Good answer by AD, it would have been a nice complement to my idea. I was going to suggest (now that you have two real integrals on an infinite domain) you can just use a transformation of variables, a map from $(-\infty,\infty)$ to $(-1,1)$ will change the interval of integration to a finite one (your integrand will change too of course) and then just use any standard technique. I was going to suggest then to just play around with the two real (transformed) integrands to figure out which technique is best for them and then just use one of the many canned routines available out there (unless you want to have a chapter of your thesis on how you came up with a new/novel way to integrate functions of this kind ;-) ). Because of the sines and cosines we suspect oscillations, I would have suggested looking at Gaussian quadrature with chebyshev nodes.
This was all before I started playing with the two real integrands myself. Holy Toledo these things are oscillatory especially if I use numbers you suggested. The graph is going across orders of magnitudes wildly fast. For example, I used $\sigma = 3700, \delta = 5000000, \nu=50001000, \kappa=10$. Mathematica couldn't even plot it. Well it spit out a picture but it was complete garbage. It cannot find an anti-derivative (there goes that) but numerically it instantly spit out an answer on the order of $10^{-14}$ so it thinks that the integral is zero which it really does look like it might be. Then I forced the numerical integrator to use a working precision of 10 digits (it carries at least 10 digits in all of its computations) and give me an answer that is at least accurate to five digits (these are...ehmmm...usually very modest goals for a world class software like mathematica) and here is what it spat out at me
"NIntegrate::mtdfb: Numerical integration with LevinRule failed. The integration continues with Method -> GaussKronrodRule. >>
NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small. >>
NIntegrate::ncvb: NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in x near {x} = {-0.4464083980}. NIntegrate obtained -5.478908926*10^-7+1.469367460*10^-6 I and 0.00001382317328. for the integral and error estimates. >>"
Just look at the integral estimate and its error in the last line. The error is like two orders of magnitude HIGHER than the error estimates. This output is from trying to integrate the original complex integral you presented so the answer is a complex number.
Now the only things I can think of is, first take comfort in the fact that this really is a hard problem. Don't feel bad for having trouble with this. If mathematica is having trouble, then it really is a tricky problem. I was going to suggest maybe some simplification like taylor expansion and then truncation but taking derivatives will be horrendous. Maybe restricting the interval of integration because of $G(x,\sigma)$ the integrands converge rather fast to zero as you go out to infinity in both directions. After chopping the interval using fourier expansion might also be interesting. It seems "natural" because of the oscillations.
Time to walk over to the math department and collaborate with the numerics people. Sorry couldn't help.
