$\newcommand{\+}{^{\dagger}}% \newcommand{\angles}[1]{\left\langle #1 \right\rangle}% \newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}% \newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}% \newcommand{\ceil}[1]{\,\left\lceil #1 \right\rceil\,}% \newcommand{\dd}{{\rm d}}% \newcommand{\down}{\downarrow}% \newcommand{\ds}[1]{\displaystyle{#1}}% \newcommand{\equalby}[1]{{#1 \atop {= \atop \vphantom{\huge A}}}}% \newcommand{\expo}[1]{\,{\rm e}^{#1}\,}% \newcommand{\fermi}{\,{\rm f}}% \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,}% \newcommand{\half}{{1 \over 2}}% \newcommand{\ic}{{\rm i}}% \newcommand{\iff}{\Longleftrightarrow} \newcommand{\imp}{\Longrightarrow}% \newcommand{\isdiv}{\,\left.\right\vert\,}% \newcommand{\ket}[1]{\left\vert #1\right\rangle}% \newcommand{\ol}[1]{\overline{#1}}% \newcommand{\pars}[1]{\left( #1 \right)}% \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\pp}{{\cal P}}% \newcommand{\root}[2][]{\,\sqrt[#1]{\,#2\,}\,}% \newcommand{\sech}{\,{\rm sech}}% \newcommand{\sgn}{\,{\rm sgn}}% \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}} \newcommand{\ul}[1]{\underline{#1}}% \newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert}$ \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 $$ \Psi\pars{z} \sim \ln\pars{z} - {1 \over 12 z} - {1 \over 12 z^{2}} + {1 \over 120 z^{4}} + {\rm O}\pars{1 \over z^{6}}\,,\quad \verts{z} \gg 1 $$
$$\color{#00f}{\large% {\rm I}\pars{\gamma} \sim -\,\half\,{\rm C} - \ln\pars{2} - \half\ln\pars{1 + \gamma}\,,\qquad\qquad\gamma\ \gg\ 1} $$