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

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Theorem

$\displaystyle \int \sinh^2 a x \cosh^2 a x \rd x = \frac {\sinh 4 a x} {32 a} - \frac x 8 + C$


Proof

\(\displaystyle \int \sinh^2 a x \cosh^2 a x \rd x\) \(=\) \(\displaystyle \int \paren {\sinh a x \cosh a x}^2 \rd x\)
\(\displaystyle \) \(=\) \(\displaystyle \int \paren {\frac {\sinh 2 a x} 2}^2 \rd x\) Double Angle Formula for Hyperbolic Sine
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 4 \int \sinh^2 2 a x \rd x\) Primitive of Constant Multiple of Function
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 4 \paren {\frac {\sinh 2 \paren {2 a x} } {4 \paren {2 a} } - \frac x 2} + C\) Primitive of $\sinh^2 a x$
\(\displaystyle \) \(=\) \(\displaystyle \frac {\sinh 4 a x} {32 a} - \frac x 8 + C\) simplifying

$\blacksquare$


Sources