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

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Theorem

$\ds \int \frac {\d x} {\sinh^2 a x \cosh a x} = -\frac 1 a \map \arctan {\sinh a x} - \frac {\csch a x} a + C$


Proof

\(\ds \int \frac {\d x} {\sinh^2 a x \cosh a x}\) \(=\) \(\ds \int \frac {\paren {\cosh^2 a x - \sinh^2 a x} \rd x} {\sinh^2 a x \cosh a x}\) Difference of Squares of Hyperbolic Cosine and Sine
\(\ds \) \(=\) \(\ds \int \frac {\cosh^2 a x \rd x} {\sinh^2 a x \cosh a x} - \int \frac {\sinh^2 a x \rd x} {\sinh^2 a x \cosh a x}\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \int \frac {\cosh a x \rd x} {\sinh^2 a x} - \int \frac {\d x} {\cosh a x}\) simplifying
\(\ds \) \(=\) \(\ds \int \frac {\cosh a x \rd x} {\sinh^2 a x} - \int \sech a x \rd x\) Definition 2 of Hyperbolic Secant
\(\ds \) \(=\) \(\ds \int \frac {\coth a x \rd x} {\sinh a x} - \int \sech a x \rd x\) Definition 2 of Hyperbolic Cotangent
\(\ds \) \(=\) \(\ds \int \csch a x \coth a x \rd x - \int \sech a x \rd x\) Definition 2 of Hyperbolic Cosecant
\(\ds \) \(=\) \(\ds \frac {-\csch a x} a - \int \sech a x \rd x\) Primitive of $\csch^n a x \coth a x$ for $n = 1$
\(\ds \) \(=\) \(\ds \frac {-\csch a x} a - \frac 1 a \map \arctan {\sinh a x}\) Primitive of $\sech a x$

$\blacksquare$


Sources