Integrating $I_{\xi}(\rho)=\int^{\infty}_{0}\frac{dt}{t}\frac{\sinh[(\xi+1)t]}{\sinh(\xi t)}\frac{[\cosh(2t)\cos(4\rho t)-1]}{\cosh(t)\sinh(2t)}$

Question

$$I_{\xi}(\rho)=\int^{\infty}_{0}\frac{dt}{t}\frac{\sinh[(\xi+1)t]}{\sinh(\xi t)}\frac{[\cosh(2t)\cos(4\rho t)-1]}{\cosh(t)\sinh(2t)}$$

I find this integral usually shown in the Bethe Ansatz literatures (such as PRL 106,217203), it is not divergent, but not smooth around t=0. Or, is this integral can only be calculated by numerics?

Answer 1

$$I_{\xi}(\rho)=\int {\sinh ((\xi +1) t)\, \text{csch}(\xi t)}\,\text{csch}(2 t) \,\text{sech}(t) \,(\cosh (2 t) \,\cos (4 \rho t)-1)\,\frac{dt}t$$

The integrand does not seem to make a problem around $0$ since its series expansion is $$\frac{4 (\xi +1)(1-4 \rho ^2)}{\xi }-\frac{t^2 \left((\xi +1) \left(4 (\xi +3) \rho ^2-\xi -16 \rho ^4+2\right)\right)}{3 \xi }+O\left(t^4\right)$$

How does behave the integrand for large values of $t$ ???

For the definite integral, I am afraid that only numerical integration could provide the result (it is not sure that it does converge).