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## 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:

The inverse hyperbolic cotangent function accepts arguments in real open intervals (-∞,-1) and (1,∞), because for all non-zero real x. Since the hyperbolic cotangent is defined through the natural exponential function , its inverse can be defined through the natural logarithm function, using the following formula, for real x, with |x|>1:

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### Properties

The derivative of the inverse hyperbolic cotangent function is:

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

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