The function
$$f(x) = x \tanh(\frac{\pi}{2}\sinh{x})$$
has useful properties in numerical calculations as it is an even function that converges extremely quickly to $x$ for large input values. Difficulty arises in calculations where the inversion of the function is needed. I am looking for a function that is even, analytic in the complex plane, invertible and behaves linearly for large and goes like $x$ for large input $x$. It would also be optimal if the function converges quickly to linear.