Primitive of x cubed over x squared plus a squared squared/Proof 1

Theorem

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

Proof
Let: