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).

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

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?