Primitive of Power of Hyperbolic Secant of a x

Theorem

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

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

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 $\operatorname{csch}^n a x$