I'm studying the limits and applicability of Abel Plana summation for different test functions (class of functions). In doing so this just pops out and couldn't handle the said integral so asked here ( I'm just a student so, please excuse me if anything is trivial):

Consider the following function :

$$F(x)=\frac{\sin^2(\Gamma(x))\Gamma'(x)}{e^{\sin^2(\Gamma(x))}}$$

Now consider the following function :

$$I(x) =-i\int_0^\infty\mathrm dy \frac{F(x + \mathrm iy) − F(x −\mathrm iy)}{\mathrm e^{2πy}-1}$$

What is the nature of the $I(x)$ as $x\rightarrow\infty$?

( Is $I(x)\rightarrow 0$ as $x\rightarrow\infty$ true?)

( Is there an analytic way to show this to be true or false?)

Some values I computed :

$x=0.3, I= -0.4596$

$x=0.5, I= 0.3347$

$x=0.7, I= 0.1407$

$x=0.9, I= 0.0706$

$x=1 , I= 0.05211$

$x=1.5, I=0.02101$

$x=2 , I= 0.02518$

$x=3, I=0.06752$

It seems that We can't do anything beyond $x=3$ numerically.

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