\[ y'(x)=\frac {3 x^2 y(x)-x^3+x^2-3 x y(x)^2+y(x)^3}{(x-1) (x+1)} \] ✓ Mathematica : cpu = 0.50343 (sec), leaf count = 238
\[\text {Solve}\left [\frac {1}{3} \log \left (\frac {\frac {3 y(x)}{x^2-1}-\frac {3 x}{x^2-1}}{3 \sqrt [3]{\frac {1}{(x-1)^3 (x+1)^3}}}+1\right )-\frac {1}{6} \log \left (\frac {\left (\frac {3 y(x)}{x^2-1}-\frac {3 x}{x^2-1}\right )^2}{9 \left (\frac {1}{(x-1)^3 (x+1)^3}\right )^{2/3}}-\frac {\frac {3 y(x)}{x^2-1}-\frac {3 x}{x^2-1}}{3 \sqrt [3]{\frac {1}{(x-1)^3 (x+1)^3}}}+1\right )+\frac {\tan ^{-1}\left (\frac {\frac {2 \left (\frac {3 y(x)}{x^2-1}-\frac {3 x}{x^2-1}\right )}{3 \sqrt [3]{\frac {1}{(x-1)^3 (x+1)^3}}}-1}{\sqrt {3}}\right )}{\sqrt {3}}=\frac {1}{2} \left (\frac {1}{\left (x^2-1\right )^3}\right )^{2/3} \left (x^2-1\right )^2 (\log (1-x)-\log (x+1))+c_1,y(x)\right ]\] ✓ Maple : cpu = 1.48 (sec), leaf count = 188
\[ \left \{ y \left ( x \right ) ={\frac {\sqrt {3}}{2} \left ( {\frac {{x}^{2}-1}{3} \left ( 3\,\tan \left ( {\it RootOf} \left ( -9\,\ln \left ( {\frac {1+x}{x-1}} \right ) \left ( {\frac {1}{ \left ( 1+x \right ) ^{3} \left ( x-1 \right ) ^{3}}} \right ) ^{2/3}{x}^{4}+18\,\ln \left ( {\frac {1+x}{x-1}} \right ) \left ( {\frac {1}{ \left ( 1+x \right ) ^{3} \left ( x-1 \right ) ^{3}}} \right ) ^{2/3}{x}^{2}-9\,\ln \left ( {\frac {1+x}{x-1}} \right ) \left ( {\frac {1}{ \left ( 1+x \right ) ^{3} \left ( x-1 \right ) ^{3}}} \right ) ^{2/3}-6\,\sqrt {3}{\it \_Z}-3\,\ln \left ( 4/3\, \left ( \left ( \tan \left ( {\it \_Z} \right ) \right ) ^{2}+1 \right ) ^{-1} \right ) -2\,\ln \left ( 3/8\,{\frac { \left ( \sqrt {3}+\tan \left ( {\it \_Z} \right ) \right ) ^{3}\sqrt {3}}{ \left ( 1+x \right ) ^{3} \left ( x-1 \right ) ^{3}}} \right ) +2\,\ln \left ( {\frac {1}{ \left ( 1+x \right ) ^{3} \left ( x-1 \right ) ^{3}}} \right ) +18\,{\it \_C1} \right ) \right ) +\sqrt {3} \right ) \sqrt [3]{{\frac {1}{ \left ( 1+x \right ) ^{3} \left ( x-1 \right ) ^{3}}}}}+{\frac {2\,\sqrt {3}x}{3}} \right ) } \right \} \]