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

Theorem

 * $\displaystyle \int \frac {\sqrt {x^2 - a^2} } {x^3} \ \mathrm d x = \frac {-\sqrt {x^2 - a^2} } {2 x^2} + \frac 1 {2 a} \operatorname{arcsec} \left\vert{\frac x a}\right\vert + C$

Proof
Let:

Also see

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