Compute values of arsinh, arcosh, artanh, arcoth, arsech, arcsch. Get numeric results, domain validation, derivative formulas, and interactive graphs.
The inverse hyperbolic functions are multi-valued, but we consider the principal values (real branch). They are frequently used in integration, differential equations, and physics.
| Function | Notation | Domain (real) | Range (principal) | Derivative | Logarithmic form |
|---|---|---|---|---|---|
| arsinh | sinh⁻¹(x) | ℝ | ℝ | 1/√(x²+1) | ln(x + √(x²+1)) |
| arcosh | cosh⁻¹(x) | [1, ∞) | [0, ∞) | 1/√(x²-1) | ln(x + √(x²-1)) |
| artanh | tanh⁻¹(x) | (-1, 1) | ℝ | 1/(1-x²) | ½ ln((1+x)/(1-x)) |
| arcoth | coth⁻¹(x) | (-∞,-1) ∪ (1,∞) | ℝ \ {0} | 1/(1-x²) | ½ ln((x+1)/(x-1)) |
| arsech | sech⁻¹(x) | (0, 1] | [0, ∞) | -1/(x√(1-x²)) | ln((1+√(1-x²))/x) |
| arcsch | csch⁻¹(x) | ℝ \ {0} | ℝ \ {0} | -1/(|x|√(1+x²)) | ln(1/x + √(1/x²+1)) |
| Integral | Result |
|---|---|
| ∫ dx / √(x²+1) | arsinh x + C |
| ∫ dx / √(x²-1) | arcosh x + C (for x>1) |
| ∫ dx / (1-x²) | artanh x + C (|x|<1) or arcoth x + C (|x|>1) |
| ∫ dx / (x√(1-x²)) | -arsech |x| + C |
| ∫ dx / (x√(1+x²)) | -arcsch |x| + C |
Inverse hyperbolic functions appear in the integration of rational functions, in the solution of Laplace's equation in Cartesian coordinates, in special relativity (rapidity), and in the description of catenary curves. They also arise in the calculation of time along a brachistochrone and in the formula for the area of a hyperbolic sector.
Calculator features: Real-time evaluation, domain checking, interactive graph, and quick examples. Uses JavaScript's built-in Math.asinh, Math.acosh, Math.atanh and derived formulas for arcoth, arsech, arcsch.
arsinh x = ln(x+√(x²+1))arcosh x = ln(x+√(x²-1))artanh x = ½ ln((1+x)/(1-x))arcoth x = ½ ln((x+1)/(x-1))arsech x = ln((1+√(1-x²))/x)arcsch x = ln(1/x + √(1/x²+1))