Primitive of x over Power of Hyperbolic Sine of a x

Theorem

 * $\displaystyle \int \frac {x \ \mathrm d x} {\sinh^n a x} = \frac {-x \cosh a x} {a \left({n - 1}\right) \sinh^{n - 1} a x} - \frac 1 {a^2 \left({n - 1}\right) \left({n - 2}\right) \sinh^{n - 2} a x} - \frac {n - 2} {n - 1} \int \frac {x \ \mathrm d x} {\sinh^{n - 2} a x} + C$

Also see

 * Primitive of $\dfrac {x \ \mathrm d x} {\cosh^n a x}$