2.4.1—Hyperbolic functions

Use cosh²x−sinh²x=1, 1−tanh²x=sech²x and coth²x−1=csch²x to replace a difficult power or quotient by a form with a known derivative.

Choose the identity that leaves a differential factor. For even powers, separate one factor when integrating; for equations, isolate a single hyperbolic function before applying an inverse.

∫sinh²x dx can use sinh²x=(cosh2x−1)/2, giving a sum of an elementary term and a linear term in x.

The sign in cosh²−sinh² is opposite to the circular identity, and applying an identity without preserving the differential can make the integral harder.