Primitive of Square of Hyperbolic Cosine of a x

Theorem

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

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

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

$=$

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

$\ds \leadsto \ \ $

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

$=$

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

$\ds $

$=$

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

simplifying

$\blacksquare$