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][1], [2][2] which talk about negative moments of positive random variables but applying the methods yields a divergent series.


  [1]: http://www.jstor.org/stable/2284399
  [2]: http://www.jstor.org/stable/2683982