text:
@testset "4.5.1.3 (d sin)^n (a+b sec)^m" begin
    (a, b, c, d, e, f, m, n, x, ) = @variables a b c d e f m n x

    #= ::Package:: =#

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

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(d*sin(e+f*x))^n*(a+b*sec(e+f*x))^m*when*a^2-b^2=0=#

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

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

    @test_int [(a + a*sec(c + d*x))*sin(c + d*x)^9, x, 5, -((a*cos(c + d*x))/d) + (2*a*cos(c + d*x)^2)/d + (4*a*cos(c + d*x)^3)/(3*d) - (3*a*cos(c + d*x)^4)/(2*d) - (6*a*cos(c + d*x)^5)/(5*d) + (2*a*cos(c + d*x)^6)/(3*d) + (4*a*cos(c + d*x)^7)/(7*d) - (a*cos(c + d*x)^8)/(8*d) - (a*cos(c + d*x)^9)/(9*d) - (a*log(cos(c + d*x)))/d]
    @test_int [(a + a*sec(c + d*x))*sin(c + d*x)^7, x, 5, -((a*cos(c + d*x))/d) + (3*a*cos(c + d*x)^2)/(2*d) + (a*cos(c + d*x)^3)/d - (3*a*cos(c + d*x)^4)/(4*d) - (3*a*cos(c + d*x)^5)/(5*d) + (a*cos(c + d*x)^6)/(6*d) + (a*cos(c + d*x)^7)/(7*d) - (a*log(cos(c + d*x)))/d]
    @test_int [(a + a*sec(c + d*x))*sin(c + d*x)^5, x, 5, -((a*cos(c + d*x))/d) + (a*cos(c + d*x)^2)/d + (2*a*cos(c + d*x)^3)/(3*d) - (a*cos(c + d*x)^4)/(4*d) - (a*cos(c + d*x)^5)/(5*d) - (a*log(cos(c + d*x)))/d]
    @test_int [(a + a*sec(c + d*x))*sin(c + d*x)^3, x, 5, -((a*cos(c + d*x))/d) + (a*cos(c + d*x)^2)/(2*d) + (a*cos(c + d*x)^3)/(3*d) - (a*log(cos(c + d*x)))/d]
    @test_int [(a + a*sec(c + d*x))*sin(c + d*x)^1, x, 4, -((a*cos(c + d*x))/d) - (a*log(cos(c + d*x)))/d]
    @test_int [csc(c + d*x)^1*(a + a*sec(c + d*x)), x, 6, (a*log(1 - cos(c + d*x)))/d - (a*log(cos(c + d*x)))/d]
    @test_int [csc(c + d*x)^3*(a + a*sec(c + d*x)), x, 5, -a^2/(2*d*(a - a*cos(c + d*x))) + (3*a*log(1 - cos(c + d*x)))/(4*d) - (a*log(cos(c + d*x)))/d + (a*log(1 + cos(c + d*x)))/(4*d)]
    @test_int [csc(c + d*x)^5*(a + a*sec(c + d*x)), x, 5, -a^3/(8*d*(a - a*cos(c + d*x))^2) - a^2/(2*d*(a - a*cos(c + d*x))) - a^2/(8*d*(a + a*cos(c + d*x))) + (11*a*log(1 - cos(c + d*x)))/(16*d) - (a*log(cos(c + d*x)))/d + (5*a*log(1 + cos(c + d*x)))/(16*d)]
    @test_int [csc(c + d*x)^7*(a + a*sec(c + d*x)), x, 5, -a^4/(24*d*(a - a*cos(c + d*x))^3) - (5*a^3)/(32*d*(a - a*cos(c + d*x))^2) - a^2/(2*d*(a - a*cos(c + d*x))) - a^3/(32*d*(a + a*cos(c + d*x))^2) - (3*a^2)/(16*d*(a + a*cos(c + d*x))) + (21*a*log(1 - cos(c + d*x)))/(32*d) - (a*log(cos(c + d*x)))/d + (11*a*log(1 + cos(c + d*x)))/(32*d)]

    @test_int [(a + a*sec(c + d*x))*sin(c + d*x)^8, x, 11, (35*a*x)/128 + (a*atanh(sin(c + d*x)))/d - (a*sin(c + d*x))/d - (35*a*cos(c + d*x)*sin(c + d*x))/(128*d) - (a*sin(c + d*x)^3)/(3*d) - (35*a*cos(c + d*x)*sin(c + d*x)^3)/(192*d) - (a*sin(c + d*x)^5)/(5*d) - (7*a*cos(c + d*x)*sin(c + d*x)^5)/(48*d) - (a*sin(c + d*x)^7)/(7*d) - (a*cos(c + d*x)*sin(c + d*x)^7)/(8*d)]
    @test_int [(a + a*sec(c + d*x))*sin(c + d*x)^6, x, 10, (5*a*x)/16 + (a*atanh(sin(c + d*x)))/d - (a*sin(c + d*x))/d - (5*a*cos(c + d*x)*sin(c + d*x))/(16*d) - (a*sin(c + d*x)^3)/(3*d) - (5*a*cos(c + d*x)*sin(c + d*x)^3)/(24*d) - (a*sin(c + d*x)^5)/(5*d) - (a*cos(c + d*x)*sin(c + d*x)^5)/(6*d)]
    @test_int [(a + a*sec(c + d*x))*sin(c + d*x)^4, x, 9, (3*a*x)/8 + (a*atanh(sin(c + d*x)))/d - (a*sin(c + d*x))/d - (3*a*cos(c + d*x)*sin(c + d*x))/(8*d) - (a*sin(c + d*x)^3)/(3*d) - (a*cos(c + d*x)*sin(c + d*x)^3)/(4*d)]
    @test_int [(a + a*sec(c + d*x))*sin(c + d*x)^2, x, 7, (a*x)/2 + (a*atanh(sin(c + d*x)))/d - (a*sin(c + d*x))/d - (a*cos(c + d*x)*sin(c + d*x))/(2*d)]
    @test_int [csc(c + d*x)^2*(a + a*sec(c + d*x)), x, 7, (a*atanh(sin(c + d*x)))/d - (a*cot(c + d*x))/d - (a*csc(c + d*x))/d]
    @test_int [csc(c + d*x)^4*(a + a*sec(c + d*x)), x, 8, (a*atanh(sin(c + d*x)))/d - (a*cot(c + d*x))/d - (a*cot(c + d*x)^3)/(3*d) - (a*csc(c + d*x))/d - (a*csc(c + d*x)^3)/(3*d)]
    @test_int [csc(c + d*x)^6*(a + a*sec(c + d*x)), x, 8, (a*atanh(sin(c + d*x)))/d - (a*cot(c + d*x))/d - (2*a*cot(c + d*x)^3)/(3*d) - (a*cot(c + d*x)^5)/(5*d) - (a*csc(c + d*x))/d - (a*csc(c + d*x)^3)/(3*d) - (a*csc(c + d*x)^5)/(5*d)]
    @test_int [csc(c + d*x)^8*(a + a*sec(c + d*x)), x, 8, (a*atanh(sin(c + d*x)))/d - (a*cot(c + d*x))/d - (a*cot(c + d*x)^3)/d - (3*a*cot(c + d*x)^5)/(5*d) - (a*cot(c + d*x)^7)/(7*d) - (a*csc(c + d*x))/d - (a*csc(c + d*x)^3)/(3*d) - (a*csc(c + d*x)^5)/(5*d) - (a*csc(c + d*x)^7)/(7*d)]
    @test_int [csc(c + d*x)^10*(a + a*sec(c + d*x)), x, 8, (a*atanh(sin(c + d*x)))/d - (a*cot(c + d*x))/d - (4*a*cot(c + d*x)^3)/(3*d) - (6*a*cot(c + d*x)^5)/(5*d) - (4*a*cot(c + d*x)^7)/(7*d) - (a*cot(c + d*x)^9)/(9*d) - (a*csc(c + d*x))/d - (a*csc(c + d*x)^3)/(3*d) - (a*csc(c + d*x)^5)/(5*d) - (a*csc(c + d*x)^7)/(7*d) - (a*csc(c + d*x)^9)/(9*d)]

    @test_int [(a + a*sec(c + d*x))^2*sin(c + d*x)^9, x, 5, (3*a^2*cos(c + d*x))/d + (4*a^2*cos(c + d*x)^2)/d - (2*a^2*cos(c + d*x)^3)/(3*d) - (3*a^2*cos(c + d*x)^4)/d - (2*a^2*cos(c + d*x)^5)/(5*d) + (4*a^2*cos(c + d*x)^6)/(3*d) + (3*a^2*cos(c + d*x)^7)/(7*d) - (a^2*cos(c + d*x)^8)/(4*d) - (a^2*cos(c + d*x)^9)/(9*d) - (2*a^2*log(cos(c + d*x)))/d + (a^2*sec(c + d*x))/d]
    @test_int [(a + a*sec(c + d*x))^2*sin(c + d*x)^7, x, 5, (2*a^2*cos(c + d*x))/d + (3*a^2*cos(c + d*x)^2)/d - (3*a^2*cos(c + d*x)^4)/(2*d) - (2*a^2*cos(c + d*x)^5)/(5*d) + (a^2*cos(c + d*x)^6)/(3*d) + (a^2*cos(c + d*x)^7)/(7*d) - (2*a^2*log(cos(c + d*x)))/d + (a^2*sec(c + d*x))/d]
    @test_int [(a + a*sec(c + d*x))^2*sin(c + d*x)^5, x, 5, (a^2*cos(c + d*x))/d + (2*a^2*cos(c + d*x)^2)/d + (a^2*cos(c + d*x)^3)/(3*d) - (a^2*cos(c + d*x)^4)/(2*d) - (a^2*cos(c + d*x)^5)/(5*d) - (2*a^2*log(cos(c + d*x)))/d + (a^2*sec(c + d*x))/d]
    @test_int [(a + a*sec(c + d*x))^2*sin(c + d*x)^3, x, 5, (a^2*cos(c + d*x)^2)/d + (a^2*cos(c + d*x)^3)/(3*d) - (2*a^2*log(cos(c + d*x)))/d + (a^2*sec(c + d*x))/d]
    @test_int [(a + a*sec(c + d*x))^2*sin(c + d*x)^1, x, 5, -((a^2*cos(c + d*x))/d) - (2*a^2*log(cos(c + d*x)))/d + (a^2*sec(c + d*x))/d]
    @test_int [csc(c + d*x)^1*(a + a*sec(c + d*x))^2, x, 5, (2*a^2*log(1 - cos(c + d*x)))/d - (2*a^2*log(cos(c + d*x)))/d + (a^2*sec(c + d*x))/d]
    @test_int [csc(c + d*x)^3*(a + a*sec(c + d*x))^2, x, 5, -(a^3/(d*(a - a*cos(c + d*x)))) + (2*a^2*log(1 - cos(c + d*x)))/d - (2*a^2*log(cos(c + d*x)))/d + (a^2*sec(c + d*x))/d]
    @test_int [csc(c + d*x)^5*(a + a*sec(c + d*x))^2, x, 5, -a^4/(4*d*(a - a*cos(c + d*x))^2) - (5*a^3)/(4*d*(a - a*cos(c + d*x))) + (17*a^2*log(1 - cos(c + d*x)))/(8*d) - (2*a^2*log(cos(c + d*x)))/d - (a^2*log(1 + cos(c + d*x)))/(8*d) + (a^2*sec(c + d*x))/d]
    @test_int [csc(c + d*x)^7*(a + a*sec(c + d*x))^2, x, 5, -a^5/(12*d*(a - a*cos(c + d*x))^3) - (3*a^4)/(8*d*(a - a*cos(c + d*x))^2) - (23*a^3)/(16*d*(a - a*cos(c + d*x))) + a^3/(16*d*(a + a*cos(c + d*x))) + (9*a^2*log(1 - cos(c + d*x)))/(4*d) - (2*a^2*log(cos(c + d*x)))/d - (a^2*log(1 + cos(c + d*x)))/(4*d) + (a^2*sec(c + d*x))/d]
    @test_int [csc(c + d*x)^9*(a + a*sec(c + d*x))^2, x, 5, -a^6/(32*d*(a - a*cos(c + d*x))^4) - (7*a^5)/(48*d*(a - a*cos(c + d*x))^3) - (15*a^4)/(32*d*(a - a*cos(c + d*x))^2) - (51*a^3)/(32*d*(a - a*cos(c + d*x))) + a^4/(64*d*(a + a*cos(c + d*x))^2) + (9*a^3)/(64*d*(a + a*cos(c + d*x))) + (303*a^2*log(1 - cos(c + d*x)))/(128*d) - (2*a^2*log(cos(c + d*x)))/d - (47*a^2*log(1 + cos(c + d*x)))/(128*d) + (a^2*sec(c + d*x))/d]

    @test_int [(a + a*sec(c + d*x))^2*sin(c + d*x)^8, x, 27, -((245*a^2*x)/128) + (2*a^2*atanh(sin(c + d*x)))/d - (2*a^2*sin(c + d*x))/d + (139*a^2*cos(c + d*x)*sin(c + d*x))/(128*d) + (11*a^2*cos(c + d*x)^3*sin(c + d*x))/(192*d) - (17*a^2*cos(c + d*x)^5*sin(c + d*x))/(48*d) + (a^2*cos(c + d*x)^7*sin(c + d*x))/(8*d) - (2*a^2*sin(c + d*x)^3)/(3*d) - (2*a^2*sin(c + d*x)^5)/(5*d) - (2*a^2*sin(c + d*x)^7)/(7*d) + (a^2*tan(c + d*x))/d]
    @test_int [(a + a*sec(c + d*x))^2*sin(c + d*x)^6, x, 18, (-25*a^2*x)/16 + (2*a^2*atanh(sin(c + d*x)))/d - (2*a^2*sin(c + d*x))/d + (7*a^2*cos(c + d*x)*sin(c + d*x))/(16*d) + (7*a^2*cos(c + d*x)^3*sin(c + d*x))/(24*d) - (a^2*cos(c + d*x)^5*sin(c + d*x))/(6*d) - (2*a^2*sin(c + d*x)^3)/(3*d) - (2*a^2*sin(c + d*x)^5)/(5*d) + (a^2*tan(c + d*x))/d]
    @test_int [(a + a*sec(c + d*x))^2*sin(c + d*x)^4, x, 14, (-9*a^2*x)/8 + (2*a^2*atanh(sin(c + d*x)))/d - (2*a^2*sin(c + d*x))/d - (a^2*cos(c + d*x)*sin(c + d*x))/(8*d) + (a^2*cos(c + d*x)^3*sin(c + d*x))/(4*d) - (2*a^2*sin(c + d*x)^3)/(3*d) + (a^2*tan(c + d*x))/d]
    @test_int [(a + a*sec(c + d*x))^2*sin(c + d*x)^2, x, 9, -(a^2*x)/2 + (2*a^2*atanh(sin(c + d*x)))/d - (2*a^2*sin(c + d*x))/d - (a^2*cos(c + d*x)*sin(c + d*x))/(2*d) + (a^2*tan(c + d*x))/d]
    @test_int [csc(c + d*x)^2*(a + a*sec(c + d*x))^2, x, 11, (2*a^2*atanh(sin(c + d*x)))/d - (2*a^2*cot(c + d*x))/d - (2*a^2*csc(c + d*x))/d + (a^2*tan(c + d*x))/d]
    @test_int [csc(c + d*x)^4*(a + a*sec(c + d*x))^2, x, 8, (2*a^2*atanh(sin(c + d*x)))/d + (10*a^2*tan(c + d*x))/(3*d) - (2*a^2*tan(c + d*x))/(d*(1 - cos(c + d*x))) - (a^4*tan(c + d*x))/(3*d*(a - a*cos(c + d*x))^2)]
    @test_int [csc(c + d*x)^6*(a + a*sec(c + d*x))^2, x, 12, (2*a^2*atanh(sin(c + d*x)))/d - (4*a^2*cot(c + d*x))/d - (5*a^2*cot(c + d*x)^3)/(3*d) - (2*a^2*cot(c + d*x)^5)/(5*d) - (2*a^2*csc(c + d*x))/d - (2*a^2*csc(c + d*x)^3)/(3*d) - (2*a^2*csc(c + d*x)^5)/(5*d) + (a^2*tan(c + d*x))/d]
    @test_int [csc(c + d*x)^8*(a + a*sec(c + d*x))^2, x, 12, (2*a^2*atanh(sin(c + d*x)))/d - (5*a^2*cot(c + d*x))/d - (3*a^2*cot(c + d*x)^3)/d - (7*a^2*cot(c + d*x)^5)/(5*d) - (2*a^2*cot(c + d*x)^7)/(7*d) - (2*a^2*csc(c + d*x))/d - (2*a^2*csc(c + d*x)^3)/(3*d) - (2*a^2*csc(c + d*x)^5)/(5*d) - (2*a^2*csc(c + d*x)^7)/(7*d) + (a^2*tan(c + d*x))/d]
    @test_int [csc(c + d*x)^10*(a + a*sec(c + d*x))^2, x, 12, (2*a^2*atanh(sin(c + d*x)))/d - (6*a^2*cot(c + d*x))/d - (14*a^2*cot(c + d*x)^3)/(3*d) - (16*a^2*cot(c + d*x)^5)/(5*d) - (9*a^2*cot(c + d*x)^7)/(7*d) - (2*a^2*cot(c + d*x)^9)/(9*d) - (2*a^2*csc(c + d*x))/d - (2*a^2*csc(c + d*x)^3)/(3*d) - (2*a^2*csc(c + d*x)^5)/(5*d) - (2*a^2*csc(c + d*x)^7)/(7*d) - (2*a^2*csc(c + d*x)^9)/(9*d) + (a^2*tan(c + d*x))/d]

    @test_int [(a + a*sec(c + d*x))^3*sin(c + d*x)^9, x, 5, (11*a^3*cos(c + d*x))/d + (3*a^3*cos(c + d*x)^2)/d - (14*a^3*cos(c + d*x)^3)/(3*d) - (7*a^3*cos(c + d*x)^4)/(2*d) + (6*a^3*cos(c + d*x)^5)/(5*d) + (11*a^3*cos(c + d*x)^6)/(6*d) + (a^3*cos(c + d*x)^7)/(7*d) - (3*a^3*cos(c + d*x)^8)/(8*d) - (a^3*cos(c + d*x)^9)/(9*d) + (a^3*log(cos(c + d*x)))/d + (3*a^3*sec(c + d*x))/d + (a^3*sec(c + d*x)^2)/(2*d)]
    @test_int [(a + a*sec(c + d*x))^3*sin(c + d*x)^7, x, 5, (8*a^3*cos(c + d*x))/d + (3*a^3*cos(c + d*x)^2)/d - (2*a^3*cos(c + d*x)^3)/d - (2*a^3*cos(c + d*x)^4)/d + (a^3*cos(c + d*x)^6)/(2*d) + (a^3*cos(c + d*x)^7)/(7*d) + (3*a^3*sec(c + d*x))/d + (a^3*sec(c + d*x)^2)/(2*d)]
    @test_int [(a + a*sec(c + d*x))^3*sin(c + d*x)^5, x, 5, (5*a^3*cos(c + d*x))/d + (5*a^3*cos(c + d*x)^2)/(2*d) - (a^3*cos(c + d*x)^3)/(3*d) - (3*a^3*cos(c + d*x)^4)/(4*d) - (a^3*cos(c + d*x)^5)/(5*d) - (a^3*log(cos(c + d*x)))/d + (3*a^3*sec(c + d*x))/d + (a^3*sec(c + d*x)^2)/(2*d)]
    @test_int [(a + a*sec(c + d*x))^3*sin(c + d*x)^3, x, 4, (2*a^3*cos(c + d*x))/d + (3*a^3*cos(c + d*x)^2)/(2*d) + (a^3*cos(c + d*x)^3)/(3*d) - (2*a^3*log(cos(c + d*x)))/d + (3*a^3*sec(c + d*x))/d + (a^3*sec(c + d*x)^2)/(2*d)]
    @test_int [(a + a*sec(c + d*x))^3*sin(c + d*x)^1, x, 5, -((a^3*cos(c + d*x))/d) - (3*a^3*log(cos(c + d*x)))/d + (3*a^3*sec(c + d*x))/d + (a^3*sec(c + d*x)^2)/(2*d)]
    @test_int [csc(c + d*x)^1*(a + a*sec(c + d*x))^3, x, 5, (4*a^3*log(1 - cos(c + d*x)))/d - (4*a^3*log(cos(c + d*x)))/d + (3*a^3*sec(c + d*x))/d + (a^3*sec(c + d*x)^2)/(2*d)]
    @test_int [csc(c + d*x)^3*(a + a*sec(c + d*x))^3, x, 5, (-2*a^4)/(d*(a - a*cos(c + d*x))) + (5*a^3*log(1 - cos(c + d*x)))/d - (5*a^3*log(cos(c + d*x)))/d + (3*a^3*sec(c + d*x))/d + (a^3*sec(c + d*x)^2)/(2*d)]
    @test_int [csc(c + d*x)^5*(a + a*sec(c + d*x))^3, x, 5, -a^5/(2*d*(a - a*cos(c + d*x))^2) - (3*a^4)/(d*(a - a*cos(c + d*x))) + (6*a^3*log(1 - cos(c + d*x)))/d - (6*a^3*log(cos(c + d*x)))/d + (3*a^3*sec(c + d*x))/d + (a^3*sec(c + d*x)^2)/(2*d)]
    @test_int [csc(c + d*x)^7*(a + a*sec(c + d*x))^3, x, 5, -a^6/(6*d*(a - a*cos(c + d*x))^3) - (7*a^5)/(8*d*(a - a*cos(c + d*x))^2) - (31*a^4)/(8*d*(a - a*cos(c + d*x))) + (111*a^3*log(1 - cos(c + d*x)))/(16*d) - (7*a^3*log(cos(c + d*x)))/d + (a^3*log(1 + cos(c + d*x)))/(16*d) + (3*a^3*sec(c + d*x))/d + (a^3*sec(c + d*x)^2)/(2*d)]
    @test_int [csc(c + d*x)^9*(a + a*sec(c + d*x))^3, x, 5, -a^7/(16*d*(a - a*cos(c + d*x))^4) - a^6/(3*d*(a - a*cos(c + d*x))^3) - (39*a^5)/(32*d*(a - a*cos(c + d*x))^2) - (75*a^4)/(16*d*(a - a*cos(c + d*x))) - a^4/(32*d*(a + a*cos(c + d*x))) + (501*a^3*log(1 - cos(c + d*x)))/(64*d) - (8*a^3*log(cos(c + d*x)))/d + (11*a^3*log(1 + cos(c + d*x)))/(64*d) + (3*a^3*sec(c + d*x))/d + (a^3*sec(c + d*x)^2)/(2*d)]

    @test_int [(a + a*sec(c + d*x))^3*sin(c + d*x)^8, x, 29, -((805*a^3*x)/128) - (a^3*atanh(sin(c + d*x)))/(2*d) + (603*a^3*cos(c + d*x)*sin(c + d*x))/(128*d) - (293*a^3*cos(c + d*x)^3*sin(c + d*x))/(192*d) - (a^3*cos(c + d*x)^5*sin(c + d*x))/(48*d) + (a^3*cos(c + d*x)^7*sin(c + d*x))/(8*d) - (a^3*sin(c + d*x)^3)/(3*d) - (2*a^3*sin(c + d*x)^5)/(5*d) - (3*a^3*sin(c + d*x)^7)/(7*d) + (3*a^3*tan(c + d*x))/d + (a^3*sec(c + d*x)*tan(c + d*x))/(2*d)]
    @test_int [(a + a*sec(c + d*x))^3*sin(c + d*x)^6, x, 18, (-85*a^3*x)/16 + (a^3*atanh(sin(c + d*x)))/(2*d) - (a^3*sin(c + d*x))/d + (43*a^3*cos(c + d*x)*sin(c + d*x))/(16*d) - (5*a^3*cos(c + d*x)^3*sin(c + d*x))/(24*d) - (a^3*cos(c + d*x)^5*sin(c + d*x))/(6*d) - (2*a^3*sin(c + d*x)^3)/(3*d) - (3*a^3*sin(c + d*x)^5)/(5*d) + (3*a^3*tan(c + d*x))/d + (a^3*sec(c + d*x)*tan(c + d*x))/(2*d)]
    @test_int [(a + a*sec(c + d*x))^3*sin(c + d*x)^4, x, 16, (-33*a^3*x)/8 + (3*a^3*atanh(sin(c + d*x)))/(2*d) - (2*a^3*sin(c + d*x))/d + (7*a^3*cos(c + d*x)*sin(c + d*x))/(8*d) + (a^3*cos(c + d*x)^3*sin(c + d*x))/(4*d) - (a^3*sin(c + d*x)^3)/d + (3*a^3*tan(c + d*x))/d + (a^3*sec(c + d*x)*tan(c + d*x))/(2*d)]
    @test_int [(a + a*sec(c + d*x))^3*sin(c + d*x)^2, x, 11, (-5*a^3*x)/2 + (5*a^3*atanh(sin(c + d*x)))/(2*d) - (3*a^3*sin(c + d*x))/d - (a^3*cos(c + d*x)*sin(c + d*x))/(2*d) + (3*a^3*tan(c + d*x))/d + (a^3*sec(c + d*x)*tan(c + d*x))/(2*d)]
    @test_int [csc(c + d*x)^2*(a + a*sec(c + d*x))^3, x, 9, (9*a^3*atanh(sin(c + d*x)))/(2*d) - (4*a^3*sin(c + d*x))/(d*(1 - cos(c + d*x))) + (3*a^3*tan(c + d*x))/d + (a^3*sec(c + d*x)*tan(c + d*x))/(2*d)]
    @test_int [csc(c + d*x)^4*(a + a*sec(c + d*x))^3, x, 11, (11*a^3*atanh(sin(c + d*x)))/(2*d) - (2*a^3*sin(c + d*x))/(3*d*(1 - cos(c + d*x))^2) - (17*a^3*sin(c + d*x))/(3*d*(1 - cos(c + d*x))) + (3*a^3*tan(c + d*x))/d + (a^3*sec(c + d*x)*tan(c + d*x))/(2*d)]
    @test_int [csc(c + d*x)^6*(a + a*sec(c + d*x))^3, x, 10, (13*a^3*atanh(sin(c + d*x)))/(2*d) + (152*a^3*tan(c + d*x))/(15*d) + (13*a^3*sec(c + d*x)*tan(c + d*x))/(2*d) - (a^6*sec(c + d*x)*tan(c + d*x))/(5*d*(a - a*cos(c + d*x))^3) - (11*a^5*sec(c + d*x)*tan(c + d*x))/(15*d*(a - a*cos(c + d*x))^2) - (76*a^6*sec(c + d*x)*tan(c + d*x))/(15*d*(a^3 - a^3*cos(c + d*x)))]
    @test_int [csc(c + d*x)^8*(a + a*sec(c + d*x))^3, x, 17, (15*a^3*atanh(sin(c + d*x)))/(2*d) - (13*a^3*cot(c + d*x))/d - (7*a^3*cot(c + d*x)^3)/d - (3*a^3*cot(c + d*x)^5)/d - (4*a^3*cot(c + d*x)^7)/(7*d) - (15*a^3*csc(c + d*x))/(2*d) - (5*a^3*csc(c + d*x)^3)/(2*d) - (3*a^3*csc(c + d*x)^5)/(2*d) - (15*a^3*csc(c + d*x)^7)/(14*d) + (a^3*csc(c + d*x)^7*sec(c + d*x)^2)/(2*d) + (3*a^3*tan(c + d*x))/d]
    @test_int [csc(c + d*x)^10*(a + a*sec(c + d*x))^3, x, 17, (17*a^3*atanh(sin(c + d*x)))/(2*d) - (16*a^3*cot(c + d*x))/d - (34*a^3*cot(c + d*x)^3)/(3*d) - (36*a^3*cot(c + d*x)^5)/(5*d) - (19*a^3*cot(c + d*x)^7)/(7*d) - (4*a^3*cot(c + d*x)^9)/(9*d) - (17*a^3*csc(c + d*x))/(2*d) - (17*a^3*csc(c + d*x)^3)/(6*d) - (17*a^3*csc(c + d*x)^5)/(10*d) - (17*a^3*csc(c + d*x)^7)/(14*d) - (17*a^3*csc(c + d*x)^9)/(18*d) + (a^3*csc(c + d*x)^9*sec(c + d*x)^2)/(2*d) + (3*a^3*tan(c + d*x))/d]

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

    @test_int [sin(c + d*x)^9/(a + a*sec(c + d*x)), x, 7, cos(c + d*x)^3/(3*a*d) - (3*cos(c + d*x)^5)/(5*a*d) + (3*cos(c + d*x)^7)/(7*a*d) - cos(c + d*x)^9/(9*a*d) + sin(c + d*x)^8/(8*a*d)]
    @test_int [sin(c + d*x)^7/(a + a*sec(c + d*x)), x, 7, cos(c + d*x)^3/(3*a*d) - (2*cos(c + d*x)^5)/(5*a*d) + cos(c + d*x)^7/(7*a*d) + sin(c + d*x)^6/(6*a*d)]
    @test_int [sin(c + d*x)^5/(a + a*sec(c + d*x)), x, 7, cos(c + d*x)^3/(3*a*d) - cos(c + d*x)^5/(5*a*d) + sin(c + d*x)^4/(4*a*d)]
    @test_int [sin(c + d*x)^3/(a + a*sec(c + d*x)), x, 6, cos(c + d*x)^3/(3*a*d) + sin(c + d*x)^2/(2*a*d)]
    @test_int [sin(c + d*x)^1/(a + a*sec(c + d*x)), x, 5, -(cos(c + d*x)/(a*d)) + log(1 + cos(c + d*x))/(a*d)]
    @test_int [csc(c + d*x)^1/(a + a*sec(c + d*x)), x, 6, -(atanh(cos(c + d*x))/(2*a*d)) + (cot(c + d*x)*csc(c + d*x))/(2*a*d) - csc(c + d*x)^2/(2*a*d)]
    @test_int [csc(c + d*x)^3/(a + a*sec(c + d*x)), x, 7, -(atanh(cos(c + d*x))/(8*a*d)) - (cot(c + d*x)*csc(c + d*x))/(8*a*d) + (cot(c + d*x)*csc(c + d*x)^3)/(4*a*d) - csc(c + d*x)^4/(4*a*d)]
    @test_int [csc(c + d*x)^5/(a + a*sec(c + d*x)), x, 8, -(atanh(cos(c + d*x))/(16*a*d)) - (cot(c + d*x)*csc(c + d*x))/(16*a*d) - (cot(c + d*x)*csc(c + d*x)^3)/(24*a*d) + (cot(c + d*x)*csc(c + d*x)^5)/(6*a*d) - csc(c + d*x)^6/(6*a*d)]

    @test_int [sin(c + d*x)^8/(a + a*sec(c + d*x)), x, 9, -((5*x)/(128*a)) - (5*cos(c + d*x)*sin(c + d*x))/(128*a*d) + (5*cos(c + d*x)^3*sin(c + d*x))/(64*a*d) + (5*cos(c + d*x)^3*sin(c + d*x)^3)/(48*a*d) + (cos(c + d*x)^3*sin(c + d*x)^5)/(8*a*d) + sin(c + d*x)^7/(7*a*d)]
    @test_int [sin(c + d*x)^6/(a + a*sec(c + d*x)), x, 8, -(x/(16*a)) - (cos(c + d*x)*sin(c + d*x))/(16*a*d) + (cos(c + d*x)^3*sin(c + d*x))/(8*a*d) + (cos(c + d*x)^3*sin(c + d*x)^3)/(6*a*d) + sin(c + d*x)^5/(5*a*d)]
    @test_int [sin(c + d*x)^4/(a + a*sec(c + d*x)), x, 7, -(x/(8*a)) - (cos(c + d*x)*sin(c + d*x))/(8*a*d) + (cos(c + d*x)^3*sin(c + d*x))/(4*a*d) + sin(c + d*x)^3/(3*a*d)]
    @test_int [sin(c + d*x)^2/(a + a*sec(c + d*x)), x, 5, -(x/(2*a)) + sin(c + d*x)/(a*d) - (cos(c + d*x)*sin(c + d*x))/(2*a*d)]
    @test_int [csc(c + d*x)^2/(a + a*sec(c + d*x)), x, 6, cot(c + d*x)^3/(3*a*d) - csc(c + d*x)^3/(3*a*d)]
    @test_int [csc(c + d*x)^4/(a + a*sec(c + d*x)), x, 7, cot(c + d*x)^3/(3*a*d) + cot(c + d*x)^5/(5*a*d) - csc(c + d*x)^5/(5*a*d)]
    @test_int [csc(c + d*x)^6/(a + a*sec(c + d*x)), x, 7, cot(c + d*x)^3/(3*a*d) + (2*cot(c + d*x)^5)/(5*a*d) + cot(c + d*x)^7/(7*a*d) - csc(c + d*x)^7/(7*a*d)]
    @test_int [csc(c + d*x)^8/(a + a*sec(c + d*x)), x, 7, cot(c + d*x)^3/(3*a*d) + (3*cot(c + d*x)^5)/(5*a*d) + (3*cot(c + d*x)^7)/(7*a*d) + cot(c + d*x)^9/(9*a*d) - csc(c + d*x)^9/(9*a*d)]
    @test_int [csc(c + d*x)^10/(a + a*sec(c + d*x)), x, 7, cot(c + d*x)^3/(3*a*d) + (4*cot(c + d*x)^5)/(5*a*d) + (6*cot(c + d*x)^7)/(7*a*d) + (4*cot(c + d*x)^9)/(9*a*d) + cot(c + d*x)^11/(11*a*d) - csc(c + d*x)^11/(11*a*d)]

    @test_int [sin(c + d*x)^11/(a + a*sec(c + d*x))^2, x, 5, (4*(a - a*cos(c + d*x))^6)/(3*a^8*d) - (4*(a - a*cos(c + d*x))^7)/(a^9*d) + (19*(a - a*cos(c + d*x))^8)/(4*a^10*d) - (25*(a - a*cos(c + d*x))^9)/(9*a^11*d) + (4*(a - a*cos(c + d*x))^10)/(5*a^12*d) - (a - a*cos(c + d*x))^11/(11*a^13*d)]
    @test_int [sin(c + d*x)^9/(a + a*sec(c + d*x))^2, x, 5, (4*(a - a*cos(c + d*x))^5)/(5*a^7*d) - (2*(a - a*cos(c + d*x))^6)/(a^8*d) + (13*(a - a*cos(c + d*x))^7)/(7*a^9*d) - (3*(a - a*cos(c + d*x))^8)/(4*a^10*d) + (a - a*cos(c + d*x))^9/(9*a^11*d)]
    @test_int [sin(c + d*x)^7/(a + a*sec(c + d*x))^2, x, 5, -(cos(c + d*x)^3/(3*a^2*d)) + cos(c + d*x)^4/(2*a^2*d) - cos(c + d*x)^6/(3*a^2*d) + cos(c + d*x)^7/(7*a^2*d)]
    @test_int [sin(c + d*x)^5/(a + a*sec(c + d*x))^2, x, 5, -(cos(c + d*x)^3/(3*a^2*d)) + cos(c + d*x)^4/(2*a^2*d) - cos(c + d*x)^5/(5*a^2*d)]
    @test_int [sin(c + d*x)^3/(a + a*sec(c + d*x))^2, x, 5, (2*cos(c + d*x))/(a^2*d) - cos(c + d*x)^2/(a^2*d) + cos(c + d*x)^3/(3*a^2*d) - (2*log(1 + cos(c + d*x)))/(a^2*d)]
    @test_int [sin(c + d*x)^1/(a + a*sec(c + d*x))^2, x, 5, -(cos(c + d*x)/(a^2*d)) + 1/(d*(a^2 + a^2*cos(c + d*x))) + (2*log(1 + cos(c + d*x)))/(a^2*d)]
    @test_int [csc(c + d*x)^1/(a + a*sec(c + d*x))^2, x, 6, -(atanh(cos(c + d*x))/(4*a^2*d)) + 1/(4*d*(a + a*cos(c + d*x))^2) - 3/(4*d*(a^2 + a^2*cos(c + d*x)))]
    @test_int [csc(c + d*x)^3/(a + a*sec(c + d*x))^2, x, 4, -((a + 2*a*cos(c + d*x))/(6*d*(1 - cos(c + d*x))*(a + a*cos(c + d*x))^3))]
    @test_int [csc(c + d*x)^5/(a + a*sec(c + d*x))^2, x, 6, atanh(cos(c + d*x))/(64*a^2*d) - 1/(64*d*(a - a*cos(c + d*x))^2) + a^2/(32*d*(a + a*cos(c + d*x))^4) - a/(48*d*(a + a*cos(c + d*x))^3) - 1/(32*d*(a + a*cos(c + d*x))^2) - 1/(64*d*(a^2 - a^2*cos(c + d*x))) - 1/(32*d*(a^2 + a^2*cos(c + d*x)))]

    @test_int [sin(c + d*x)^8/(a + a*sec(c + d*x))^2, x, 16, (11*x)/(128*a^2) + (11*cos(c + d*x)*sin(c + d*x))/(128*a^2*d) - (7*cos(c + d*x)^3*sin(c + d*x))/(64*a^2*d) - (cos(c + d*x)^5*sin(c + d*x))/(16*a^2*d) - (cos(c + d*x)^3*sin(c + d*x)^3)/(6*a^2*d) - (cos(c + d*x)^5*sin(c + d*x)^3)/(8*a^2*d) - (2*sin(c + d*x)^5)/(5*a^2*d) + (2*sin(c + d*x)^7)/(7*a^2*d)]
    @test_int [sin(c + d*x)^6/(a + a*sec(c + d*x))^2, x, 7, (3*x)/(16*a^2) - (3*cos(c + d*x)*sin(c + d*x))/(16*a^2*d) - (cos(c + d*x)*sin(c + d*x)^3)/(8*a^2*d) - ((a - a*cos(c + d*x))^3*sin(c + d*x)^3)/(6*a^5*d) - sin(c + d*x)^5/(10*a^2*d)]
    @test_int [sin(c + d*x)^4/(a + a*sec(c + d*x))^2, x, 11, (7*x)/(8*a^2) - (2*sin(c + d*x))/(a^2*d) + (7*cos(c + d*x)*sin(c + d*x))/(8*a^2*d) + (cos(c + d*x)^3*sin(c + d*x))/(4*a^2*d) + (2*sin(c + d*x)^3)/(3*a^2*d)]
    @test_int [sin(c + d*x)^2/(a + a*sec(c + d*x))^2, x, 9, -((5*x)/(2*a^2)) + (2*sin(c + d*x))/(a^2*d) - (cos(c + d*x)*sin(c + d*x))/(2*a^2*d) + (2*sin(c + d*x))/(a^2*d*(1 + cos(c + d*x)))]
    @test_int [csc(c + d*x)^2/(a + a*sec(c + d*x))^2, x, 11, -(cot(c + d*x)^3/(3*a^2*d)) - (2*cot(c + d*x)^5)/(5*a^2*d) - (2*csc(c + d*x)^3)/(3*a^2*d) + (2*csc(c + d*x)^5)/(5*a^2*d)]
    @test_int [csc(c + d*x)^4/(a + a*sec(c + d*x))^2, x, 13, -(cot(c + d*x)^3/(3*a^2*d)) - (3*cot(c + d*x)^5)/(5*a^2*d) - (2*cot(c + d*x)^7)/(7*a^2*d) - (2*csc(c + d*x)^5)/(5*a^2*d) + (2*csc(c + d*x)^7)/(7*a^2*d)]
    @test_int [csc(c + d*x)^6/(a + a*sec(c + d*x))^2, x, 13, -(cot(c + d*x)^3/(3*a^2*d)) - (4*cot(c + d*x)^5)/(5*a^2*d) - (5*cot(c + d*x)^7)/(7*a^2*d) - (2*cot(c + d*x)^9)/(9*a^2*d) - (2*csc(c + d*x)^7)/(7*a^2*d) + (2*csc(c + d*x)^9)/(9*a^2*d)]
    @test_int [csc(c + d*x)^8/(a + a*sec(c + d*x))^2, x, 13, -(cot(c + d*x)^3/(3*a^2*d)) - cot(c + d*x)^5/(a^2*d) - (9*cot(c + d*x)^7)/(7*a^2*d) - (7*cot(c + d*x)^9)/(9*a^2*d) - (2*cot(c + d*x)^11)/(11*a^2*d) - (2*csc(c + d*x)^9)/(9*a^2*d) + (2*csc(c + d*x)^11)/(11*a^2*d)]

    @test_int [sin(c + d*x)^11/(a + a*sec(c + d*x))^3, x, 5, (2*(a - a*cos(c + d*x))^6)/(3*a^9*d) - (16*(a - a*cos(c + d*x))^7)/(7*a^10*d) + (25*(a - a*cos(c + d*x))^8)/(8*a^11*d) - (19*(a - a*cos(c + d*x))^9)/(9*a^12*d) + (7*(a - a*cos(c + d*x))^10)/(10*a^13*d) - (a - a*cos(c + d*x))^11/(11*a^14*d)]
    @test_int [sin(c + d*x)^9/(a + a*sec(c + d*x))^3, x, 5, -cos(c + d*x)^4/(4*a^3*d) + (3*cos(c + d*x)^5)/(5*a^3*d) - cos(c + d*x)^6/(3*a^3*d) - (2*cos(c + d*x)^7)/(7*a^3*d) + (3*cos(c + d*x)^8)/(8*a^3*d) - cos(c + d*x)^9/(9*a^3*d)]
    @test_int [sin(c + d*x)^7/(a + a*sec(c + d*x))^3, x, 5, -cos(c + d*x)^4/(4*a^3*d) + (3*cos(c + d*x)^5)/(5*a^3*d) - cos(c + d*x)^6/(2*a^3*d) + cos(c + d*x)^7/(7*a^3*d)]
    @test_int [sin(c + d*x)^5/(a + a*sec(c + d*x))^3, x, 5, (-4*cos(c + d*x))/(a^3*d) + (2*cos(c + d*x)^2)/(a^3*d) - (4*cos(c + d*x)^3)/(3*a^3*d) + (3*cos(c + d*x)^4)/(4*a^3*d) - cos(c + d*x)^5/(5*a^3*d) + (4*log(1 + cos(c + d*x)))/(a^3*d)]
    @test_int [sin(c + d*x)^3/(a + a*sec(c + d*x))^3, x, 5, (5*cos(c + d*x))/(a^3*d) - (3*cos(c + d*x)^2)/(2*a^3*d) + cos(c + d*x)^3/(3*a^3*d) - 2/(d*(a^3 + a^3*cos(c + d*x))) - (7*log(1 + cos(c + d*x)))/(a^3*d)]
    @test_int [sin(c + d*x)/(a + a*sec(c + d*x))^3, x, 5, -(cos(c + d*x)/(a^3*d)) - 1/(2*a*d*(a + a*cos(c + d*x))^2) + 3/(d*(a^3 + a^3*cos(c + d*x))) + (3*log(1 + cos(c + d*x)))/(a^3*d)]
    @test_int [csc(c + d*x)/(a + a*sec(c + d*x))^3, x, 6, -atanh(cos(c + d*x))/(8*a^3*d) - 1/(6*d*(a + a*cos(c + d*x))^3) + 5/(8*a*d*(a + a*cos(c + d*x))^2) - 7/(8*d*(a^3 + a^3*cos(c + d*x)))]
    @test_int [csc(c + d*x)^3/(a + a*sec(c + d*x))^3, x, 5, atanh(cos(c + d*x))/(32*a^3*d) - a/(16*d*(a + a*cos(c + d*x))^4) + 1/(6*d*(a + a*cos(c + d*x))^3) - 3/(32*a*d*(a + a*cos(c + d*x))^2) - 1/(32*d*(a^3 - a^3*cos(c + d*x))) - 1/(16*d*(a^3 + a^3*cos(c + d*x)))]
    @test_int [csc(c + d*x)^5/(a + a*sec(c + d*x))^3, x, 6, (3*atanh(cos(c + d*x)))/(128*a^3*d) - 1/(128*a*d*(a - a*cos(c + d*x))^2) - a^2/(40*d*(a + a*cos(c + d*x))^5) + (3*a)/(64*d*(a + a*cos(c + d*x))^4) - 1/(64*a*d*(a + a*cos(c + d*x))^2) - 3/(128*d*(a^3 + a^3*cos(c + d*x)))]

    @test_int [sin(c + d*x)^8/(a + a*sec(c + d*x))^3, x, 19, (-29*x)/(128*a^3) - (29*cos(c + d*x)*sin(c + d*x))/(128*a^3*d) - (29*cos(c + d*x)^3*sin(c + d*x))/(192*a^3*d) + (23*cos(c + d*x)^5*sin(c + d*x))/(48*a^3*d) + (cos(c + d*x)^7*sin(c + d*x))/(8*a^3*d) + (4*sin(c + d*x)^3)/(3*a^3*d) - (7*sin(c + d*x)^5)/(5*a^3*d) + (3*sin(c + d*x)^7)/(7*a^3*d)]
    @test_int [sin(c + d*x)^6/(a + a*sec(c + d*x))^3, x, 15, -((23*x)/(16*a^3)) + (4*sin(c + d*x))/(a^3*d) - (23*cos(c + d*x)*sin(c + d*x))/(16*a^3*d) - (23*cos(c + d*x)^3*sin(c + d*x))/(24*a^3*d) - (cos(c + d*x)^5*sin(c + d*x))/(6*a^3*d) - (7*sin(c + d*x)^3)/(3*a^3*d) + (3*sin(c + d*x)^5)/(5*a^3*d)]
    @test_int [sin(c + d*x)^4/(a + a*sec(c + d*x))^3, x, 13, (51*x)/(8*a^3) - (7*sin(c + d*x))/(a^3*d) + (19*cos(c + d*x)*sin(c + d*x))/(8*a^3*d) + (cos(c + d*x)^3*sin(c + d*x))/(4*a^3*d) - (4*sin(c + d*x))/(a^3*d*(1 + cos(c + d*x))) + sin(c + d*x)^3/(a^3*d)]
    @test_int [sin(c + d*x)^2/(a + a*sec(c + d*x))^3, x, 10, -((11*x)/(2*a^3)) + (3*sin(c + d*x))/(a^3*d) - (cos(c + d*x)*sin(c + d*x))/(2*a^3*d) - (2*sin(c + d*x))/(3*a^3*d*(1 + cos(c + d*x))^2) + (19*sin(c + d*x))/(3*a^3*d*(1 + cos(c + d*x)))]
    @test_int [csc(c + d*x)^2/(a + a*sec(c + d*x))^3, x, 15, (3*cot(c + d*x)^5)/(5*a^3*d) + (4*cot(c + d*x)^7)/(7*a^3*d) - csc(c + d*x)^3/(a^3*d) + (7*csc(c + d*x)^5)/(5*a^3*d) - (4*csc(c + d*x)^7)/(7*a^3*d)]
    @test_int [csc(c + d*x)^4/(a + a*sec(c + d*x))^3, x, 16, (3*cot(c + d*x)^5)/(5*a^3*d) + cot(c + d*x)^7/(a^3*d) + (4*cot(c + d*x)^9)/(9*a^3*d) - (3*csc(c + d*x)^5)/(5*a^3*d) + csc(c + d*x)^7/(a^3*d) - (4*csc(c + d*x)^9)/(9*a^3*d)]
    @test_int [csc(c + d*x)^6/(a + a*sec(c + d*x))^3, x, 16, (3*cot(c + d*x)^5)/(5*a^3*d) + (10*cot(c + d*x)^7)/(7*a^3*d) + (11*cot(c + d*x)^9)/(9*a^3*d) + (4*cot(c + d*x)^11)/(11*a^3*d) - (3*csc(c + d*x)^7)/(7*a^3*d) + (7*csc(c + d*x)^9)/(9*a^3*d) - (4*csc(c + d*x)^11)/(11*a^3*d)]
    @test_int [csc(c + d*x)^8/(a + a*sec(c + d*x))^3, x, 16, (3*cot(c + d*x)^5)/(5*a^3*d) + (13*cot(c + d*x)^7)/(7*a^3*d) + (7*cot(c + d*x)^9)/(3*a^3*d) + (15*cot(c + d*x)^11)/(11*a^3*d) + (4*cot(c + d*x)^13)/(13*a^3*d) - csc(c + d*x)^9/(3*a^3*d) + (7*csc(c + d*x)^11)/(11*a^3*d) - (4*csc(c + d*x)^13)/(13*a^3*d)]

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

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

    @test_int [(a + a*sec(c + d*x))*(e*sin(c + d*x))^(5/2), x, 11, -((a*e^(5/2)*atan(sqrt(e*sin(c + d*x))/sqrt(e)))/d) + (a*e^(5/2)*atanh(sqrt(e*sin(c + d*x))/sqrt(e)))/d + (6*a*e^2*Elliptic.E((1/2)*(c - pi/2 + d*x), 2)*sqrt(e*sin(c + d*x)))/(5*d*sqrt(sin(c + d*x))) - (2*a*e*(e*sin(c + d*x))^(3/2))/(3*d) - (2*a*e*cos(c + d*x)*(e*sin(c + d*x))^(3/2))/(5*d)]
    @test_int [(a + a*sec(c + d*x))*(e*sin(c + d*x))^(3/2), x, 11, (a*e^(3/2)*atan(sqrt(e*sin(c + d*x))/sqrt(e)))/d + (a*e^(3/2)*atanh(sqrt(e*sin(c + d*x))/sqrt(e)))/d + (2*a*e^2*Elliptic.F((1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(3*d*sqrt(e*sin(c + d*x))) - (2*a*e*sqrt(e*sin(c + d*x)))/d - (2*a*e*cos(c + d*x)*sqrt(e*sin(c + d*x)))/(3*d)]
    @test_int [(a + a*sec(c + d*x))*sqrt(e*sin(c + d*x)), x, 9, -((a*sqrt(e)*atan(sqrt(e*sin(c + d*x))/sqrt(e)))/d) + (a*sqrt(e)*atanh(sqrt(e*sin(c + d*x))/sqrt(e)))/d + (2*a*Elliptic.E((1/2)*(c - pi/2 + d*x), 2)*sqrt(e*sin(c + d*x)))/(d*sqrt(sin(c + d*x)))]
    @test_int [(a + a*sec(c + d*x))/sqrt(e*sin(c + d*x)), x, 9, (a*atan(sqrt(e*sin(c + d*x))/sqrt(e)))/(d*sqrt(e)) + (a*atanh(sqrt(e*sin(c + d*x))/sqrt(e)))/(d*sqrt(e)) + (2*a*Elliptic.F((1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(d*sqrt(e*sin(c + d*x)))]
    @test_int [(a + a*sec(c + d*x))/(e*sin(c + d*x))^(3/2), x, 11, -((a*atan(sqrt(e*sin(c + d*x))/sqrt(e)))/(d*e^(3/2))) + (a*atanh(sqrt(e*sin(c + d*x))/sqrt(e)))/(d*e^(3/2)) - (2*a)/(d*e*sqrt(e*sin(c + d*x))) - (2*a*cos(c + d*x))/(d*e*sqrt(e*sin(c + d*x))) - (2*a*Elliptic.E((1/2)*(c - pi/2 + d*x), 2)*sqrt(e*sin(c + d*x)))/(d*e^2*sqrt(sin(c + d*x)))]
    @test_int [(a + a*sec(c + d*x))/(e*sin(c + d*x))^(5/2), x, 11, (a*atan(sqrt(e*sin(c + d*x))/sqrt(e)))/(d*e^(5/2)) + (a*atanh(sqrt(e*sin(c + d*x))/sqrt(e)))/(d*e^(5/2)) - (2*a)/(3*d*e*(e*sin(c + d*x))^(3/2)) - (2*a*cos(c + d*x))/(3*d*e*(e*sin(c + d*x))^(3/2)) + (2*a*Elliptic.F((1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(3*d*e^2*sqrt(e*sin(c + d*x)))]

    @test_int [(a + a*sec(c + d*x))^2*(e*sin(c + d*x))^(5/2), x, 15, -((2*a^2*e^(5/2)*atan(sqrt(e*sin(c + d*x))/sqrt(e)))/d) + (2*a^2*e^(5/2)*atanh(sqrt(e*sin(c + d*x))/sqrt(e)))/d - (9*a^2*e^2*Elliptic.E((1/2)*(c - pi/2 + d*x), 2)*sqrt(e*sin(c + d*x)))/(5*d*sqrt(sin(c + d*x))) - (4*a^2*e*(e*sin(c + d*x))^(3/2))/(3*d) - (2*a^2*e*cos(c + d*x)*(e*sin(c + d*x))^(3/2))/(5*d) + (a^2*e*sec(c + d*x)*(e*sin(c + d*x))^(3/2))/d]
    @test_int [(a + a*sec(c + d*x))^2*(e*sin(c + d*x))^(3/2), x, 15, (2*a^2*e^(3/2)*atan(sqrt(e*sin(c + d*x))/sqrt(e)))/d + (2*a^2*e^(3/2)*atanh(sqrt(e*sin(c + d*x))/sqrt(e)))/d - (a^2*e^2*Elliptic.F((1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(3*d*sqrt(e*sin(c + d*x))) - (4*a^2*e*sqrt(e*sin(c + d*x)))/d - (2*a^2*e*cos(c + d*x)*sqrt(e*sin(c + d*x)))/(3*d) + (a^2*e*sec(c + d*x)*sqrt(e*sin(c + d*x)))/d]
    @test_int [(a + a*sec(c + d*x))^2*sqrt(e*sin(c + d*x)), x, 13, -((2*a^2*sqrt(e)*atan(sqrt(e*sin(c + d*x))/sqrt(e)))/d) + (2*a^2*sqrt(e)*atanh(sqrt(e*sin(c + d*x))/sqrt(e)))/d + (a^2*Elliptic.E((1/2)*(c - pi/2 + d*x), 2)*sqrt(e*sin(c + d*x)))/(d*sqrt(sin(c + d*x))) + (a^2*sec(c + d*x)*(e*sin(c + d*x))^(3/2))/(d*e)]
    @test_int [(a + a*sec(c + d*x))^2/sqrt(e*sin(c + d*x)), x, 13, (2*a^2*atan(sqrt(e*sin(c + d*x))/sqrt(e)))/(d*sqrt(e)) + (2*a^2*atanh(sqrt(e*sin(c + d*x))/sqrt(e)))/(d*sqrt(e)) + (3*a^2*Elliptic.F((1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(d*sqrt(e*sin(c + d*x))) + (a^2*sec(c + d*x)*sqrt(e*sin(c + d*x)))/(d*e)]
    @test_int [(a + a*sec(c + d*x))^2/(e*sin(c + d*x))^(3/2), x, 16, -((2*a^2*atan(sqrt(e*sin(c + d*x))/sqrt(e)))/(d*e^(3/2))) + (2*a^2*atanh(sqrt(e*sin(c + d*x))/sqrt(e)))/(d*e^(3/2)) - (4*a^2)/(d*e*sqrt(e*sin(c + d*x))) - (2*a^2*cos(c + d*x))/(d*e*sqrt(e*sin(c + d*x))) - (2*a^2*sec(c + d*x))/(d*e*sqrt(e*sin(c + d*x))) - (5*a^2*Elliptic.E((1/2)*(c - pi/2 + d*x), 2)*sqrt(e*sin(c + d*x)))/(d*e^2*sqrt(sin(c + d*x))) + (3*a^2*sec(c + d*x)*(e*sin(c + d*x))^(3/2))/(d*e^3)]
    @test_int [(a + a*sec(c + d*x))^2/(e*sin(c + d*x))^(5/2), x, 16, (2*a^2*atan(sqrt(e*sin(c + d*x))/sqrt(e)))/(d*e^(5/2)) + (2*a^2*atanh(sqrt(e*sin(c + d*x))/sqrt(e)))/(d*e^(5/2)) - (4*a^2)/(3*d*e*(e*sin(c + d*x))^(3/2)) - (2*a^2*cos(c + d*x))/(3*d*e*(e*sin(c + d*x))^(3/2)) - (2*a^2*sec(c + d*x))/(3*d*e*(e*sin(c + d*x))^(3/2)) + (7*a^2*Elliptic.F((1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(3*d*e^2*sqrt(e*sin(c + d*x))) + (5*a^2*sec(c + d*x)*sqrt(e*sin(c + d*x)))/(3*d*e^3)]

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

    @test_int [(e*sin(c + d*x))^(7/2)/(a + a*sec(c + d*x)), x, 8, -((4*e^4*Elliptic.F((1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(21*a*d*sqrt(e*sin(c + d*x)))) - (2*e^3*cos(c + d*x)*sqrt(e*sin(c + d*x)))/(21*a*d) + (2*e^3*cos(c + d*x)^3*sqrt(e*sin(c + d*x)))/(7*a*d) + (2*e*(e*sin(c + d*x))^(5/2))/(5*a*d)]
    @test_int [(e*sin(c + d*x))^(5/2)/(a + a*sec(c + d*x)), x, 7, -((4*e^2*Elliptic.E((1/2)*(c - pi/2 + d*x), 2)*sqrt(e*sin(c + d*x)))/(5*a*d*sqrt(sin(c + d*x)))) + (2*e*(e*sin(c + d*x))^(3/2))/(3*a*d) - (2*e*cos(c + d*x)*(e*sin(c + d*x))^(3/2))/(5*a*d)]
    @test_int [(e*sin(c + d*x))^(3/2)/(a + a*sec(c + d*x)), x, 7, -((4*e^2*Elliptic.F((1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(3*a*d*sqrt(e*sin(c + d*x)))) + (2*e*sqrt(e*sin(c + d*x)))/(a*d) - (2*e*cos(c + d*x)*sqrt(e*sin(c + d*x)))/(3*a*d)]
    @test_int [sqrt(e*sin(c + d*x))/(a + a*sec(c + d*x)), x, 7, -((2*e)/(a*d*sqrt(e*sin(c + d*x)))) + (2*e*cos(c + d*x))/(a*d*sqrt(e*sin(c + d*x))) + (4*Elliptic.E((1/2)*(c - pi/2 + d*x), 2)*sqrt(e*sin(c + d*x)))/(a*d*sqrt(sin(c + d*x)))]
    @test_int [1/((a + a*sec(c + d*x))*sqrt(e*sin(c + d*x))), x, 7, -((2*e)/(3*a*d*(e*sin(c + d*x))^(3/2))) + (2*e*cos(c + d*x))/(3*a*d*(e*sin(c + d*x))^(3/2)) + (4*Elliptic.F((1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(3*a*d*sqrt(e*sin(c + d*x)))]
    @test_int [1/((a + a*sec(c + d*x))*(e*sin(c + d*x))^(3/2)), x, 8, -((2*e)/(5*a*d*(e*sin(c + d*x))^(5/2))) + (2*e*cos(c + d*x))/(5*a*d*(e*sin(c + d*x))^(5/2)) - (4*cos(c + d*x))/(5*a*d*e*sqrt(e*sin(c + d*x))) - (4*Elliptic.E((1/2)*(c - pi/2 + d*x), 2)*sqrt(e*sin(c + d*x)))/(5*a*d*e^2*sqrt(sin(c + d*x)))]
    @test_int [1/((a + a*sec(c + d*x))*(e*sin(c + d*x))^(5/2)), x, 8, -((2*e)/(7*a*d*(e*sin(c + d*x))^(7/2))) + (2*e*cos(c + d*x))/(7*a*d*(e*sin(c + d*x))^(7/2)) - (4*cos(c + d*x))/(21*a*d*e*(e*sin(c + d*x))^(3/2)) + (4*Elliptic.F((1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(21*a*d*e^2*sqrt(e*sin(c + d*x)))]

    @test_int [(e*sin(c + d*x))^(7/2)/(a + a*sec(c + d*x))^2, x, 14, (52*e^4*Elliptic.F((1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(21*a^2*d*sqrt(e*sin(c + d*x))) - (4*e^3*sqrt(e*sin(c + d*x)))/(a^2*d) + (26*e^3*cos(c + d*x)*sqrt(e*sin(c + d*x)))/(21*a^2*d) + (2*e^3*cos(c + d*x)^3*sqrt(e*sin(c + d*x)))/(7*a^2*d) + (4*e*(e*sin(c + d*x))^(5/2))/(5*a^2*d)]
    @test_int [(e*sin(c + d*x))^(5/2)/(a + a*sec(c + d*x))^2, x, 14, (4*e^3)/(a^2*d*sqrt(e*sin(c + d*x))) - (2*e^3*cos(c + d*x))/(a^2*d*sqrt(e*sin(c + d*x))) - (2*e^3*cos(c + d*x)^3)/(a^2*d*sqrt(e*sin(c + d*x))) - (44*e^2*Elliptic.E((1/2)*(c - pi/2 + d*x), 2)*sqrt(e*sin(c + d*x)))/(5*a^2*d*sqrt(sin(c + d*x))) + (4*e*(e*sin(c + d*x))^(3/2))/(3*a^2*d) - (12*e*cos(c + d*x)*(e*sin(c + d*x))^(3/2))/(5*a^2*d)]
    @test_int [(e*sin(c + d*x))^(3/2)/(a + a*sec(c + d*x))^2, x, 14, (4*e^3)/(3*a^2*d*(e*sin(c + d*x))^(3/2)) - (2*e^3*cos(c + d*x))/(3*a^2*d*(e*sin(c + d*x))^(3/2)) - (2*e^3*cos(c + d*x)^3)/(3*a^2*d*(e*sin(c + d*x))^(3/2)) - (4*e^2*Elliptic.F((1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(a^2*d*sqrt(e*sin(c + d*x))) + (4*e*sqrt(e*sin(c + d*x)))/(a^2*d) - (4*e*cos(c + d*x)*sqrt(e*sin(c + d*x)))/(3*a^2*d)]
    @test_int [sqrt(e*sin(c + d*x))/(a + a*sec(c + d*x))^2, x, 15, (4*e^3)/(5*a^2*d*(e*sin(c + d*x))^(5/2)) - (2*e^3*cos(c + d*x))/(5*a^2*d*(e*sin(c + d*x))^(5/2)) - (2*e^3*cos(c + d*x)^3)/(5*a^2*d*(e*sin(c + d*x))^(5/2)) - (4*e)/(a^2*d*sqrt(e*sin(c + d*x))) + (16*e*cos(c + d*x))/(5*a^2*d*sqrt(e*sin(c + d*x))) + (28*Elliptic.E((1/2)*(c - pi/2 + d*x), 2)*sqrt(e*sin(c + d*x)))/(5*a^2*d*sqrt(sin(c + d*x)))]
    @test_int [1/((a + a*sec(c + d*x))^2*sqrt(e*sin(c + d*x))), x, 15, (4*e^3)/(7*a^2*d*(e*sin(c + d*x))^(7/2)) - (2*e^3*cos(c + d*x))/(7*a^2*d*(e*sin(c + d*x))^(7/2)) - (2*e^3*cos(c + d*x)^3)/(7*a^2*d*(e*sin(c + d*x))^(7/2)) - (4*e)/(3*a^2*d*(e*sin(c + d*x))^(3/2)) + (16*e*cos(c + d*x))/(21*a^2*d*(e*sin(c + d*x))^(3/2)) + (20*Elliptic.F((1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(21*a^2*d*sqrt(e*sin(c + d*x)))]
    @test_int [1/((a + a*sec(c + d*x))^2*(e*sin(c + d*x))^(3/2)), x, 17, (4*e^3)/(9*a^2*d*(e*sin(c + d*x))^(9/2)) - (2*e^3*cos(c + d*x))/(9*a^2*d*(e*sin(c + d*x))^(9/2)) - (2*e^3*cos(c + d*x)^3)/(9*a^2*d*(e*sin(c + d*x))^(9/2)) - (4*e)/(5*a^2*d*(e*sin(c + d*x))^(5/2)) + (16*e*cos(c + d*x))/(45*a^2*d*(e*sin(c + d*x))^(5/2)) - (4*cos(c + d*x))/(15*a^2*d*e*sqrt(e*sin(c + d*x))) - (4*Elliptic.E((1/2)*(c - pi/2 + d*x), 2)*sqrt(e*sin(c + d*x)))/(15*a^2*d*e^2*sqrt(sin(c + d*x)))]
    @test_int [1/((a + a*sec(c + d*x))^2*(e*sin(c + d*x))^(5/2)), x, 17, (4*e^3)/(11*a^2*d*(e*sin(c + d*x))^(11/2)) - (2*e^3*cos(c + d*x))/(11*a^2*d*(e*sin(c + d*x))^(11/2)) - (2*e^3*cos(c + d*x)^3)/(11*a^2*d*(e*sin(c + d*x))^(11/2)) - (4*e)/(7*a^2*d*(e*sin(c + d*x))^(7/2)) + (16*e*cos(c + d*x))/(77*a^2*d*(e*sin(c + d*x))^(7/2)) - (4*cos(c + d*x))/(231*a^2*d*e*(e*sin(c + d*x))^(3/2)) + (4*Elliptic.F((1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(231*a^2*d*e^2*sqrt(e*sin(c + d*x)))]

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

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

    @test_int [(a + a*sec(c + d*x))^3*(e*sin(c + d*x))^m, x, 9, (a^3*cos(c + d*x)*HypergeometricFunctions._₂F₁(1/2, (1 + m)/2, (3 + m)/2, sin(c + d*x)^2)*(e*sin(c + d*x))^(1 + m))/(d*e*(1 + m)*sqrt(cos(c + d*x)^2)) + (3*a^3*HypergeometricFunctions._₂F₁(1, (1 + m)/2, (3 + m)/2, sin(c + d*x)^2)*(e*sin(c + d*x))^(1 + m))/(d*e*(1 + m)) + (a^3*HypergeometricFunctions._₂F₁(2, (1 + m)/2, (3 + m)/2, sin(c + d*x)^2)*(e*sin(c + d*x))^(1 + m))/(d*e*(1 + m)) + (1/(d*e*(1 + m)))*(3*a^3*sqrt(cos(c + d*x)^2)*HypergeometricFunctions._₂F₁(3/2, (1 + m)/2, (3 + m)/2, sin(c + d*x)^2)*sec(c + d*x)*(e*sin(c + d*x))^(1 + m))]
    @test_int [(a + a*sec(c + d*x))^2*(e*sin(c + d*x))^m, x, 7, (a^2*cos(c + d*x)*HypergeometricFunctions._₂F₁(1/2, (1 + m)/2, (3 + m)/2, sin(c + d*x)^2)*(e*sin(c + d*x))^(1 + m))/(d*e*(1 + m)*sqrt(cos(c + d*x)^2)) + (2*a^2*HypergeometricFunctions._₂F₁(1, (1 + m)/2, (3 + m)/2, sin(c + d*x)^2)*(e*sin(c + d*x))^(1 + m))/(d*e*(1 + m)) + (1/(d*e*(1 + m)))*(a^2*sqrt(cos(c + d*x)^2)*HypergeometricFunctions._₂F₁(3/2, (1 + m)/2, (3 + m)/2, sin(c + d*x)^2)*sec(c + d*x)*(e*sin(c + d*x))^(1 + m))]
    @test_int [(a + a*sec(c + d*x))^1*(e*sin(c + d*x))^m, x, 5, (a*cos(c + d*x)*HypergeometricFunctions._₂F₁(1/2, (1 + m)/2, (3 + m)/2, sin(c + d*x)^2)*(e*sin(c + d*x))^(1 + m))/(d*e*(1 + m)*sqrt(cos(c + d*x)^2)) + (a*HypergeometricFunctions._₂F₁(1, (1 + m)/2, (3 + m)/2, sin(c + d*x)^2)*(e*sin(c + d*x))^(1 + m))/(d*e*(1 + m))]
    @test_int [(e*sin(c + d*x))^m/(a + a*sec(c + d*x))^1, x, 5, -((e*(e*sin(c + d*x))^(-1 + m))/(a*d*(1 - m))) + (e*cos(c + d*x)*HypergeometricFunctions._₂F₁(-(1/2), (1/2)*(-1 + m), (1 + m)/2, sin(c + d*x)^2)*(e*sin(c + d*x))^(-1 + m))/(a*d*(1 - m)*sqrt(cos(c + d*x)^2))]
    @test_int [(e*sin(c + d*x))^m/(a + a*sec(c + d*x))^2, x, 9, (2*e^3*(e*sin(c + d*x))^(-3 + m))/(a^2*d*(3 - m)) - (e^3*cos(c + d*x)*HypergeometricFunctions._₂F₁(-(3/2), (1/2)*(-3 + m), (1/2)*(-1 + m), sin(c + d*x)^2)*(e*sin(c + d*x))^(-3 + m))/(a^2*d*(3 - m)*sqrt(cos(c + d*x)^2)) - (e^3*cos(c + d*x)*HypergeometricFunctions._₂F₁(-(1/2), (1/2)*(-3 + m), (1/2)*(-1 + m), sin(c + d*x)^2)*(e*sin(c + d*x))^(-3 + m))/(a^2*d*(3 - m)*sqrt(cos(c + d*x)^2)) - (2*e*(e*sin(c + d*x))^(-1 + m))/(a^2*d*(1 - m))]
    @test_int [(e*sin(c + d*x))^m/(a + a*sec(c + d*x))^3, x, 12, -((4*e^5*(e*sin(c + d*x))^(-5 + m))/(a^3*d*(5 - m))) + (e^5*cos(c + d*x)*HypergeometricFunctions._₂F₁(-(5/2), (1/2)*(-5 + m), (1/2)*(-3 + m), sin(c + d*x)^2)*(e*sin(c + d*x))^(-5 + m))/(a^3*d*(5 - m)*sqrt(cos(c + d*x)^2)) + (3*e^5*cos(c + d*x)*HypergeometricFunctions._₂F₁(-(3/2), (1/2)*(-5 + m), (1/2)*(-3 + m), sin(c + d*x)^2)*(e*sin(c + d*x))^(-5 + m))/(a^3*d*(5 - m)*sqrt(cos(c + d*x)^2)) + (7*e^3*(e*sin(c + d*x))^(-3 + m))/(a^3*d*(3 - m)) - (3*e*(e*sin(c + d*x))^(-1 + m))/(a^3*d*(1 - m))]

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

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

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

    @test_int [(a + a*sec(c + d*x))^n*sin(c + d*x)^7, x, 4, If(13>=8, -(((3 - n)*(8 - n)*(16 - n)*HypergeometricFunctions._₂F₁(6, 4 + n, 5 + n, 1 + sec(c + d*x))*(a + a*sec(c + d*x))^(4 + n))/(42*a^4*d*(1 - n)*(4 + n))) - (cos(c + d*x)^7*(1 - sec(c + d*x))^2*(a + a*sec(c + d*x))^(4 + n))/(a^4*d*(1 - n)) + (cos(c + d*x)^7*(a + a*sec(c + d*x))^(4 + n)*(6*(8 - n) - (108 - 25*n + n^2)*sec(c + d*x)))/(42*a^4*d*(1 - n)), -(((3 - n)*(8 - n)*(16 - n)*HypergeometricFunctions._₂F₁(6, 4 + n, 5 + n, 1 + sec(c + d*x))*(a + a*sec(c + d*x))^(4 + n))/(42*a^4*d*(4 - 3*n - n^2))) - (cos(c + d*x)^7*(1 - sec(c + d*x))^2*(a + a*sec(c + d*x))^(4 + n))/(a^4*d*(1 - n)) + (cos(c + d*x)^7*(a + a*sec(c + d*x))^(4 + n)*(6*(8 - n) - (108 - 25*n + n^2)*sec(c + d*x)))/(42*a^4*d*(1 - n)))]
    @test_int [(a + a*sec(c + d*x))^n*sin(c + d*x)^5, x, 4, ((12 - n)*cos(c + d*x)^4*(a + a*sec(c + d*x))^(3 + n))/(20*a^3*d) - (cos(c + d*x)^5*(a + a*sec(c + d*x))^(3 + n))/(5*a^3*d) + ((32 - 13*n + n^2)*HypergeometricFunctions._₂F₁(4, 3 + n, 4 + n, 1 + sec(c + d*x))*(a + a*sec(c + d*x))^(3 + n))/(20*a^3*d*(3 + n))]
    @test_int [(a + a*sec(c + d*x))^n*sin(c + d*x)^3, x, 3, (cos(c + d*x)^3*(a + a*sec(c + d*x))^(2 + n))/(3*a^2*d) - ((4 - n)*HypergeometricFunctions._₂F₁(3, 2 + n, 3 + n, 1 + sec(c + d*x))*(a + a*sec(c + d*x))^(2 + n))/(3*a^2*d*(2 + n))]
    @test_int [(a + a*sec(c + d*x))^n*sin(c + d*x)^1, x, 2, (HypergeometricFunctions._₂F₁(2, 1 + n, 2 + n, 1 + sec(c + d*x))*(a + a*sec(c + d*x))^(1 + n))/(a*d*(1 + n))]
    @test_int [csc(c + d*x)^1*(a + a*sec(c + d*x))^n, x, 2, -((HypergeometricFunctions._₂F₁(1, n, 1 + n, (1/2)*(1 + sec(c + d*x)))*(a + a*sec(c + d*x))^n)/(2*d*n))]
    @test_int [csc(c + d*x)^3*(a + a*sec(c + d*x))^n, x, 4, -((a*(2 - n)*(a + a*sec(c + d*x))^(-1 + n))/(4*d*(1 - n))) + (a*(a + a*sec(c + d*x))^(-1 + n))/(2*d*(1 - sec(c + d*x))) - ((2 + n)*HypergeometricFunctions._₂F₁(1, n, 1 + n, (1/2)*(1 + sec(c + d*x)))*(a + a*sec(c + d*x))^n)/(8*d*n)]
    @test_int [csc(c + d*x)^5*(a + a*sec(c + d*x))^n, x, 5, (a^2*(12 + 9*n + n^2)*HypergeometricFunctions._₂F₁(1, -2 + n, -1 + n, (1/2)*(1 + sec(c + d*x)))*(a + a*sec(c + d*x))^(-2 + n))/(16*d*(2 - n)) + (a^2*(3 + n)*sec(c + d*x)^2*(a + a*sec(c + d*x))^(-2 + n))/(4*d*(1 - n)*(1 - sec(c + d*x))^2) - (a^2*sec(c + d*x)^3*(a + a*sec(c + d*x))^(-2 + n))/(d*(1 - n)*(1 - sec(c + d*x))^2) - (a^2*(a + a*sec(c + d*x))^(-2 + n)*(12 + 4*n - 7*n^2 - n^3 - 2*(1 - n)*(6 + n)*sec(c + d*x)))/(8*d*(2 - 3*n + n^2)*(1 - sec(c + d*x)))]

    @test_int [(a + a*sec(c + d*x))^n*sin(c + d*x)^4, x, 11, -((AppellF1(1 - n, -(1/2), 1/2 - n, 2 - n, cos(c + d*x), -cos(c + d*x))*(1 + cos(c + d*x))^(1/2 - n)*(n - n*cos(c + d*x))*cot(c + d*x)*(a + a*sec(c + d*x))^n)/(d*(1 - n)*sqrt(1 - cos(c + d*x)))) - (cos(c + d*x)*(a + a*sec(c + d*x))^n*sin(c + d*x))/d + (2^(1/2 + n)*AppellF1(1/2, -4 + n, 1/2 - n, 3/2, 1 - cos(c + d*x), (1/2)*(1 - cos(c + d*x)))*cos(c + d*x)^n*(1 + cos(c + d*x))^(-(1/2) - n)*(a + a*sec(c + d*x))^n*sin(c + d*x))/d]
    @test_int [(a + a*sec(c + d*x))^n*sin(c + d*x)^2, x, 6, -((AppellF1(1 - n, -(1/2), -(1/2) - n, 2 - n, cos(c + d*x), -cos(c + d*x))*sqrt(1 - cos(c + d*x))*(1 + cos(c + d*x))^(1/2 - n)*cot(c + d*x)*(a + a*sec(c + d*x))^n)/(d*(1 - n)))]
    @test_int [csc(c + d*x)^2*(a + a*sec(c + d*x))^n, x, 4, -((cot(c + d*x)*(a + a*sec(c + d*x))^n)/d) + (2^(-(1/2) + n)*n*HypergeometricFunctions._₂F₁(1/2, 3/2 - n, 3/2, (1/2)*(1 - sec(c + d*x)))*(1 + sec(c + d*x))^(-(1/2) - n)*(a + a*sec(c + d*x))^n*tan(c + d*x))/d]
    @test_int [csc(c + d*x)^4*(a + a*sec(c + d*x))^n, x, 7, If(13>=8, ((2 - n + n^2)*cos(c + d*x)*(a + a*sec(c + d*x))^n*sin(c + d*x))/(d*(3 - 2*n)*(1 - 4*n^2)*(1 - cos(c + d*x))^2) - (a^4*cos(c + d*x)*(a + a*sec(c + d*x))^n*sin(c + d*x))/(d*(3 - 2*n)*(a - a*cos(c + d*x))^2*(a + a*cos(c + d*x))^2) - (a^3*(4 - n)*cos(c + d*x)*(a + a*sec(c + d*x))^n*sin(c + d*x))/(d*(3 - 8*n + 4*n^2)*(a - a*cos(c + d*x))^2*(a + a*cos(c + d*x))) + (n*(7 - 3*n - n^2)*cos(c + d*x)*((1 + cos(c + d*x))/(1 - cos(c + d*x)))^(-(1/2) - n)*HypergeometricFunctions._₂F₁(-(1/2) - n, 1 - n, 2 - n, -((2*cos(c + d*x))/(1 - cos(c + d*x))))*(a + a*sec(c + d*x))^n*sin(c + d*x))/(d*(1 - 2*n)*(3 - 2*n)*(1 - n)*(1 + 2*n)*(1 - cos(c + d*x))^2), ((2 - n + n^2)*cos(c + d*x)*(a + a*sec(c + d*x))^n*sin(c + d*x))/(d*(3 - 2*n)*(1 - 4*n^2)*(1 - cos(c + d*x))^2) - (a^4*cos(c + d*x)*(a + a*sec(c + d*x))^n*sin(c + d*x))/(d*(3 - 2*n)*(a - a*cos(c + d*x))^2*(a + a*cos(c + d*x))^2) - (a^3*(4 - n)*cos(c + d*x)*(a + a*sec(c + d*x))^n*sin(c + d*x))/(d*(3 - 8*n + 4*n^2)*(a - a*cos(c + d*x))^2*(a + a*cos(c + d*x))) + (n*(7 - 3*n - n^2)*cos(c + d*x)*((1 + cos(c + d*x))/(1 - cos(c + d*x)))^(-(1/2) - n)*HypergeometricFunctions._₂F₁(-(1/2) - n, 1 - n, 2 - n, -((2*cos(c + d*x))/(1 - cos(c + d*x))))*(a + a*sec(c + d*x))^n*sin(c + d*x))/(d*(3 - 5*n - 10*n^2 + 20*n^3 - 8*n^4)*(1 - cos(c + d*x))^2))]

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

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

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

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

    @test_int [sin(c + d*x)^7*(a + b*sec(c + d*x)), x, 5, -((a*cos(c + d*x))/d) + (3*b*cos(c + d*x)^2)/(2*d) + (a*cos(c + d*x)^3)/d - (3*b*cos(c + d*x)^4)/(4*d) - (3*a*cos(c + d*x)^5)/(5*d) + (b*cos(c + d*x)^6)/(6*d) + (a*cos(c + d*x)^7)/(7*d) - (b*log(cos(c + d*x)))/d]
    @test_int [sin(c + d*x)^5*(a + b*sec(c + d*x)), x, 5, -((a*cos(c + d*x))/d) + (b*cos(c + d*x)^2)/d + (2*a*cos(c + d*x)^3)/(3*d) - (b*cos(c + d*x)^4)/(4*d) - (a*cos(c + d*x)^5)/(5*d) - (b*log(cos(c + d*x)))/d]
    @test_int [sin(c + d*x)^3*(a + b*sec(c + d*x)), x, 5, -((a*cos(c + d*x))/d) + (b*cos(c + d*x)^2)/(2*d) + (a*cos(c + d*x)^3)/(3*d) - (b*log(cos(c + d*x)))/d]
    @test_int [sin(c + d*x)^1*(a + b*sec(c + d*x)), x, 4, -((a*cos(c + d*x))/d) - (b*log(cos(c + d*x)))/d]
    @test_int [csc(c + d*x)^1*(a + b*sec(c + d*x)), x, 5, -((a*atanh(cos(c + d*x)))/d) + (b*log(tan(c + d*x)))/d]
    @test_int [csc(c + d*x)^3*(a + b*sec(c + d*x)), x, 7, -((a*atanh(cos(c + d*x)))/(2*d)) - (b*cot(c + d*x)^2)/(2*d) - (a*cot(c + d*x)*csc(c + d*x))/(2*d) + (b*log(tan(c + d*x)))/d]
    @test_int [csc(c + d*x)^5*(a + b*sec(c + d*x)), x, 9, -((3*a*atanh(cos(c + d*x)))/(8*d)) - (b*cot(c + d*x)^2)/d - (b*cot(c + d*x)^4)/(4*d) - (3*a*cot(c + d*x)*csc(c + d*x))/(8*d) - (a*cot(c + d*x)*csc(c + d*x)^3)/(4*d) + (b*log(tan(c + d*x)))/d]
    @test_int [csc(c + d*x)^7*(a + b*sec(c + d*x)), x, 10, -((5*a*atanh(cos(c + d*x)))/(16*d)) - (3*b*cot(c + d*x)^2)/(2*d) - (3*b*cot(c + d*x)^4)/(4*d) - (b*cot(c + d*x)^6)/(6*d) - (5*a*cot(c + d*x)*csc(c + d*x))/(16*d) - (5*a*cot(c + d*x)*csc(c + d*x)^3)/(24*d) - (a*cot(c + d*x)*csc(c + d*x)^5)/(6*d) + (b*log(tan(c + d*x)))/d]

    @test_int [sin(c + d*x)^6*(a + b*sec(c + d*x)), x, 10, (5*a*x)/16 + (b*atanh(sin(c + d*x)))/d - (b*sin(c + d*x))/d - (5*a*cos(c + d*x)*sin(c + d*x))/(16*d) - (b*sin(c + d*x)^3)/(3*d) - (5*a*cos(c + d*x)*sin(c + d*x)^3)/(24*d) - (b*sin(c + d*x)^5)/(5*d) - (a*cos(c + d*x)*sin(c + d*x)^5)/(6*d)]
    @test_int [sin(c + d*x)^4*(a + b*sec(c + d*x)), x, 9, (3*a*x)/8 + (b*atanh(sin(c + d*x)))/d - (b*sin(c + d*x))/d - (3*a*cos(c + d*x)*sin(c + d*x))/(8*d) - (b*sin(c + d*x)^3)/(3*d) - (a*cos(c + d*x)*sin(c + d*x)^3)/(4*d)]
    @test_int [sin(c + d*x)^2*(a + b*sec(c + d*x)), x, 7, (a*x)/2 + (b*atanh(sin(c + d*x)))/d - (b*sin(c + d*x))/d - (a*cos(c + d*x)*sin(c + d*x))/(2*d)]
    @test_int [csc(c + d*x)^2*(a + b*sec(c + d*x)), x, 7, (b*atanh(sin(c + d*x)))/d - (a*cot(c + d*x))/d - (b*csc(c + d*x))/d]
    @test_int [csc(c + d*x)^4*(a + b*sec(c + d*x)), x, 8, (b*atanh(sin(c + d*x)))/d - (a*cot(c + d*x))/d - (a*cot(c + d*x)^3)/(3*d) - (b*csc(c + d*x))/d - (b*csc(c + d*x)^3)/(3*d)]
    @test_int [csc(c + d*x)^6*(a + b*sec(c + d*x)), x, 8, (b*atanh(sin(c + d*x)))/d - (a*cot(c + d*x))/d - (2*a*cot(c + d*x)^3)/(3*d) - (a*cot(c + d*x)^5)/(5*d) - (b*csc(c + d*x))/d - (b*csc(c + d*x)^3)/(3*d) - (b*csc(c + d*x)^5)/(5*d)]

    @test_int [sin(c + d*x)^5*(a + b*sec(c + d*x))^2, x, 5, -(((a^2 - 2*b^2)*cos(c + d*x))/d) + (2*a*b*cos(c + d*x)^2)/d + ((2*a^2 - b^2)*cos(c + d*x)^3)/(3*d) - (a*b*cos(c + d*x)^4)/(2*d) - (a^2*cos(c + d*x)^5)/(5*d) - (2*a*b*log(cos(c + d*x)))/d + (b^2*sec(c + d*x))/d]
    @test_int [sin(c + d*x)^3*(a + b*sec(c + d*x))^2, x, 5, -(((a^2 - b^2)*cos(c + d*x))/d) + (a*b*cos(c + d*x)^2)/d + (a^2*cos(c + d*x)^3)/(3*d) - (2*a*b*log(cos(c + d*x)))/d + (b^2*sec(c + d*x))/d]
    @test_int [sin(c + d*x)^1*(a + b*sec(c + d*x))^2, x, 5, -((a^2*cos(c + d*x))/d) - (2*a*b*log(cos(c + d*x)))/d + (b^2*sec(c + d*x))/d]
    @test_int [csc(c + d*x)^1*(a + b*sec(c + d*x))^2, x, 5, ((a + b)^2*log(1 - cos(c + d*x)))/(2*d) - (2*a*b*log(cos(c + d*x)))/d - ((a - b)^2*log(1 + cos(c + d*x)))/(2*d) + (b^2*sec(c + d*x))/d]
    @test_int [csc(c + d*x)^3*(a + b*sec(c + d*x))^2, x, 6, -(((2*a*b + (a^2 + b^2)*cos(c + d*x))*csc(c + d*x)^2)/(2*d)) + ((a + b)*(a + 3*b)*log(1 - cos(c + d*x)))/(4*d) - (2*a*b*log(cos(c + d*x)))/d - ((a - 3*b)*(a - b)*log(1 + cos(c + d*x)))/(4*d) + (b^2*sec(c + d*x))/d]

    @test_int [sin(c + d*x)^6*(a + b*sec(c + d*x))^2, x, 12, (5/16)*(a^2 - 6*b^2)*x + (2*a*b*atanh(sin(c + d*x)))/d - (2*a*b*sin(c + d*x))/d - ((11*a^2 - 18*b^2)*cos(c + d*x)*sin(c + d*x))/(16*d) + ((13*a^2 - 6*b^2)*cos(c + d*x)^3*sin(c + d*x))/(24*d) - (a^2*cos(c + d*x)^5*sin(c + d*x))/(6*d) - (2*a*b*sin(c + d*x)^3)/(3*d) - (2*a*b*sin(c + d*x)^5)/(5*d) + (b^2*tan(c + d*x))/d]
    @test_int [sin(c + d*x)^4*(a + b*sec(c + d*x))^2, x, 7, (3/8)*(a^2 - 4*b^2)*x + (2*a*b*atanh(sin(c + d*x)))/d - (b*(28*a^2 + b^2)*sin(c + d*x))/(6*a*d) - ((39*a^2 + 2*b^2)*cos(c + d*x)*sin(c + d*x))/(24*d) - ((12*a^2 + b^2)*(b + a*cos(c + d*x))^2*sin(c + d*x))/(12*a*b*d) + ((b + a*cos(c + d*x))^3*sin(c + d*x))/(4*a*d) + ((b + a*cos(c + d*x))^3*tan(c + d*x))/(b*d)]
    @test_int [sin(c + d*x)^2*(a + b*sec(c + d*x))^2, x, 10, (a^2*x)/2 - b^2*x + (2*a*b*atanh(sin(c + d*x)))/d - (2*a*b*sin(c + d*x))/d - (a^2*cos(c + d*x)*sin(c + d*x))/(2*d) + (b^2*tan(c + d*x))/d]
    @test_int [csc(c + d*x)^2*(a + b*sec(c + d*x))^2, x, 8, (2*a*b*atanh(sin(c + d*x)))/d - ((a^2 + b^2)*cot(c + d*x))/d - (2*a*b*csc(c + d*x))/d + (b^2*tan(c + d*x))/d]
    @test_int [csc(c + d*x)^4*(a + b*sec(c + d*x))^2, x, 9, (2*a*b*atanh(sin(c + d*x)))/d - ((a^2 + 2*b^2)*cot(c + d*x))/d - ((a^2 + b^2)*cot(c + d*x)^3)/(3*d) - (2*a*b*csc(c + d*x))/d - (2*a*b*csc(c + d*x)^3)/(3*d) + (b^2*tan(c + d*x))/d]
    @test_int [csc(c + d*x)^6*(a + b*sec(c + d*x))^2, x, 9, (2*a*b*atanh(sin(c + d*x)))/d - ((a^2 + 3*b^2)*cot(c + d*x))/d - ((2*a^2 + 3*b^2)*cot(c + d*x)^3)/(3*d) - ((a^2 + b^2)*cot(c + d*x)^5)/(5*d) - (2*a*b*csc(c + d*x))/d - (2*a*b*csc(c + d*x)^3)/(3*d) - (2*a*b*csc(c + d*x)^5)/(5*d) + (b^2*tan(c + d*x))/d]

    @test_int [sin(c + d*x)^5*(a + b*sec(c + d*x))^3, x, 5, -((a*(a^2 - 6*b^2)*cos(c + d*x))/d) + (b*(6*a^2 - b^2)*cos(c + d*x)^2)/(2*d) + (a*(2*a^2 - 3*b^2)*cos(c + d*x)^3)/(3*d) - (3*a^2*b*cos(c + d*x)^4)/(4*d) - (a^3*cos(c + d*x)^5)/(5*d) - (b*(3*a^2 - 2*b^2)*log(cos(c + d*x)))/d + (3*a*b^2*sec(c + d*x))/d + (b^3*sec(c + d*x)^2)/(2*d)]
    @test_int [sin(c + d*x)^3*(a + b*sec(c + d*x))^3, x, 4, -((a*(a^2 - 3*b^2)*cos(c + d*x))/d) + (3*a^2*b*cos(c + d*x)^2)/(2*d) + (a^3*cos(c + d*x)^3)/(3*d) - (b*(3*a^2 - b^2)*log(cos(c + d*x)))/d + (3*a*b^2*sec(c + d*x))/d + (b^3*sec(c + d*x)^2)/(2*d)]
    @test_int [sin(c + d*x)^1*(a + b*sec(c + d*x))^3, x, 5, -((a^3*cos(c + d*x))/d) - (3*a^2*b*log(cos(c + d*x)))/d + (3*a*b^2*sec(c + d*x))/d + (b^3*sec(c + d*x)^2)/(2*d)]
    @test_int [csc(c + d*x)^1*(a + b*sec(c + d*x))^3, x, 5, ((a + b)^3*log(1 - cos(c + d*x)))/(2*d) - (b*(3*a^2 + b^2)*log(cos(c + d*x)))/d - ((a - b)^3*log(1 + cos(c + d*x)))/(2*d) + (3*a*b^2*sec(c + d*x))/d + (b^3*sec(c + d*x)^2)/(2*d)]
    @test_int [csc(c + d*x)^3*(a + b*sec(c + d*x))^3, x, 6, -((a^2*(b*(3 + b^2/a^2) + a*(1 + (3*b^2)/a^2)*cos(c + d*x))*csc(c + d*x)^2)/(2*d)) + ((a + b)^2*(a + 4*b)*log(1 - cos(c + d*x)))/(4*d) - (b*(3*a^2 + 2*b^2)*log(cos(c + d*x)))/d - ((a - 4*b)*(a - b)^2*log(1 + cos(c + d*x)))/(4*d) + (3*a*b^2*sec(c + d*x))/d + (b^3*sec(c + d*x)^2)/(2*d)]

    @test_int [sin(c + d*x)^6*(a + b*sec(c + d*x))^3, x, 21, (5*a^3*x)/16 - (45/8)*a*b^2*x + (3*a^2*b*atanh(sin(c + d*x)))/d - (5*b^3*atanh(sin(c + d*x)))/(2*d) - (3*a^2*b*sin(c + d*x))/d + (5*b^3*sin(c + d*x))/(2*d) - (5*a^3*cos(c + d*x)*sin(c + d*x))/(16*d) - (a^2*b*sin(c + d*x)^3)/d + (5*b^3*sin(c + d*x)^3)/(6*d) - (5*a^3*cos(c + d*x)*sin(c + d*x)^3)/(24*d) - (3*a^2*b*sin(c + d*x)^5)/(5*d) - (a^3*cos(c + d*x)*sin(c + d*x)^5)/(6*d) + (45*a*b^2*tan(c + d*x))/(8*d) - (15*a*b^2*sin(c + d*x)^2*tan(c + d*x))/(8*d) - (3*a*b^2*sin(c + d*x)^4*tan(c + d*x))/(4*d) + (b^3*sin(c + d*x)^3*tan(c + d*x)^2)/(2*d)]
    @test_int [sin(c + d*x)^4*(a + b*sec(c + d*x))^3, x, 8, (3/8)*a*(a^2 - 12*b^2)*x + (3*b*(2*a^2 - b^2)*atanh(sin(c + d*x)))/(2*d) - (b*(17*a^2 - b^2)*sin(c + d*x))/(2*d) - (a*(21*a^2 - 2*b^2)*cos(c + d*x)*sin(c + d*x))/(8*d) - ((6*a^2 - b^2)*(b + a*cos(c + d*x))^2*sin(c + d*x))/(4*b*d) - ((4*a^2 - b^2)*(b + a*cos(c + d*x))^3*sin(c + d*x))/(4*b^2*d) + (a*(b + a*cos(c + d*x))^4*tan(c + d*x))/(b^2*d) + ((b + a*cos(c + d*x))^4*sec(c + d*x)*tan(c + d*x))/(2*b*d)]
    @test_int [sin(c + d*x)^2*(a + b*sec(c + d*x))^3, x, 8, (1/2)*a*(a^2 - 6*b^2)*x + (b*(6*a^2 - b^2)*atanh(sin(c + d*x)))/(2*d) - (15*a^2*b*sin(c + d*x))/(2*d) - (5*a^3*cos(c + d*x)*sin(c + d*x))/(2*d) + (3*a*(b + a*cos(c + d*x))^2*tan(c + d*x))/(2*d) + ((b + a*cos(c + d*x))^3*sec(c + d*x)*tan(c + d*x))/(2*d)]
    @test_int [csc(c + d*x)^2*(a + b*sec(c + d*x))^3, x, 15, (3*a^2*b*atanh(sin(c + d*x)))/d + (3*b^3*atanh(sin(c + d*x)))/(2*d) - (a^3*cot(c + d*x))/d - (3*a*b^2*cot(c + d*x))/d - (3*a^2*b*csc(c + d*x))/d - (3*b^3*csc(c + d*x))/(2*d) + (b^3*csc(c + d*x)*sec(c + d*x)^2)/(2*d) + (3*a*b^2*tan(c + d*x))/d]
    @test_int [csc(c + d*x)^4*(a + b*sec(c + d*x))^3, x, 17, (3*a^2*b*atanh(sin(c + d*x)))/d + (5*b^3*atanh(sin(c + d*x)))/(2*d) - (a^3*cot(c + d*x))/d - (6*a*b^2*cot(c + d*x))/d - (a^3*cot(c + d*x)^3)/(3*d) - (a*b^2*cot(c + d*x)^3)/d - (3*a^2*b*csc(c + d*x))/d - (5*b^3*csc(c + d*x))/(2*d) - (a^2*b*csc(c + d*x)^3)/d - (5*b^3*csc(c + d*x)^3)/(6*d) + (b^3*csc(c + d*x)^3*sec(c + d*x)^2)/(2*d) + (3*a*b^2*tan(c + d*x))/d]
    @test_int [csc(c + d*x)^6*(a + b*sec(c + d*x))^3, x, 17, (3*a^2*b*atanh(sin(c + d*x)))/d + (7*b^3*atanh(sin(c + d*x)))/(2*d) - (a^3*cot(c + d*x))/d - (9*a*b^2*cot(c + d*x))/d - (2*a^3*cot(c + d*x)^3)/(3*d) - (3*a*b^2*cot(c + d*x)^3)/d - (a^3*cot(c + d*x)^5)/(5*d) - (3*a*b^2*cot(c + d*x)^5)/(5*d) - (3*a^2*b*csc(c + d*x))/d - (7*b^3*csc(c + d*x))/(2*d) - (a^2*b*csc(c + d*x)^3)/d - (7*b^3*csc(c + d*x)^3)/(6*d) - (3*a^2*b*csc(c + d*x)^5)/(5*d) - (7*b^3*csc(c + d*x)^5)/(10*d) + (b^3*csc(c + d*x)^5*sec(c + d*x)^2)/(2*d) + (3*a*b^2*tan(c + d*x))/d]

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

    @test_int [sin(c + d*x)^7/(a + b*sec(c + d*x)), x, 5, -(((a^2 - b^2)^3*cos(c + d*x))/(a^7*d)) - (b*(3*a^4 - 3*a^2*b^2 + b^4)*cos(c + d*x)^2)/(2*a^6*d) + ((3*a^4 - 3*a^2*b^2 + b^4)*cos(c + d*x)^3)/(3*a^5*d) + (b*(3*a^2 - b^2)*cos(c + d*x)^4)/(4*a^4*d) - ((3*a^2 - b^2)*cos(c + d*x)^5)/(5*a^3*d) - (b*cos(c + d*x)^6)/(6*a^2*d) + cos(c + d*x)^7/(7*a*d) + (b*(a^2 - b^2)^3*log(b + a*cos(c + d*x)))/(a^8*d)]
    @test_int [sin(c + d*x)^5/(a + b*sec(c + d*x)), x, 5, -(((a^2 - b^2)^2*cos(c + d*x))/(a^5*d)) - (b*(2*a^2 - b^2)*cos(c + d*x)^2)/(2*a^4*d) + ((2*a^2 - b^2)*cos(c + d*x)^3)/(3*a^3*d) + (b*cos(c + d*x)^4)/(4*a^2*d) - cos(c + d*x)^5/(5*a*d) + (b*(a^2 - b^2)^2*log(b + a*cos(c + d*x)))/(a^6*d)]
    @test_int [sin(c + d*x)^3/(a + b*sec(c + d*x)), x, 5, -(((a^2 - b^2)*cos(c + d*x))/(a^3*d)) - (b*cos(c + d*x)^2)/(2*a^2*d) + cos(c + d*x)^3/(3*a*d) + (b*(a^2 - b^2)*log(b + a*cos(c + d*x)))/(a^4*d)]
    @test_int [sin(c + d*x)^1/(a + b*sec(c + d*x)), x, 5, -(cos(c + d*x)/(a*d)) + (b*log(b + a*cos(c + d*x)))/(a^2*d)]
    @test_int [csc(c + d*x)^1/(a + b*sec(c + d*x)), x, 4, log(1 - cos(c + d*x))/(2*(a + b)*d) - log(1 + cos(c + d*x))/(2*(a - b)*d) + (b*log(b + a*cos(c + d*x)))/((a^2 - b^2)*d)]
    @test_int [csc(c + d*x)^3/(a + b*sec(c + d*x)), x, 6, ((b - a*cos(c + d*x))*csc(c + d*x)^2)/(2*(a^2 - b^2)*d) + (a*log(1 - cos(c + d*x)))/(4*(a + b)^2*d) - (a*log(1 + cos(c + d*x)))/(4*(a - b)^2*d) + (a^2*b*log(b + a*cos(c + d*x)))/((a^2 - b^2)^2*d)]
    @test_int [csc(c + d*x)^5/(a + b*sec(c + d*x)), x, 7, ((4*a^2*b - a*(3*a^2 + b^2)*cos(c + d*x))*csc(c + d*x)^2)/(8*(a^2 - b^2)^2*d) + ((b - a*cos(c + d*x))*csc(c + d*x)^4)/(4*(a^2 - b^2)*d) + (a*(3*a + b)*log(1 - cos(c + d*x)))/(16*(a + b)^3*d) - (a*(3*a - b)*log(1 + cos(c + d*x)))/(16*(a - b)^3*d) + (a^4*b*log(b + a*cos(c + d*x)))/((a^2 - b^2)^3*d)]

    @test_int [sin(c + d*x)^6/(a + b*sec(c + d*x)), x, 7, ((5*a^6 - 30*a^4*b^2 + 40*a^2*b^4 - 16*b^6)*x)/(16*a^7) - (2*(a - b)^(5/2)*b*(a + b)^(5/2)*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/(a^7*d) + ((16*b*(a^2 - b^2)^2 - a*(5*a^4 - 14*a^2*b^2 + 8*b^4)*cos(c + d*x))*sin(c + d*x))/(16*a^6*d) + ((8*b*(a^2 - b^2) - a*(5*a^2 - 6*b^2)*cos(c + d*x))*sin(c + d*x)^3)/(24*a^4*d) + ((6*b - 5*a*cos(c + d*x))*sin(c + d*x)^5)/(30*a^2*d)]
    @test_int [sin(c + d*x)^4/(a + b*sec(c + d*x)), x, 6, ((3*a^4 - 12*a^2*b^2 + 8*b^4)*x)/(8*a^5) - (2*(a - b)^(3/2)*b*(a + b)^(3/2)*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/(a^5*d) + ((8*b*(a^2 - b^2) - a*(3*a^2 - 4*b^2)*cos(c + d*x))*sin(c + d*x))/(8*a^4*d) + ((4*b - 3*a*cos(c + d*x))*sin(c + d*x)^3)/(12*a^2*d)]
    @test_int [sin(c + d*x)^2/(a + b*sec(c + d*x)), x, 5, ((a^2 - 2*b^2)*x)/(2*a^3) - (2*sqrt(a - b)*b*sqrt(a + b)*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/(a^3*d) + ((2*b - a*cos(c + d*x))*sin(c + d*x))/(2*a^2*d)]
    @test_int [csc(c + d*x)^2/(a + b*sec(c + d*x)), x, 5, -((2*a*b*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(3/2)*(a + b)^(3/2)*d)) + ((b - a*cos(c + d*x))*csc(c + d*x))/((a^2 - b^2)*d)]
    @test_int [csc(c + d*x)^4/(a + b*sec(c + d*x)), x, 6, -((2*a^3*b*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(5/2)*(a + b)^(5/2)*d)) + ((3*a^2*b - a*(2*a^2 + b^2)*cos(c + d*x))*csc(c + d*x))/(3*(a^2 - b^2)^2*d) + ((b - a*cos(c + d*x))*csc(c + d*x)^3)/(3*(a^2 - b^2)*d)]
    @test_int [csc(c + d*x)^6/(a + b*sec(c + d*x)), x, 7, -((2*a^5*b*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(7/2)*(a + b)^(7/2)*d)) + ((15*a^4*b - a*(8*a^4 + 9*a^2*b^2 - 2*b^4)*cos(c + d*x))*csc(c + d*x))/(15*(a^2 - b^2)^3*d) + ((5*a^2*b - a*(4*a^2 + b^2)*cos(c + d*x))*csc(c + d*x)^3)/(15*(a^2 - b^2)^2*d) + ((b - a*cos(c + d*x))*csc(c + d*x)^5)/(5*(a^2 - b^2)*d)]

    @test_int [sin(c + d*x)^7/(a + b*sec(c + d*x))^2, x, 5, -(((a^2 - 7*b^2)*(a^2 - b^2)^2*cos(c + d*x))/(a^8*d)) - (3*b*(a^2 - b^2)^2*cos(c + d*x)^2)/(a^7*d) + ((3*a^4 - 9*a^2*b^2 + 5*b^4)*cos(c + d*x)^3)/(3*a^6*d) + (b*(3*a^2 - 2*b^2)*cos(c + d*x)^4)/(2*a^5*d) - (3*(a^2 - b^2)*cos(c + d*x)^5)/(5*a^4*d) - (b*cos(c + d*x)^6)/(3*a^3*d) + cos(c + d*x)^7/(7*a^2*d) + (b^2*(a^2 - b^2)^3)/(a^9*d*(b + a*cos(c + d*x))) + (2*b*(a^2 - 4*b^2)*(a^2 - b^2)^2*log(b + a*cos(c + d*x)))/(a^9*d)]
    @test_int [sin(c + d*x)^5/(a + b*sec(c + d*x))^2, x, 5, -(((a^4 - 6*a^2*b^2 + 5*b^4)*cos(c + d*x))/(a^6*d)) - (2*b*(a^2 - b^2)*cos(c + d*x)^2)/(a^5*d) + ((2*a^2 - 3*b^2)*cos(c + d*x)^3)/(3*a^4*d) + (b*cos(c + d*x)^4)/(2*a^3*d) - cos(c + d*x)^5/(5*a^2*d) + (b^2*(a^2 - b^2)^2)/(a^7*d*(b + a*cos(c + d*x))) + (2*b*(a^4 - 4*a^2*b^2 + 3*b^4)*log(b + a*cos(c + d*x)))/(a^7*d)]
    @test_int [sin(c + d*x)^3/(a + b*sec(c + d*x))^2, x, 5, -(((a^2 - 3*b^2)*cos(c + d*x))/(a^4*d)) - (b*cos(c + d*x)^2)/(a^3*d) + cos(c + d*x)^3/(3*a^2*d) + (b^2*(a^2 - b^2))/(a^5*d*(b + a*cos(c + d*x))) + (2*b*(a^2 - 2*b^2)*log(b + a*cos(c + d*x)))/(a^5*d)]
    @test_int [sin(c + d*x)^1/(a + b*sec(c + d*x))^2, x, 5, -(cos(c + d*x)/(a^2*d)) + b^2/(a^3*d*(b + a*cos(c + d*x))) + (2*b*log(b + a*cos(c + d*x)))/(a^3*d)]
    @test_int [csc(c + d*x)^1/(a + b*sec(c + d*x))^2, x, 5, b^2/(a*(a^2 - b^2)*d*(b + a*cos(c + d*x))) + log(1 - cos(c + d*x))/(2*(a + b)^2*d) - log(1 + cos(c + d*x))/(2*(a - b)^2*d) + (2*a*b*log(b + a*cos(c + d*x)))/((a^2 - b^2)^2*d)]
    @test_int [csc(c + d*x)^3/(a + b*sec(c + d*x))^2, x, 6, (a*b^2)/((a^2 - b^2)^2*d*(b + a*cos(c + d*x))) + ((2*a*b - (a^2 + b^2)*cos(c + d*x))*csc(c + d*x)^2)/(2*(a^2 - b^2)^2*d) + ((a - b)*log(1 - cos(c + d*x)))/(4*(a + b)^3*d) - ((a + b)*log(1 + cos(c + d*x)))/(4*(a - b)^3*d) + (2*a*b*(a^2 + b^2)*log(b + a*cos(c + d*x)))/((a^2 - b^2)^3*d)]
    @test_int [csc(c + d*x)^5/(a + b*sec(c + d*x))^2, x, 7, (a^3*b^2)/((a^2 - b^2)^3*d*(b + a*cos(c + d*x))) + ((8*a*b*(a^2 + b^2) - (3*a^4 + 12*a^2*b^2 + b^4)*cos(c + d*x))*csc(c + d*x)^2)/(8*(a^2 - b^2)^3*d) + ((2*a*b - (a^2 + b^2)*cos(c + d*x))*csc(c + d*x)^4)/(4*(a^2 - b^2)^2*d) + ((3*a^2 - 4*a*b - b^2)*log(1 - cos(c + d*x)))/(16*(a + b)^4*d) - ((3*a^2 + 4*a*b - b^2)*log(1 + cos(c + d*x)))/(16*(a - b)^4*d) + (2*a^3*b*(a^2 + 2*b^2)*log(b + a*cos(c + d*x)))/((a^2 - b^2)^4*d)]

    @test_int [sin(c + d*x)^6/(a + b*sec(c + d*x))^2, x, 10, ((5*a^6 - 90*a^4*b^2 + 200*a^2*b^4 - 112*b^6)*x)/(16*a^8) - (2*(a - b)^(3/2)*b*(a + b)^(3/2)*(2*a^2 - 7*b^2)*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/(a^8*d) + (b*(61*a^4 - 170*a^2*b^2 + 105*b^4)*sin(c + d*x))/(15*a^7*d) - ((27*a^4 - 86*a^2*b^2 + 56*b^4)*cos(c + d*x)*sin(c + d*x))/(16*a^6*d) + ((15*a^4 - 52*a^2*b^2 + 35*b^4)*cos(c + d*x)^2*sin(c + d*x))/(15*a^5*b*d) - ((16*a^4 - 61*a^2*b^2 + 42*b^4)*cos(c + d*x)^3*sin(c + d*x))/(24*a^4*b^2*d) - (cos(c + d*x)^3*sin(c + d*x))/(3*b*d*(b + a*cos(c + d*x))) + (a*cos(c + d*x)^4*sin(c + d*x))/(6*b^2*d*(b + a*cos(c + d*x))) + ((5*a^4 - 20*a^2*b^2 + 14*b^4)*cos(c + d*x)^4*sin(c + d*x))/(10*a^3*b^2*d*(b + a*cos(c + d*x))) + (7*b*cos(c + d*x)^5*sin(c + d*x))/(30*a^2*d*(b + a*cos(c + d*x))) - (cos(c + d*x)^6*sin(c + d*x))/(6*a*d*(b + a*cos(c + d*x)))]
    @test_int [sin(c + d*x)^4/(a + b*sec(c + d*x))^2, x, 8, ((3*a^4 - 36*a^2*b^2 + 40*b^4)*x)/(8*a^6) - (2*sqrt(a - b)*b*sqrt(a + b)*(2*a^2 - 5*b^2)*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/(a^6*d) + (b*(11*a^2 - 15*b^2)*sin(c + d*x))/(3*a^5*d) - ((13*a^2 - 20*b^2)*cos(c + d*x)*sin(c + d*x))/(8*a^4*d) + ((3*a^2 - 5*b^2)*cos(c + d*x)^2*sin(c + d*x))/(3*a^3*b*d) + (cos(c + d*x)^3*sin(c + d*x))/(4*a^2*d) - ((a^2 - b^2)*cos(c + d*x)^3*sin(c + d*x))/(a^2*b*d*(b + a*cos(c + d*x)))]
    @test_int [sin(c + d*x)^2/(a + b*sec(c + d*x))^2, x, 8, ((a^2 - 6*b^2)*x)/(2*a^4) - (2*b*(2*a^2 - 3*b^2)*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/(a^4*sqrt(a - b)*sqrt(a + b)*d) + (3*b*sin(c + d*x))/(a^3*d) - (3*cos(c + d*x)*sin(c + d*x))/(2*a^2*d) + (cos(c + d*x)^2*sin(c + d*x))/(a*d*(b + a*cos(c + d*x)))]
    @test_int [csc(c + d*x)^2/(a + b*sec(c + d*x))^2, x, 11, -((4*a^2*b*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(5/2)*(a + b)^(5/2)*d)) - (2*b^3*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(5/2)*(a + b)^(5/2)*d) - sin(c + d*x)/(2*(a + b)^2*d*(1 - cos(c + d*x))) + sin(c + d*x)/(2*(a - b)^2*d*(1 + cos(c + d*x))) + (a*b^2*sin(c + d*x))/((a^2 - b^2)^2*d*(b + a*cos(c + d*x)))]
    @test_int [csc(c + d*x)^4/(a + b*sec(c + d*x))^2, x, 15, -((2*a^2*b^3*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(7/2)*(a + b)^(7/2)*d)) - (4*a^2*b*(a^2 + b^2)*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(7/2)*(a + b)^(7/2)*d) - sin(c + d*x)/(12*(a + b)^2*d*(1 - cos(c + d*x))^2) - ((a - b)*sin(c + d*x))/(4*(a + b)^3*d*(1 - cos(c + d*x))) - sin(c + d*x)/(12*(a + b)^2*d*(1 - cos(c + d*x))) + sin(c + d*x)/(12*(a - b)^2*d*(1 + cos(c + d*x))^2) + sin(c + d*x)/(12*(a - b)^2*d*(1 + cos(c + d*x))) + ((a + b)*sin(c + d*x))/(4*(a - b)^3*d*(1 + cos(c + d*x))) + (a^3*b^2*sin(c + d*x))/((a^2 - b^2)^3*d*(b + a*cos(c + d*x)))]

    @test_int [sin(c + d*x)^7/(a + b*sec(c + d*x))^3, x, 5, -(((a^6 - 18*a^4*b^2 + 45*a^2*b^4 - 28*b^6)*cos(c + d*x))/(a^9*d)) - (3*b*(3*a^4 - 10*a^2*b^2 + 7*b^4)*cos(c + d*x)^2)/(2*a^8*d) + ((a^4 - 6*a^2*b^2 + 5*b^4)*cos(c + d*x)^3)/(a^7*d) + (b*(9*a^2 - 10*b^2)*cos(c + d*x)^4)/(4*a^6*d) - (3*(a^2 - 2*b^2)*cos(c + d*x)^5)/(5*a^5*d) - (b*cos(c + d*x)^6)/(2*a^4*d) + cos(c + d*x)^7/(7*a^3*d) - (b^3*(a^2 - b^2)^3)/(2*a^10*d*(b + a*cos(c + d*x))^2) + (3*b^2*(a^2 - 3*b^2)*(a^2 - b^2)^2)/(a^10*d*(b + a*cos(c + d*x))) + (3*b*(a^2 - b^2)*(a^4 - 9*a^2*b^2 + 12*b^4)*log(b + a*cos(c + d*x)))/(a^10*d)]
    @test_int [sin(c + d*x)^5/(a + b*sec(c + d*x))^3, x, 5, -(((a^4 - 12*a^2*b^2 + 15*b^4)*cos(c + d*x))/(a^7*d)) - (b*(3*a^2 - 5*b^2)*cos(c + d*x)^2)/(a^6*d) + (2*(a^2 - 3*b^2)*cos(c + d*x)^3)/(3*a^5*d) + (3*b*cos(c + d*x)^4)/(4*a^4*d) - cos(c + d*x)^5/(5*a^3*d) - (b^3*(a^2 - b^2)^2)/(2*a^8*d*(b + a*cos(c + d*x))^2) + (b^2*(3*a^4 - 10*a^2*b^2 + 7*b^4))/(a^8*d*(b + a*cos(c + d*x))) + (b*(3*a^4 - 20*a^2*b^2 + 21*b^4)*log(b + a*cos(c + d*x)))/(a^8*d)]
    @test_int [sin(c + d*x)^3/(a + b*sec(c + d*x))^3, x, 5, -(((a^2 - 6*b^2)*cos(c + d*x))/(a^5*d)) - (3*b*cos(c + d*x)^2)/(2*a^4*d) + cos(c + d*x)^3/(3*a^3*d) - (b^3*(a^2 - b^2))/(2*a^6*d*(b + a*cos(c + d*x))^2) + (b^2*(3*a^2 - 5*b^2))/(a^6*d*(b + a*cos(c + d*x))) + (b*(3*a^2 - 10*b^2)*log(b + a*cos(c + d*x)))/(a^6*d)]
    @test_int [sin(c + d*x)^1/(a + b*sec(c + d*x))^3, x, 5, -(cos(c + d*x)/(a^3*d)) - b^3/(2*a^4*d*(b + a*cos(c + d*x))^2) + (3*b^2)/(a^4*d*(b + a*cos(c + d*x))) + (3*b*log(b + a*cos(c + d*x)))/(a^4*d)]
    @test_int [csc(c + d*x)^1/(a + b*sec(c + d*x))^3, x, 5, -(b^3/(2*a^2*(a^2 - b^2)*d*(b + a*cos(c + d*x))^2)) + (b^2*(3*a^2 - b^2))/(a^2*(a^2 - b^2)^2*d*(b + a*cos(c + d*x))) + log(1 - cos(c + d*x))/(2*(a + b)^3*d) - log(1 + cos(c + d*x))/(2*(a - b)^3*d) + (b*(3*a^2 + b^2)*log(b + a*cos(c + d*x)))/((a^2 - b^2)^3*d)]
    @test_int [csc(c + d*x)^3/(a + b*sec(c + d*x))^3, x, 5, -(b^3/(2*(a^2 - b^2)^2*d*(b + a*cos(c + d*x))^2)) + (b^2*(3*a^2 + b^2))/((a^2 - b^2)^3*d*(b + a*cos(c + d*x))) + ((b*(3*a^2 + b^2) - a*(a^2 + 3*b^2)*cos(c + d*x))*csc(c + d*x)^2)/(2*(a^2 - b^2)^3*d) + ((a - 2*b)*log(1 - cos(c + d*x)))/(4*(a + b)^4*d) - ((a + 2*b)*log(1 + cos(c + d*x)))/(4*(a - b)^4*d) + (b*(3*a^4 + 8*a^2*b^2 + b^4)*log(b + a*cos(c + d*x)))/((a^2 - b^2)^4*d)]
    @test_int [csc(c + d*x)^5/(a + b*sec(c + d*x))^3, x, 7, -((a^2*b^3)/(2*(a^2 - b^2)^3*d*(b + a*cos(c + d*x))^2)) + (3*a^2*b^2*(a^2 + b^2))/((a^2 - b^2)^4*d*(b + a*cos(c + d*x))) + ((4*b*(3*a^4 + 8*a^2*b^2 + b^4) - 3*a*(a^4 + 10*a^2*b^2 + 5*b^4)*cos(c + d*x))*csc(c + d*x)^2)/(8*(a^2 - b^2)^4*d) + ((b*(3*a^2 + b^2) - a*(a^2 + 3*b^2)*cos(c + d*x))*csc(c + d*x)^4)/(4*(a^2 - b^2)^3*d) + (3*a*(a - 3*b)*log(1 - cos(c + d*x)))/(16*(a + b)^5*d) - (3*a*(a + 3*b)*log(1 + cos(c + d*x)))/(16*(a - b)^5*d) + (3*a^2*b*(a^4 + 5*a^2*b^2 + 2*b^4)*log(b + a*cos(c + d*x)))/((a^2 - b^2)^5*d)]

    @test_int [sin(c + d*x)^6/(a + b*sec(c + d*x))^3, x, 11, ((5*a^6 - 180*a^4*b^2 + 600*a^2*b^4 - 448*b^6)*x)/(16*a^9) - (sqrt(a - b)*b*sqrt(a + b)*(6*a^4 - 47*a^2*b^2 + 56*b^4)*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/(a^9*d) + (b*(213*a^4 - 985*a^2*b^2 + 840*b^4)*sin(c + d*x))/(30*a^8*d) - ((43*a^4 - 244*a^2*b^2 + 224*b^4)*cos(c + d*x)*sin(c + d*x))/(16*a^7*d) + ((45*a^4 - 291*a^2*b^2 + 280*b^4)*cos(c + d*x)^2*sin(c + d*x))/(30*a^6*b*d) - ((24*a^4 - 169*a^2*b^2 + 168*b^4)*cos(c + d*x)^3*sin(c + d*x))/(24*a^5*b^2*d) - (cos(c + d*x)^4*sin(c + d*x))/(4*b*d*(b + a*cos(c + d*x))^2) + (a*cos(c + d*x)^5*sin(c + d*x))/(10*b^2*d*(b + a*cos(c + d*x))^2) + ((9*a^4 - 60*a^2*b^2 + 56*b^4)*cos(c + d*x)^5*sin(c + d*x))/(60*a^3*b^2*d*(b + a*cos(c + d*x))^2) + (4*b*cos(c + d*x)^6*sin(c + d*x))/(15*a^2*d*(b + a*cos(c + d*x))^2) - (cos(c + d*x)^7*sin(c + d*x))/(6*a*d*(b + a*cos(c + d*x))^2) + ((15*a^4 - 110*a^2*b^2 + 112*b^4)*cos(c + d*x)^4*sin(c + d*x))/(20*a^4*b^2*d*(b + a*cos(c + d*x)))]
    @test_int [sin(c + d*x)^4/(a + b*sec(c + d*x))^3, x, 9, (3*(a^4 - 24*a^2*b^2 + 40*b^4)*x)/(8*a^7) - (3*b*(2*a^4 - 11*a^2*b^2 + 10*b^4)*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/(a^7*sqrt(a - b)*sqrt(a + b)*d) + (b*(13*a^2 - 30*b^2)*sin(c + d*x))/(2*a^6*d) - (3*(7*a^2 - 20*b^2)*cos(c + d*x)*sin(c + d*x))/(8*a^5*d) + ((3*a^2 - 10*b^2)*cos(c + d*x)^2*sin(c + d*x))/(2*a^4*b*d) - ((4*a^2 - 15*b^2)*cos(c + d*x)^3*sin(c + d*x))/(4*a^3*b^2*d) - ((a^2 - b^2)*cos(c + d*x)^4*sin(c + d*x))/(2*a^2*b*d*(b + a*cos(c + d*x))^2) + ((2*a^2 - 7*b^2)*cos(c + d*x)^4*sin(c + d*x))/(2*a^2*b^2*d*(b + a*cos(c + d*x)))]
    @test_int [sin(c + d*x)^2/(a + b*sec(c + d*x))^3, x, 9, ((a^2 - 12*b^2)*x)/(2*a^5) - (b*(6*a^4 - 19*a^2*b^2 + 12*b^4)*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/(a^5*(a - b)^(3/2)*(a + b)^(3/2)*d) + (b*(11*a^2 - 12*b^2)*sin(c + d*x))/(2*a^4*(a^2 - b^2)*d) - ((5*a^2 - 6*b^2)*cos(c + d*x)*sin(c + d*x))/(2*a^3*(a^2 - b^2)*d) + (cos(c + d*x)^3*sin(c + d*x))/(2*a*d*(b + a*cos(c + d*x))^2) + ((3*a^2 - 4*b^2)*cos(c + d*x)^2*sin(c + d*x))/(2*a^2*(a^2 - b^2)*d*(b + a*cos(c + d*x)))]
    @test_int [csc(c + d*x)^2/(a + b*sec(c + d*x))^3, x, 16, -((2*b^3*(3*a^2 - b^2)*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/(a*(a - b)^(7/2)*(a + b)^(7/2)*d)) - (2*a*b*(3*a^2 + b^2)*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(7/2)*(a + b)^(7/2)*d) - (b^3*(a^2 + 2*b^2)*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/(a*(a - b)^(7/2)*(a + b)^(7/2)*d) - sin(c + d*x)/(2*(a + b)^3*d*(1 - cos(c + d*x))) + sin(c + d*x)/(2*(a - b)^3*d*(1 + cos(c + d*x))) - (b^3*sin(c + d*x))/(2*(a^2 - b^2)^2*d*(b + a*cos(c + d*x))^2) + (3*b^4*sin(c + d*x))/(2*(a^2 - b^2)^3*d*(b + a*cos(c + d*x))) + (b^2*(3*a^2 - b^2)*sin(c + d*x))/((a^2 - b^2)^3*d*(b + a*cos(c + d*x)))]
    @test_int [csc(c + d*x)^4/(a + b*sec(c + d*x))^3, x, 20, -((2*a*b^3*(3*a^2 + b^2)*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(9/2)*(a + b)^(9/2)*d)) - (a*b^3*(a^2 + 2*b^2)*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(9/2)*(a + b)^(9/2)*d) - (2*a*b*(3*a^4 + 8*a^2*b^2 + b^4)*atanh((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(9/2)*(a + b)^(9/2)*d) - sin(c + d*x)/(12*(a + b)^3*d*(1 - cos(c + d*x))^2) - ((a - 2*b)*sin(c + d*x))/(4*(a + b)^4*d*(1 - cos(c + d*x))) - sin(c + d*x)/(12*(a + b)^3*d*(1 - cos(c + d*x))) + sin(c + d*x)/(12*(a - b)^3*d*(1 + cos(c + d*x))^2) + sin(c + d*x)/(12*(a - b)^3*d*(1 + cos(c + d*x))) + ((a + 2*b)*sin(c + d*x))/(4*(a - b)^4*d*(1 + cos(c + d*x))) - (a^2*b^3*sin(c + d*x))/(2*(a^2 - b^2)^3*d*(b + a*cos(c + d*x))^2) + (3*a^2*b^4*sin(c + d*x))/(2*(a^2 - b^2)^4*d*(b + a*cos(c + d*x))) + (a^2*b^2*(3*a^2 + b^2)*sin(c + d*x))/((a^2 - b^2)^4*d*(b + a*cos(c + d*x)))]

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

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

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

    @test_int [(e*sin(c + d*x))^(7/2)/(a + b*sec(c + d*x)), x, 15, -((b*(a^2 - b^2)^(5/4)*e^(7/2)*atan((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/(a^(9/2)*d)) - (b*(a^2 - b^2)^(5/4)*e^(7/2)*atanh((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/(a^(9/2)*d) + (2*(5*a^4 - 28*a^2*b^2 + 21*b^4)*e^4*Elliptic.F((1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(21*a^5*d*sqrt(e*sin(c + d*x))) + (b^2*(a^2 - b^2)^2*e^4*Elliptic.Pi((2*a)/(a - sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(a^5*(a^2 - b^2 - a*sqrt(a^2 - b^2))*d*sqrt(e*sin(c + d*x))) + (b^2*(a^2 - b^2)^2*e^4*Elliptic.Pi((2*a)/(a + sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(a^5*(a^2 - b^2 + a*sqrt(a^2 - b^2))*d*sqrt(e*sin(c + d*x))) + (2*e^3*(21*b*(a^2 - b^2) - a*(5*a^2 - 7*b^2)*cos(c + d*x))*sqrt(e*sin(c + d*x)))/(21*a^4*d) + (2*e*(7*b - 5*a*cos(c + d*x))*(e*sin(c + d*x))^(5/2))/(35*a^2*d)]
    @test_int [(e*sin(c + d*x))^(5/2)/(a + b*sec(c + d*x)), x, 14, (b*(a^2 - b^2)^(3/4)*e^(5/2)*atan((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/(a^(7/2)*d) - (b*(a^2 - b^2)^(3/4)*e^(5/2)*atanh((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/(a^(7/2)*d) - (b^2*(a^2 - b^2)*e^3*Elliptic.Pi((2*a)/(a - sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(a^4*(a - sqrt(a^2 - b^2))*d*sqrt(e*sin(c + d*x))) - (b^2*(a^2 - b^2)*e^3*Elliptic.Pi((2*a)/(a + sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(a^4*(a + sqrt(a^2 - b^2))*d*sqrt(e*sin(c + d*x))) + (2*(3*a^2 - 5*b^2)*e^2*Elliptic.E((1/2)*(c - pi/2 + d*x), 2)*sqrt(e*sin(c + d*x)))/(5*a^3*d*sqrt(sin(c + d*x))) + (2*e*(5*b - 3*a*cos(c + d*x))*(e*sin(c + d*x))^(3/2))/(15*a^2*d)]
    @test_int [(e*sin(c + d*x))^(3/2)/(a + b*sec(c + d*x)), x, 14, -((b*(a^2 - b^2)^(1/4)*e^(3/2)*atan((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/(a^(5/2)*d)) - (b*(a^2 - b^2)^(1/4)*e^(3/2)*atanh((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/(a^(5/2)*d) + (2*(a^2 - 3*b^2)*e^2*Elliptic.F((1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(3*a^3*d*sqrt(e*sin(c + d*x))) + (b^2*(a^2 - b^2)*e^2*Elliptic.Pi((2*a)/(a - sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(a^3*(a^2 - b^2 - a*sqrt(a^2 - b^2))*d*sqrt(e*sin(c + d*x))) + (b^2*(a^2 - b^2)*e^2*Elliptic.Pi((2*a)/(a + sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(a^3*(a^2 - b^2 + a*sqrt(a^2 - b^2))*d*sqrt(e*sin(c + d*x))) + (2*e*(3*b - a*cos(c + d*x))*sqrt(e*sin(c + d*x)))/(3*a^2*d)]
    @test_int [(e*sin(c + d*x))^(1/2)/(a + b*sec(c + d*x)), x, 13, (b*sqrt(e)*atan((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/(a^(3/2)*(a^2 - b^2)^(1/4)*d) - (b*sqrt(e)*atanh((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/(a^(3/2)*(a^2 - b^2)^(1/4)*d) - (b^2*e*Elliptic.Pi((2*a)/(a - sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(a^2*(a - sqrt(a^2 - b^2))*d*sqrt(e*sin(c + d*x))) - (b^2*e*Elliptic.Pi((2*a)/(a + sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(a^2*(a + sqrt(a^2 - b^2))*d*sqrt(e*sin(c + d*x))) + (2*Elliptic.E((1/2)*(c - pi/2 + d*x), 2)*sqrt(e*sin(c + d*x)))/(a*d*sqrt(sin(c + d*x)))]
    @test_int [1/(e*sin(c + d*x))^(1/2)/(a + b*sec(c + d*x)), x, 13, -((b*atan((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/(sqrt(a)*(a^2 - b^2)^(3/4)*d*sqrt(e))) - (b*atanh((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/(sqrt(a)*(a^2 - b^2)^(3/4)*d*sqrt(e)) + (2*Elliptic.F((1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(a*d*sqrt(e*sin(c + d*x))) + (b^2*Elliptic.Pi((2*a)/(a - sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(a*(a^2 - b^2 - a*sqrt(a^2 - b^2))*d*sqrt(e*sin(c + d*x))) + (b^2*Elliptic.Pi((2*a)/(a + sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(a*(a^2 - b^2 + a*sqrt(a^2 - b^2))*d*sqrt(e*sin(c + d*x)))]
    @test_int [1/(e*sin(c + d*x))^(3/2)/(a + b*sec(c + d*x)), x, 14, (sqrt(a)*b*atan((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/((a^2 - b^2)^(5/4)*d*e^(3/2)) - (sqrt(a)*b*atanh((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/((a^2 - b^2)^(5/4)*d*e^(3/2)) + (2*(b - a*cos(c + d*x)))/((a^2 - b^2)*d*e*sqrt(e*sin(c + d*x))) - (b^2*Elliptic.Pi((2*a)/(a - sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/((a^2 - b^2)*(a - sqrt(a^2 - b^2))*d*e*sqrt(e*sin(c + d*x))) - (b^2*Elliptic.Pi((2*a)/(a + sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/((a^2 - b^2)*(a + sqrt(a^2 - b^2))*d*e*sqrt(e*sin(c + d*x))) - (2*a*Elliptic.E((1/2)*(c - pi/2 + d*x), 2)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)*d*e^2*sqrt(sin(c + d*x)))]
    @test_int [1/(e*sin(c + d*x))^(5/2)/(a + b*sec(c + d*x)), x, 14, -((a^(3/2)*b*atan((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/((a^2 - b^2)^(7/4)*d*e^(5/2))) - (a^(3/2)*b*atanh((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/((a^2 - b^2)^(7/4)*d*e^(5/2)) + (2*(b - a*cos(c + d*x)))/(3*(a^2 - b^2)*d*e*(e*sin(c + d*x))^(3/2)) + (2*a*Elliptic.F((1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(3*(a^2 - b^2)*d*e^2*sqrt(e*sin(c + d*x))) + (a*b^2*Elliptic.Pi((2*a)/(a - sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/((a^2 - b^2)*(a^2 - b^2 - a*sqrt(a^2 - b^2))*d*e^2*sqrt(e*sin(c + d*x))) + (a*b^2*Elliptic.Pi((2*a)/(a + sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/((a^2 - b^2)*(a^2 - b^2 + a*sqrt(a^2 - b^2))*d*e^2*sqrt(e*sin(c + d*x)))]
    @test_int [1/(e*sin(c + d*x))^(7/2)/(a + b*sec(c + d*x)), x, 15, (a^(5/2)*b*atan((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/((a^2 - b^2)^(9/4)*d*e^(7/2)) - (a^(5/2)*b*atanh((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/((a^2 - b^2)^(9/4)*d*e^(7/2)) + (2*(b - a*cos(c + d*x)))/(5*(a^2 - b^2)*d*e*(e*sin(c + d*x))^(5/2)) + (2*(5*a^2*b - a*(3*a^2 + 2*b^2)*cos(c + d*x)))/(5*(a^2 - b^2)^2*d*e^3*sqrt(e*sin(c + d*x))) - (a^2*b^2*Elliptic.Pi((2*a)/(a - sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/((a^2 - b^2)^2*(a - sqrt(a^2 - b^2))*d*e^3*sqrt(e*sin(c + d*x))) - (a^2*b^2*Elliptic.Pi((2*a)/(a + sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/((a^2 - b^2)^2*(a + sqrt(a^2 - b^2))*d*e^3*sqrt(e*sin(c + d*x))) - (2*a*(3*a^2 + 2*b^2)*Elliptic.E((1/2)*(c - pi/2 + d*x), 2)*sqrt(e*sin(c + d*x)))/(5*(a^2 - b^2)^2*d*e^4*sqrt(sin(c + d*x)))]

    @test_int [(e*sin(c + d*x))^(9/2)/(a + b*sec(c + d*x))^2, x, 35, -((7*b^3*(a^2 - b^2)^(3/4)*e^(9/2)*atan((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/(2*a^(13/2)*d)) + (2*b*(a^2 - b^2)^(7/4)*e^(9/2)*atan((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/(a^(13/2)*d) + (7*b^3*(a^2 - b^2)^(3/4)*e^(9/2)*atanh((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/(2*a^(13/2)*d) - (2*b*(a^2 - b^2)^(7/4)*e^(9/2)*atanh((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/(a^(13/2)*d) + (7*b^4*(a^2 - b^2)*e^5*Elliptic.Pi((2*a)/(a - sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(2*a^7*(a - sqrt(a^2 - b^2))*d*sqrt(e*sin(c + d*x))) - (2*b^2*(a^2 - b^2)^2*e^5*Elliptic.Pi((2*a)/(a - sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(a^7*(a - sqrt(a^2 - b^2))*d*sqrt(e*sin(c + d*x))) + (7*b^4*(a^2 - b^2)*e^5*Elliptic.Pi((2*a)/(a + sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(2*a^7*(a + sqrt(a^2 - b^2))*d*sqrt(e*sin(c + d*x))) - (2*b^2*(a^2 - b^2)^2*e^5*Elliptic.Pi((2*a)/(a + sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(a^7*(a + sqrt(a^2 - b^2))*d*sqrt(e*sin(c + d*x))) + (14*e^4*Elliptic.E((1/2)*(c - pi/2 + d*x), 2)*sqrt(e*sin(c + d*x)))/(15*a^2*d*sqrt(sin(c + d*x))) - (7*b^2*(3*a^2 - 5*b^2)*e^4*Elliptic.E((1/2)*(c - pi/2 + d*x), 2)*sqrt(e*sin(c + d*x)))/(5*a^6*d*sqrt(sin(c + d*x))) - (4*b^2*(8*a^2 - 5*b^2)*e^4*Elliptic.E((1/2)*(c - pi/2 + d*x), 2)*sqrt(e*sin(c + d*x)))/(5*a^6*d*sqrt(sin(c + d*x))) - (14*e^3*cos(c + d*x)*(e*sin(c + d*x))^(3/2))/(45*a^2*d) - (7*b^2*e^3*(5*b - 3*a*cos(c + d*x))*(e*sin(c + d*x))^(3/2))/(15*a^5*d) + (4*b*e^3*(5*(a^2 - b^2) + 3*a*b*cos(c + d*x))*(e*sin(c + d*x))^(3/2))/(15*a^5*d) + (4*b*e*(e*sin(c + d*x))^(7/2))/(7*a^3*d) - (2*e*cos(c + d*x)*(e*sin(c + d*x))^(7/2))/(9*a^2*d) + (b^2*e*(e*sin(c + d*x))^(7/2))/(a^3*d*(b + a*cos(c + d*x)))]
    @test_int [(e*sin(c + d*x))^(7/2)/(a + b*sec(c + d*x))^2, x, 35, (5*b^3*(a^2 - b^2)^(1/4)*e^(7/2)*atan((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/(2*a^(11/2)*d) - (2*b*(a^2 - b^2)^(5/4)*e^(7/2)*atan((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/(a^(11/2)*d) + (5*b^3*(a^2 - b^2)^(1/4)*e^(7/2)*atanh((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/(2*a^(11/2)*d) - (2*b*(a^2 - b^2)^(5/4)*e^(7/2)*atanh((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/(a^(11/2)*d) + (10*e^4*Elliptic.F((1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(21*a^2*d*sqrt(e*sin(c + d*x))) - (5*b^2*(a^2 - 3*b^2)*e^4*Elliptic.F((1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(3*a^6*d*sqrt(e*sin(c + d*x))) - (4*b^2*(4*a^2 - 3*b^2)*e^4*Elliptic.F((1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(3*a^6*d*sqrt(e*sin(c + d*x))) - (5*b^4*(a^2 - b^2)*e^4*Elliptic.Pi((2*a)/(a - sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(2*a^6*(a^2 - b^2 - a*sqrt(a^2 - b^2))*d*sqrt(e*sin(c + d*x))) + (2*b^2*(a^2 - b^2)^2*e^4*Elliptic.Pi((2*a)/(a - sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(a^6*(a^2 - b^2 - a*sqrt(a^2 - b^2))*d*sqrt(e*sin(c + d*x))) - (5*b^4*(a^2 - b^2)*e^4*Elliptic.Pi((2*a)/(a + sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(2*a^6*(a^2 - b^2 + a*sqrt(a^2 - b^2))*d*sqrt(e*sin(c + d*x))) + (2*b^2*(a^2 - b^2)^2*e^4*Elliptic.Pi((2*a)/(a + sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(a^6*(a^2 - b^2 + a*sqrt(a^2 - b^2))*d*sqrt(e*sin(c + d*x))) - (10*e^3*cos(c + d*x)*sqrt(e*sin(c + d*x)))/(21*a^2*d) - (5*b^2*e^3*(3*b - a*cos(c + d*x))*sqrt(e*sin(c + d*x)))/(3*a^5*d) + (4*b*e^3*(3*(a^2 - b^2) + a*b*cos(c + d*x))*sqrt(e*sin(c + d*x)))/(3*a^5*d) + (4*b*e*(e*sin(c + d*x))^(5/2))/(5*a^3*d) - (2*e*cos(c + d*x)*(e*sin(c + d*x))^(5/2))/(7*a^2*d) + (b^2*e*(e*sin(c + d*x))^(5/2))/(a^3*d*(b + a*cos(c + d*x)))]
    @test_int [(e*sin(c + d*x))^(5/2)/(a + b*sec(c + d*x))^2, x, 32, -((3*b^3*e^(5/2)*atan((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/(2*a^(9/2)*(a^2 - b^2)^(1/4)*d)) + (2*b*(a^2 - b^2)^(3/4)*e^(5/2)*atan((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/(a^(9/2)*d) + (3*b^3*e^(5/2)*atanh((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/(2*a^(9/2)*(a^2 - b^2)^(1/4)*d) - (2*b*(a^2 - b^2)^(3/4)*e^(5/2)*atanh((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/(a^(9/2)*d) + (3*b^4*e^3*Elliptic.Pi((2*a)/(a - sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(2*a^5*(a - sqrt(a^2 - b^2))*d*sqrt(e*sin(c + d*x))) - (2*b^2*(a^2 - b^2)*e^3*Elliptic.Pi((2*a)/(a - sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(a^5*(a - sqrt(a^2 - b^2))*d*sqrt(e*sin(c + d*x))) + (3*b^4*e^3*Elliptic.Pi((2*a)/(a + sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(2*a^5*(a + sqrt(a^2 - b^2))*d*sqrt(e*sin(c + d*x))) - (2*b^2*(a^2 - b^2)*e^3*Elliptic.Pi((2*a)/(a + sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(a^5*(a + sqrt(a^2 - b^2))*d*sqrt(e*sin(c + d*x))) + (6*e^2*Elliptic.E((1/2)*(c - pi/2 + d*x), 2)*sqrt(e*sin(c + d*x)))/(5*a^2*d*sqrt(sin(c + d*x))) - (7*b^2*e^2*Elliptic.E((1/2)*(c - pi/2 + d*x), 2)*sqrt(e*sin(c + d*x)))/(a^4*d*sqrt(sin(c + d*x))) + (4*b*e*(e*sin(c + d*x))^(3/2))/(3*a^3*d) - (2*e*cos(c + d*x)*(e*sin(c + d*x))^(3/2))/(5*a^2*d) + (b^2*e*(e*sin(c + d*x))^(3/2))/(a^3*d*(b + a*cos(c + d*x)))]
    @test_int [(e*sin(c + d*x))^(3/2)/(a + b*sec(c + d*x))^2, x, 32, (b^3*e^(3/2)*atan((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/(2*a^(7/2)*(a^2 - b^2)^(3/4)*d) - (2*b*(a^2 - b^2)^(1/4)*e^(3/2)*atan((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/(a^(7/2)*d) + (b^3*e^(3/2)*atanh((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/(2*a^(7/2)*(a^2 - b^2)^(3/4)*d) - (2*b*(a^2 - b^2)^(1/4)*e^(3/2)*atanh((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/(a^(7/2)*d) + (2*e^2*Elliptic.F((1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(3*a^2*d*sqrt(e*sin(c + d*x))) - (5*b^2*e^2*Elliptic.F((1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(a^4*d*sqrt(e*sin(c + d*x))) - (b^4*e^2*Elliptic.Pi((2*a)/(a - sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(2*a^4*(a^2 - b^2 - a*sqrt(a^2 - b^2))*d*sqrt(e*sin(c + d*x))) + (2*b^2*(a^2 - b^2)*e^2*Elliptic.Pi((2*a)/(a - sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(a^4*(a^2 - b^2 - a*sqrt(a^2 - b^2))*d*sqrt(e*sin(c + d*x))) - (b^4*e^2*Elliptic.Pi((2*a)/(a + sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(2*a^4*(a^2 - b^2 + a*sqrt(a^2 - b^2))*d*sqrt(e*sin(c + d*x))) + (2*b^2*(a^2 - b^2)*e^2*Elliptic.Pi((2*a)/(a + sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(a^4*(a^2 - b^2 + a*sqrt(a^2 - b^2))*d*sqrt(e*sin(c + d*x))) + (4*b*e*sqrt(e*sin(c + d*x)))/(a^3*d) - (2*e*cos(c + d*x)*sqrt(e*sin(c + d*x)))/(3*a^2*d) + (b^2*e*sqrt(e*sin(c + d*x)))/(a^3*d*(b + a*cos(c + d*x)))]
    @test_int [(e*sin(c + d*x))^(1/2)/(a + b*sec(c + d*x))^2, x, 27, (b^3*sqrt(e)*atan((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/(2*a^(5/2)*(a^2 - b^2)^(5/4)*d) + (2*b*sqrt(e)*atan((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/(a^(5/2)*(a^2 - b^2)^(1/4)*d) - (b^3*sqrt(e)*atanh((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/(2*a^(5/2)*(a^2 - b^2)^(5/4)*d) - (2*b*sqrt(e)*atanh((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/(a^(5/2)*(a^2 - b^2)^(1/4)*d) - (2*b^2*e*Elliptic.Pi((2*a)/(a - sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(a^3*(a - sqrt(a^2 - b^2))*d*sqrt(e*sin(c + d*x))) - (b^4*e*Elliptic.Pi((2*a)/(a - sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(2*a^3*(a^2 - b^2)*(a - sqrt(a^2 - b^2))*d*sqrt(e*sin(c + d*x))) - (2*b^2*e*Elliptic.Pi((2*a)/(a + sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(a^3*(a + sqrt(a^2 - b^2))*d*sqrt(e*sin(c + d*x))) - (b^4*e*Elliptic.Pi((2*a)/(a + sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(2*a^3*(a^2 - b^2)*(a + sqrt(a^2 - b^2))*d*sqrt(e*sin(c + d*x))) + (2*Elliptic.E((1/2)*(c - pi/2 + d*x), 2)*sqrt(e*sin(c + d*x)))/(a^2*d*sqrt(sin(c + d*x))) - (b^2*Elliptic.E((1/2)*(c - pi/2 + d*x), 2)*sqrt(e*sin(c + d*x)))/(a^2*(a^2 - b^2)*d*sqrt(sin(c + d*x))) + (b^2*(e*sin(c + d*x))^(3/2))/(a*(a^2 - b^2)*d*e*(b + a*cos(c + d*x)))]
    @test_int [1/(e*sin(c + d*x))^(1/2)/(a + b*sec(c + d*x))^2, x, 27, -((3*b^3*atan((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/(2*a^(3/2)*(a^2 - b^2)^(7/4)*d*sqrt(e))) - (2*b*atan((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/(a^(3/2)*(a^2 - b^2)^(3/4)*d*sqrt(e)) - (3*b^3*atanh((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/(2*a^(3/2)*(a^2 - b^2)^(7/4)*d*sqrt(e)) - (2*b*atanh((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/(a^(3/2)*(a^2 - b^2)^(3/4)*d*sqrt(e)) + (2*Elliptic.F((1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(a^2*d*sqrt(e*sin(c + d*x))) + (b^2*Elliptic.F((1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(a^2*(a^2 - b^2)*d*sqrt(e*sin(c + d*x))) + (2*b^2*Elliptic.Pi((2*a)/(a - sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(a^2*(a^2 - b^2 - a*sqrt(a^2 - b^2))*d*sqrt(e*sin(c + d*x))) + (3*b^4*Elliptic.Pi((2*a)/(a - sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(2*a^2*(a^2 - b^2)*(a^2 - b^2 - a*sqrt(a^2 - b^2))*d*sqrt(e*sin(c + d*x))) + (2*b^2*Elliptic.Pi((2*a)/(a + sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(a^2*(a^2 - b^2 + a*sqrt(a^2 - b^2))*d*sqrt(e*sin(c + d*x))) + (3*b^4*Elliptic.Pi((2*a)/(a + sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(2*a^2*(a^2 - b^2)*(a^2 - b^2 + a*sqrt(a^2 - b^2))*d*sqrt(e*sin(c + d*x))) + (b^2*sqrt(e*sin(c + d*x)))/(a*(a^2 - b^2)*d*e*(b + a*cos(c + d*x)))]
    @test_int [1/(e*sin(c + d*x))^(3/2)/(a + b*sec(c + d*x))^2, x, 33, (5*b^3*atan((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/(2*sqrt(a)*(a^2 - b^2)^(9/4)*d*e^(3/2)) + (2*b*atan((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/(sqrt(a)*(a^2 - b^2)^(5/4)*d*e^(3/2)) - (5*b^3*atanh((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/(2*sqrt(a)*(a^2 - b^2)^(9/4)*d*e^(3/2)) - (2*b*atanh((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/(sqrt(a)*(a^2 - b^2)^(5/4)*d*e^(3/2)) - (2*cos(c + d*x))/(a^2*d*e*sqrt(e*sin(c + d*x))) + b^2/(a*(a^2 - b^2)*d*e*(b + a*cos(c + d*x))*sqrt(e*sin(c + d*x))) + (4*b*(a - b*cos(c + d*x)))/(a^2*(a^2 - b^2)*d*e*sqrt(e*sin(c + d*x))) + (b^2*(5*a*b - (3*a^2 + 2*b^2)*cos(c + d*x)))/(a^2*(a^2 - b^2)^2*d*e*sqrt(e*sin(c + d*x))) - (5*b^4*Elliptic.Pi((2*a)/(a - sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(2*a*(a^2 - b^2)^2*(a - sqrt(a^2 - b^2))*d*e*sqrt(e*sin(c + d*x))) - (2*b^2*Elliptic.Pi((2*a)/(a - sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(a*(a^2 - b^2)*(a - sqrt(a^2 - b^2))*d*e*sqrt(e*sin(c + d*x))) - (5*b^4*Elliptic.Pi((2*a)/(a + sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(2*a*(a^2 - b^2)^2*(a + sqrt(a^2 - b^2))*d*e*sqrt(e*sin(c + d*x))) - (2*b^2*Elliptic.Pi((2*a)/(a + sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(a*(a^2 - b^2)*(a + sqrt(a^2 - b^2))*d*e*sqrt(e*sin(c + d*x))) - (2*Elliptic.E((1/2)*(c - pi/2 + d*x), 2)*sqrt(e*sin(c + d*x)))/(a^2*d*e^2*sqrt(sin(c + d*x))) - (4*b^2*Elliptic.E((1/2)*(c - pi/2 + d*x), 2)*sqrt(e*sin(c + d*x)))/(a^2*(a^2 - b^2)*d*e^2*sqrt(sin(c + d*x))) - (b^2*(3*a^2 + 2*b^2)*Elliptic.E((1/2)*(c - pi/2 + d*x), 2)*sqrt(e*sin(c + d*x)))/(a^2*(a^2 - b^2)^2*d*e^2*sqrt(sin(c + d*x)))]
    @test_int [1/(e*sin(c + d*x))^(5/2)/(a + b*sec(c + d*x))^2, x, 33, -((7*sqrt(a)*b^3*atan((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/(2*(a^2 - b^2)^(11/4)*d*e^(5/2))) - (2*sqrt(a)*b*atan((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/((a^2 - b^2)^(7/4)*d*e^(5/2)) - (7*sqrt(a)*b^3*atanh((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/(2*(a^2 - b^2)^(11/4)*d*e^(5/2)) - (2*sqrt(a)*b*atanh((sqrt(a)*sqrt(e*sin(c + d*x)))/((a^2 - b^2)^(1/4)*sqrt(e))))/((a^2 - b^2)^(7/4)*d*e^(5/2)) - (2*cos(c + d*x))/(3*a^2*d*e*(e*sin(c + d*x))^(3/2)) + b^2/(a*(a^2 - b^2)*d*e*(b + a*cos(c + d*x))*(e*sin(c + d*x))^(3/2)) + (4*b*(a - b*cos(c + d*x)))/(3*a^2*(a^2 - b^2)*d*e*(e*sin(c + d*x))^(3/2)) + (b^2*(7*a*b - (5*a^2 + 2*b^2)*cos(c + d*x)))/(3*a^2*(a^2 - b^2)^2*d*e*(e*sin(c + d*x))^(3/2)) + (2*Elliptic.F((1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(3*a^2*d*e^2*sqrt(e*sin(c + d*x))) + (4*b^2*Elliptic.F((1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(3*a^2*(a^2 - b^2)*d*e^2*sqrt(e*sin(c + d*x))) + (b^2*(5*a^2 + 2*b^2)*Elliptic.F((1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(3*a^2*(a^2 - b^2)^2*d*e^2*sqrt(e*sin(c + d*x))) + (7*b^4*Elliptic.Pi((2*a)/(a - sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(2*(a^2 - b^2)^2*(a^2 - b^2 - a*sqrt(a^2 - b^2))*d*e^2*sqrt(e*sin(c + d*x))) + (2*b^2*Elliptic.Pi((2*a)/(a - sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/((a^2 - b^2)*(a^2 - b^2 - a*sqrt(a^2 - b^2))*d*e^2*sqrt(e*sin(c + d*x))) + (7*b^4*Elliptic.Pi((2*a)/(a + sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(2*(a^2 - b^2)^2*(a^2 - b^2 + a*sqrt(a^2 - b^2))*d*e^2*sqrt(e*sin(c + d*x))) + (2*b^2*Elliptic.Pi((2*a)/(a + sqrt(a^2 - b^2)), (1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/((a^2 - b^2)*(a^2 - b^2 + a*sqrt(a^2 - b^2))*d*e^2*sqrt(e*sin(c + d*x)))]

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

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

    #= [sin(e + f*x)^6*sqrt(a + b*sec(e + f*x)), x, 11, -(((a - b)*sqrt(a + b)*(3248*a^4 - 1416*a^2*b^2 + 315*b^4)*cos(e + f*x)^(3/2)*csc(e + f*x)*Elliptic.E(asin(sqrt(b + a*cos(e + f*x))/(sqrt(a + b)*sqrt(cos(e + f*x)))), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b)))*sqrt(a + b*sec(e + f*x)))/(7680*a^5*f*sqrt(b + a*cos(e + f*x)))) + (1/(7680*a^5*f*sqrt(b + a*cos(e + f*x))))*sqrt(a + b)*(10080*a^5 - 3248*a^4*b - 928*a^3*b^2 + 1416*a^2*b^3 + 210*a*b^4 - 315*b^5)*cos(e + f*x)^(3/2)*csc(e + f*x)*Elliptic.F(asin(sqrt(b + a*cos(e + f*x))/(sqrt(a + b)*sqrt(cos(e + f*x)))), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b)))*sqrt(a + b*sec(e + f*x)) - (sqrt(a + b)*(320*a^6 + 240*a^4*b^2 - 100*a^2*b^4 + 21*b^6)*cos(e + f*x)^(3/2)*csc(e + f*x)*Elliptic.Pi((a + b)/a, asin(sqrt(b + a*cos(e + f*x))/(sqrt(a + b)*sqrt(cos(e + f*x)))), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b)))*sqrt(a + b*sec(e + f*x)))/(512*a^6*f*sqrt(b + a*cos(e + f*x))) - (b*(3248*a^4 - 1416*a^2*b^2 + 315*b^4)*sqrt(a + b*sec(e + f*x))*sin(e + f*x))/(7680*a^5*f) - ((880*a^4 - 416*a^2*b^2 + 105*b^4)*cos(e + f*x)*sqrt(a + b*sec(e + f*x))*sin(e + f*x))/(1280*a^4*f) - (2*cos(e + f*x)*(b + a*cos(e + f*x))*sqrt(a + b*sec(e + f*x))*sin(e + f*x))/(b*f) + ((1920*a^4 - 428*a^2*b^2 + 105*b^4)*cos(e + f*x)*(b + a*cos(e + f*x))*sqrt(a + b*sec(e + f*x))*sin(e + f*x))/(960*a^4*b*f) + (8*a*cos(e + f*x)^2*(b + a*cos(e + f*x))*sqrt(a + b*sec(e + f*x))*sin(e + f*x))/(3*b^2*f) - ((1280*a^4 - 260*a^2*b^2 + 63*b^4)*cos(e + f*x)^2*(b + a*cos(e + f*x))*sqrt(a + b*sec(e + f*x))*sin(e + f*x))/(480*a^3*b^2*f) + (3*b*cos(e + f*x)^3*(b + a*cos(e + f*x))*sqrt(a + b*sec(e + f*x))*sin(e + f*x))/(20*a^2*f) - (cos(e + f*x)^4*(b + a*cos(e + f*x))*sqrt(a + b*sec(e + f*x))*sin(e + f*x))/(6*a*f)]
    @test_int [sin(e + f*x)^4*sqrt(a + b*sec(e + f*x)), x, 9, -(((a - b)*sqrt(a + b)*(68*a^2 - 15*b^2)*cos(e + f*x)^(3/2)*csc(e + f*x)*Elliptic.E(asin(sqrt(b + a*cos(e + f*x))/(sqrt(a + b)*sqrt(cos(e + f*x)))), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b)))*sqrt(a + b*sec(e + f*x)))/(192*a^3*f*sqrt(b + a*cos(e + f*x)))) + (sqrt(a + b)*(264*a^3 - 68*a^2*b - 10*a*b^2 + 15*b^3)*cos(e + f*x)^(3/2)*csc(e + f*x)*Elliptic.F(asin(sqrt(b + a*cos(e + f*x))/(sqrt(a + b)*sqrt(cos(e + f*x)))), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b)))*sqrt(a + b*sec(e + f*x)))/(192*a^3*f*sqrt(b + a*cos(e + f*x))) - (sqrt(a + b)*(48*a^4 + 24*a^2*b^2 - 5*b^4)*cos(e + f*x)^(3/2)*csc(e + f*x)*Elliptic.Pi((a + b)/a, asin(sqrt(b + a*cos(e + f*x))/(sqrt(a + b)*sqrt(cos(e + f*x)))), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b)))*sqrt(a + b*sec(e + f*x)))/(64*a^4*f*sqrt(b + a*cos(e + f*x))) - (b*(68*a^2 - 15*b^2)*sqrt(a + b*sec(e + f*x))*sin(e + f*x))/(192*a^3*f) - (5*(4*a^2 - b^2)*cos(e + f*x)*sqrt(a + b*sec(e + f*x))*sin(e + f*x))/(32*a^2*f) - (5*b*cos(e + f*x)*(b + a*cos(e + f*x))*sqrt(a + b*sec(e + f*x))*sin(e + f*x))/(24*a^2*f) + (cos(e + f*x)^2*(b + a*cos(e + f*x))*sqrt(a + b*sec(e + f*x))*sin(e + f*x))/(4*a*f)]
    @test_int [sin(e + f*x)^2*sqrt(a + b*sec(e + f*x)), x, 9, -(((a - b)*sqrt(a + b)*cot(e + f*x)*Elliptic.E(asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(4*a*f)) + ((6*a - b)*sqrt(a + b)*cot(e + f*x)*Elliptic.F(asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(4*a*f) - (sqrt(a + b)*(4*a^2 + b^2)*cot(e + f*x)*Elliptic.Pi((a + b)/a, asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(4*a^2*f) - (b*sqrt(a + b*sec(e + f*x))*sin(e + f*x))/(4*a*f) - (cos(e + f*x)*sqrt(a + b*sec(e + f*x))*sin(e + f*x))/(2*f)] =#
    @test_int [sin(e + f*x)^0*sqrt(a + b*sec(e + f*x)), x, 1, -((2*cot(e + f*x)*Elliptic.Pi(a/(a + b), asin(sqrt(a + b)/sqrt(a + b*sec(e + f*x))), (a - b)/(a + b))*sqrt(-((b*(1 - sec(e + f*x)))/(a + b*sec(e + f*x))))*sqrt((b*(1 + sec(e + f*x)))/(a + b*sec(e + f*x)))*(a + b*sec(e + f*x)))/(sqrt(a + b)*f))]
    @test_int [csc(e + f*x)^2*sqrt(a + b*sec(e + f*x)), x, 2, (sqrt(a + b)*cot(e + f*x)*Elliptic.F(asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/f - (cot(e + f*x)*sqrt(a + b*sec(e + f*x)))/f]
    #= [csc(e + f*x)^4*sqrt(a + b*sec(e + f*x)), x, 0, 0]
    @test_int [csc(e + f*x)^6*sqrt(a + b*sec(e + f*x)), x, 0, 0] =#

    #= [sin(e + f*x)^4*(a + b*sec(e + f*x))^(3/2), x, 10, -(((a - b)*sqrt(a + b)*(236*a^2 + 3*b^2)*cos(e + f*x)^(3/2)*csc(e + f*x)*Elliptic.E(asin(sqrt(b + a*cos(e + f*x))/(sqrt(a + b)*sqrt(cos(e + f*x)))), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b)))*sqrt(a + b*sec(e + f*x)))/(64*a^2*f*sqrt(b + a*cos(e + f*x)))) + (sqrt(a + b)*(216*a^3 - 236*a^2*b + 2*a*b^2 - 3*b^3)*cos(e + f*x)^(3/2)*csc(e + f*x)*Elliptic.F(asin(sqrt(b + a*cos(e + f*x))/(sqrt(a + b)*sqrt(cos(e + f*x)))), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b)))*sqrt(a + b*sec(e + f*x)))/(64*a^2*f*sqrt(b + a*cos(e + f*x))) - (3*sqrt(a + b)*(16*a^4 - 24*a^2*b^2 + b^4)*cos(e + f*x)^(3/2)*csc(e + f*x)*Elliptic.Pi((a + b)/a, asin(sqrt(b + a*cos(e + f*x))/(sqrt(a + b)*sqrt(cos(e + f*x)))), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b)))*sqrt(a + b*sec(e + f*x)))/(64*a^3*f*sqrt(b + a*cos(e + f*x))) - (b*(236*a^2 + 3*b^2)*sqrt(a + b*sec(e + f*x))*sin(e + f*x))/(64*a^2*f) - (3*(28*a^2 + b^2)*cos(e + f*x)*sqrt(a + b*sec(e + f*x))*sin(e + f*x))/(32*a*f) - ((16*a^2 + b^2)*cos(e + f*x)*(b + a*cos(e + f*x))*sqrt(a + b*sec(e + f*x))*sin(e + f*x))/(8*a*b*f) + (2*(b + a*cos(e + f*x))^2*sqrt(a + b*sec(e + f*x))*sin(e + f*x))/(b*f) + (cos(e + f*x)*(b + a*cos(e + f*x))^2*sqrt(a + b*sec(e + f*x))*sin(e + f*x))/(4*a*f)]
    @test_int [sin(e + f*x)^2*(a + b*sec(e + f*x))^(3/2), x, 10, -((13*(a - b)*sqrt(a + b)*cos(e + f*x)^(3/2)*csc(e + f*x)*Elliptic.E(asin(sqrt(b + a*cos(e + f*x))/(sqrt(a + b)*sqrt(cos(e + f*x)))), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b)))*sqrt(a + b*sec(e + f*x)))/(4*f*sqrt(b + a*cos(e + f*x)))) + ((14*a - 13*b)*sqrt(a + b)*cos(e + f*x)^(3/2)*csc(e + f*x)*Elliptic.F(asin(sqrt(b + a*cos(e + f*x))/(sqrt(a + b)*sqrt(cos(e + f*x)))), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b)))*sqrt(a + b*sec(e + f*x)))/(4*f*sqrt(b + a*cos(e + f*x))) - (sqrt(a + b)*(4*a^2 - 3*b^2)*cos(e + f*x)^(3/2)*csc(e + f*x)*Elliptic.Pi((a + b)/a, asin(sqrt(b + a*cos(e + f*x))/(sqrt(a + b)*sqrt(cos(e + f*x)))), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b)))*sqrt(a + b*sec(e + f*x)))/(4*a*f*sqrt(b + a*cos(e + f*x))) - (13*b*sqrt(a + b*sec(e + f*x))*sin(e + f*x))/(4*f) - (5*a*cos(e + f*x)*sqrt(a + b*sec(e + f*x))*sin(e + f*x))/(2*f) + (2*(b + a*cos(e + f*x))*sqrt(a + b*sec(e + f*x))*sin(e + f*x))/f] =#
    @test_int [sin(e + f*x)^0*(a + b*sec(e + f*x))^(3/2), x, 5, -((2*(a - b)*sqrt(a + b)*cot(e + f*x)*Elliptic.E(asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/f) + (2*(2*a - b)*sqrt(a + b)*cot(e + f*x)*Elliptic.F(asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/f - (2*a*sqrt(a + b)*cot(e + f*x)*Elliptic.Pi((a + b)/a, asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/f]
    @test_int [csc(e + f*x)^2*(a + b*sec(e + f*x))^(3/2), x, 4, -((3*(a - b)*sqrt(a + b)*cot(e + f*x)*Elliptic.E(asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/f) + (3*(a - b)*sqrt(a + b)*cot(e + f*x)*Elliptic.F(asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/f - (cot(e + f*x)*(a + b*sec(e + f*x))^(3/2))/f]
    #= [csc(e + f*x)^4*(a + b*sec(e + f*x))^(3/2), x, 0, 0]
    @test_int [csc(e + f*x)^6*(a + b*sec(e + f*x))^(3/2), x, 0, 0] =#

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

    #= [sin(e + f*x)^4/sqrt(a + b*sec(e + f*x)), x, 9, ((a - b)*sqrt(a + b)*(188*a^2 - 105*b^2)*sqrt(cos(e + f*x))*sqrt(b + a*cos(e + f*x))*csc(e + f*x)*Elliptic.E(asin(sqrt(b + a*cos(e + f*x))/(sqrt(a + b)*sqrt(cos(e + f*x)))), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(192*a^4*f*sqrt(a + b*sec(e + f*x))) - (sqrt(a + b)*(120*a^3 - 188*a^2*b - 70*a*b^2 + 105*b^3)*sqrt(cos(e + f*x))*sqrt(b + a*cos(e + f*x))*csc(e + f*x)*Elliptic.F(asin(sqrt(b + a*cos(e + f*x))/(sqrt(a + b)*sqrt(cos(e + f*x)))), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(192*a^4*f*sqrt(a + b*sec(e + f*x))) - (sqrt(a + b)*(48*a^4 - 72*a^2*b^2 + 35*b^4)*sqrt(cos(e + f*x))*sqrt(b + a*cos(e + f*x))*csc(e + f*x)*Elliptic.Pi((a + b)/a, asin(sqrt(b + a*cos(e + f*x))/(sqrt(a + b)*sqrt(cos(e + f*x)))), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(64*a^5*f*sqrt(a + b*sec(e + f*x))) - (5*(12*a^2 - 7*b^2)*(b + a*cos(e + f*x))*sin(e + f*x))/(96*a^3*f*sqrt(a + b*sec(e + f*x))) - (7*b*cos(e + f*x)*(b + a*cos(e + f*x))*sin(e + f*x))/(24*a^2*f*sqrt(a + b*sec(e + f*x))) + (cos(e + f*x)^2*(b + a*cos(e + f*x))*sin(e + f*x))/(4*a*f*sqrt(a + b*sec(e + f*x))) + (b*(188*a^2 - 105*b^2)*(b + a*cos(e + f*x))*tan(e + f*x))/(192*a^4*f*sqrt(a + b*sec(e + f*x)))]
    @test_int [sin(e + f*x)^2/sqrt(a + b*sec(e + f*x)), x, 9, (3*(a - b)*sqrt(a + b)*sqrt(cos(e + f*x))*sqrt(b + a*cos(e + f*x))*csc(e + f*x)*Elliptic.E(asin(sqrt(b + a*cos(e + f*x))/(sqrt(a + b)*sqrt(cos(e + f*x)))), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(4*a^2*f*sqrt(a + b*sec(e + f*x))) - ((2*a - 3*b)*sqrt(a + b)*sqrt(cos(e + f*x))*sqrt(b + a*cos(e + f*x))*csc(e + f*x)*Elliptic.F(asin(sqrt(b + a*cos(e + f*x))/(sqrt(a + b)*sqrt(cos(e + f*x)))), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(4*a^2*f*sqrt(a + b*sec(e + f*x))) - (sqrt(a + b)*(4*a^2 - 3*b^2)*sqrt(cos(e + f*x))*sqrt(b + a*cos(e + f*x))*csc(e + f*x)*Elliptic.Pi((a + b)/a, asin(sqrt(b + a*cos(e + f*x))/(sqrt(a + b)*sqrt(cos(e + f*x)))), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(4*a^3*f*sqrt(a + b*sec(e + f*x))) - ((b + a*cos(e + f*x))*sin(e + f*x))/(2*a*f*sqrt(a + b*sec(e + f*x))) + (3*b*(b + a*cos(e + f*x))*tan(e + f*x))/(4*a^2*f*sqrt(a + b*sec(e + f*x)))] =#
    @test_int [sin(e + f*x)^0/sqrt(a + b*sec(e + f*x)), x, 1, -((2*sqrt(a + b)*cot(e + f*x)*Elliptic.Pi((a + b)/a, asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(a*f))]
    @test_int [csc(e + f*x)^2/sqrt(a + b*sec(e + f*x)), x, 6, (cot(e + f*x)*Elliptic.E(asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(sqrt(a + b)*f) - (cot(e + f*x)*Elliptic.F(asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(sqrt(a + b)*f) - cot(e + f*x)/(f*sqrt(a + b*sec(e + f*x))) + (b^2*tan(e + f*x))/((a^2 - b^2)*f*sqrt(a + b*sec(e + f*x)))]
    #= [csc(e + f*x)^4/sqrt(a + b*sec(e + f*x)), x, 0, 0]
    @test_int [csc(e + f*x)^6/sqrt(a + b*sec(e + f*x)), x, 0, 0] =#

    #= [sin(e + f*x)^4/(a + b*sec(e + f*x))^(3/2), x, 10, (3*(a - b)*sqrt(a + b)*(92*a^2 - 105*b^2)*sqrt(cos(e + f*x))*sqrt(b + a*cos(e + f*x))*csc(e + f*x)*Elliptic.E(asin(sqrt(b + a*cos(e + f*x))/(sqrt(a + b)*sqrt(cos(e + f*x)))), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(64*a^5*f*sqrt(a + b*sec(e + f*x))) - (3*sqrt(a + b)*(56*a^3 - 92*a^2*b - 70*a*b^2 + 105*b^3)*sqrt(cos(e + f*x))*sqrt(b + a*cos(e + f*x))*csc(e + f*x)*Elliptic.F(asin(sqrt(b + a*cos(e + f*x))/(sqrt(a + b)*sqrt(cos(e + f*x)))), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(64*a^5*f*sqrt(a + b*sec(e + f*x))) - (3*sqrt(a + b)*(16*a^4 - 120*a^2*b^2 + 105*b^4)*sqrt(cos(e + f*x))*sqrt(b + a*cos(e + f*x))*csc(e + f*x)*Elliptic.Pi((a + b)/a, asin(sqrt(b + a*cos(e + f*x))/(sqrt(a + b)*sqrt(cos(e + f*x)))), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(64*a^6*f*sqrt(a + b*sec(e + f*x))) - (2*(a^2 - b^2)*cos(e + f*x)^2*sin(e + f*x))/(a^2*b*f*sqrt(a + b*sec(e + f*x))) - (21*(4*a^2 - 5*b^2)*(b + a*cos(e + f*x))*sin(e + f*x))/(32*a^4*f*sqrt(a + b*sec(e + f*x))) + ((16*a^2 - 21*b^2)*cos(e + f*x)*(b + a*cos(e + f*x))*sin(e + f*x))/(8*a^3*b*f*sqrt(a + b*sec(e + f*x))) + (cos(e + f*x)^2*(b + a*cos(e + f*x))*sin(e + f*x))/(4*a^2*f*sqrt(a + b*sec(e + f*x))) + (3*b*(92*a^2 - 105*b^2)*(b + a*cos(e + f*x))*tan(e + f*x))/(64*a^5*f*sqrt(a + b*sec(e + f*x)))]
    @test_int [sin(e + f*x)^2/(a + b*sec(e + f*x))^(3/2), x, 10, (15*(a - b)*sqrt(a + b)*sqrt(cos(e + f*x))*sqrt(b + a*cos(e + f*x))*csc(e + f*x)*Elliptic.E(asin(sqrt(b + a*cos(e + f*x))/(sqrt(a + b)*sqrt(cos(e + f*x)))), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(4*a^3*f*sqrt(a + b*sec(e + f*x))) - (5*(2*a - 3*b)*sqrt(a + b)*sqrt(cos(e + f*x))*sqrt(b + a*cos(e + f*x))*csc(e + f*x)*Elliptic.F(asin(sqrt(b + a*cos(e + f*x))/(sqrt(a + b)*sqrt(cos(e + f*x)))), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(4*a^3*f*sqrt(a + b*sec(e + f*x))) - (sqrt(a + b)*(4*a^2 - 15*b^2)*sqrt(cos(e + f*x))*sqrt(b + a*cos(e + f*x))*csc(e + f*x)*Elliptic.Pi((a + b)/a, asin(sqrt(b + a*cos(e + f*x))/(sqrt(a + b)*sqrt(cos(e + f*x)))), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(4*a^4*f*sqrt(a + b*sec(e + f*x))) + (2*cos(e + f*x)*sin(e + f*x))/(a*f*sqrt(a + b*sec(e + f*x))) - (5*(b + a*cos(e + f*x))*sin(e + f*x))/(2*a^2*f*sqrt(a + b*sec(e + f*x))) + (15*b*(b + a*cos(e + f*x))*tan(e + f*x))/(4*a^3*f*sqrt(a + b*sec(e + f*x)))] =#
    @test_int [sin(e + f*x)^0/(a + b*sec(e + f*x))^(3/2), x, 6, (2*cot(e + f*x)*Elliptic.E(asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(a*sqrt(a + b)*f) - (2*cot(e + f*x)*Elliptic.F(asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(a*sqrt(a + b)*f) - (2*sqrt(a + b)*cot(e + f*x)*Elliptic.Pi((a + b)/a, asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(a^2*f) + (2*b^2*tan(e + f*x))/(a*(a^2 - b^2)*f*sqrt(a + b*sec(e + f*x)))]
    @test_int [csc(e + f*x)^2/(a + b*sec(e + f*x))^(3/2), x, 6, (4*a*cot(e + f*x)*Elliptic.E(asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/((a - b)*(a + b)^(3/2)*f) - ((3*a - b)*cot(e + f*x)*Elliptic.F(asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/((a - b)*(a + b)^(3/2)*f) - cot(e + f*x)/(f*(a + b*sec(e + f*x))^(3/2)) + (b^2*tan(e + f*x))/((a^2 - b^2)*f*(a + b*sec(e + f*x))^(3/2)) + (4*a*b^2*tan(e + f*x))/((a^2 - b^2)^2*f*sqrt(a + b*sec(e + f*x)))]
    #= [csc(e + f*x)^4/(a + b*sec(e + f*x))^(3/2), x, 0, 0]
    @test_int [csc(e + f*x)^6/(a + b*sec(e + f*x))^(3/2), x, 0, 0] =#

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

    @test_int [(e*sin(c + d*x))^m*(a + b*sec(c + d*x))^3, x, 9, (a^3*cos(c + d*x)*HypergeometricFunctions._₂F₁(1/2, (1 + m)/2, (3 + m)/2, sin(c + d*x)^2)*(e*sin(c + d*x))^(1 + m))/(d*e*(1 + m)*sqrt(cos(c + d*x)^2)) + (3*a^2*b*HypergeometricFunctions._₂F₁(1, (1 + m)/2, (3 + m)/2, sin(c + d*x)^2)*(e*sin(c + d*x))^(1 + m))/(d*e*(1 + m)) + (b^3*HypergeometricFunctions._₂F₁(2, (1 + m)/2, (3 + m)/2, sin(c + d*x)^2)*(e*sin(c + d*x))^(1 + m))/(d*e*(1 + m)) + (3*a*b^2*sqrt(cos(c + d*x)^2)*HypergeometricFunctions._₂F₁(3/2, (1 + m)/2, (3 + m)/2, sin(c + d*x)^2)*sec(c + d*x)*(e*sin(c + d*x))^(1 + m))/(d*e*(1 + m))]
    @test_int [(e*sin(c + d*x))^m*(a + b*sec(c + d*x))^2, x, 9, (a^2*cos(c + d*x)*HypergeometricFunctions._₂F₁(1/2, (1 + m)/2, (3 + m)/2, sin(c + d*x)^2)*sin(c + d*x)*(e*sin(c + d*x))^m)/(d*(1 + m)*sqrt(cos(c + d*x)^2)) + (2*a*b*HypergeometricFunctions._₂F₁(1, (1 + m)/2, (3 + m)/2, sin(c + d*x)^2)*(e*sin(c + d*x))^(1 + m))/(d*e*(1 + m)) + (b^2*sqrt(cos(c + d*x)^2)*HypergeometricFunctions._₂F₁(3/2, (1 + m)/2, (3 + m)/2, sin(c + d*x)^2)*(e*sin(c + d*x))^m*tan(c + d*x))/(d*(1 + m))]
    @test_int [(e*sin(c + d*x))^m*(a + b*sec(c + d*x))^1, x, 5, (a*cos(c + d*x)*HypergeometricFunctions._₂F₁(1/2, (1 + m)/2, (3 + m)/2, sin(c + d*x)^2)*(e*sin(c + d*x))^(1 + m))/(d*e*(1 + m)*sqrt(cos(c + d*x)^2)) + (b*HypergeometricFunctions._₂F₁(1, (1 + m)/2, (3 + m)/2, sin(c + d*x)^2)*(e*sin(c + d*x))^(1 + m))/(d*e*(1 + m))]
    @test_int [(e*sin(c + d*x))^m/(a + b*sec(c + d*x))^1, x, 4, -((b*e*AppellF1(1 - m, (1 - m)/2, (1 - m)/2, 2 - m, -((a - b)/(b + a*cos(c + d*x))), (a + b)/(b + a*cos(c + d*x)))*(-((a*(1 - cos(c + d*x)))/(b + a*cos(c + d*x))))^((1 - m)/2)*((a*(1 + cos(c + d*x)))/(b + a*cos(c + d*x)))^((1 - m)/2)*(e*sin(c + d*x))^(-1 + m))/(a^2*d*(1 - m))) + (cos(c + d*x)*HypergeometricFunctions._₂F₁(1/2, (1 + m)/2, (3 + m)/2, sin(c + d*x)^2)*(e*sin(c + d*x))^(1 + m))/(a*d*e*(1 + m)*sqrt(cos(c + d*x)^2))]
    @test_int [(e*sin(c + d*x))^m/(a + b*sec(c + d*x))^2, x, 6, -((2*b*e*AppellF1(1 - m, (1 - m)/2, (1 - m)/2, 2 - m, -((a - b)/(b + a*cos(c + d*x))), (a + b)/(b + a*cos(c + d*x)))*(-((a*(1 - cos(c + d*x)))/(b + a*cos(c + d*x))))^((1 - m)/2)*((a*(1 + cos(c + d*x)))/(b + a*cos(c + d*x)))^((1 - m)/2)*(e*sin(c + d*x))^(-1 + m))/(a^3*d*(1 - m))) + (b^2*e*AppellF1(2 - m, (1 - m)/2, (1 - m)/2, 3 - m, -((a - b)/(b + a*cos(c + d*x))), (a + b)/(b + a*cos(c + d*x)))*(-((a*(1 - cos(c + d*x)))/(b + a*cos(c + d*x))))^((1 - m)/2)*((a*(1 + cos(c + d*x)))/(b + a*cos(c + d*x)))^((1 - m)/2)*(e*sin(c + d*x))^(-1 + m))/(a^3*d*(2 - m)*(b + a*cos(c + d*x))) + (cos(c + d*x)*HypergeometricFunctions._₂F₁(1/2, (1 + m)/2, (3 + m)/2, sin(c + d*x)^2)*(e*sin(c + d*x))^(1 + m))/(a^2*d*e*(1 + m)*sqrt(cos(c + d*x)^2))]
    @test_int [(e*sin(c + d*x))^m/(a + b*sec(c + d*x))^3, x, 7, -((3*b*e*AppellF1(1 - m, (1 - m)/2, (1 - m)/2, 2 - m, -((a - b)/(b + a*cos(c + d*x))), (a + b)/(b + a*cos(c + d*x)))*(-((a*(1 - cos(c + d*x)))/(b + a*cos(c + d*x))))^((1 - m)/2)*((a*(1 + cos(c + d*x)))/(b + a*cos(c + d*x)))^((1 - m)/2)*(e*sin(c + d*x))^(-1 + m))/(a^4*d*(1 - m))) - (b^3*e*AppellF1(3 - m, (1 - m)/2, (1 - m)/2, 4 - m, -((a - b)/(b + a*cos(c + d*x))), (a + b)/(b + a*cos(c + d*x)))*(-((a*(1 - cos(c + d*x)))/(b + a*cos(c + d*x))))^((1 - m)/2)*((a*(1 + cos(c + d*x)))/(b + a*cos(c + d*x)))^((1 - m)/2)*(e*sin(c + d*x))^(-1 + m))/(a^4*d*(3 - m)*(b + a*cos(c + d*x))^2) + (3*b^2*e*AppellF1(2 - m, (1 - m)/2, (1 - m)/2, 3 - m, -((a - b)/(b + a*cos(c + d*x))), (a + b)/(b + a*cos(c + d*x)))*(-((a*(1 - cos(c + d*x)))/(b + a*cos(c + d*x))))^((1 - m)/2)*((a*(1 + cos(c + d*x)))/(b + a*cos(c + d*x)))^((1 - m)/2)*(e*sin(c + d*x))^(-1 + m))/(a^4*d*(2 - m)*(b + a*cos(c + d*x))) + (cos(c + d*x)*HypergeometricFunctions._₂F₁(1/2, (1 + m)/2, (3 + m)/2, sin(c + d*x)^2)*(e*sin(c + d*x))^(1 + m))/(a^3*d*e*(1 + m)*sqrt(cos(c + d*x)^2))]

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

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

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

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

    @test_int [sin(c + d*x)^4*(a + b*sec(c + d*x))^n, x, 0, Unintegrable((a + b*sec(c + d*x))^n*sin(c + d*x)^4, x)]
    @test_int [sin(c + d*x)^2*(a + b*sec(c + d*x))^n, x, 0, Unintegrable((a + b*sec(c + d*x))^n*sin(c + d*x)^2, x)]
    @test_int [csc(c + d*x)^2*(a + b*sec(c + d*x))^n, x, 4, -((cot(c + d*x)*(a + b*sec(c + d*x))^n)/d) + (sqrt(2)*b*n*AppellF1(1/2, 1/2, 1 - n, 3/2, (1/2)*(1 - sec(c + d*x)), (b*(1 - sec(c + d*x)))/(a + b))*(a + b*sec(c + d*x))^n*tan(c + d*x))/(((a + b*sec(c + d*x))/(a + b))^n*((a + b)*d*sqrt(1 + sec(c + d*x))))]
    @test_int [csc(c + d*x)^4*(a + b*sec(c + d*x))^n, x, -1, -((3*AppellF1(-(1/2), 5/2, -n, 1/2, (1/2)*(1 - sec(c + d*x)), (b*(1 - sec(c + d*x)))/(a + b))*cot(c + d*x)*sqrt(1 + sec(c + d*x))*(a + b*sec(c + d*x))^n)/(((a + b*sec(c + d*x))/(a + b))^n*(2*sqrt(2)*d))) - (AppellF1(-(3/2), 5/2, -n, -(1/2), (1/2)*(1 - sec(c + d*x)), (b*(1 - sec(c + d*x)))/(a + b))*cot(c + d*x)^3*(1 + sec(c + d*x))^(3/2)*(a + b*sec(c + d*x))^n)/(((a + b*sec(c + d*x))/(a + b))^n*(6*sqrt(2)*d)) + (AppellF1(1/2, 3/2, -n, 3/2, (1/2)*(1 - sec(c + d*x)), (b*(1 - sec(c + d*x)))/(a + b))*(a + b*sec(c + d*x))^n*tan(c + d*x))/(((a + b*sec(c + d*x))/(a + b))^n*(sqrt(2)*d*sqrt(1 + sec(c + d*x)))) + (AppellF1(1/2, 5/2, -n, 3/2, (1/2)*(1 - sec(c + d*x)), (b*(1 - sec(c + d*x)))/(a + b))*(a + b*sec(c + d*x))^n*tan(c + d*x))/(((a + b*sec(c + d*x))/(a + b))^n*(2*sqrt(2)*d*sqrt(1 + sec(c + d*x))))]

    @test_int [sin(c + d*x)^(3/2)*(a + b*sec(c + d*x))^n, x, 0, Unintegrable((a + b*sec(c + d*x))^n*sin(c + d*x)^(3/2), x)]
    @test_int [sin(c + d*x)^(1/2)*(a + b*sec(c + d*x))^n, x, 0, Unintegrable((a + b*sec(c + d*x))^n*sqrt(sin(c + d*x)), x)]
    @test_int [1/sin(c + d*x)^(1/2)*(a + b*sec(c + d*x))^n, x, 0, Unintegrable((a + b*sec(c + d*x))^n/sqrt(sin(c + d*x)), x)]
    @test_int [1/sin(c + d*x)^(3/2)*(a + b*sec(c + d*x))^n, x, 0, Unintegrable((a + b*sec(c + d*x))^n/sin(c + d*x)^(3/2), x)]

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

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(d*csc(e+f*x))^n*(a+b*sec(e+f*x))^m*when*a^2-b^2=0=#

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

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

    @test_int [(e*csc(c + d*x))^(5/2)*(a + a*sec(c + d*x)), x, 11, -((2*a*e^2*cot(c + d*x)*sqrt(e*csc(c + d*x)))/(3*d)) - (2*a*e^2*csc(c + d*x)*sqrt(e*csc(c + d*x)))/(3*d) + (a*e^2*atan(sqrt(sin(c + d*x)))*sqrt(e*csc(c + d*x))*sqrt(sin(c + d*x)))/d + (a*e^2*atanh(sqrt(sin(c + d*x)))*sqrt(e*csc(c + d*x))*sqrt(sin(c + d*x)))/d + (2*a*e^2*sqrt(e*csc(c + d*x))*Elliptic.F((1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(3*d)]
    @test_int [(e*csc(c + d*x))^(3/2)*(a + a*sec(c + d*x)), x, 11, -((2*a*e*sqrt(e*csc(c + d*x)))/d) - (2*a*e*cos(c + d*x)*sqrt(e*csc(c + d*x)))/d - (a*e*atan(sqrt(sin(c + d*x)))*sqrt(e*csc(c + d*x))*sqrt(sin(c + d*x)))/d + (a*e*atanh(sqrt(sin(c + d*x)))*sqrt(e*csc(c + d*x))*sqrt(sin(c + d*x)))/d - (2*a*e*sqrt(e*csc(c + d*x))*Elliptic.E((1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/d]
    @test_int [sqrt(e*csc(c + d*x))*(a + a*sec(c + d*x)), x, 9, (a*atan(sqrt(sin(c + d*x)))*sqrt(e*csc(c + d*x))*sqrt(sin(c + d*x)))/d + (a*atanh(sqrt(sin(c + d*x)))*sqrt(e*csc(c + d*x))*sqrt(sin(c + d*x)))/d + (2*a*sqrt(e*csc(c + d*x))*Elliptic.F((1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/d]
    @test_int [(a + a*sec(c + d*x))/sqrt(e*csc(c + d*x)), x, 9, -((a*atan(sqrt(sin(c + d*x))))/(d*sqrt(e*csc(c + d*x))*sqrt(sin(c + d*x)))) + (a*atanh(sqrt(sin(c + d*x))))/(d*sqrt(e*csc(c + d*x))*sqrt(sin(c + d*x))) + (2*a*Elliptic.E((1/2)*(c - pi/2 + d*x), 2))/(d*sqrt(e*csc(c + d*x))*sqrt(sin(c + d*x)))]
    @test_int [(a + a*sec(c + d*x))/(e*csc(c + d*x))^(3/2), x, 11, -((2*a)/(d*e*sqrt(e*csc(c + d*x)))) - (2*a*cos(c + d*x))/(3*d*e*sqrt(e*csc(c + d*x))) + (a*atan(sqrt(sin(c + d*x))))/(d*e*sqrt(e*csc(c + d*x))*sqrt(sin(c + d*x))) + (a*atanh(sqrt(sin(c + d*x))))/(d*e*sqrt(e*csc(c + d*x))*sqrt(sin(c + d*x))) + (2*a*Elliptic.F((1/2)*(c - pi/2 + d*x), 2))/(3*d*e*sqrt(e*csc(c + d*x))*sqrt(sin(c + d*x)))]
    @test_int [(a + a*sec(c + d*x))/(e*csc(c + d*x))^(5/2), x, 11, -((a*atan(sqrt(sin(c + d*x))))/(d*e^2*sqrt(e*csc(c + d*x))*sqrt(sin(c + d*x)))) + (a*atanh(sqrt(sin(c + d*x))))/(d*e^2*sqrt(e*csc(c + d*x))*sqrt(sin(c + d*x))) + (6*a*Elliptic.E((1/2)*(c - pi/2 + d*x), 2))/(5*d*e^2*sqrt(e*csc(c + d*x))*sqrt(sin(c + d*x))) - (2*a*sin(c + d*x))/(3*d*e^2*sqrt(e*csc(c + d*x))) - (2*a*cos(c + d*x)*sin(c + d*x))/(5*d*e^2*sqrt(e*csc(c + d*x)))]

    @test_int [(e*csc(c + d*x))^(5/2)*(a + a*sec(c + d*x))^2, x, 15, -((2*a^2*e^2*cot(c + d*x)*sqrt(e*csc(c + d*x)))/(3*d)) - (4*a^2*e^2*csc(c + d*x)*sqrt(e*csc(c + d*x)))/(3*d) - (2*a^2*e^2*csc(c + d*x)*sqrt(e*csc(c + d*x))*sec(c + d*x))/(3*d) + (2*a^2*e^2*atan(sqrt(sin(c + d*x)))*sqrt(e*csc(c + d*x))*sqrt(sin(c + d*x)))/d + (2*a^2*e^2*atanh(sqrt(sin(c + d*x)))*sqrt(e*csc(c + d*x))*sqrt(sin(c + d*x)))/d + (7*a^2*e^2*sqrt(e*csc(c + d*x))*Elliptic.F((1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(3*d) + (5*a^2*e^2*sqrt(e*csc(c + d*x))*tan(c + d*x))/(3*d)]
    @test_int [(e*csc(c + d*x))^(3/2)*(a + a*sec(c + d*x))^2, x, 15, -((4*a^2*e*sqrt(e*csc(c + d*x)))/d) - (2*a^2*e*cos(c + d*x)*sqrt(e*csc(c + d*x)))/d - (2*a^2*e*sqrt(e*csc(c + d*x))*sec(c + d*x))/d - (2*a^2*e*atan(sqrt(sin(c + d*x)))*sqrt(e*csc(c + d*x))*sqrt(sin(c + d*x)))/d + (2*a^2*e*atanh(sqrt(sin(c + d*x)))*sqrt(e*csc(c + d*x))*sqrt(sin(c + d*x)))/d - (5*a^2*e*sqrt(e*csc(c + d*x))*Elliptic.E((1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/d + (3*a^2*e*sqrt(e*csc(c + d*x))*sin(c + d*x)*tan(c + d*x))/d]
    @test_int [sqrt(e*csc(c + d*x))*(a + a*sec(c + d*x))^2, x, 12, (2*a^2*atan(sqrt(sin(c + d*x)))*sqrt(e*csc(c + d*x))*sqrt(sin(c + d*x)))/d + (2*a^2*atanh(sqrt(sin(c + d*x)))*sqrt(e*csc(c + d*x))*sqrt(sin(c + d*x)))/d + (3*a^2*sqrt(e*csc(c + d*x))*Elliptic.F((1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/d + (a^2*sqrt(e*csc(c + d*x))*tan(c + d*x))/d]
    @test_int [(a + a*sec(c + d*x))^2/sqrt(e*csc(c + d*x)), x, 12, -((2*a^2*atan(sqrt(sin(c + d*x))))/(d*sqrt(e*csc(c + d*x))*sqrt(sin(c + d*x)))) + (2*a^2*atanh(sqrt(sin(c + d*x))))/(d*sqrt(e*csc(c + d*x))*sqrt(sin(c + d*x))) + (a^2*Elliptic.E((1/2)*(c - pi/2 + d*x), 2))/(d*sqrt(e*csc(c + d*x))*sqrt(sin(c + d*x))) + (a^2*tan(c + d*x))/(d*sqrt(e*csc(c + d*x)))]
    @test_int [(a + a*sec(c + d*x))^2/(e*csc(c + d*x))^(3/2), x, 14, -((4*a^2)/(d*e*sqrt(e*csc(c + d*x)))) - (2*a^2*cos(c + d*x))/(3*d*e*sqrt(e*csc(c + d*x))) + (a^2*sec(c + d*x))/(d*e*sqrt(e*csc(c + d*x))) + (2*a^2*atan(sqrt(sin(c + d*x))))/(d*e*sqrt(e*csc(c + d*x))*sqrt(sin(c + d*x))) + (2*a^2*atanh(sqrt(sin(c + d*x))))/(d*e*sqrt(e*csc(c + d*x))*sqrt(sin(c + d*x))) - (a^2*Elliptic.F((1/2)*(c - pi/2 + d*x), 2))/(3*d*e*sqrt(e*csc(c + d*x))*sqrt(sin(c + d*x)))]
    @test_int [(a + a*sec(c + d*x))^2/(e*csc(c + d*x))^(5/2), x, 14, -((2*a^2*atan(sqrt(sin(c + d*x))))/(d*e^2*sqrt(e*csc(c + d*x))*sqrt(sin(c + d*x)))) + (2*a^2*atanh(sqrt(sin(c + d*x))))/(d*e^2*sqrt(e*csc(c + d*x))*sqrt(sin(c + d*x))) - (9*a^2*Elliptic.E((1/2)*(c - pi/2 + d*x), 2))/(5*d*e^2*sqrt(e*csc(c + d*x))*sqrt(sin(c + d*x))) - (4*a^2*sin(c + d*x))/(3*d*e^2*sqrt(e*csc(c + d*x))) - (2*a^2*cos(c + d*x)*sin(c + d*x))/(5*d*e^2*sqrt(e*csc(c + d*x))) + (a^2*tan(c + d*x))/(d*e^2*sqrt(e*csc(c + d*x)))]

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

    @test_int [(e*csc(c + d*x))^(5/2)/(a + a*sec(c + d*x)), x, 8, -((4*e^2*cot(c + d*x)*sqrt(e*csc(c + d*x)))/(21*a*d)) + (2*e^2*cot(c + d*x)*csc(c + d*x)^2*sqrt(e*csc(c + d*x)))/(7*a*d) - (2*e^2*csc(c + d*x)^3*sqrt(e*csc(c + d*x)))/(7*a*d) + (4*e^2*sqrt(e*csc(c + d*x))*Elliptic.F((1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(21*a*d)]
    @test_int [(e*csc(c + d*x))^(3/2)/(a + a*sec(c + d*x)), x, 8, -((4*e*cos(c + d*x)*sqrt(e*csc(c + d*x)))/(5*a*d)) + (2*e*cot(c + d*x)*csc(c + d*x)*sqrt(e*csc(c + d*x)))/(5*a*d) - (2*e*csc(c + d*x)^2*sqrt(e*csc(c + d*x)))/(5*a*d) - (4*e*sqrt(e*csc(c + d*x))*Elliptic.E((1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(5*a*d)]
    @test_int [sqrt(e*csc(c + d*x))/(a + a*sec(c + d*x)), x, 7, (2*cot(c + d*x)*sqrt(e*csc(c + d*x)))/(3*a*d) - (2*csc(c + d*x)*sqrt(e*csc(c + d*x)))/(3*a*d) + (4*sqrt(e*csc(c + d*x))*Elliptic.F((1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(3*a*d)]
    @test_int [1/(sqrt(e*csc(c + d*x))*(a + a*sec(c + d*x))), x, 7, (2*cot(c + d*x))/(a*d*sqrt(e*csc(c + d*x))) - (2*csc(c + d*x))/(a*d*sqrt(e*csc(c + d*x))) + (4*Elliptic.E((1/2)*(c - pi/2 + d*x), 2))/(a*d*sqrt(e*csc(c + d*x))*sqrt(sin(c + d*x)))]
    @test_int [1/((e*csc(c + d*x))^(3/2)*(a + a*sec(c + d*x))), x, 7, 2/(a*d*e*sqrt(e*csc(c + d*x))) - (2*cos(c + d*x))/(3*a*d*e*sqrt(e*csc(c + d*x))) - (4*Elliptic.F((1/2)*(c - pi/2 + d*x), 2))/(3*a*d*e*sqrt(e*csc(c + d*x))*sqrt(sin(c + d*x)))]
    @test_int [1/((e*csc(c + d*x))^(5/2)*(a + a*sec(c + d*x))), x, 7, -((4*Elliptic.E((1/2)*(c - pi/2 + d*x), 2))/(5*a*d*e^2*sqrt(e*csc(c + d*x))*sqrt(sin(c + d*x)))) + (2*sin(c + d*x))/(3*a*d*e^2*sqrt(e*csc(c + d*x))) - (2*cos(c + d*x)*sin(c + d*x))/(5*a*d*e^2*sqrt(e*csc(c + d*x)))]
    @test_int [1/((e*csc(c + d*x))^(7/2)*(a + a*sec(c + d*x))), x, 8, -((2*cos(c + d*x))/(21*a*d*e^3*sqrt(e*csc(c + d*x)))) + (2*cos(c + d*x)^3)/(7*a*d*e^3*sqrt(e*csc(c + d*x))) - (4*Elliptic.F((1/2)*(c - pi/2 + d*x), 2))/(21*a*d*e^3*sqrt(e*csc(c + d*x))*sqrt(sin(c + d*x))) + (2*sin(c + d*x)^2)/(5*a*d*e^3*sqrt(e*csc(c + d*x)))]

    @test_int [(e*csc(c + d*x))^(5/2)/(a + a*sec(c + d*x))^2, x, 16, -((4*e^2*cot(c + d*x)*sqrt(e*csc(c + d*x)))/(231*a^2*d)) + (16*e^2*cot(c + d*x)*csc(c + d*x)^2*sqrt(e*csc(c + d*x)))/(77*a^2*d) - (2*e^2*cot(c + d*x)^3*csc(c + d*x)^2*sqrt(e*csc(c + d*x)))/(11*a^2*d) - (4*e^2*csc(c + d*x)^3*sqrt(e*csc(c + d*x)))/(7*a^2*d) - (2*e^2*cot(c + d*x)*csc(c + d*x)^4*sqrt(e*csc(c + d*x)))/(11*a^2*d) + (4*e^2*csc(c + d*x)^5*sqrt(e*csc(c + d*x)))/(11*a^2*d) + (4*e^2*sqrt(e*csc(c + d*x))*Elliptic.F((1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(231*a^2*d)]
    @test_int [(e*csc(c + d*x))^(3/2)/(a + a*sec(c + d*x))^2, x, 16, -((4*e*cos(c + d*x)*sqrt(e*csc(c + d*x)))/(15*a^2*d)) + (16*e*cot(c + d*x)*csc(c + d*x)*sqrt(e*csc(c + d*x)))/(45*a^2*d) - (2*e*cot(c + d*x)^3*csc(c + d*x)*sqrt(e*csc(c + d*x)))/(9*a^2*d) - (4*e*csc(c + d*x)^2*sqrt(e*csc(c + d*x)))/(5*a^2*d) - (2*e*cot(c + d*x)*csc(c + d*x)^3*sqrt(e*csc(c + d*x)))/(9*a^2*d) + (4*e*csc(c + d*x)^4*sqrt(e*csc(c + d*x)))/(9*a^2*d) - (4*e*sqrt(e*csc(c + d*x))*Elliptic.E((1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(15*a^2*d)]
    @test_int [sqrt(e*csc(c + d*x))/(a + a*sec(c + d*x))^2, x, 14, (16*cot(c + d*x)*sqrt(e*csc(c + d*x)))/(21*a^2*d) - (2*cot(c + d*x)^3*sqrt(e*csc(c + d*x)))/(7*a^2*d) - (4*csc(c + d*x)*sqrt(e*csc(c + d*x)))/(3*a^2*d) - (2*cot(c + d*x)*csc(c + d*x)^2*sqrt(e*csc(c + d*x)))/(7*a^2*d) + (4*csc(c + d*x)^3*sqrt(e*csc(c + d*x)))/(7*a^2*d) + (20*sqrt(e*csc(c + d*x))*Elliptic.F((1/2)*(c - pi/2 + d*x), 2)*sqrt(sin(c + d*x)))/(21*a^2*d)]
    @test_int [1/(sqrt(e*csc(c + d*x))*(a + a*sec(c + d*x))^2), x, 14, (16*cot(c + d*x))/(5*a^2*d*sqrt(e*csc(c + d*x))) - (2*cot(c + d*x)^3)/(5*a^2*d*sqrt(e*csc(c + d*x))) - (4*csc(c + d*x))/(a^2*d*sqrt(e*csc(c + d*x))) - (2*cot(c + d*x)*csc(c + d*x)^2)/(5*a^2*d*sqrt(e*csc(c + d*x))) + (4*csc(c + d*x)^3)/(5*a^2*d*sqrt(e*csc(c + d*x))) + (28*Elliptic.E((1/2)*(c - pi/2 + d*x), 2))/(5*a^2*d*sqrt(e*csc(c + d*x))*sqrt(sin(c + d*x)))]
    @test_int [1/((e*csc(c + d*x))^(3/2)*(a + a*sec(c + d*x))^2), x, 13, 4/(a^2*d*e*sqrt(e*csc(c + d*x))) - (4*cos(c + d*x))/(3*a^2*d*e*sqrt(e*csc(c + d*x))) - (2*cos(c + d*x)*cot(c + d*x)^2)/(3*a^2*d*e*sqrt(e*csc(c + d*x))) - (2*cot(c + d*x)*csc(c + d*x))/(3*a^2*d*e*sqrt(e*csc(c + d*x))) + (4*csc(c + d*x)^2)/(3*a^2*d*e*sqrt(e*csc(c + d*x))) - (4*Elliptic.F((1/2)*(c - pi/2 + d*x), 2))/(a^2*d*e*sqrt(e*csc(c + d*x))*sqrt(sin(c + d*x)))]
    @test_int [1/((e*csc(c + d*x))^(5/2)*(a + a*sec(c + d*x))^2), x, 13, -((2*cot(c + d*x))/(a^2*d*e^2*sqrt(e*csc(c + d*x)))) - (2*cos(c + d*x)^2*cot(c + d*x))/(a^2*d*e^2*sqrt(e*csc(c + d*x))) + (4*csc(c + d*x))/(a^2*d*e^2*sqrt(e*csc(c + d*x))) - (44*Elliptic.E((1/2)*(c - pi/2 + d*x), 2))/(5*a^2*d*e^2*sqrt(e*csc(c + d*x))*sqrt(sin(c + d*x))) + (4*sin(c + d*x))/(3*a^2*d*e^2*sqrt(e*csc(c + d*x))) - (12*cos(c + d*x)*sin(c + d*x))/(5*a^2*d*e^2*sqrt(e*csc(c + d*x)))]
    @test_int [1/((e*csc(c + d*x))^(7/2)*(a + a*sec(c + d*x))^2), x, 13, -(4/(a^2*d*e^3*sqrt(e*csc(c + d*x)))) + (26*cos(c + d*x))/(21*a^2*d*e^3*sqrt(e*csc(c + d*x))) + (2*cos(c + d*x)^3)/(7*a^2*d*e^3*sqrt(e*csc(c + d*x))) + (52*Elliptic.F((1/2)*(c - pi/2 + d*x), 2))/(21*a^2*d*e^3*sqrt(e*csc(c + d*x))*sqrt(sin(c + d*x))) + (4*sin(c + d*x)^2)/(5*a^2*d*e^3*sqrt(e*csc(c + d*x)))]

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

max_stars_repo_path: test/4 Trig functions/4.5 Secant/4.5.1.3 (d sin)^n (a+b sec)^m.jl