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

Theorem

 * $\displaystyle \int \frac {\sqrt {x^2 + a^2} } {x^3} \ \mathrm d x = \frac {-\sqrt {x^2 + a^2} } {2 x^2} - \frac 1 {2 a} \ln \left({\frac {a + \sqrt {x^2 + a^2} } x}\right) + C$

Proof
Let:

Also see

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