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

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## Theorem

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

## Proof

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

$\blacksquare$