## Definitions

### General

The inverse hyperbolic secant function, in modern notation written as arsech(x) or arcsech(x) or sech^{-1} x, gives the value t (hyperbolic angle), so that:

\[\mathrm{sech}\,{t} = x \]

The inverse hyperbolic secant function accepts arguments in real interval (0,1], because \( 0\lt \mathrm{sech}\,{x} \le 1 \) for all real x. Since the hyperbolic secant is defined through the natural exponential function \(\mathrm{e}^x\), its inverse can be defined through the natural logarithm function, using the following formula, for real x, with 0<x≤1:

\[ \mathrm{arsech}\,{x} = \ln\left(\frac{1+\sqrt{1-x^2}}{x}\right) \]

### Properties

The derivative of the inverse hyperbolic secant function is:

\[ \left(\mathrm{arsech}\,{x}\right)' = \frac{-1}{x\sqrt{1-x^2}}\quad, x\in(0,1)\]

The integral of the inverse hyperbolic secant function is given by:

\[ \int \mathrm{arsech}\,{x}\, \mathrm{d}x = x\, \mathrm{arsech}\,{x} -2\arctan\sqrt{ \frac{1-x}{1+x} } + C \quad, x\in (0,1] \]