## Definitions

### General

The inverse hyperbolic cotangent function, in modern notation written as arcoth(x) or arccoth(x) or coth^{-1} x, gives the value t (hyperbolic angle), so that:

\[\coth {t} = x \]

The inverse hyperbolic cotangent function accepts arguments in real open intervals (-∞,-1) and (1,∞), because \( |\coth{x}|\gt 1 \) for all non-zero real x. Since the hyperbolic cotangent 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 |x|>1:

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

### Properties

The derivative of the inverse hyperbolic cotangent function is:

\[ \left(\mathrm{arcoth}\,{x}\right)' = \frac{1}{1-x^2}\quad, \left\{x\in\mathbb{R} \,|\, |x| \gt 1 \right\} \]

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

\[ \int \mathrm{arcoth}\,{x}\, \mathrm{d}x = x\, \mathrm{arcoth}\,{x} + \frac{\ln\left(x^2-1\right)}{2} + C \quad, \left\{x\in\mathbb{R} \,| \, |x| \gt 1 \right\} \]