Primitive of Reciprocal of x by x cubed plus a cubed

Theorem

 * $\displaystyle \int \frac {\mathrm d x} {x \left({x^3 + a^3}\right)} = \frac 1 {3 a^3} \ln \left\vert{\frac {x^3} {x^3 + a^3} }\right\vert + C$

Proof
From Primitive of Reciprocal of x by Power of x plus Power of a:
 * $\displaystyle \int \frac {\mathrm d x} {x \left({x^n + a^n}\right)} = \frac 1 {n a^n} \ln \left\vert{\frac {x^n} {x^n + a^n} }\right\vert + C$

Setting $n = 3$:

directly.