The following integral $$\int\limits_0^\infty \frac{\cos{\left(\frac{1}{2}\sqrt{3}s\right)}}{\sqrt{\cosh{s}-\cos{\theta}}}\,ds$$ can be found in the paper
Tevian Dray and Gerard 't Hooft, The gravitational shock wave of a massless particle, Nuclear Physics B 253 (1985) 173--188, doi:10.1016/0550-3213(85)90525-5.
They write that they "have not attempted to perform the integration explicitly". Was this integral ever calculated explicitly?