Primitive of Function of Root of a squared plus x squared

Theorem

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

where $x = a \tan u$.

Proof
First note that:

Then: