Inverse hyperbolic sine ($\mathrm{arsinh}x=\ln (x+\sqrt{1+x^2})$)

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.