Primitive of x by Square of Hyperbolic Tangent of a x

Theorem

 * $\ds \int x \tanh^2 a x \rd x = \frac {x^2} 2 - \frac {x \tanh a x} a + \frac 1 {a^2} \ln \size {\cosh a x} + C$

Proof
With a view to expressing the primitive in the form:
 * $\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$

let:

and let:

Then:

Also see

 * Primitive of $x \sinh^2 a x$
 * Primitive of $x \cosh^2 a x$
 * Primitive of $x \coth^2 a x$
 * Primitive of $x \sech^2 a x$
 * Primitive of $x \csch^2 a x$