Primitive of x squared by Hyperbolic Sine of a x

Theorem

 * $\displaystyle \int x^2 \sinh a x \ \mathrm d x = \left({\frac {x^2} a + \frac 2 {a^3} }\right) \cosh a x - \frac {2 x \sinh a x} {a^2} + C$

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

let:

and let:

Then:

Also see

 * Primitive of $x^2 \cosh a x$