\section{Primitive of Reciprocal of x squared by x cubed plus a cubed squared}
Tags: Primitives involving x cubed plus a cubed

\begin{theorem}
:$\displaystyle \int \frac {\mathrm d x} {x^2 \left({x^3 + a^3}\right)^2} = \frac {-1} {a^6 x} - \frac {x^2} {3 a^6 \left({x^3 + a^3}\right)} - \frac 4 {3 a^6} \int \frac {x \ \mathrm d x} {x^3 + a^3}$
\end{theorem}

\begin{proof}
{{begin-eqn}}
{{eqn | l = \int \frac {\mathrm d x} {x^2 \left({x^3 + a^3}\right)^2}
      | r = \int \frac {a^3 \ \mathrm d x} {a^3 x^2 \left({x^3 + a^3}\right)^2}
      | c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $a^3$
}}
{{eqn | r = \int \frac {\left({x^3 + a^3 - x^3}\right) \ \mathrm d x} {a^3 x^2 \left({x^3 + a^3}\right)^2}
      | c = 
}}
{{eqn | r = \frac 1 {a^3} \int \frac {\left({x^3 + a^3}\right) \ \mathrm d x} {x^2 \left({x^3 + a^3}\right)^2} - \frac 1 {a^3} \int \frac {x^3 \ \mathrm d x} {x^2 \left({x^3 + a^3}\right)^2}
      | c = [[Linear Combination of Integrals]]
}}
{{eqn | r = \frac 1 {a^3} \int \frac {\mathrm d x} {x^2 \left({x^3 + a^3}\right)} - \frac 1 {a^3} \int \frac {x \ \mathrm d x} {\left({x^3 + a^3}\right)^2}
      | c = simplifying
}}
{{eqn | r = \frac 1 {a^3} \left({\frac {-1} {a^3 x} - \frac 1 {a^3} \int \frac {x \ \mathrm d x} {\left({x^3 + a^3}\right)} }\right) - \frac 1 {a^3} \int \frac {x \ \mathrm d x} {\left({x^3 + a^3}\right)^2}
      | c = [[Primitive of Reciprocal of x squared by x cubed plus a cubed/Lemma|Primitive of $\dfrac 1 {x^2 \left({x^3 + a^3}\right)}$: Lemma]]
}}
{{eqn | r = \frac {-1} {a^6 x} - \frac 1 {a^6} \int \frac {x \ \mathrm d x} {\left({x^3 + a^3}\right)} - \frac 1 {a^3} \int \frac {x \ \mathrm d x} {\left({x^3 + a^3}\right)^2}
      | c = simplifying
}}
{{end-eqn}}
Now consider:
{{begin-eqn}}
{{eqn | l = \int \frac {x \ \mathrm d x} {\left({x^3 + a^3}\right)^2}
      | r = \int \frac {x^3 \ \mathrm d x} {x^2 \left({x^3 + a^3}\right)^2}
      | c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $x^2$
}}
{{eqn | r = \int \frac {\left({x^3 + a^3 - a^3}\right) \ \mathrm d x} {x^2 \left({x^3 + a^3}\right)^2}
      | c = 
}}
{{eqn | r = \int \frac {\left({x^3 + a^3}\right) \ \mathrm d x} {x^2 \left({x^3 + a^3}\right)^2} - \int \frac {a^3 \ \mathrm d x} {x^2 \left({x^3 + a^3}\right)^2}
      | c = [[Linear Combination of Integrals]]
}}
{{eqn | r = \int \frac {\mathrm d x} {x^2 \left({x^3 + a^3}\right)} - a^3 \int \frac {\mathrm d x} {x^2 \left({x^3 + a^3}\right)^2}
      | c = simplifying
}}
{{end-eqn}}
{{proof wanted|seems to be going nowhere}}
\end{proof}


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