I would like to know if there's a way to compute or approximate the following expectation:

$$\mathbb{E}[(c+e^X)^{-n}]$$

where $X=N(0,\sigma^2)$ and $n,c>0$ (you can also assume that $n$ is a natural number if necessary). After googling a bit I found the following articles: 1, 2 which talk about negative moments of positive random variables but applying the methods yields a divergent series.