Primitive of Reciprocal of x by x fourth plus a fourth

Theorem

 * $\displaystyle \int \frac {\mathrm d x} {x \left({x^4 + a^4}\right)} = \frac 1 {4 a^4} \ln \left({\frac {x^4} {x^4 + a^4} }\right)$

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 $n = 4$:

directly.