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

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Theorem

$\displaystyle \int \frac {\mathrm d x} {\sinh^2 a x \cosh^2 a x} = \frac {-2 \coth 2 a x} a + C$


Proof

\(\displaystyle \int \frac {\mathrm d x} {\sinh^2 a x \cosh^2 a x}\) \(=\) \(\displaystyle \int \frac {\mathrm d x} {\left({\sinh a x \cosh a x}\right)^2}\)
\(\displaystyle \) \(=\) \(\displaystyle \int \frac {\mathrm d x} {\left({\dfrac {\sinh^2 2 a x} 2}\right)^2}\) Double Angle Formula for Hyperbolic Sine
\(\displaystyle \) \(=\) \(\displaystyle 4 \int \frac {\mathrm d x} {\sinh^2 2 a x}\) Primitive of Constant Multiple of Function
\(\displaystyle \) \(=\) \(\displaystyle 4 \int \operatorname{csch}^2 2 a x \ \mathrm d x\) Definition of Hyperbolic Cosecant
\(\displaystyle \) \(=\) \(\displaystyle 4 \frac {-\coth 2 a x} {2 a} + C\) Primitive of $\operatorname{csch}^2 a x$
\(\displaystyle \) \(=\) \(\displaystyle \frac {-2 \coth 2 a x} a + C\) simplifying

$\blacksquare$


Sources