Is there any way to integrate $\frac1{f(x)}$ in terms of the integral of $f(x)$? Is there any way to find $$\int \frac1{f(x)}\mathrm dx$$ in terms of $\int f(x) \mathrm dx$, $f(x)$ and its derivatives?

In general, no. $\int x\exp(-x)\mathrm dx$ is elementary, while $\int \frac{\exp(x)}{x}\mathrm dx$ is not expressible in terms of elementary functions.
