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

Theorem

 * $\displaystyle \int \frac {\mathrm d x} {x^3 \left({x^2 + a^2}\right)^2} = -\frac 1 {2 a^4 x^2} - \frac 1 {2 a^4 \left({x^2 + a^2}\right)} - \frac 1 {a^6} \ln \left({\frac {x^2} {x^2 + a^2} }\right) + C$

Proof
Let: