(I want to apologise in advance for this messy integral)
Consider the following integral :
$$ I(x) = \int_0^\infty\frac{\sinh(2e^{(\alpha_1-x)}\sin(\alpha_2-y)).\sin(2e^{(\alpha_1-x)}\cos(\alpha_2-y))}{e^{2πy}-1} \, dy $$
Here,
\begin{align*} \alpha_1(x,y) &= 0.5\log(x^2+y^2)(x-c)-y\tan^{-1}(y/x) \\ \alpha_2(x,y) &= 0.5\log(x^2+y^2)y+(x-c)\tan^{-1}(y/x) \end{align*}
And $c$ is a constant.
I want to know the nature of "(sharp) upper bound" on $I(x)$ as $x\to\infty$ ?
How should I start?
( Is it true that $I(x)=o(x^\epsilon) $ for every $\epsilon>0$ ?)
Any advice on guessing of the possible growth is welcome.