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

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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 $\operatorname{csch} a x$

$\blacksquare$


Sources