I am trying to solve the integral involving trigonometry and inverse hyperbolic functions, like the following: $I=\int_{-\infty}^{\infty} f(x)\frac{\sin(dx/2)^2}{(dx/2)}dx,\;\; (d>0),$ where $f(x)=\frac{\mbox{acrsinh}(x)}{x}=\frac{\log(x+\sqrt{1+x^2})}{x}.$
Can I solve this integral using Contour integration? It can be done when $f(x)=1$. But the technique seems not working for above $f(x)$? How can I solve this integral analytically? Thanks a lot!
Regards, Free