I have an integral to solve
$$ I(\mu_1,\mu_2,\mu_3,\ldots,\mu_n)= \int_{-\infty}^{\infty} \prod_{i=1}^n \frac{\tau_i\alpha_i \exp(\tau_i(x-\mu_i))}{\left(1+\exp(\tau_i(x-\mu_i))\right)^{\alpha_i+1}}dx. $$
I would be happy with just a good approximation of $I(\mu_1,\mu_2,\mu_3,\ldots,\mu_n)$ as a function (Of course an analytical solution would be better but I doubt that it exists...).
From numerical evaluations, I know that the product is a skewed monomodal distribution (its components are also skewed monomodal distributions, i.e., Generalized Logistic Distributions of Type I) and the indefinite integral appears to always have a sigmoid shape. However, I have not managed to find the maximum, thus a Laplace approximation has not been feasible (yet?).