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

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Theorem

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


Proof

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

$\blacksquare$


Sources