I was wondering whether anyone could think of a continuous distribution $F$, say, such that there exists a closed-form expression (or something that does not require the numerical approximation of an $n$-dimensional integral) for the expectation of $$\dfrac{\exp(a_k+x_k)}{\sum_{i=1}^n\exp(a_i+x_i)}$$ where $1\leq k\leq n$, $\{a_i\}_{i=1}^n$ is a set of constants, and all random variables $x_1,x_2,\dots,x_n$ are drawn from $F$ in an iid fashion.
Thanks!