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

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Theorem

$\displaystyle \int \frac {\mathrm d x} {\sinh^2 a x \cosh a x} = -\frac 1 a \arctan \left({\sinh a x}\right) - \frac {\operatorname{csch} a x} a + C$


Proof

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

$\blacksquare$


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