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.
Tweedie, M.C.K. (1957): "Statistical properties of inverse Gaussian distributions, I", Annals of Mathematical Statistics, 28, 696-705.