Primitive of Power of Hyperbolic Tangent of a x

Theorem

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

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

$=$

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

$\ds $

$=$

$\ds \int \tanh^{n - 2} a x \paren {1 - \sech^2 a x} \rd x$

$\ds $

$=$

$\ds -\int \tanh^{n - 2} a x \sech^2 a x \rd x + \int \tanh^{n - 2} \rd x$

$\ds $

$=$

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

$\blacksquare$