\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) ={\frac {y \left ( x \right ) }{x} \left ( -1-\ln \left ( {\frac { \left ( 1+x \right ) \left ( x-1 \right ) }{x}} \right ) +\ln \left ( {\frac { \left ( 1+x \right ) \left ( x-1 \right ) }{x}} \right ) xy \left ( x \right ) \right ) }=0} \]
Mathematica: cpu = 135.603219 (sec), leaf count = 128 \[ \left \{\left \{y(x)\to \frac {e^{\text {Li}_2(-x)+\text {Li}_2(x)} x^{\log (1-x)-\frac {\log (x)}{2}+\log (x+1)-\log \left (x-\frac {1}{x}\right )-1}}{c_1-\int _1^x \log \left (\frac {(K[1]-1) (K[1]+1)}{K[1]}\right ) \exp \left (\text {Li}_2(-K[1])+\text {Li}_2(K[1])-\frac {1}{2} \log (K[1]) \left (-2 \log (1-K[1])+\log (K[1])-2 \log (K[1]+1)+2 \log \left (K[1]-\frac {1}{K[1]}\right )+2\right )\right ) \, dK[1]}\right \}\right \} \]
Maple: cpu = 0.110 (sec), leaf count = 106 \[ \left \{ y \left ( x \right ) ={\frac {{{\rm e}^{{\it dilog} \left ( 1+x \right ) }}{x}^{\ln \left ( 1+x \right ) }}{{{\rm e}^{{\it dilog} \left ( x \right ) }}x}{{\rm e}^{-{\frac { \left ( \ln \left ( x \right ) \right ) ^{2}}{2}}}} \left ( \int \!-{\frac {{{\rm e}^{{\it dilog} \left ( 1+x \right ) }}{x}^{\ln \left ( 1+x \right ) }}{{{\rm e}^{ {\it dilog} \left ( x \right ) }}x}{{\rm e}^{-{\frac { \left ( \ln \left ( x \right ) \right ) ^{2}}{2}}}}\ln \left ( {\frac { \left ( 1+x \right ) \left ( x-1 \right ) }{x}} \right ) \left ( {x}^{\ln \left ( { \frac { \left ( 1+x \right ) \left ( x-1 \right ) }{x}} \right ) } \right ) ^{-1}}\,{\rm d}x+{\it \_C1} \right ) ^{-1} \left ( {x}^{\ln \left ( {\frac { \left ( 1+x \right ) \left ( x-1 \right ) }{x}} \right ) } \right ) ^{-1}} \right \} \]