## Inverse hyperbolic tangent calculator

This tool evaluates the inverse hyperbolic tangent of a number: artanh(x). Enter the argument x below.

 x = Result: artanh(x) =

## Definitions

### General

The inverse hyperbolic tangent function, in modern notation written as artanh(x) or arctanh(x) or tanh-1 x, gives the value t (hyperbolic angle), so that:

$\tanh {t} = x$

The inverse hyperbolic tangent function accepts arguments in real open interval (-1,1), because $$|\tanh{x}|\lt 1, \forall x\in \mathbb{R}$$. Since the hyperbolic tangent 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:

$\mathrm{artanh}\,{x} = \frac{1}{2}\ln\left(\frac{1+x}{1-x}\right)$

### Properties

The derivative of the inverse hyperbolic tangent function is:

$\left(\mathrm{artanh}\,{x}\right)' = \frac{1}{1-x^2}\quad, \left\{x\in\mathbb{R} | -1\lt x \lt 1 \right\}$

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

$\int \mathrm{artanh}\,{x}\, \mathrm{d}x = x\, \mathrm{artanh}\,{x} + \frac{\ln\left(x^2-1\right)}{2} + C \quad, \left\{x\in\mathbb{R} | -1\lt x \lt 1\right\}$

### Identities

$\begin{split} & \tanh (\mathrm{artanh}\,{x}) &= x \\ \\ & \sinh (\mathrm{artanh}\,{x}) &= \frac{x}{\sqrt{1-x^2}} \\ \\ & \cosh (\mathrm{artanh}\,{x}) &= \frac{1}{\sqrt{1-x^2}} \\ \\ & \mathrm{artanh}\,{x}+\mathrm{artanh}\,{y} & = \mathrm{artanh}\,\left( \frac{x+y}{1+xy} \right) \\ \\ & \mathrm{artanh}\,{x}-\mathrm{artanh}\,{y} & = \mathrm{artanh}\,\left( \frac{x-y}{1-xy} \right) \\ \\ \end{split}$