Consider an entire function $f : \mathbb{C} \rightarrow \mathbb{C}$! We search the function $$ g: (a,b) \rightarrow \mathbb{C},$$ which solves the following equation locally: $g'(t)=f(g(t))$ and $g(0)=f(x_0)$.

I can compute the inverse $G$ of $g$, if $f(x_0) \neq 0$, i.e. $$ G(y) = \int\limits_{f(x_0)}^y \frac{d s}{f(s)}.$$

I also known how to compute the Taylor expansion recursively, whose radius of convergence is positive (see below). Also we can give suitable approximations of the solution in terms of Picard iterations. I am not interested in such a solution!

Is there an alternative to this integral expression?