Primitive of x by Hyperbolic Tangent of a x

Theorem

 * $\displaystyle \int x \tanh a x \rd x = \frac 1 {a^2} \paren {\frac {\paren {a x}^3} 3 - \frac {\paren {a x}^5} {15} + \frac {2 \paren {a x}^7} {105} + \cdots + \frac { 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + \cdots} + C$

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

Also see

 * Primitive of $x \sinh a x$
 * Primitive of $x \cosh a x$
 * Primitive of $x \coth a x$
 * Primitive of $x \sech a x$
 * Primitive of $x \csch a x$