Primitive of Power of Hyperbolic Cotangent of a x

Theorem

$\ds \int \coth^n a x \rd x = \frac {-\coth^{n - 1} a x} {a \paren {n - 1} } + \int \coth^{n - 2} a x \rd x + C$

$\ds \int \coth^n a x \rd x$

$=$

$\ds \int \coth^{n - 2} a x \coth^2 a x \rd x$

$\ds $

$=$

$\ds \int \coth^{n - 2} a x \paren {1 + \csch^2 a x} \rd x$

$\ds $

$=$

$\ds \int \coth^{n - 2} a x \csch^2 a x \rd x + \int \coth^{n - 2} \rd x$

$\ds $

$=$

$\ds \frac {-\coth^{n - 1} a x} {a \paren {n - 1} } + \int \coth^{n - 2} a x \rd x + C$

$\blacksquare$