## Definitions

### General

The inverse hyperbolic sine function, in modern notation written as arsinh(t) or arcsinh(t) or sinh^{-1} t, gives the value x (hyperbolic angle), so that:

\[\sinh {x} = t \]

The inverse hyperbolic sine function accepts arguments from the whole real space. Since the hyperbolic sine 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 any real x:

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

### Properties

The derivative of the inverse hyperbolic sine function is:

\[ \left(\mathrm{arsinh}\,{x}\right)' = \frac{1}{\sqrt{1+x^2}}\]

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

\[ \int \mathrm{arsinh}\,{x}\, \mathrm{d}x = x\, \mathrm{arsinh}\,{x} - \sqrt{1+x^2} + C \]

### Identities

\[ \begin{split} & \sinh (\mathrm{arsinh}\,{x}) &= x \\ \\ & \cosh (\mathrm{arsinh}\,{x}) &= \sqrt{1+x^2} \\ \\ & \tanh (\mathrm{arsinh}\,{x}) &= \frac{x}{\sqrt{1+x^2}} \\ \\ & \mathrm{arsinh}\,{x}+\mathrm{arsinh}\,{y} & = \mathrm{arsinh}\,\left( x\sqrt{1+y^2} + y\sqrt{1+x^2} \right) \\ \\ & \mathrm{arsinh}\,{x}-\mathrm{arsinh}\,{y} & = \mathrm{arsinh}\,\left( x\sqrt{1+y^2} - y\sqrt{1+x^2} \right) \\ \\ & \mathrm{arsinh}\,\left(\tan{x}\right) & = \mathrm{artanh}\,\left(\sin{x}\right) \\ \\ \end{split} \]