Primitive of Power of Hyperbolic Secant of a x

Theorem

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

Proof
With a view to expressing the primitive in the form:
 * $\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$

let:

and let:

Then:

Also see

 * Primitive of $\sinh^n a x$
 * Primitive of $\cosh^n a x$
 * Primitive of $\tanh^n a x$
 * Primitive of $\coth^n a x$
 * Primitive of $\csch^n a x$