We have independent Bernoulli random variables $X_k$ with $\Pr(X_k=1) = p_k$ and a linear combination of them $U = \sum_k a_k X_k$. The general question is about computing expectation of a given function $f: R \to R$,
$E_U f(U)$,
in particular for $f(u) = 1/(1+e^u)$. By computing I mean an accurate numerical approximation. A brute force approach, for example, would be to sample $X$.
A more specific question: given the characteristic function, in our case it is
$\phi_{U}(t) = \prod_{k}(1-p_k + p_k e^{i a_k t})$,
are there numerical procedures to approximate the expectation $E_U f(U)$? It seems that there should be, as it is a one-dimensional integration...
Maybe I should also mention that moments of $U$ are easy to compute, which gives an option to approximate $E f(U)$ using Taylor series, https://en.wikipedia.org/wiki/Taylor_expansions_for_the_moments_of_functions_of_random_variables. But there is the following concern. The series approximate $f$ locally around the mean, $\bar U$, and is accurate when $E(U- \bar U)^n$ is small, but it need not be the case. I tried it numerically, these central moments seem to grow unbounded.