Primitive of x by Inverse Hyperbolic Sine of x over a

Theorem

 * $\ds \int x \arsinh \frac x a \rd x = \paren {\frac {x^2} 2 + \frac {a^2} 4} \arsinh \frac x a - \frac {x \sqrt {x^2 + a^2} } 4 + C$

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 $x \arcosh \dfrac x a$


 * Primitive of $x \artanh \dfrac x a$


 * Primitive of $x \arcoth \dfrac x a$


 * Primitive of $x \arsech \dfrac x a$


 * Primitive of $x \arcsch \dfrac x a$