Multiplicative inverse of the power series $e^x - c$ for $c \neq 1$.

We know that the power series $f(x)= e^x -c \in \mathbb C[[x]]$ for $c \neq 1$, has a multiplicative inverse, since it's constant coefficient is non-zero. I was wondering whether the inverse is known in a closed form. Something like inverse of $\frac{e^x-1}{x}$, which is interesting as it generates the Bernoulli numbers.

Wolfram Alpha can find the inverse in terms of $c$.