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

Theorem

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

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 $\dfrac 1 {x^2} \arcosh \dfrac x a$


 * Primitive of $\dfrac 1 {x^2} \artanh \dfrac x a$


 * Primitive of $\dfrac 1 {x^2} \arcoth \dfrac x a$