Given $k$ independent variables $x_1, x_2, \ldots, x_k$ following normal distributions, where $x_i$ follows $\mathcal N(\mu_i, \sigma_i)$, what is the distribution of the "softmax" of $x_1$, defined as $$\frac{\exp(x_1)}{\sum_i \exp(x_i)}?$$
The questions here https://stats.stackexchange.com/questions/315476 and Expectation of a multivariate Gaussian after going through a Softmax seem related, but I cannot find a way to apply the results there.