Primitive of x over Hyperbolic Sine of a x

Theorem

$\ds \int \frac {x \rd x} {\sinh a x}$

$=$

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

$\ds $

$=$

$\ds \dfrac 1 {a^2} \paren {a x - \dfrac {\paren {a x}^3} {18} + \dfrac {7 \paren {a x}^5} {1800} - \cdots} + C$

where $B_{2 n}$ denotes the $2 n$th Bernoulli number.

$\ds \int \frac {x \rd x} {\sinh a x}$

$=$

$\ds \int x \csch a x \rd x$

$\ds $

$=$

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

$\blacksquare$