Primitive of Root of x squared plus a squared over x

Theorem

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

Proof
Let: