Primitive of x by Square of Hyperbolic Sine of a x

Theorem

 * $\displaystyle \int x \sinh^2 a x \rd x = \dfrac {x \sinh 2 a x} {4 a} - \frac {\cosh 2 a x} {8 a^2} - \frac {x^2} 4 + C$

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

let:

and let:

Then:

Also see

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