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\begin{align}
{\rm I}\pars{\gamma}
&\equiv
{1 \over 4}\,\pars{1 + \gamma}^{2}
\int_{-1}^{1}\int_{-1}^{1}\ln\pars{\verts{x - y}}\verts{xy}^{\gamma}\,\dd x\,\dd y
\\[3mm]&=
{1 \over 4}\,\pars{1 + \gamma}^{2}
\int_{-1}^{1}\int_{0}^{1}\ln\pars{\verts{x^{2} - y^{2}}}\verts{xy}^{\gamma}
\,\dd x\,\dd y
\\[3mm]&=
\half\,\pars{1 + \gamma}^{2}
\int_{0}^{1}\int_{0}^{1}\ln\pars{\verts{x^{2} - y^{2}}}\pars{xy}^{\gamma}
\,\dd x\,\dd y
\\[3mm]&=
\half\,\pars{1 + \gamma}^{2}
\int_{0}^{1}\dd x\int_{0}^{1}\braces{%
2\ln\pars{x} + \ln\pars{\verts{1 - \bracks{y \over x}^{2}}}}
x^{2\gamma + 1}\pars{y \over x}^{\gamma}
\,{\dd y \over x}
\\[3mm]&=
\pars{1 + \gamma}^{2}
\underbrace{\int_{0}^{1}\dd x\,x^{2\gamma + 1}\ln\pars{x}\int_{0}^{1/x}y^{\gamma}\,\dd y}
_{\ds{\equiv\ {\cal F}_{1}\pars{\gamma}}}
\\[3mm]&\phantom{=}+
\half\,\pars{1 + \gamma}^{2}
\underbrace{\int_{0}^{1}\dd x\,x^{2\gamma + 1}\int_{0}^{1/x}
\ln\pars{\verts{1 - y^{2}}}y^{\gamma}\,\dd y}
_{\ds{\equiv\ {\cal F}_{2}\pars{\gamma}}}\tag{1}
\end{align}
\begin{align}
{\cal F}_{1}\pars{\gamma}&=
\int_{0}^{1}x^{2\gamma + 1}\ln\pars{x}\,{\dd x \over \pars{1 + \gamma}x^{\gamma + 1}}
={1 \over 1 + \gamma}\lim_{\mu \to \gamma}\partiald{}{\mu}\int_{0}^{1}x^{\mu}
\,\dd x
=
{1 \over 1 + \gamma}\lim_{\mu \to \gamma}\partiald{}{\mu}\pars{1 \over \mu + 1}
\\[3mm]&=
-\,{1 \over \pars{1 + \gamma}^{3}}
\end{align}
Whit this result, ${\rm I}\pars{\gamma}$
$\pars{~\mbox{see expression}\ \pars{1} ~}$ is reduced to
$$
{\rm I}\pars{\gamma}=-\,{1 \over 1 + \gamma}
+ \half\,\pars{1 + \gamma}^{2}\,{\cal F}_{2}\pars{\gamma}\tag{2}
$$
where ${\cal F}_{2}\pars{\gamma}$ is defined in $\pars{1}$.
\begin{align}
{\cal F}_{2}\pars{\gamma}&=
\int_{0}^{\infty}\,\dd y\,\ln\pars{\verts{1 - y^{2}}}y^{\gamma}
\int_{0}^{1}\dd x\,x^{2\gamma + 1}\Theta\pars{{1 \over y} - x}\,\dd x
\\[3mm]&=
\int_{0}^{\infty}\,\dd y\,\ln\pars{\verts{1 - y^{2}}}y^{\gamma}\bracks{%
\Theta\pars{1 - y}\int_{0}^{1}\dd x\,x^{2\gamma + 1}\,\dd x
+
\Theta\pars{y - 1}\int_{0}^{1/y}\dd x\,x^{2\gamma + 1}\,\dd x}
\\[3mm]&=
\int_{0}^{\infty}\,\dd y\,\ln\pars{\verts{1 - y^{2}}}y^{\gamma}\bracks{%
\Theta\pars{1 - y}\,{1 \over 2\pars{1 + \gamma}} +
\Theta\pars{y - 1}\,{1 \over 2\pars{1 + \gamma}y^{2\gamma + 2}}}
\\[3mm]&={1 \over 2\pars{1 + \gamma}}\bracks{%
\int_{0}^{1}\ln\pars{1 - y^{2}}y^{\gamma}\,\dd y
+
\int_{1}^{\infty}\ln\pars{y^{2} - 1}y^{-\gamma - 2}\,\dd y}
\\[3mm]&={1 \over 2\pars{1 + \gamma}}\braces{%
\int_{0}^{1}\ln\pars{1 - y^{2}}y^{\gamma}\,\dd y
+
\int_{0}^{1}\bracks{\ln\pars{1 - y^{2}} - 2\ln\pars{y}}y^{\gamma}\,\dd y}
\\[3mm]&={1 \over 1 + \gamma}\bracks{%
\int_{0}^{1}\ln\pars{1 - y^{2}}y^{\gamma}\,\dd y
-
\int_{0}^{1}\ln\pars{y}y^{\gamma}\,\dd y}
\end{align}
Since
$\ds{\int_{0}^{1}\ln\pars{y}y^{\gamma}\,\dd y = \lim_{\mu \to \gamma}\partiald{}{\mu}\int_{0}^{1}y^{\mu}\,\dd y = -\,{1 \over \pars{1 + \gamma}^{2}}}$,
${\cal F}_{2}\pars{\gamma}$ is reduced to:
$$
{\cal F}_{2}\pars{\gamma}=
{1 \over \pars{1 + \gamma}^{3}}
+
{1 \over 1 + \gamma}\int_{0}^{1}\ln\pars{1 - y^{2}}y^{\gamma}\,\dd y
$$
$$
\mbox{and}\ \pars{~\mbox{see expression}\ \pars{2}~}\quad
{\rm I}\pars{\gamma}=-\,{1 \over 2\pars{1 + \gamma}}
+
\half\,\pars{1 + \gamma}\int_{0}^{1}\ln\pars{1 - y^{2}}y^{\gamma}\,\dd y\tag{3}
$$
The last integral $\pars{~\mbox{in}\ \pars{3}~}$ is evaluated as follows
$\pars{~{\rm B}\pars{x,y}\ \mbox{and}\ \Gamma\pars{z}\ \mbox{are the}\ Beta\ \mbox{and}\ Gamma\ \mbox{functions, respectively}~}$:
\begin{align}
\int_{0}^{1}\!\!\!\!\!\ln\pars{1 - y^{2}}y^{\gamma}\,\dd y
&=\half\int_{0}^{1}\ln\pars{1 - y}y^{\pars{\gamma - 1}/2}\,\dd y
=\half\lim_{\mu \to 0}\partiald{}{\mu}\int_{0}^{1}\pars{1 - y}^{\mu}
y^{\pars{\gamma - 1}/2}\,\dd y
\\[3mm]&=
\half\lim_{\mu \to 0}\partiald{{\rm B}\pars{\mu + 1,\bracks{\gamma + 1}/2}}{\mu}
=\half\lim_{\mu \to 0}\partiald{}{\mu}\bracks{%
{\Gamma\pars{\mu + 1}\Gamma\pars{\gamma/2 + 1/2}
\over \Gamma\pars{\mu + \gamma/2 + 3/2}}}
\\[3mm]&=\half\,\Gamma\pars{\half + \half\,\gamma}\lim_{\mu \to 0}\bracks{%
{\Gamma\pars{\mu + 1}\Psi\pars{\mu + 1} \over \Gamma\pars{\mu + \gamma/2 + 3/2}}
-
{\Gamma\pars{\mu + 1}\Psi\pars{\mu + \gamma/2 + 3/2} \over \Gamma\pars{\mu + \gamma/2 + 3/2}}}
\\[3mm]&=\half\,
\overbrace{\Gamma\pars{1/2 + \gamma/2} \over \Gamma\pars{3/2 + \gamma/2}}
^{\ds{=\ {2 \over 1 + \gamma}}}
\bracks{\Psi\pars{1} - \Psi\pars{{3 \over 2} + \half\,\gamma}}
=
-\,{{\bf C} + \Psi\pars{3/2 + \gamma/2} \over 1 + \gamma}
\end{align}
where $\Psi\pars{z}$ is the $\it digamma$ function and ${\bf C}$ is the
$\it\mbox{Euler-Mascheroni constant}$.
With this result and expression $\pars{3}$ we arrive to:
\begin{align}
\color{#00f}{\large{\rm I}\pars{\gamma}}
&=
\color{#00f}{\large-\,\half\bracks{%
{1 \over 1 + \gamma} + {\rm C} + \Psi\pars{{3 \over 2} + \half\,\gamma}}}
\\[3mm]&=\color{#00f}{\large%
-\,\half\bracks{%
{3 \over 1 + \gamma} + {\rm C} + \Psi\pars{\half\,\bracks{1 + \gamma}}}}
\end{align}
Also
$$
\Psi\pars{z} \sim \ln\pars{z} - {1 \over 12 z} - {1 \over 12 z^{2}}
+ {1 \over 120 z^{4}} + {\rm O}\pars{1 \over z^{6}}\,,\quad
\verts{z} \gg 1
$$
$$\color{#00f}{\large%
{\rm I}\pars{\gamma}
\sim
-\,\half\,{\rm C} - \ln\pars{2} -
\half\ln\pars{1 + \gamma}\,,\qquad\qquad\gamma\ \gg\ 1}
$$
