# Primitive of Reciprocal of Square of Hyperbolic Sine of a x by Square of Hyperbolic Cosine of a x

## Theorem

$\displaystyle \int \frac {\mathrm d x} {\sinh^2 a x \cosh^2 a x} = \frac {-2 \coth 2 a x} a + C$

## Proof

 $\displaystyle \int \frac {\mathrm d x} {\sinh^2 a x \cosh^2 a x}$ $=$ $\displaystyle \int \frac {\mathrm d x} {\left({\sinh a x \cosh a x}\right)^2}$ $\displaystyle$ $=$ $\displaystyle \int \frac {\mathrm d x} {\left({\dfrac {\sinh^2 2 a x} 2}\right)^2}$ Double Angle Formula for Hyperbolic Sine $\displaystyle$ $=$ $\displaystyle 4 \int \frac {\mathrm d x} {\sinh^2 2 a x}$ Primitive of Constant Multiple of Function $\displaystyle$ $=$ $\displaystyle 4 \int \operatorname{csch}^2 2 a x \ \mathrm d x$ Definition of Hyperbolic Cosecant $\displaystyle$ $=$ $\displaystyle 4 \frac {-\coth 2 a x} {2 a} + C$ Primitive of $\operatorname{csch}^2 a x$ $\displaystyle$ $=$ $\displaystyle \frac {-2 \coth 2 a x} a + C$ simplifying

$\blacksquare$