The following integrals arise in fluid flow problems.
Let $\xi>0$, $-2\pi<\eta_1<0$, $0<\eta_2\le \pi$, and $\eta_1<\eta<\eta_2$. Then evaluate \begin{align*} &\int_0^{\infty} \frac{\sin(\lambda\xi)\cosh[\lambda(\pi+\eta_1)]\sinh[\lambda(\eta_2-\eta)]\,d\lambda}{\sinh(\lambda\pi)\sinh[\lambda(\eta_2-\eta_1)]}, \\ &\int_0^{\infty} \frac{\cos(\lambda\xi)\sinh[\lambda(\pi+\eta_1)]\sinh[\lambda(\eta_2-\eta)]\,d\lambda}{\sinh(\lambda\pi)\sinh[\lambda(\eta_2-\eta_1)]}. \end{align*} Most likely such integrals would have to be evaluated using contour integration. Mathematica is able to evaluate special cases, but the general case mentioned above appears difficult.