I've been looking at nonelementary integrals of the form $\frac{1}{f(x) + g(x)}$, where $f$ and $g$ are simple but different enough to be interesting. Mathematica can't evaluate any of these integrals, even in terms of known special functions:

$$\int \frac{dx}{e^x - x}$$ $$\int \frac{dx}{x e^x -1}$$ $$\int \frac{dx}{x + \log(x)}$$ $$\int \frac{dx}{\log(x) + \log(x + 1)}$$

Have integrals like these been investigated much? I've tried several other integrals of this form without success. I can't get anywhere by hand, and I can't find any research on integrals like these. I had a hunch that introducing Lambert W functions might help, but it doesn't seem to.