Is there any trick to solve integrals of the form $\int^{\infty}_{-\infty}|(a f_a(x) + b f_b(x))|^2 |(c f_a(x) - d f_b(x))|^2 dx $

Question

I have an integral of the form

$I = \int^{\infty}_{-\infty}|(a f_a(x) + b f_b(x))|^2 |(c f_a(x) - d f_b(x))|^2 dx $

where,

$f_a(x) = e^{(i\omega2+\Gamma1)(t1-x)}\Theta[x-t1]$ and $f_b(x) = e^{(i\omega2+\Gamma2)(t2-x)}\Theta[x-t2]$ and all the coefficients beside $i$ are real and $\Theta$ is a Heaviside function.

The integral is a simple exponential integration and I could get it solved in mathematica however there are 16 integrals and it is hard to get any insight from the solution. Is there a way to simplify the integral and obtain a more compact neat solution?

Thanks for your time.