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

Theorem

 * $\ds \int \frac {\paren {\sqrt {x^2 - a^2} }^3} x \rd x = \frac {\paren {\sqrt {x^2 - a^2} }^3} 3 - a^2 \sqrt {x^2 - a^2} + a^3 \arcsec \size {\frac x a} + C$

for $\size x \ge a$.

Proof
Let:

Also see

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