Primitive of Function of Root of x squared minus a squared

Theorem

 * $\displaystyle \int F \left({\sqrt {x^2 - a^2}}\right) \ \mathrm d x = a \int \sec u \tan u \ F \left({a \tan u}\right) \ \mathrm d u$

where $x = a \sec u$.

Proof
First note that:

Then: