Primitive of Reciprocal of x by Root of x squared plus a squared

Theorem

 * $\displaystyle \int \frac {\mathrm d x} {x \sqrt {x^2 + a^2} } = -\frac 1 a \operatorname{csch}^{-1} {\frac x a} + C$

Proof
Let:

Also see

 * Primitive of Reciprocal of Variable times Root of Variable Squared minus Constant Squared