Consider $$ f(x) = \frac{x}{2} + \tanh (\beta x) \, ,\qquad x\in [-1,1]\, , \, \beta > 0 \, , $$ Main question: Is there a closed form/elementary functions form of either $(f^{-1})'$ or $f^{-1}$?
Computing $\frac{1}{f'} = \left(\frac{1}{2} + \beta \, {\rm sech} ^2 (\beta x) \right)^{-1}$ is easy enough, but this is not precisely what I'm looking for.
If there isn't any closed form, I will definitely be interested in any function is "like" $f$. for which we can compute the inverse derivative. This is obviously softer than the main question.