Primitive of x by Hyperbolic Tangent of a x

Theorem

 * $\displaystyle \int x \tanh a x \ \mathrm d x = \frac 1 {a^2} \left({\frac {\left({a x}\right)^3} 3 - \frac {\left({a x}\right)^5} {15} + \frac {2 \left({a x}\right)^7} {105} + \cdots + \frac { 2^{2 n} \left({2^{2 n} - 1}\right) B_{2 n} \left({a x}\right)^{2 n + 1} } {\left({2 n + 1}\right)!} + \cdots}\right) + C$

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 $x \csch a x$