)))/(a + b))*cos(e + f*x)*(a + b*sin(e + f*x))^m)/(((a + b*sin(e + f*x))/(a + b))^m*(b^2*f*(2 + m)*sqrt(1 + sin(e + f*x))))]
    @test_int [(a + b*sin(e + f*x))^m*(c + d*sin(e + f*x))^1, x, 7, -((sqrt(2)*(a + b)*d*AppellF1(1/2, 1/2, -1 - m, 3/2, (1/2)*(1 - sin(e + f*x)), (b*(1 - sin(e + f*x)))/(a + b))*cos(e + f*x)*(a + b*sin(e + f*x))^m)/(((a + b*sin(e + f*x))/(a + b))^m*(b*f*sqrt(1 + sin(e + f*x))))) - (sqrt(2)*(b*c - a*d)*AppellF1(1/2, 1/2, -m, 3/2, (1/2)*(1 - sin(e + f*x)), (b*(1 - sin(e + f*x)))/(a + b))*cos(e + f*x)*(a + b*sin(e + f*x))^m)/(((a + b*sin(e + f*x))/(a + b))^m*(b*f*sqrt(1 + sin(e + f*x))))]
    @test_int [(a + b*sin(e + f*x))^m*(c + d*sin(e + f*x))^0, x, 3, -((sqrt(2)*AppellF1(1/2, 1/2, -m, 3/2, (1/2)*(1 - sin(e + f*x)), (b*(1 - sin(e + f*x)))/(a + b))*cos(e + f*x)*(a + b*sin(e + f*x))^m)/(((a + b*sin(e + f*x))/(a + b))^m*(f*sqrt(1 + sin(e + f*x)))))]
    @test_int [(a + b*sin(e + f*x))^m/(c + d*sin(e + f*x))^1, x, 0, Unintegrable((a + b*sin(e + f*x))^m/(c + d*sin(e + f*x)), x)]
    @test_int [(a + b*sin(e + f*x))^m/(c + d*sin(e + f*x))^2, x, 0, Unintegrable((a + b*sin(e + f*x))^m/(c + d*sin(e + f*x))^2, x)]
    @test_int [(a + b*sin(e + f*x))^m/(c + d*sin(e + f*x))^3, x, 0, Unintegrable((a + b*sin(e + f*x))^m/(c + d*sin(e + f*x))^3, x)]

    @test_int [(a + b*sin(e + f*x))^m*(c + d*sin(e + f*x))^(5/2), x, 0, Unintegrable((a + b*sin(e + f*x))^m*(c + d*sin(e + f*x))^(5/2), x)]
    @test_int [(a + b*sin(e + f*x))^m*(c + d*sin(e + f*x))^(3/2), x, 0, Unintegrable((a + b*sin(e + f*x))^m*(c + d*sin(e + f*x))^(3/2), x)]
    @test_int [(a + b*sin(e + f*x))^m*(c + d*sin(e + f*x))^(1/2), x, 0, Unintegrable((a + b*sin(e + f*x))^m*sqrt(c + d*sin(e + f*x)), x)]
    @test_int [(a + b*sin(e + f*x))^m/(c + d*sin(e + f*x))^(1/2), x, 0, Unintegrable((a + b*sin(e + f*x))^m/sqrt(c + d*sin(e + f*x)), x)]
    @test_int [(a + b*sin(e + f*x))^m/(c + d*sin(e + f*x))^(3/2), x, 0, Unintegrable((a + b*sin(e + f*x))^m/(c + d*sin(e + f*x))^(3/2), x)]
    @test_int [(a + b*sin(e + f*x))^m/(c + d*sin(e + f*x))^(5/2), x, 0, Unintegrable((a + b*sin(e + f*x))^m/(c + d*sin(e + f*x))^(5/2), x)]

    #= ::Title::Closed:: =#
    #=Integrands*of*the*form*(a+b*sin(e+f*x))^m*(c*(d*sin(e+f*x))^p)^n=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(a+a*sin(e+f*x))^m*(c*(d*sin(e+f*x))^p)^n=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+a*sin(e+f*x))^m*(d*csc(e+f*x))^n*with*n*symbolic=#

    @test_int [(d*csc(e + f*x))^n*(a + a*sin(e + f*x))^3, x, 8, If(13>=8, (a^3*d^3*(1 - 2*n)*cot(e + f*x)*(d*csc(e + f*x))^(-3 + n))/(f*(1 - n)*(2 - n)) + (d^3*cot(e + f*x)*(d*csc(e + f*x))^(-3 + n)*(a^3 + a^3*csc(e + f*x)))/(f*(1 - n)) + (a^3*d^3*(5 - 4*n)*cos(e + f*x)*(d*csc(e + f*x))^(-3 + n)*HypergeometricFunctions._₂F₁(1/2, (3 - n)/2, (5 - n)/2, sin(e + f*x)^2))/(f*(1 - n)*(3 - n)*sqrt(cos(e + f*x)^2)) + (a^3*d^4*(11 - 4*n)*cos(e + f*x)*(d*csc(e + f*x))^(-4 + n)*HypergeometricFunctions._₂F₁(1/2, (4 - n)/2, (6 - n)/2, sin(e + f*x)^2))/(f*(2 - n)*(4 - n)*sqrt(cos(e + f*x)^2)), (a^3*d^3*(1 - 2*n)*cot(e + f*x)*(d*csc(e + f*x))^(-3 + n))/(f*(2 - 3*n + n^2)) + (d^3*cot(e + f*x)*(d*csc(e + f*x))^(-3 + n)*(a^3 + a^3*csc(e + f*x)))/(f*(1 - n)) + (a^3*d^3*(5 - 4*n)*cos(e + f*x)*(d*csc(e + f*x))^(-3 + n)*HypergeometricFunctions._₂F₁(1/2, (3 - n)/2, (5 - n)/2, sin(e + f*x)^2))/(f*(3 - 4*n + n^2)*sqrt(cos(e + f*x)^2)) + (a^3*d^4*(11 - 4*n)*cos(e + f*x)*(d*csc(e + f*x))^(-4 + n)*HypergeometricFunctions._₂F₁(1/2, (4 - n)/2, (6 - n)/2, sin(e + f*x)^2))/(f*(8 - 6*n + n^2)*sqrt(cos(e + f*x)^2)))]
    @test_int [(d*csc(e + f*x))^n*(a + a*sin(e + f*x))^2, x, 7, If(13>=8, (a^2*d^2*cot(e + f*x)*(d*csc(e + f*x))^(-2 + n))/(f*(1 - n)) + (2*a^2*d^2*cos(e + f*x)*(d*csc(e + f*x))^(-2 + n)*HypergeometricFunctions._₂F₁(1/2, (2 - n)/2, (4 - n)/2, sin(e + f*x)^2))/(f*(2 - n)*sqrt(cos(e + f*x)^2)) + (a^2*d^3*(3 - 2*n)*cos(e + f*x)*(d*csc(e + f*x))^(-3 + n)*HypergeometricFunctions._₂F₁(1/2, (3 - n)/2, (5 - n)/2, sin(e + f*x)^2))/(f*(1 - n)*(3 - n)*sqrt(cos(e + f*x)^2)), (a^2*d^2*cot(e + f*x)*(d*csc(e + f*x))^(-2 + n))/(f*(1 - n)) + (2*a^2*d^2*cos(e + f*x)*(d*csc(e + f*x))^(-2 + n)*HypergeometricFunctions._₂F₁(1/2, (2 - n)/2, (4 - n)/2, sin(e + f*x)^2))/(f*(2 - n)*sqrt(cos(e + f*x)^2)) + (a^2*d^3*(3 - 2*n)*cos(e + f*x)*(d*csc(e + f*x))^(-3 + n)*HypergeometricFunctions._₂F₁(1/2, (3 - n)/2, (5 - n)/2, sin(e + f*x)^2))/(f*(3 - 4*n + n^2)*sqrt(cos(e + f*x)^2)))]
    @test_int [(d*csc(e + f*x))^n*(a + a*sin(e + f*x))^1, x, 6, (a*d*cos(e + f*x)*(d*csc(e + f*x))^(-1 + n)*HypergeometricFunctions._₂F₁(1/2, (1 - n)/2, (3 - n)/2, sin(e + f*x)^2))/(f*(1 - n)*sqrt(cos(e + f*x)^2)) + (a*d^2*cos(e + f*x)*(d*csc(e + f*x))^(-2 + n)*HypergeometricFunctions._₂F₁(1/2, (2 - n)/2, (4 - n)/2, sin(e + f*x)^2))/(f*(2 - n)*sqrt(cos(e + f*x)^2))]
    @test_int [(d*csc(e + f*x))^n/(a + a*sin(e + f*x))^1, x, 7, -((cot(e + f*x)*(d*csc(e + f*x))^n)/(f*(a + a*csc(e + f*x)))) + (d*n*cos(e + f*x)*(d*csc(e + f*x))^(-1 + n)*HypergeometricFunctions._₂F₁(1/2, (1 - n)/2, (3 - n)/2, sin(e + f*x)^2))/(a*f*(1 - n)*sqrt(cos(e + f*x)^2)) + (cos(e + f*x)*(d*csc(e + f*x))^n*HypergeometricFunctions._₂F₁(1/2, -(n/2), (2 - n)/2, sin(e + f*x)^2))/(a*f*sqrt(cos(e + f*x)^2))]
    @test_int [(d*csc(e + f*x))^n/(a + a*sin(e + f*x))^2, x, 8, -((2*n*cot(e + f*x)*(d*csc(e + f*x))^(2 + n))/(3*a^2*d^2*f*(1 + csc(e + f*x)))) + (cot(e + f*x)*(d*csc(e + f*x))^(2 + n))/(3*d^2*f*(a + a*csc(e + f*x))^2) + (2*n*cos(e + f*x)*(d*csc(e + f*x))^(2 + n)*HypergeometricFunctions._₂F₁(1/2, (1/2)*(-2 - n), -(n/2), sin(e + f*x)^2))/(3*a^2*d^2*f*sqrt(cos(e + f*x)^2)) - ((1 + 2*n)*cos(e + f*x)*(d*csc(e + f*x))^(1 + n)*HypergeometricFunctions._₂F₁(1/2, (1/2)*(-1 - n), (1 - n)/2, sin(e + f*x)^2))/(3*a^2*d*f*sqrt(cos(e + f*x)^2))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+a*sin(e+f*x))^m*(c*(d*sin(e+f*x))^p)^n*with*n*and*p*symbolic=#

    @test_int [(a + a*sin(e + f*x))^m*(c*(d*sin(e + f*x))^p)^n, x, 5, -((2^(1/2 + m)*AppellF1(1/2, (-n)*p, 1/2 - m, 3/2, 1 - sin(e + f*x), (1/2)*(1 - sin(e + f*x)))*cos(e + f*x)*(c*(d*sin(e + f*x))^p)^n*(1 + sin(e + f*x))^(-(1/2) - m)*(a + a*sin(e + f*x))^m)/(sin(e + f*x)^(n*p)*f))]

    @test_int [(c*(d*sin(e + f*x))^p)^n*(a + a*sin(e + f*x))^3, x, 7, -((a^3*(7 + 2*n*p)*cos(e + f*x)*sin(e + f*x)*(c*(d*sin(e + f*x))^p)^n)/(f*(2 + n*p)*(3 + n*p))) + (a^3*(5 + 4*n*p)*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, (1/2)*(1 + n*p), (1/2)*(3 + n*p), sin(e + f*x)^2)*sin(e + f*x)*(c*(d*sin(e + f*x))^p)^n)/(f*(1 + n*p)*(2 + n*p)*sqrt(cos(e + f*x)^2)) + (a^3*(11 + 4*n*p)*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, (1/2)*(2 + n*p), (1/2)*(4 + n*p), sin(e + f*x)^2)*sin(e + f*x)^2*(c*(d*sin(e + f*x))^p)^n)/(f*(2 + n*p)*(3 + n*p)*sqrt(cos(e + f*x)^2)) - (cos(e + f*x)*sin(e + f*x)*(c*(d*sin(e + f*x))^p)^n*(a^3 + a^3*sin(e + f*x)))/(f*(3 + n*p))]
    @test_int [(c*(d*sin(e + f*x))^p)^n*(a + a*sin(e + f*x))^2, x, 5, -((a^2*cos(e + f*x)*sin(e + f*x)*(c*(d*sin(e + f*x))^p)^n)/(f*(2 + n*p))) + (a^2*(3 + 2*n*p)*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, (1/2)*(1 + n*p), (1/2)*(3 + n*p), sin(e + f*x)^2)*sin(e + f*x)*(c*(d*sin(e + f*x))^p)^n)/(f*(1 + n*p)*(2 + n*p)*sqrt(cos(e + f*x)^2)) + (2*a^2*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, (1/2)*(2 + n*p), (1/2)*(4 + n*p), sin(e + f*x)^2)*sin(e + f*x)^2*(c*(d*sin(e + f*x))^p)^n)/(f*(2 + n*p)*sqrt(cos(e + f*x)^2))]
    @test_int [(c*(d*sin(e + f*x))^p)^n*(a + a*sin(e + f*x))^1, x, 4, (a*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, (1/2)*(1 + n*p), (1/2)*(3 + n*p), sin(e + f*x)^2)*sin(e + f*x)*(c*(d*sin(e + f*x))^p)^n)/(f*(1 + n*p)*sqrt(cos(e + f*x)^2)) + (a*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, (1/2)*(2 + n*p), (1/2)*(4 + n*p), sin(e + f*x)^2)*sin(e + f*x)^2*(c*(d*sin(e + f*x))^p)^n)/(f*(2 + n*p)*sqrt(cos(e + f*x)^2))]
    @test_int [(c*(d*sin(e + f*x))^p)^n/(a + a*sin(e + f*x))^1, x, 5, (cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, (n*p)/2, (1/2)*(2 + n*p), sin(e + f*x)^2)*(c*(d*sin(e + f*x))^p)^n)/(a*f*sqrt(cos(e + f*x)^2)) - (n*p*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, (1/2)*(1 + n*p), (1/2)*(3 + n*p), sin(e + f*x)^2)*sin(e + f*x)*(c*(d*sin(e + f*x))^p)^n)/(a*f*(1 + n*p)*sqrt(cos(e + f*x)^2)) - (cos(e + f*x)*(c*(d*sin(e + f*x))^p)^n)/(f*(a + a*sin(e + f*x)))]
    @test_int [(c*(d*sin(e + f*x))^p)^n/(a + a*sin(e + f*x))^2, x, 6, -((n*p*(1 - 2*n*p)*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, (1/2)*(1 + n*p), (1/2)*(3 + n*p), sin(e + f*x)^2)*sin(e + f*x)*(c*(d*sin(e + f*x))^p)^n)/(3*a^2*f*(1 + n*p)*sqrt(cos(e + f*x)^2))) + (2*(1 - n^2*p^2)*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, (1/2)*(2 + n*p), (1/2)*(4 + n*p), sin(e + f*x)^2)*sin(e + f*x)^2*(c*(d*sin(e + f*x))^p)^n)/(3*a^2*f*(2 + n*p)*sqrt(cos(e + f*x)^2)) + (2*(1 - n*p)*cos(e + f*x)*sin(e + f*x)*(c*(d*sin(e + f*x))^p)^n)/(3*a^2*f*(1 + sin(e + f*x))) + (cos(e + f*x)*sin(e + f*x)*(c*(d*sin(e + f*x))^p)^n)/(3*f*(a + a*sin(e + f*x))^2)]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(a+b*sin(e+f*x))^m*(c*(d*sin(e+f*x))^p)^n=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+b*sin(e+f*x))^m*(d*csc(e+f*x))^n*with*n*symbolic=#

    @test_int [(d*csc(e + f*x))^n*(a + b*sin(e + f*x))^3, x, 8, If(13>=8, (a^2*b*d^3*(1 - 2*n)*cot(e + f*x)*(d*csc(e + f*x))^(-3 + n))/(f*(1 - n)*(2 - n)) + (a^2*d^3*cot(e + f*x)*(d*csc(e + f*x))^(-3 + n)*(b + a*csc(e + f*x)))/(f*(1 - n)) + (a*d^3*(3*b^2*(1 - n) + a^2*(2 - n))*cos(e + f*x)*(d*csc(e + f*x))^(-3 + n)*HypergeometricFunctions._₂F₁(1/2, (3 - n)/2, (5 - n)/2, sin(e + f*x)^2))/(f*(1 - n)*(3 - n)*sqrt(cos(e + f*x)^2)) + (b*d^4*(b^2*(2 - n) + 3*a^2*(3 - n))*cos(e + f*x)*(d*csc(e + f*x))^(-4 + n)*HypergeometricFunctions._₂F₁(1/2, (4 - n)/2, (6 - n)/2, sin(e + f*x)^2))/(f*(2 - n)*(4 - n)*sqrt(cos(e + f*x)^2)), (a^2*b*d^3*(1 - 2*n)*cot(e + f*x)*(d*csc(e + f*x))^(-3 + n))/(f*(2 - 3*n + n^2)) + (a^2*d^3*cot(e + f*x)*(d*csc(e + f*x))^(-3 + n)*(b + a*csc(e + f*x)))/(f*(1 - n)) + (a*d^3*(3*b^2*(1 - n) + a^2*(2 - n))*cos(e + f*x)*(d*csc(e + f*x))^(-3 + n)*HypergeometricFunctions._₂F₁(1/2, (3 - n)/2, (5 - n)/2, sin(e + f*x)^2))/(f*(3 - 4*n + n^2)*sqrt(cos(e + f*x)^2)) + (b*d^4*(b^2*(2 - n) + 3*a^2*(3 - n))*cos(e + f*x)*(d*csc(e + f*x))^(-4 + n)*HypergeometricFunctions._₂F₁(1/2, (4 - n)/2, (6 - n)/2, sin(e + f*x)^2))/(f*(8 - 6*n + n^2)*sqrt(cos(e + f*x)^2)))]
    @test_int [(d*csc(e + f*x))^n*(a + b*sin(e + f*x))^2, x, 7, If(13>=8, (a^2*d^2*cot(e + f*x)*(d*csc(e + f*x))^(-2 + n))/(f*(1 - n)) + (2*a*b*d^2*cos(e + f*x)*(d*csc(e + f*x))^(-2 + n)*HypergeometricFunctions._₂F₁(1/2, (2 - n)/2, (4 - n)/2, sin(e + f*x)^2))/(f*(2 - n)*sqrt(cos(e + f*x)^2)) + (d^3*(b^2*(1 - n) + a^2*(2 - n))*cos(e + f*x)*(d*csc(e + f*x))^(-3 + n)*HypergeometricFunctions._₂F₁(1/2, (3 - n)/2, (5 - n)/2, sin(e + f*x)^2))/(f*(1 - n)*(3 - n)*sqrt(cos(e + f*x)^2)), (a^2*d^2*cot(e + f*x)*(d*csc(e + f*x))^(-2 + n))/(f*(1 - n)) + (2*a*b*d^2*cos(e + f*x)*(d*csc(e + f*x))^(-2 + n)*HypergeometricFunctions._₂F₁(1/2, (2 - n)/2, (4 - n)/2, sin(e + f*x)^2))/(f*(2 - n)*sqrt(cos(e + f*x)^2)) + (d^3*(b^2*(1 - n) + a^2*(2 - n))*cos(e + f*x)*(d*csc(e + f*x))^(-3 + n)*HypergeometricFunctions._₂F₁(1/2, (3 - n)/2, (5 - n)/2, sin(e + f*x)^2))/(f*(3 - 4*n + n^2)*sqrt(cos(e + f*x)^2)))]
    @test_int [(d*csc(e + f*x))^n*(a + b*sin(e + f*x))^1, x, 6, (a*d*cos(e + f*x)*(d*csc(e + f*x))^(-1 + n)*HypergeometricFunctions._₂F₁(1/2, (1 - n)/2, (3 - n)/2, sin(e + f*x)^2))/(f*(1 - n)*sqrt(cos(e + f*x)^2)) + (b*d^2*cos(e + f*x)*(d*csc(e + f*x))^(-2 + n)*HypergeometricFunctions._₂F₁(1/2, (2 - n)/2, (4 - n)/2, sin(e + f*x)^2))/(f*(2 - n)*sqrt(cos(e + f*x)^2))]
    @test_int [(d*csc(e + f*x))^n/(a + b*sin(e + f*x))^1, x, 7, (b*AppellF1(1/2, n/2, 1, 3/2, cos(e + f*x)^2, -((b^2*cos(e + f*x)^2)/(a^2 - b^2)))*cos(e + f*x)*(d*csc(e + f*x))^(1 + n)*sin(e + f*x)*(sin(e + f*x)^2)^(n/2))/((a^2 - b^2)*d*f) - (a*AppellF1(1/2, (1 + n)/2, 1, 3/2, cos(e + f*x)^2, -((b^2*cos(e + f*x)^2)/(a^2 - b^2)))*cos(e + f*x)*(d*csc(e + f*x))^(1 + n)*(sin(e + f*x)^2)^((1 + n)/2))/((a^2 - b^2)*d*f)]
    @test_int [(d*csc(e + f*x))^n/(a + b*sin(e + f*x))^2, x, 10, -((b^2*AppellF1(1/2, (1/2)*(-1 + n), 2, 3/2, cos(e + f*x)^2, -((b^2*cos(e + f*x)^2)/(a^2 - b^2)))*cos(e + f*x)*(d*csc(e + f*x))^(2 + n)*sin(e + f*x)^3*(sin(e + f*x)^2)^((1/2)*(-1 + n)))/((a^2 - b^2)^2*d^2*f)) - (a^2*AppellF1(1/2, (1 + n)/2, 2, 3/2, cos(e + f*x)^2, -((b^2*cos(e + f*x)^2)/(a^2 - b^2)))*cos(e + f*x)*(d*csc(e + f*x))^(2 + n)*sin(e + f*x)*(sin(e + f*x)^2)^((1 + n)/2))/((a^2 - b^2)^2*d^2*f) + (2*a*b*AppellF1(1/2, n/2, 2, 3/2, cos(e + f*x)^2, -((b^2*cos(e + f*x)^2)/(a^2 - b^2)))*cos(e + f*x)*(d*csc(e + f*x))^(2 + n)*(sin(e + f*x)^2)^((2 + n)/2))/((a^2 - b^2)^2*d^2*f)]
    @test_int [(d*csc(e + f*x))^n/(a + b*sin(e + f*x))^3, x, 12, -((3*a*b^2*AppellF1(1/2, (1/2)*(-1 + n), 3, 3/2, cos(e + f*x)^2, -((b^2*cos(e + f*x)^2)/(a^2 - b^2)))*cos(e + f*x)*(d*csc(e + f*x))^(3 + n)*sin(e + f*x)^4*(sin(e + f*x)^2)^((1/2)*(-1 + n)))/((a^2 - b^2)^3*d^3*f)) + (b^3*AppellF1(1/2, (1/2)*(-2 + n), 3, 3/2, cos(e + f*x)^2, -((b^2*cos(e + f*x)^2)/(a^2 - b^2)))*cos(e + f*x)*(d*csc(e + f*x))^(3 + n)*sin(e + f*x)^3*(sin(e + f*x)^2)^(n/2))/((a^2 - b^2)^3*d^3*f) + (3*a^2*b*AppellF1(1/2, n/2, 3, 3/2, cos(e + f*x)^2, -((b^2*cos(e + f*x)^2)/(a^2 - b^2)))*cos(e + f*x)*(d*csc(e + f*x))^(3 + n)*sin(e + f*x)^3*(sin(e + f*x)^2)^(n/2))/((a^2 - b^2)^3*d^3*f) - (a^3*AppellF1(1/2, (1 + n)/2, 3, 3/2, cos(e + f*x)^2, -((b^2*cos(e + f*x)^2)/(a^2 - b^2)))*cos(e + f*x)*(d*csc(e + f*x))^(3 + n)*(sin(e + f*x)^2)^((3 + n)/2))/((a^2 - b^2)^3*d^3*f)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+b*sin(e+f*x))^m*(c*(d*sin(e+f*x))^p)^n*with*n*and*p*symbolic=#

    @test_int [(a + b*sin(e + f*x))^m*(c*(d*sin(e + f*x))^p)^n, x, 1, ((c*(d*sin(e + f*x))^p)^n*Unintegrable((d*sin(e + f*x))^(n*p)*(a + b*sin(e + f*x))^m, x))/(d*sin(e + f*x))^(n*p)]

    @test_int [(c*(d*sin(e + f*x))^p)^n*(a + b*sin(e + f*x))^3, x, 6, -((a*b^2*(7 + 2*n*p)*cos(e + f*x)*sin(e + f*x)*(c*(d*sin(e + f*x))^p)^n)/(f*(2 + n*p)*(3 + n*p))) + (a*(3*b^2*(1 + n*p) + a^2*(2 + n*p))*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, (1/2)*(1 + n*p), (1/2)*(3 + n*p), sin(e + f*x)^2)*sin(e + f*x)*(c*(d*sin(e + f*x))^p)^n)/(f*(1 + n*p)*(2 + n*p)*sqrt(cos(e + f*x)^2)) + (b*(b^2*(2 + n*p) + 3*a^2*(3 + n*p))*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, (1/2)*(2 + n*p), (1/2)*(4 + n*p), sin(e + f*x)^2)*sin(e + f*x)^2*(c*(d*sin(e + f*x))^p)^n)/(f*(2 + n*p)*(3 + n*p)*sqrt(cos(e + f*x)^2)) - (b^2*cos(e + f*x)*sin(e + f*x)*(c*(d*sin(e + f*x))^p)^n*(a + b*sin(e + f*x)))/(f*(3 + n*p))]
    @test_int [(c*(d*sin(e + f*x))^p)^n*(a + b*sin(e + f*x))^2, x, 5, -((b^2*cos(e + f*x)*sin(e + f*x)*(c*(d*sin(e + f*x))^p)^n)/(f*(2 + n*p))) + ((b^2*(1 + n*p) + a^2*(2 + n*p))*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, (1/2)*(1 + n*p), (1/2)*(3 + n*p), sin(e + f*x)^2)*sin(e + f*x)*(c*(d*sin(e + f*x))^p)^n)/(f*(1 + n*p)*(2 + n*p)*sqrt(cos(e + f*x)^2)) + (2*a*b*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, (1/2)*(2 + n*p), (1/2)*(4 + n*p), sin(e + f*x)^2)*sin(e + f*x)^2*(c*(d*sin(e + f*x))^p)^n)/(f*(2 + n*p)*sqrt(cos(e + f*x)^2))]
    @test_int [(c*(d*sin(e + f*x))^p)^n*(a + b*sin(e + f*x))^1, x, 4, (a*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, (1/2)*(1 + n*p), (1/2)*(3 + n*p), sin(e + f*x)^2)*sin(e + f*x)*(c*(d*sin(e + f*x))^p)^n)/(f*(1 + n*p)*sqrt(cos(e + f*x)^2)) + (b*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, (1/2)*(2 + n*p), (1/2)*(4 + n*p), sin(e + f*x)^2)*sin(e + f*x)^2*(c*(d*sin(e + f*x))^p)^n)/(f*(2 + n*p)*sqrt(cos(e + f*x)^2))]
    @test_int [(c*(d*sin(e + f*x))^p)^n/(a + b*sin(e + f*x))^1, x, 6, (b*AppellF1(1/2, -((n*p)/2), 1, 3/2, cos(e + f*x)^2, -((b^2*cos(e + f*x)^2)/(a^2 - b^2)))*cos(e + f*x)*(c*(d*sin(e + f*x))^p)^n)/((sin(e + f*x)^2)^((n*p)/2)*((a^2 - b^2)*f)) - (a*AppellF1(1/2, (1/2)*(1 - n*p), 1, 3/2, cos(e + f*x)^2, -((b^2*cos(e + f*x)^2)/(a^2 - b^2)))*cot(e + f*x)*(sin(e + f*x)^2)^((1/2)*(1 - n*p))*(c*(d*sin(e + f*x))^p)^n)/((a^2 - b^2)*f)]
    @test_int [(c*(d*sin(e + f*x))^p)^n/(a + b*sin(e + f*x))^2, x, 11, (2*a*b*AppellF1(1/2, -((n*p)/2), 2, 3/2, cos(e + f*x)^2, -((b^2*cos(e + f*x)^2)/(a^2 - b^2)))*cos(e + f*x)*(c*(d*sin(e + f*x))^p)^n)/((sin(e + f*x)^2)^((n*p)/2)*((a^2 - b^2)^2*f)) - (b^2*AppellF1(1/2, (1/2)*(-1 - n*p), 2, 3/2, cos(e + f*x)^2, -((b^2*cos(e + f*x)^2)/(a^2 - b^2)))*cos(e + f*x)*sin(e + f*x)*(sin(e + f*x)^2)^((1/2)*(-1 - n*p))*(c*(d*sin(e + f*x))^p)^n)/((a^2 - b^2)^2*f) - (a^2*AppellF1(1/2, (1/2)*(1 - n*p), 2, 3/2, cos(e + f*x)^2, -((b^2*cos(e + f*x)^2)/(a^2 - b^2)))*cot(e + f*x)*(sin(e + f*x)^2)^((1/2)*(1 - n*p))*(c*(d*sin(e + f*x))^p)^n)/((a^2 - b^2)^2*f)]
    @test_int [(c*(d*sin(e + f*x))^p)^n/(a + b*sin(e + f*x))^3, x, 14, (3*a^2*b*AppellF1(1/2, -((n*p)/2), 3, 3/2, cos(e + f*x)^2, -((b^2*cos(e + f*x)^2)/(a^2 - b^2)))*cos(e + f*x)*(c*(d*sin(e + f*x))^p)^n)/((sin(e + f*x)^2)^((n*p)/2)*((a^2 - b^2)^3*f)) + (b^3*AppellF1(1/2, (1/2)*(-2 - n*p), 3, 3/2, cos(e + f*x)^2, -((b^2*cos(e + f*x)^2)/(a^2 - b^2)))*cos(e + f*x)*(c*(d*sin(e + f*x))^p)^n)/((sin(e + f*x)^2)^((n*p)/2)*((a^2 - b^2)^3*f)) - (3*a*b^2*AppellF1(1/2, (1/2)*(-1 - n*p), 3, 3/2, cos(e + f*x)^2, -((b^2*cos(e + f*x)^2)/(a^2 - b^2)))*cos(e + f*x)*sin(e + f*x)*(sin(e + f*x)^2)^((1/2)*(-1 - n*p))*(c*(d*sin(e + f*x))^p)^n)/((a^2 - b^2)^3*f) - (a^3*AppellF1(1/2, (1/2)*(1 - n*p), 3, 3/2, cos(e + f*x)^2, -((b^2*cos(e + f*x)^2)/(a^2 - b^2)))*cot(e + f*x)*(sin(e + f*x)^2)^((1/2)*(1 - n*p))*(c*(d*sin(e + f*x))^p)^n)/((a^2 - b^2)^3*f)]
    end

max_stars_repo_path: test/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.jl