Primitive of Reciprocal of x fourth plus a fourth

Theorem

 * $\displaystyle \int \frac {\mathrm d x} {x^4 + a^4} = \frac 1 {4 a^3 \sqrt 2} \ln \left({\frac {x^2 + a x \sqrt 2 + a^2} {x^2 - a x \sqrt 2 + a^2} }\right) - \frac 1 {2 a^3 \sqrt 2} \left({\arctan \left({1 - \frac {x \sqrt 2} a}\right) - \arctan \left({1 + \frac {x \sqrt 2} a}\right)}\right)$

Proof
Then:

Similarly:

Thus: