Primitive of Reciprocal of Hyperbolic Sine of a x by Hyperbolic Cosine of a x minus 1

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Theorem

$\displaystyle \int \frac {\d x} {\sinh a x \paren {\cosh a x - 1} } = \frac {-1} {2 a} \ln \size {\tanh \frac {a x} 2} - \frac 1 {2 a \paren {\cosh a x - 1} } + C$


Proof

\(\displaystyle \int \frac {\d x} {\sinh a x \paren {\cosh a x - 1} }\) \(=\) \(\displaystyle \int \frac {\map \sinh {a x} \rd x} {\sinh^2 a x \cosh a x - \sinh^2 a x}\) multiplying through $\dfrac {\sinh a x} {\sinh a x}$
\(\displaystyle \) \(=\) \(\displaystyle \int \frac {\map \sinh {a x} \rd x} {\cosh a x \paren {\cosh^2 a x - 1} + 1 - \cosh^2 a x}\) Difference of Squares of Hyperbolic Cosine and Sine
\(\displaystyle \) \(=\) \(\displaystyle \int \frac {\map \sinh {a x} \rd x} {\cosh^3 a x - \cosh^2 a x - \cosh a x + 1}\)
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 a \int \frac {\d t} {t^3 - t^2 - t + 1}\) Integration by Substitution: $t \to \map \cosh {a x}$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {4 a} \int \frac 1 {t + 1} \rd t - \frac 1 {4 a} \int \frac 1 {t - 1} \rd t + \frac 1 {2 a} \int \frac 1 {\paren {t - 1}^2} \rd t\) Partial fraction decomposition
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {4 a} \map \ln {t + 1} - \frac 1 {4 a} \map \ln {t - 1} - \frac 1 {2 a \paren {t - 1} } + C\) Primitive of Reciprocal
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {4 a} \map \ln { \frac {\cosh ax + 1} {\cosh ax - 1} } - \frac 1 {2 a \paren {\cosh a x - 1} } + C\) substituting back $t \to \cosh ax$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {4 a} \map \ln {\frac {2 \cosh^2 \frac {a x} 2} {2 \sinh^2 \frac {a x} 2} } - \frac 1 {2 a \paren {\cosh ax - 1} } + C\) Double angle formula for hyperbolic cosine
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {4 a} \map \ln {\coth^2 \frac {a x} 2} - \frac 1 {2 a \paren {\cosh a x - 1} } + C\) Definition of Hyperbolic Cotangent
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {2 a} \ln \size {\coth \frac {a x} 2} - \frac 1 {2 a \paren {\cosh a x - 1} } + C\) Logarithm of power
\(\displaystyle \) \(=\) \(\displaystyle -\frac 1 {2 a} \ln \size {\tanh \frac {a x} 2} - \frac 1 {2 a \paren {\cosh a x - 1} } + C\) Logarithm of Reciprocal

$\blacksquare$


Sources