Primitive of x cubed over a squared minus x squared

Theorem

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

for $x^2 < a^2$.

Proof
Let:

Also see

 * Primitive of $\dfrac {x^3} {x^2 - a^2}$