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

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Theorem

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


Proof

\(\ds \int \frac {\cosh^2 a x \rd x} {\sinh a x}\) \(=\) \(\ds \int \frac {\paren {\sinh^2 a x + 1} \rd x} {\sinh a x}\) Difference of Squares of Hyperbolic Cosine and Sine
\(\ds \) \(=\) \(\ds \int \frac {\sinh^2 a x \rd x} {\sinh a x} + \int \frac {\d x} {\sinh a x}\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \int \sinh a x \rd x + \int \csch a x \rd x\) Definition 2 of Hyperbolic Cosecant
\(\ds \) \(=\) \(\ds \frac {\cosh a x} a + \int \csch a x \rd x\) Primitive of $\sinh a x$
\(\ds \) \(=\) \(\ds \frac {\cosh a x} a + \frac 1 a \ln \size {\tanh \frac {a x} 2} + C\) Primitive of $\csch a x$

$\blacksquare$


Sources