Primitive of Hyperbolic Sine of a x over Power of x

Theorem

 * $\ds \int \frac {\sinh a x \rd x} {x^n} = \frac {-\sinh a x} {\paren {n - 1} x^{n - 1} } + \frac a {n - 1} \int \frac {\cosh a x \rd x} {x^{n - 1} } + C$

Proof
With a view to expressing the problem 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 $\dfrac {\cosh a x} {x^n}$