\[ y'(x)=\frac {y(x) \left (x y(x) \log \left (\frac {(x-1) (x+1)}{x}\right )-\log \left (\frac {(x-1) (x+1)}{x}\right )-1\right )}{x} \] ✓ Mathematica : cpu = 403.502 (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 = 0.243 (sec), leaf count = 106
\[ \left \{ y \left ( x \right ) ={\frac {{{\rm e}^{{\it dilog} \left ( 1+x \right ) }}{x}^{\ln \left ( 1+x \right ) }}{x{{\rm e}^{{\it dilog} \left ( x \right ) }}}{{\rm e}^{-{\frac { \left ( \ln \left ( x \right ) \right ) ^{2}}{2}}}} \left ( \int \!-{\frac {{{\rm e}^{{\it dilog} \left ( 1+x \right ) }}{x}^{\ln \left ( 1+x \right ) }}{x{{\rm e}^{{\it dilog} \left ( x \right ) }}}{{\rm e}^{-{\frac { \left ( \ln \left ( x \right ) \right ) ^{2}}{2}}}}\ln \left ( {\frac { \left ( x-1 \right ) \left ( 1+x \right ) }{x}} \right ) \left ( {x}^{\ln \left ( {\frac { \left ( x-1 \right ) \left ( 1+x \right ) }{x}} \right ) } \right ) ^{-1}}\,{\rm d}x+{\it \_C1} \right ) ^{-1} \left ( {x}^{\ln \left ( {\frac { \left ( x-1 \right ) \left ( 1+x \right ) }{x}} \right ) } \right ) ^{-1}} \right \} \]