Inverse hyperbolic functions ($\mathrm{arsinh},\mathrm{arcosh},\mathrm{artanh}$)

The inverse hyperbolic functions invert the hyperbolic functions, with real branches on their standard ranges and complex branches obtained through analytic continuation. For example, $\mathrm{arsinh}z=\log (z+\sqrt{z^2+1})$ on a chosen branch; the multivalued inverse hyperbolic sine records all inverse values.

The real inverse of the hyperbolic sine is defined on all real numbers and has derivative $1/\sqrt{1+x^2}$. It appears in arc length and in integrals involving square roots of sums of squares, where a hyperbolic substitution can simplify the expression.