state your exact density and I'll take a look at it
The link below shows you the density functions of Inverse_Gaussian_distribution. Thanks
Inverse Gaussian distribution - Wikipedia, the free encyclopedia
I'd be curious where this comes from, like 1/Normal.
But this is only a positive rv.
In any case I might try to find the MGF thats on that page.
You would have to complete the nasty square in the exponent by including the xt.
I'll assume you want something this unusual for a scholarly purpose, in whch case you'll be needing a reference. The following reference contains what you're looking for:
Tweedie, M.C.K. (1957): "Statistical properties of inverse Gaussian distributions, I", Annals of Mathematical Statistics, 28, 696-705.