# Evaluation of integrals

I wast to evaluate the following integrals:

$$\begin{multline} \label{eqlens12} \int_0^{\infty}\frac{(1+4\lambda^2)}{(1+\lambda^2)\left[\lambda\sin(2\eta_2)+\sinh(2\lambda\eta_2)\right]} \\ \left\{\lambda\sin(2\eta_2)+\sinh(2\lambda\eta_2)+\left[1+2\lambda^2\sin^2\eta_2-\cosh(2\lambda\eta_2)\right]\tanh(\lambda\pi)\right\}\,d\lambda. \end{multline}$$ $$$$\label{eqlens11} \int_0^{\infty}\frac{(1+4\lambda^2)\left\{1+(1+2\lambda^2)\left[3\cosh(\lambda\pi)-\cosh(3\lambda\pi)\right] -3\cosh(2\lambda\pi)\right\}\,d\lambda}{2(1+\lambda^2)\cosh(\lambda\pi)\left[1+2\lambda^2-\cosh(3\lambda\pi)\right]}.$$$$ $$$$\int_0^{\infty}(4\lambda^2+1)\left[1-\tanh(\lambda\eta_2)\tanh(\lambda\pi)\right]\,d\lambda.$$$$ In the first and last integrals $$\pi>\eta_2>0$$. Would be grateful if anybody can help.

• Have you tried with WolframAlpha? – Nosrati Dec 14 '18 at 12:53
• Yes, of course. It is able to evaluate the integrals for special values of $\eta_2$ (especially in the last integral, it is able to evaluate for $\eta_2=\pi/4,\pi/2,\pi$ etc. (the upper limit for $\eta_2$ should $\le \pi$ and not $<pi$), but I am not able to find the general expression from these particular cases. – Jog Dec 15 '18 at 3:31