#### 2.742   ODE No. 742

$y'(x)=-\frac {(-\cos (y(x))+x+1) \cos (y(x))}{(x+1) (x \sin (y(x))-1)}$ Mathematica : cpu = 3.07936 (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 = 1.654 (sec), leaf count = 239

$\left \{ y \left ( x \right ) =\arctan \left ( {\frac {1}{{{\it \_C1}}^{2}-2\,{\it \_C1}\,\ln \left ( 1+x \right ) + \left ( \ln \left ( 1+x \right ) \right ) ^{2}+1} \left ( \left ( -\ln \left ( 1+x \right ) +{\it \_C1} \right ) \sqrt { \left ( \ln \left ( 1+x \right ) \right ) ^{2}-2\,{\it \_C1}\,\ln \left ( 1+x \right ) +{{\it \_C1}}^{2}-{x}^{2}+1}+x \right ) },{\frac {1}{{{\it \_C1}}^{2}-2\,{\it \_C1}\,\ln \left ( 1+x \right ) + \left ( \ln \left ( 1+x \right ) \right ) ^{2}+1} \left ( \ln \left ( 1+x \right ) x-{\it \_C1}\,x+\sqrt { \left ( \ln \left ( 1+x \right ) \right ) ^{2}-2\,{\it \_C1}\,\ln \left ( 1+x \right ) +{{\it \_C1}}^{2}-{x}^{2}+1} \right ) } \right ) ,y \left ( x \right ) =\arctan \left ( {\frac {1}{{{\it \_C1}}^{2}-2\,{\it \_C1}\,\ln \left ( 1+x \right ) + \left ( \ln \left ( 1+x \right ) \right ) ^{2}+1} \left ( \left ( \ln \left ( 1+x \right ) -{\it \_C1} \right ) \sqrt { \left ( \ln \left ( 1+x \right ) \right ) ^{2}-2\,{\it \_C1}\,\ln \left ( 1+x \right ) +{{\it \_C1}}^{2}-{x}^{2}+1}+x \right ) },{\frac {1}{{{\it \_C1}}^{2}-2\,{\it \_C1}\,\ln \left ( 1+x \right ) + \left ( \ln \left ( 1+x \right ) \right ) ^{2}+1} \left ( \ln \left ( 1+x \right ) x-{\it \_C1}\,x-\sqrt { \left ( \ln \left ( 1+x \right ) \right ) ^{2}-2\,{\it \_C1}\,\ln \left ( 1+x \right ) +{{\it \_C1}}^{2}-{x}^{2}+1} \right ) } \right ) \right \}$