Well, it doesn't look like the indefinite integral has a closed form. I don't know how to do it with residue integration because it doesn't look like it has any poles (the integral should be 2*pi*i times the sum of the residues of all poles having positive imaginary components), but maybe someone has some ideas. I want:
integral from -infinity to +infinity of: exp(-C1*exp(-2*x)-C2*x^2) dx
As a function of C1 and C2. exp(y) is e to the y. If only there was only one parameter, of course, say C1, I could just solve it numerically for C1=1 and use a change of variables to find it as a function of variable C1 but alas there is C1 and C2. If anyone has any ideas that would be great.