Primitive of Reciprocal of x fourth plus a fourth/Lemma 1

Lemma for Primitive of Reciprocal of $x^4 + a^4$

 * $\displaystyle \int \frac {\d x} {x^2 + a x \sqrt 2 + a^2} = \frac {\sqrt 2} a \, \map \arctan {1 + \frac {x \sqrt 2} a}$

Proof
The discriminant of $x^2 + a x \sqrt 2 + a^2$ is:

Thus: