Inverse Hyperbolic Function Calculator
The inverse hyperbolic functions reverse sinh, cosh and tanh, and each has a neat closed form written with the natural logarithm and a square root. They turn up constantly in integration and in relativity, where the inverse hyperbolic tangent links ordinary velocity to rapidity. Enter a value x and this calculator returns arsinh, arcosh and artanh, each only where it is defined: arsinh for all x, arcosh for x at least 1, and artanh for x strictly between minus 1 and 1. Out-of-domain results show n/a.
Inverse hyperbolic formulas
arsinh(x) = ln(x + sqrt(x^2 + 1)), all real x
arcosh(x) = ln(x + sqrt(x^2 - 1)), for x >= 1
artanh(x) = 0.5 * ln((1 + x) / (1 - x)), for -1 < x < 1
Each value comes from the exact logarithmic form. The calculator checks the domain first and returns n/a when x falls outside the valid range for a given function.
About inverse hyperbolic functions
- arsinh is odd: arsinh(-x) equals minus arsinh(x).
- arcosh(1) is 0, the smallest valid input for the inverse cosh.
- artanh(0) is 0 and the function is odd on its interval.
- The integral of 1 over sqrt(x^2 + 1) is arsinh(x).
- In relativity, rapidity equals artanh of velocity divided by the speed of light.
Inverse hyperbolic functions: frequently asked questions
What are the inverse hyperbolic functions?
They undo the hyperbolic functions. arsinh undoes sinh, arcosh undoes cosh, and artanh undoes tanh. Each has a closed logarithmic form, for example arsinh(x) = ln(x + sqrt(x^2 + 1)).
What are the domains of these functions?
arsinh is defined for all real x. arcosh is defined only for x at least 1. artanh is defined only for x strictly between -1 and 1. The calculator returns n/a outside these domains.
What are the logarithmic formulas?
arsinh(x) = ln(x + sqrt(x^2 + 1)); arcosh(x) = ln(x + sqrt(x^2 - 1)) for x at least 1; artanh(x) = one half times ln((1 + x) / (1 - x)) for x between -1 and 1.
Why does artanh blow up near plus or minus one?
As x approaches 1 the term (1 - x) in the denominator approaches zero, so the logarithm and the result grow without bound. At exactly plus or minus 1 the function is undefined.
Where are inverse hyperbolic functions used?
They appear in integration (many integrals evaluate to arsinh or artanh), in relativity where artanh relates velocity to rapidity, and in solving equations involving hyperbolic functions.
Official sources
- NIST Digital Library of Mathematical Functions: Inverse Hyperbolic Functions.
- NIST Digital Library of Mathematical Functions: Hyperbolic Functions: Definitions.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.