Primitive of Reciprocal of Square of Hyperbolic Cosine of a x minus 1

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Theorem

$\ds \int \frac {\d x} {\paren {\cosh a x - 1}^2} = \frac 1 {2 a} \coth \frac {a x} 2 - \frac 1 {6 a} \coth^3 \frac {a x} 2 + C$


Proof

\(\ds \int \frac {\d x} {\paren {\cosh a x - 1}^2}\) \(=\) \(\ds \int \paren {\frac 1 2 \csch^2 \frac {a x} 2}^2 \rd x\) Reciprocal of Hyperbolic Cosine Minus One
\(\ds \) \(=\) \(\ds \frac 1 4 \int \csch^4 \frac {a x} 2 \rd x\) simplifying
\(\ds \) \(=\) \(\ds \frac 1 4 \paren {\frac {-\csch^2 \dfrac {a x} 2 \coth \dfrac {a x} 2} {\dfrac {3 a} 2} - \frac 2 3 \int \csch^2 \frac {a x} 2 \rd x} + C\) Primitive of $\csch^n a x$
\(\ds \) \(=\) \(\ds \frac {-1} {6 a} \csch^2 \frac {a x} 2 \coth \dfrac {a x} 2 - \frac 1 6 \int \csch^2 \frac {a x} 2 \rd x + C\) simplifying
\(\ds \) \(=\) \(\ds \frac {-1} {6 a} \csch^2 \frac {a x} 2 \coth \dfrac {a x} 2 - \frac 1 6 \paren {\frac {-2} a \coth \frac {a x} 2} + C\) Primitive of $\csch^2 a x$
\(\ds \) \(=\) \(\ds \frac {-1} {6 a} \paren {\coth^2 \frac {a x} 2 - 1} \coth \dfrac {a x} 2 + \frac 2 {6 a} \coth \frac {a x} 2 + C\) Difference of Squares of Hyperbolic Cotangent and Cosecant
\(\ds \) \(=\) \(\ds \frac 1 {2 a} \coth \frac {a x} 2 - \frac 1 {6 a} \coth^3 \frac {a x} 2 + C\) simplifying

$\blacksquare$


Also see


Sources

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