arctansinh

Arctansinh is the function defined by f(x) = arctan(sinh x), the composition of the hyperbolic sine with the inverse tangent. The domain is all real numbers, and the range is the open interval (-π/2, π/2), since sinh maps R onto R and arctan maps R onto (-π/2, π/2). The function is continuous, strictly increasing, and odd, with smooth derivatives for all x.

The derivative of f is f'(x) = cosh x / (1 + sinh^2 x) = cosh x / cosh^2 x = 1 / cosh x = sech x. This positive derivative confirms monotonicity and shows that the slope decays as |x| grows. Higher derivatives can be obtained by standard differentiation of the composed functions.

Asymptotically, f(x) approaches the endpoints of its range: f(x) → π/2 as x → ∞ and f(x) → -π/2 as x → -∞.

Inverse function and domain of invertibility: Since f is a bijection from R onto (-π/2, π/2), it has an inverse f^{-1}: (-π/2, π/2) → R. The inverse is given by f^{-1}(y) = arsinh(tan y) (often written as asinh(tan y)), because y = arctan(sinh x) implies tan y = sinh x, hence x = arsinh(tan y).

Relation to integrals: The derivative identity f'(x) = sech x implies ∫ sech x dx = arctan(sinh x) + C.

See also: arctan, sinh, asinh (inverse hyperbolic sine).