## Definitions

### General

The inverse hyperbolic cosecant function, in modern notation written as arcsch(x) or arccsch(x) or csch^{-1} x, gives the value t (hyperbolic angle), so that:

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

The inverse hyperbolic cosecant function accepts as arguments all real numbers except zero. Since the hyperbolic cosecant 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:

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

### Properties

The derivative of the inverse hyperbolic cosecant function is:

\[ \left(\mathrm{arcsch}\,{x}\right)' = \frac{-1}{|x|\sqrt{1+x^2}}\quad, x\ne 0 \]

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

\[ \int \mathrm{arcsch}\,{x}\, \mathrm{d}x = x\, \mathrm{arcsch}\,{x} + \mathrm{arcoth}\,\left(\sqrt{\frac{1}{x^2}+1} \right) + C \quad, x\ne 0 \]