# Primitive of Square of Hyperbolic Cosine of a x over Hyperbolic Sine of a x

## Theorem

$\displaystyle \int \frac {\cosh^2 a x \ \mathrm d x} {\sinh a x} = \frac {\cosh a x} a + \frac 1 a \ln \left\vert{\tanh \frac {a x} 2}\right\vert + C$

## Proof

 $\displaystyle \int \frac {\cosh^2 a x \ \mathrm d x} {\sinh a x}$ $=$ $\displaystyle \int \frac {\left({\sinh^2 a x + 1}\right) \ \mathrm d x} {\sinh a x}$ Difference of Squares of Hyperbolic Cosine and Sine $\displaystyle$ $=$ $\displaystyle \int \frac {\left({\sinh^2 a x}\right) \ \mathrm d x} {\sinh a x} + \int \frac {\mathrm d x} {\sinh a x}$ Linear Combination of Integrals $\displaystyle$ $=$ $\displaystyle \int \sinh a x \ \mathrm d x + \int \operatorname{csch} a x \ \mathrm d x$ Definition of Hyperbolic Cosecant $\displaystyle$ $=$ $\displaystyle \frac {\cosh a x} a + \int \operatorname{csch} a x \ \mathrm d x$ Primitive of $\sinh a x$ $\displaystyle$ $=$ $\displaystyle \frac {\cosh a x} a + \frac 1 a \ln \left\vert{\tanh \frac {a x} 2}\right\vert + C$ Primitive of $\operatorname{csch} a x$

$\blacksquare$