Primitive of Square of Hyperbolic Sine of a x

Theorem

$\ds \int \sinh^2 a x \rd x = \dfrac {\sinh a x \cosh a x} {2 a} - \frac x 2 + C$

$\ds \int \sinh^2 a x \rd x = \frac {\sinh 2 a x} {4 a} - \frac x 2 + C$

$\ds \int \sinh^2 x \rd x$

$=$

$\ds \frac {\sinh x \cosh x - x} 2 + C$

$\ds \leadsto \ \ $

$\ds \int \sinh^2 a x \rd x$

$=$

$\ds \frac 1 a \paren {\frac {\sinh a x \cosh a x - a x} 2} + C$

$\ds $

$=$

$\ds \frac {\sinh a x \cosh a x} {2 a} - \frac x 2 + C$

simplifying

$\blacksquare$