I am trying to exactly solve the following integral:

Integral from 0 to infinity of

dw[e^(-Tw^2)(wcos(aw)+Hsin(aw))(wcos(bw)+Hsin(bw))/(w^2+H^2)],

where T, a, b, and H are real numbers that are each greater than 0.

I have tried Mathematica 14.0 but could not find an antiderivative using it.

Any help to find an exact answer will be very much appreciated.