\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) ={\frac { \left ( y \left ( x \right ) \right ) ^{3}-3\,x \left ( y \left ( x \right ) \right ) ^{2}+3\,{x}^{2}y \left ( x \right ) -{x}^{3}+{x}^{2}}{ \left ( 1+x \right ) \left ( x-1 \right ) }}=0} \]
Mathematica: cpu = 0.255032 (sec), leaf count = 238 \[ \text {Solve}\left [\frac {1}{3} \log \left (\frac {\frac {3 y(x)}{x^2-1}-\frac {3 x}{x^2-1}}{3 \sqrt [3]{\frac {1}{(x-1)^3 (x+1)^3}}}+1\right )-\frac {1}{6} \log \left (\frac {\left (\frac {3 y(x)}{x^2-1}-\frac {3 x}{x^2-1}\right )^2}{9 \left (\frac {1}{(x-1)^3 (x+1)^3}\right )^{2/3}}-\frac {\frac {3 y(x)}{x^2-1}-\frac {3 x}{x^2-1}}{3 \sqrt [3]{\frac {1}{(x-1)^3 (x+1)^3}}}+1\right )+\frac {\tan ^{-1}\left (\frac {\frac {2 \left (\frac {3 y(x)}{x^2-1}-\frac {3 x}{x^2-1}\right )}{3 \sqrt [3]{\frac {1}{(x-1)^3 (x+1)^3}}}-1}{\sqrt {3}}\right )}{\sqrt {3}}=c_1+\frac {1}{2} \left (\frac {1}{\left (x^2-1\right )^3}\right )^{2/3} \left (x^2-1\right )^2 (\log (1-x)-\log (x+1)),y(x)\right ] \]
Maple: cpu = 0.203 (sec), leaf count = 469 \[ \left \{ y \left ( x \right ) ={\frac {\sqrt {3}}{6} \left ( \sqrt [3]{{ \frac {1}{ \left ( 1+x \right ) ^{3} \left ( x-1 \right ) ^{3}}}}\sqrt {3} {x}^{2}+3\,\sqrt [3]{{\frac {1}{ \left ( 1+x \right ) ^{3} \left ( x-1 \right ) ^{3}}}}\tan \left ( {\it RootOf} \left ( -18\,\ln \left ( 1+x \right ) \left ( {\frac {1}{ \left ( 1+x \right ) ^{3} \left ( x-1 \right ) ^{3}}} \right ) ^{2/3}{x}^{4}+18\, \left ( {\frac {1}{ \left ( 1 +x \right ) ^{3} \left ( x-1 \right ) ^{3}}} \right ) ^{2/3}\ln \left ( x- 1 \right ) {x}^{4}+36\,\ln \left ( 1+x \right ) \left ( {\frac {1}{ \left ( 1+x \right ) ^{3} \left ( x-1 \right ) ^{3}}} \right ) ^{2/3}{x}^{ 2}-36\, \left ( {\frac {1}{ \left ( 1+x \right ) ^{3} \left ( x-1 \right ) ^{3}}} \right ) ^{2/3}\ln \left ( x-1 \right ) {x}^{2}-18\,\ln \left ( 1 +x \right ) \left ( {\frac {1}{ \left ( 1+x \right ) ^{3} \left ( x-1 \right ) ^{3}}} \right ) ^{2/3}+18\,\ln \left ( x-1 \right ) \left ( { \frac {1}{ \left ( 1+x \right ) ^{3} \left ( x-1 \right ) ^{3}}} \right ) ^ {2/3}-12\,{\it \_Z}\,\sqrt {3}-4\,\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 ) +4\,\ln \left ( {\frac {1}{ \left ( 1+x \right ) ^{3} \left ( x-1 \right ) ^{3}}} \right ) -6\,\ln \left ( 4/3\, \left ( \left ( \tan \left ( {\it \_Z} \right ) \right ) ^{2}+1 \right ) ^{-1} \right ) +36\,{\it \_C1} \right ) \right ) {x}^{2}-\sqrt [3]{{\frac {1}{ \left ( 1+x \right ) ^{3 } \left ( x-1 \right ) ^{3}}}}\sqrt {3}-3\,\tan \left ( {\it RootOf} \left ( -18\,\ln \left ( 1+x \right ) \left ( {\frac {1}{ \left ( 1+x \right ) ^{3} \left ( x-1 \right ) ^{3}}} \right ) ^{2/3}{x}^{4}+18\, \left ( {\frac {1}{ \left ( 1+x \right ) ^{3} \left ( x-1 \right ) ^{3}}} \right ) ^{2/3}\ln \left ( x-1 \right ) {x}^{4}+36\,\ln \left ( 1+x \right ) \left ( {\frac {1}{ \left ( 1+x \right ) ^{3} \left ( x-1 \right ) ^{3}}} \right ) ^{2/3}{x}^{2}-36\, \left ( {\frac {1}{ \left ( 1 +x \right ) ^{3} \left ( x-1 \right ) ^{3}}} \right ) ^{2/3}\ln \left ( x- 1 \right ) {x}^{2}-18\,\ln \left ( 1+x \right ) \left ( {\frac {1}{ \left ( 1+x \right ) ^{3} \left ( x-1 \right ) ^{3}}} \right ) ^{2/3}+18\, \ln \left ( x-1 \right ) \left ( {\frac {1}{ \left ( 1+x \right ) ^{3} \left ( x-1 \right ) ^{3}}} \right ) ^{2/3}-12\,{\it \_Z}\,\sqrt {3}-4\, \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 ) +4\,\ln \left ( {\frac {1}{ \left ( 1+x \right ) ^{3} \left ( x-1 \right ) ^{3}}} \right ) -6\,\ln \left ( 4/3\, \left ( \left ( \tan \left ( {\it \_Z} \right ) \right ) ^{2}+1 \right ) ^{-1} \right ) +36\,{\it \_C1} \right ) \right ) \sqrt [3]{{\frac {1}{ \left ( 1+x \right ) ^{3} \left ( x-1 \right ) ^{3}}}}+2\,\sqrt {3}x \right ) } \right \} \]