Primitive of x by Secant of a x

Theorem

 * $\displaystyle \int x \sec a x \ \mathrm d x = \frac 1 {a^2} \left({\frac {\left({a x}\right)^2} 2 + \frac {\left({a x}\right)^4} 8 + \frac {5 \left({a x}\right)^6} {144} + \cdots + \frac {E_n \left({a x}\right)^{2 n + 2} } {\left({2 n + 2}\right) \left({2 n}\right)!} + \cdots}\right) + C = \frac{1}{a^2} \sum_{n \mathop = 0}^\infty \frac{E_{n} \left({ax}\right)^{2n+2} }{\left({2n+2}\right)\left({2n}\right)!} + C $

where $E_n$ is the $n$th Euler number.

Also see

 * Primitive of $x \sin a x$
 * Primitive of $x \cos a x$
 * Primitive of $x \tan a x$
 * Primitive of $x \cot a x$
 * Primitive of $x \csc a x$