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 B253(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?