Hyperbolic Tangent Half-Angle Substitution

Theorem

 * $\displaystyle \int F \left({\sinh x, \cosh x}\right)\ \mathrm d x = 2 \int F \left({\frac {2 u} {1 - u^2}, \frac {1 + u^2} {1 - u^2} }\right) \frac {\mathrm d u} {1 - u^2}$

where $u = \tanh \dfrac x 2$.

Proof
The result follows from Integration by Substitution.

Also see

 * Weierstrass Substitution