What's the fastest solution to this definite integral
$$ I = \int_{\theta_1}^{\theta_2} \int_{\omega_1}^{\omega_2} \cos [\omega(\alpha + \beta \cos \theta)+\gamma] d\omega d\theta\; ?$$
This problem happens in the eigenfilter design problem, cf. $[1,2]$.
This double integral can be reduced to calculating the single integrals
$$ I = I_\theta(\omega_2)-I_\theta(\omega_1)
= \int_{\theta_1}^{\theta_2}f(\omega_2,\theta)d\theta-\int_{\theta_1}^{\theta_2}f(\omega_1,\theta)d\theta$$
with
$$f(\omega,\theta)=\frac{\sin[\omega(\alpha + \beta \cos \theta)+\gamma]}{\alpha + \beta \cos \theta}.$$
Till this point, one conventionally uses a means of numerical tools, such as matlab, scipy toolkit, torchquad $[3]$, etc., to calculate the integration, at the cost of high computing intensity. But, because the eigenfilter design is a long process consisting of millions of such integrals, it takes too long time. Can you give any advise to accelerate this process? Is there a close form solution to this integral? Thanks a lot.
References
$[1]$ Doclo, S., & Moonen, M. ($2003$). Design of far-field and near-field broadband beamformers using eigenfilters. Signal Processing, $\mathbf{83}$ ($12$), $2641-2673$.
$[2]$ Doclo, S. ($2003$). Multi-microphone noise reduction and dereverberation techniques for speech applications.