Primitive of x by Hyperbolic Secant of a x

Theorem

$\ds \int x \sech a x \rd x$

$=$

$\ds \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {E_{2 n} \paren {a x}^{2 n + 2} } {\paren {2 n + 2} \paren {2 n}!} + C$

$\ds $

$=$

$\ds \frac 1 {a^2} \paren {\frac {\paren {a x}^2} 2 - \frac {\paren {a x}^4} 8 + \frac {5 \paren {a x}^6} {144} - \cdots } + C$

where $E_{2 n}$ denotes the $2 n$th Euler number.

$\ds \int x \sech a x \rd x$

$=$

$\ds \frac 1 {a^2} \int \theta \sech \theta \rd \theta$

$\ds $

$=$

$\ds \frac 1 {a^2} \int \theta \sum_{n \mathop = 0}^\infty \frac{E_{2 n} \, \theta^{2 n} } {\paren {2 n}!} \rd \theta$

$\ds $

$=$

$\ds \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {E_{2 n} } {\paren {2 n}!} \int \theta^{2 n + 1} \rd \theta$

$\ds $

$=$

$\ds \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {E_{2 n} \paren {a x}^{2 n + 2} } {\paren {2 n + 2} \paren {2 n}!} + C$

$\blacksquare$