\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) =-{\frac {\cos \left ( y \left ( x \right ) \right ) \left ( x-\cos \left ( y \left ( x \right ) \right ) +1 \right ) }{ \left ( x\sin \left ( y \left ( x \right ) \right ) -1 \right ) \left ( 1+x \right ) }}=0} \]
Mathematica: cpu = 5.755231 (sec), leaf count = 3913 \[ \left \{\left \{y(x)\to -\sec ^{-1}\left (\frac {c_1 x^3}{x^2-1}+\frac {\log (x+1) x^3}{x^2-1}-\frac {c_1^3 x^3}{\left (x^2-1\right ) \left (c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1\right )}-\frac {\log ^3(x+1) x^3}{\left (x^2-1\right ) \left (c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1\right )}-\frac {3 c_1 \log ^2(x+1) x^3}{\left (x^2-1\right ) \left (c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1\right )}-\frac {c_1 x^3}{\left (x^2-1\right ) \left (c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1\right )}-\frac {3 c_1^2 \log (x+1) x^3}{\left (x^2-1\right ) \left (c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1\right )}-\frac {\log (x+1) x^3}{\left (x^2-1\right ) \left (c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1\right )}-\frac {c_1^2 \sqrt {-x^2+c_1^2+\log ^2(x+1)+2 c_1 \log (x+1)+1} x^2}{\left (x^2-1\right ) \left (c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1\right )}-\frac {\log ^2(x+1) \sqrt {-x^2+c_1^2+\log ^2(x+1)+2 c_1 \log (x+1)+1} x^2}{\left (x^2-1\right ) \left (c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1\right )}-\frac {2 c_1 \log (x+1) \sqrt {-x^2+c_1^2+\log ^2(x+1)+2 c_1 \log (x+1)+1} x^2}{\left (x^2-1\right ) \left (c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1\right )}-\frac {\sqrt {-x^2+c_1^2+\log ^2(x+1)+2 c_1 \log (x+1)+1} x^2}{\left (x^2-1\right ) \left (c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1\right )}+\frac {c_1 x}{x^2-1}-c_1 x+\frac {\log (x+1) x}{x^2-1}-\log (x+1) x+\frac {c_1^3 x}{c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1}+\frac {\log ^3(x+1) x}{c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1}+\frac {3 c_1 \log ^2(x+1) x}{c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1}+\frac {c_1 x}{c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1}+\frac {3 c_1^2 \log (x+1) x}{c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1}+\frac {\log (x+1) x}{c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1}+\frac {c_1^2 \sqrt {-x^2+c_1^2+\log ^2(x+1)+2 c_1 \log (x+1)+1}}{c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1}+\frac {\log ^2(x+1) \sqrt {-x^2+c_1^2+\log ^2(x+1)+2 c_1 \log (x+1)+1}}{c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1}+\frac {2 c_1 \log (x+1) \sqrt {-x^2+c_1^2+\log ^2(x+1)+2 c_1 \log (x+1)+1}}{c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1}+\frac {\sqrt {-x^2+c_1^2+\log ^2(x+1)+2 c_1 \log (x+1)+1}}{c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1}\right )\right \},\left \{y(x)\to \sec ^{-1}\left (\frac {c_1 x^3}{x^2-1}+\frac {\log (x+1) x^3}{x^2-1}-\frac {c_1^3 x^3}{\left (x^2-1\right ) \left (c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1\right )}-\frac {\log ^3(x+1) x^3}{\left (x^2-1\right ) \left (c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1\right )}-\frac {3 c_1 \log ^2(x+1) x^3}{\left (x^2-1\right ) \left (c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1\right )}-\frac {c_1 x^3}{\left (x^2-1\right ) \left (c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1\right )}-\frac {3 c_1^2 \log (x+1) x^3}{\left (x^2-1\right ) \left (c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1\right )}-\frac {\log (x+1) x^3}{\left (x^2-1\right ) \left (c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1\right )}-\frac {c_1^2 \sqrt {-x^2+c_1^2+\log ^2(x+1)+2 c_1 \log (x+1)+1} x^2}{\left (x^2-1\right ) \left (c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1\right )}-\frac {\log ^2(x+1) \sqrt {-x^2+c_1^2+\log ^2(x+1)+2 c_1 \log (x+1)+1} x^2}{\left (x^2-1\right ) \left (c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1\right )}-\frac {2 c_1 \log (x+1) \sqrt {-x^2+c_1^2+\log ^2(x+1)+2 c_1 \log (x+1)+1} x^2}{\left (x^2-1\right ) \left (c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1\right )}-\frac {\sqrt {-x^2+c_1^2+\log ^2(x+1)+2 c_1 \log (x+1)+1} x^2}{\left (x^2-1\right ) \left (c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1\right )}+\frac {c_1 x}{x^2-1}-c_1 x+\frac {\log (x+1) x}{x^2-1}-\log (x+1) x+\frac {c_1^3 x}{c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1}+\frac {\log ^3(x+1) x}{c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1}+\frac {3 c_1 \log ^2(x+1) x}{c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1}+\frac {c_1 x}{c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1}+\frac {3 c_1^2 \log (x+1) x}{c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1}+\frac {\log (x+1) x}{c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1}+\frac {c_1^2 \sqrt {-x^2+c_1^2+\log ^2(x+1)+2 c_1 \log (x+1)+1}}{c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1}+\frac {\log ^2(x+1) \sqrt {-x^2+c_1^2+\log ^2(x+1)+2 c_1 \log (x+1)+1}}{c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1}+\frac {2 c_1 \log (x+1) \sqrt {-x^2+c_1^2+\log ^2(x+1)+2 c_1 \log (x+1)+1}}{c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1}+\frac {\sqrt {-x^2+c_1^2+\log ^2(x+1)+2 c_1 \log (x+1)+1}}{c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1}\right )\right \},\left \{y(x)\to -\sec ^{-1}\left (\frac {c_1 x^3}{x^2-1}+\frac {\log (x+1) x^3}{x^2-1}-\frac {c_1^3 x^3}{\left (x^2-1\right ) \left (c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1\right )}-\frac {\log ^3(x+1) x^3}{\left (x^2-1\right ) \left (c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1\right )}-\frac {3 c_1 \log ^2(x+1) x^3}{\left (x^2-1\right ) \left (c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1\right )}-\frac {c_1 x^3}{\left (x^2-1\right ) \left (c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1\right )}-\frac {3 c_1^2 \log (x+1) x^3}{\left (x^2-1\right ) \left (c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1\right )}-\frac {\log (x+1) x^3}{\left (x^2-1\right ) \left (c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1\right )}+\frac {c_1^2 \sqrt {-x^2+c_1^2+\log ^2(x+1)+2 c_1 \log (x+1)+1} x^2}{\left (x^2-1\right ) \left (c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1\right )}+\frac {\log ^2(x+1) \sqrt {-x^2+c_1^2+\log ^2(x+1)+2 c_1 \log (x+1)+1} x^2}{\left (x^2-1\right ) \left (c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1\right )}+\frac {2 c_1 \log (x+1) \sqrt {-x^2+c_1^2+\log ^2(x+1)+2 c_1 \log (x+1)+1} x^2}{\left (x^2-1\right ) \left (c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1\right )}+\frac {\sqrt {-x^2+c_1^2+\log ^2(x+1)+2 c_1 \log (x+1)+1} x^2}{\left (x^2-1\right ) \left (c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1\right )}+\frac {c_1 x}{x^2-1}-c_1 x+\frac {\log (x+1) x}{x^2-1}-\log (x+1) x+\frac {c_1^3 x}{c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1}+\frac {\log ^3(x+1) x}{c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1}+\frac {3 c_1 \log ^2(x+1) x}{c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1}+\frac {c_1 x}{c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1}+\frac {3 c_1^2 \log (x+1) x}{c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1}+\frac {\log (x+1) x}{c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1}-\frac {c_1^2 \sqrt {-x^2+c_1^2+\log ^2(x+1)+2 c_1 \log (x+1)+1}}{c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1}-\frac {\log ^2(x+1) \sqrt {-x^2+c_1^2+\log ^2(x+1)+2 c_1 \log (x+1)+1}}{c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1}-\frac {2 c_1 \log (x+1) \sqrt {-x^2+c_1^2+\log ^2(x+1)+2 c_1 \log (x+1)+1}}{c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1}-\frac {\sqrt {-x^2+c_1^2+\log ^2(x+1)+2 c_1 \log (x+1)+1}}{c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1}\right )\right \},\left \{y(x)\to \sec ^{-1}\left (\frac {c_1 x^3}{x^2-1}+\frac {\log (x+1) x^3}{x^2-1}-\frac {c_1^3 x^3}{\left (x^2-1\right ) \left (c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1\right )}-\frac {\log ^3(x+1) x^3}{\left (x^2-1\right ) \left (c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1\right )}-\frac {3 c_1 \log ^2(x+1) x^3}{\left (x^2-1\right ) \left (c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1\right )}-\frac {c_1 x^3}{\left (x^2-1\right ) \left (c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1\right )}-\frac {3 c_1^2 \log (x+1) x^3}{\left (x^2-1\right ) \left (c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1\right )}-\frac {\log (x+1) x^3}{\left (x^2-1\right ) \left (c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1\right )}+\frac {c_1^2 \sqrt {-x^2+c_1^2+\log ^2(x+1)+2 c_1 \log (x+1)+1} x^2}{\left (x^2-1\right ) \left (c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1\right )}+\frac {\log ^2(x+1) \sqrt {-x^2+c_1^2+\log ^2(x+1)+2 c_1 \log (x+1)+1} x^2}{\left (x^2-1\right ) \left (c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1\right )}+\frac {2 c_1 \log (x+1) \sqrt {-x^2+c_1^2+\log ^2(x+1)+2 c_1 \log (x+1)+1} x^2}{\left (x^2-1\right ) \left (c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1\right )}+\frac {\sqrt {-x^2+c_1^2+\log ^2(x+1)+2 c_1 \log (x+1)+1} x^2}{\left (x^2-1\right ) \left (c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1\right )}+\frac {c_1 x}{x^2-1}-c_1 x+\frac {\log (x+1) x}{x^2-1}-\log (x+1) x+\frac {c_1^3 x}{c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1}+\frac {\log ^3(x+1) x}{c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1}+\frac {3 c_1 \log ^2(x+1) x}{c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1}+\frac {c_1 x}{c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1}+\frac {3 c_1^2 \log (x+1) x}{c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1}+\frac {\log (x+1) x}{c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1}-\frac {c_1^2 \sqrt {-x^2+c_1^2+\log ^2(x+1)+2 c_1 \log (x+1)+1}}{c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1}-\frac {\log ^2(x+1) \sqrt {-x^2+c_1^2+\log ^2(x+1)+2 c_1 \log (x+1)+1}}{c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1}-\frac {2 c_1 \log (x+1) \sqrt {-x^2+c_1^2+\log ^2(x+1)+2 c_1 \log (x+1)+1}}{c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1}-\frac {\sqrt {-x^2+c_1^2+\log ^2(x+1)+2 c_1 \log (x+1)+1}}{c_1^2+2 \log (x+1) c_1+\log ^2(x+1)+1}\right )\right \}\right \} \]
Maple: cpu = 1.185 (sec), leaf count = 259 \[ \left \{ y \left ( x \right ) =\arctan \left ( -{\frac {-\ln \left ( 1+x \right ) +{\it \_C1}}{{{\it \_C1}}^{2}-2\,{\it \_C1}\,\ln \left ( 1+x \right ) + \left ( \ln \left ( 1+x \right ) \right ) ^{2}+1} \left ( - \ln \left ( 1+x \right ) x+{\it \_C1}\,x+\sqrt { \left ( \ln \left ( 1+x \right ) \right ) ^{2}-2\,{\it \_C1}\,\ln \left ( 1+x \right ) +{{\it \_C1}}^{2}-{x}^{2}+1} \right ) }+x,-{\frac {1}{{{\it \_C1}}^{2}-2\,{ \it \_C1}\,\ln \left ( 1+x \right ) + \left ( \ln \left ( 1+x \right ) \right ) ^{2}+1} \left ( -\ln \left ( 1+x \right ) x+{\it \_C1}\,x+ \sqrt { \left ( \ln \left ( 1+x \right ) \right ) ^{2}-2\,{\it \_C1}\, \ln \left ( 1+x \right ) +{{\it \_C1}}^{2}-{x}^{2}+1} \right ) } \right ) ,y \left ( x \right ) =\arctan \left ( {\frac {-\ln \left ( 1+x \right ) +{\it \_C1}}{{{\it \_C1}}^{2}-2\,{\it \_C1}\,\ln \left ( 1+x \right ) + \left ( \ln \left ( 1+x \right ) \right ) ^{2}+1} \left ( \ln \left ( 1+x \right ) x-{\it \_C1}\,x+\sqrt { \left ( \ln \left ( 1+x \right ) \right ) ^{2}-2\,{\it \_C1}\,\ln \left ( 1+x \right ) +{{\it \_C1}}^{2}-{x}^{2}+1} \right ) }+x,{\frac {1}{{{\it \_C1}}^{2}-2\,{\it \_C1}\,\ln \left ( 1+x \right ) + \left ( \ln \left ( 1+x \right ) \right ) ^{2}+1} \left ( \ln \left ( 1+x \right ) x-{\it \_C1}\,x+\sqrt { \left ( \ln \left ( 1+x \right ) \right ) ^{2}-2\,{\it \_C1}\,\ln \left ( 1+x \right ) +{{\it \_C1}}^{2}-{x}^{2}+1} \right ) } \right ) \right \} \]