I'd like to find the class of functions that may appear as definite integrals of a rational function of $x$ and $\exp(-x)$ from $0$ to $\infty$. For that I imagine there might be an algorithm to reduce such integrals to a finite set of ``master'' integrals which can then be handled case-by-case.

More precisely, I'm interested in integrals of the type

$$\int_0^\infty \frac{P(x,\exp(-x))}{Q(x,\exp(-x))} dx$$

where $P$ and $Q$ are polynomials in two variables (such that the integral exists). The above definite integral will be a function of the parameters of the two polynomials. More precisely, if

$$P(x,y) = \sum_{ij} P_{ij} x^i y^j$$ $$Q(x,y) = \sum_{ij} Q_{ij} x^i y^j$$

then the above integral will be a function $f(P_{ij}, Q_{ij})$. What I'm interested in is the class of functions $f$.

Are there any general statements on integrals of the above type?