Let $f :(-1,1) \to \mathbb{R};\ \ f(x)=\sum_{n=0}^\infty a_n x^n$ be an analytical function expressible as a power series.

Also, let

$$g : (-1,1) \to \mathbb{R}; \ \ g(x)=\frac{d}{dx} \log{f(x)} = \frac{\sum_{n=1}^\infty n a_n x^{n-1}}{\sum_{n=0}^\infty a_n x^n} =\sum_{n=0}^\infty d_n x^n$$

Assume that $\log f(x)$ is defined for all $\lvert x\rvert<1$. Is it possible to obtain a non-recursive expression for the coefficients $d_n$?

I came across some other answers related to quotients of power series (like this one or this other one), but they all rely on recursive expressions. I wonder whether there is a closed form for expressing those coefficients.