Examples of integrals found using hyperbolic substitutions I've read  that various types of integrals usually determined by involving $\tan$ and $\sec$ into the mix can sometimes be found more easily using hyperbolic functions.
As I'm not very familiar with the latter family, could someone give me some examples?
Very simple example: $$\int \frac 1 {1+x^2}\,\mathrm dx.$$
 A: 
Very simple example: $$\int \frac 1 {1+x^2}\,\mathrm dx.$$

Substituting in $x=\sinh\theta$ results in $\int\mathrm{sech}\,\theta\,\mathrm d\theta,$ which can be derived using trickery or using the substitution $u=e^\theta.$ But since this is a standard integral and subtituting in $x=\tan\theta$ much quicklier gives the answer $\arctan(x),$ this is is a non-example of hyperbolic substitution being easier than trigonometric substitution!
Otherwise, whenever the substitution $x=\tan\theta$ or $x=\sec\theta$ is called for, $x=\sinh\theta$ or $x=\pm\cosh\theta,$ respectively, are invariably preferable, due to nicer principal domains and being easier to differentiate and integrate. Here are some examples:





integrand
substitution
result




1
$\displaystyle\frac{x^2}{\sqrt{x^2+4}}$
$x=2\sinh\theta\quad\color{green}✔$
$4\int\sinh^2\theta\,\mathrm d\theta$



$\displaystyle\frac{x^2}{\sqrt{x^2+4}}$
$x=2\tan\theta\quad\color{red}✗$
$4\int\tan^2\theta\sec\theta\,\mathrm d\theta$


2
$\displaystyle\frac{\sqrt{x^2-4}}{x^2}\quad(x\leq-2)$
$x=-2\cosh\theta\quad\color{green}✔$
$-\int\tanh^2\theta\,\mathrm d\theta$



$\displaystyle\frac{\sqrt{x^2-4}}{x^2}\quad(x\leq-2)$
$x=2\sec\theta\quad\color{red}✗$
$-\int\tan^2\theta\cos\theta\,\mathrm d\theta$


3
$\sqrt{9-x^2}$
$x=3\sin\theta\quad\color{green}✔$
$9\int\cos^2\theta\,\mathrm d\theta$



$\sqrt{9-x^2}$
$x=3\tanh\theta\quad\color{red}✗$
$9\int\mathrm{sech^3}\,\theta\,\mathrm d\theta$



