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 $\operatorname{csch} a x$

$\blacksquare$


Sources