Primitive of Power of Hyperbolic Secant of a x by Hyperbolic Tangent of a x

Theorem

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

for $n \ne 0$.

$\ds z$

$=$

$\ds \sech a x$

$\ds \leadsto \ \ $

$\ds \frac {\d z} {\d x}$

$=$

$\ds -a \sech a x \tanh a x$

$\ds \leadsto \ \ $

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

$=$

$\ds \int \frac {-z^{n - 1} \rd z} a$

$\ds $

$=$

$\ds \frac {-1} a \int z^{n - 1} \rd z$

$\ds $

$=$

$\ds \frac {-1} a \frac {z^n} n$

$\ds $

$=$

$\ds \frac {-\sech^n a x} {n a} + C$

substituting for $z$

$\blacksquare$