Inverse Hyperbolic Sine (arcsinh) Calculator

This inverse hyperbolic sine calculator returns arcsinh(x), also written asinh(x), the value whose hyperbolic sine is x. It is defined directly in terms of the natural logarithm as arcsinh(x) = ln(x + the square root of (x squared plus one)). Because hyperbolic sine takes every real value, the inverse accepts any real number as input and returns a single real result, so there are no domain restrictions to worry about. Inverse hyperbolic functions come up when integrating certain expressions, in physics, and in statistics as a variance stabilising transform. Enter a value of x and read the result, with a check showing that the hyperbolic sine of the answer returns your input. The definition, a worked example and the assumptions are below.

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Value of x

Check: sinh(result)

Input x

The formula

The inverse hyperbolic sine has a closed form in logarithms: arcsinh(x) = ln(x + sqrt(x^2 + 1)). It is defined for every real x and is an odd function.

Worked example

arcsinh(1) = ln(1 + sqrt(2)), about 0.881374. The hyperbolic sine of 0.881374 returns 1. Enter 1 above to confirm.

Frequently asked questions

What is the formula for arcsinh?

arcsinh(x) = ln(x + sqrt(x^2 + 1)), a closed form using the natural logarithm.

What inputs are allowed?

Any real number. Hyperbolic sine covers all real values, so its inverse has no domain restriction.

Is arcsinh the same as 1 / sinh?

No. arcsinh is the inverse function of sinh, not its reciprocal. The reciprocal of sinh is the hyperbolic cosecant.

Who this calculator is for

This calculator is for students, engineers and anyone needing the inverse hyperbolic sine.

What this calculator assumes

You enter a real number x.
arcsinh is computed from its logarithmic form.
Results are rounded for display.

Formula and sources

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