Does any one know how to calculate the following integration?

$$
\int_{\mathbb{R}} \left(\exp(z \: e^{-y^2})-1\right)^2 dy=?,\quad z>0.
$$

This post is related to my previous question [here][1] , based on which one can have a upper bound. Can any one give an exact solution for this integration?

Thanks a lot!
Anand


  [1]: https://mathoverflow.net/questions/161068/has-anyone-seen-this-series