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

## Theorem

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

## Proof

 $\ds \int \frac {\mathrm d x} {\sinh a x \cosh a x}$ $=$ $\ds \int \frac {\operatorname{sech} a x \ \mathrm d x} {\sinh a x}$ Definition of Hyperbolic Secant $\ds$ $=$ $\ds \int \frac {\operatorname{sech}^2 a x \ \mathrm d x} {\sinh a x \operatorname{sech} a x}$ multiplying top and bottom by $\operatorname{sech} a x$ $\ds$ $=$ $\ds \int \frac {\operatorname{sech}^2 a x \ \mathrm d x} {\frac {\sinh a x} {\cosh a x} }$ Definition of Hyperbolic Secant $\ds$ $=$ $\ds \int \frac {\operatorname{sech}^2 a x \ \mathrm d x} {\tanh a x}$ Definition of Hyperbolic Tangent $\ds$ $=$ $\ds \frac 1 a \ln \left\vert{\tanh a x}\right\vert + C$ Primitive of $\dfrac {\operatorname{sech}^2 a x} {\tanh a x}$

$\blacksquare$