Gradshteyn&Ryzhik 3.554.5 states that: $$\int_0^{\infty } \frac1x \biggl( 2qe^{-x}-\frac{\sinh (q x)}{\sinh \left(\frac{x}{2}\right)} \biggr) \, dx=\log\bigl(\cos (\pi q) \bigr)+2 \log \left(\Gamma \bigl(q+\frac12 \bigr)\right)-\log (\pi )~, \quad q^2<\frac{1}{2}$$ It looks like to have relation with the Binet representation of log-gamma function, but I haven't figure out how to solve this, and would like you to help. Any kind of approach is appreciated.$$$$ Note that a related problem in 3.559 is: $$\int_0^{\infty } \frac{ e^{-x} }{x} \left(e^{ (2-a) x} \frac{(1-a x) (1-e^{-x}) - x e^{-x}}{4 \sinh ^2\left(\frac{x}{2}\right)}+a-\frac{1}{2}\right) \, dx \\ =a+\log (\Gamma (a))-\frac{1}{2}-\frac{1}{2} \log (2 \pi )~, \qquad a>0$$ Maybe this can be solved together.
from Hot Weekly Questions - Mathematics Stack Exchange
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