Primitive of Hyperbolic Cosecant Function/Logarithm Form

Theorem

$\ds \int \csch x \rd x = -\ln \size {\csch x + \coth x} + C$

where $\csch x + \coth x \ne 0$.

Let:

$\ds u$

$=$

$\ds \coth x + \csch x$

$\ds \leadsto \ \ $

$\ds \frac {\d u} {\d x}$

$=$

$\ds \frac \d {\d x} \coth x + \frac \d {\d x} \csch x$

$\ds $

$=$

$\ds -\csch^2 x + \frac \d {\d x} \csch x$

$\ds $

$=$

$\ds -\csch^2 x - \csch x \coth x$

$\ds $

$=$

$\ds -\csch x \paren {\csch x + \coth x}$

factorising

Then:

$\ds \int \csch x \rd x$

$=$

$\ds \int \frac {\csch x \paren {\csch x + \coth x} } {\csch x + \coth x} \rd x$

multiplying top and bottom by $\csch x + \coth x$

$\ds $

$=$

$\ds -\int \frac {-\csch x \paren {\csch x + \coth x} } {\csch x + \coth x} \rd x$

$\ds $

$=$

$\ds -\ln \size {\csch x + \coth x} + C$

$\blacksquare$