Primitive of Inverse Hyperbolic Sine of x over a over x squared

Theorem

 * $\ds \int \frac {\sinh^{-1} \dfrac x a \rd x} {x^2} = \frac {-\sinh^{-1} \dfrac x a} x - \frac 1 a \map \ln {\frac a x + \frac {\sqrt {a^2 + x^2} } {\size x} }$

Proof
With a view to expressing the primitive in the form:
 * $\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$

let:

and let:

Then:

Also see

 * Primitive of $\dfrac {\cosh^{-1} \frac x a} {x^2}$


 * Primitive of $\dfrac {\tanh^{-1} \frac x a} {x^2}$


 * Primitive of $\dfrac {\coth^{-1} \frac x a} {x^2}$