Primitive of Root of x squared minus a squared cubed over x

Theorem

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

Proof
Let:

Also see

 * Primitive of $\dfrac{\left({\sqrt {x^2 + a^2} }\right)^3} x$
 * Primitive of $\dfrac{\left({\sqrt {a^2 - x^2} }\right)^3} x$