Computation of exponential integrals

Let $P\subset\mathbb{R}^d$ be a convex full-dimensional polytope and $f:\mathbb{R}^d\mapsto\mathbb{R}$ be a linear function. The computational complexity of the integral $\int_P\exp\{f(x)\}d\,x$ is studied. It is shown that these integrals are subjected to certain non-trivial algebraic relations that makes it possible to design polynomial-time algorithms for some polytopes. Applications of exponential integrals to computation of volume and to non-linear programming are given.