I would like to know if there is a way of finding the inverse function of $f(x)=e^{-\varepsilon x}\sinh(x)$ with $-1<\varepsilon<0$.
It seems there is no simple way even if we consider Lambert or Gudermann-like functions.



Expression of the inverse function of $f(x)=e^{-\varepsilon x}\sinh(x)$
$$y=e^{- \epsilon x} \sinh (x)=\sum\_{n=0}^\infty \frac{(1-\epsilon )^n+(-1)^{n+1} (1+\epsilon )^n}{2 n!}\,x^n$$ Now use [series reversion](https://mathworld.wolfram.com/SeriesReversion.html) using the explicit formula for the $n^{\text{th}}$ term as given by Morse and Feshbach.


This will give
$$x=y+\sum\_{n=2}^\infty a\_n y^n$$ the first coefficients being
$$\left(
\begin{array}{cc}
 2 & \epsilon \\
 3 & \frac{1}{6} (3 \epsilon -1) (3 \epsilon +1) \\
 4 & \frac{2}{3} \epsilon (2 \epsilon -1) (2 \epsilon +1) \\
 5 & \frac{1}{120} (5 \epsilon -3) (5 \epsilon -1) (5 \epsilon
 +1) (5 \epsilon +3) \\
 6 & \frac{2}{15} \epsilon (3 \epsilon -2) (3 \epsilon -1) (3
 \epsilon +1) (3 \epsilon +2) \\
\end{array}
\right)$$


