Primitive of Reciprocal of x squared by x squared plus a squared squared

Theorem

 * $\displaystyle \int \frac {\d x} {x^2 \paren {x^2 + a^2}^2} = -\frac 1 {a^4 x} - \frac x {2 a^4 \paren {x^2 + a^2} } - \frac 3 {2 a^5} \arctan \frac x a + C$

Proof
Let: