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

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Theorem

$\ds \int \frac {\d x} {\sinh a x \cosh^2 a x} = \frac 1 a \ln \size {\tanh \frac {a x} 2} + \frac {\sech a x} a + C$


Proof

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

$\blacksquare$


Sources