\[ y'(x)=-\frac {(-\cos (y(x))+x+1) \cos (y(x))}{(x+1) (x \sin (y(x))-1)} \] ✓ Mathematica : cpu = 2.87251 (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 = 1.87 (sec), leaf count = 239
\[y \relax (x ) = \arctan \left (\frac {\left (c_{1}-\ln \left (1+x \right )\right ) \sqrt {\ln \left (1+x \right )^{2}-2 c_{1} \ln \left (1+x \right )+c_{1}^{2}-x^{2}+1}+x}{c_{1}^{2}-2 c_{1} \ln \left (1+x \right )+\ln \left (1+x \right )^{2}+1}, \frac {\ln \left (1+x \right ) x -x c_{1}+\sqrt {\ln \left (1+x \right )^{2}-2 c_{1} \ln \left (1+x \right )+c_{1}^{2}-x^{2}+1}}{c_{1}^{2}-2 c_{1} \ln \left (1+x \right )+\ln \left (1+x \right )^{2}+1}\right )\]