In what follows $\Gamma$ is the Euler gamma function, and the (possibly complex) constants appearing are nice enough so that there are no poles or convergence issues. I am stuck with an integral of the type
\begin{equation} \int_0^\infty \frac{\Gamma(x-s)\Gamma(-x-s)}{\Gamma(x+s)\Gamma(-x+s)} x \mathrm{tanh}(x) \mathrm{d}x. \end{equation}
How can we deal with this kind of integral? Can I express it in terms of Gamma functions? I am trying to toy with many duplication and product formulas for the gamma function, as well as some known Mellin transforms, but nothing seems to appear because I don't know how to deal with the hyperbolic tangent. Even for a single gamma function or a simple quotient \begin{equation} \int_0^\infty \frac{\Gamma(a+x)}{\Gamma(b-x)} x \mathrm{tanh}(x) \mathrm{d}x \qquad \text{or} \qquad \int_0^\infty \Gamma(x+a) x \mathrm{tanh}(x) \mathrm{d}x. \end{equation}
Thanks in advance for any direction!