#### 2.850   ODE No. 850

$y'(x)=\text {\_F1}(y(x)-\log (\sin (x))+\log (\cos (x)+1))+\csc (x)$ Mathematica : cpu = 0.227882 (sec), leaf count = 1485

$\text {Solve}\left [\int _1^x-\frac {\left (\cot ^2(K[1])+\csc (K[1]) \cot (K[1])+1\right ) \sin (K[1]) (\csc (K[1])+\text {\_F1}(\log (\cos (K[1])+1)-\log (\sin (K[1]))+y(x)))}{-\cot ^2(K[1])+\text {\_F1}(\log (\cos (K[1])+1)-\log (\sin (K[1]))+y(x)) \cot (K[1])+\csc ^2(K[1])+\csc (K[1]) \text {\_F1}(\log (\cos (K[1])+1)-\log (\sin (K[1]))+y(x))-1}dK[1]+\int _1^{y(x)}-\frac {\sin (x) \left (\int _1^x\left (\frac {\left (\cot ^2(K[1])+\csc (K[1]) \cot (K[1])+1\right ) \sin (K[1]) (\csc (K[1])+\text {\_F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))) \left (\cot (K[1]) \text {\_F1}'(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))+\csc (K[1]) \text {\_F1}'(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))\right )}{\left (-\cot ^2(K[1])+\text {\_F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1]))) \cot (K[1])+\csc ^2(K[1])+\csc (K[1]) \text {\_F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))-1\right ){}^2}-\frac {\left (\cot ^2(K[1])+\csc (K[1]) \cot (K[1])+1\right ) \sin (K[1]) \text {\_F1}'(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))}{-\cot ^2(K[1])+\text {\_F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1]))) \cot (K[1])+\csc ^2(K[1])+\csc (K[1]) \text {\_F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))-1}\right )dK[1] \csc ^3(x)+\text {\_F1}(K[2]+\log (\cos (x)+1)-\log (\sin (x))) \int _1^x\left (\frac {\left (\cot ^2(K[1])+\csc (K[1]) \cot (K[1])+1\right ) \sin (K[1]) (\csc (K[1])+\text {\_F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))) \left (\cot (K[1]) \text {\_F1}'(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))+\csc (K[1]) \text {\_F1}'(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))\right )}{\left (-\cot ^2(K[1])+\text {\_F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1]))) \cot (K[1])+\csc ^2(K[1])+\csc (K[1]) \text {\_F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))-1\right ){}^2}-\frac {\left (\cot ^2(K[1])+\csc (K[1]) \cot (K[1])+1\right ) \sin (K[1]) \text {\_F1}'(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))}{-\cot ^2(K[1])+\text {\_F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1]))) \cot (K[1])+\csc ^2(K[1])+\csc (K[1]) \text {\_F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))-1}\right )dK[1] \csc ^2(x)-\cot (x) \csc (x)-\cot ^2(x) \int _1^x\left (\frac {\left (\cot ^2(K[1])+\csc (K[1]) \cot (K[1])+1\right ) \sin (K[1]) (\csc (K[1])+\text {\_F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))) \left (\cot (K[1]) \text {\_F1}'(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))+\csc (K[1]) \text {\_F1}'(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))\right )}{\left (-\cot ^2(K[1])+\text {\_F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1]))) \cot (K[1])+\csc ^2(K[1])+\csc (K[1]) \text {\_F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))-1\right ){}^2}-\frac {\left (\cot ^2(K[1])+\csc (K[1]) \cot (K[1])+1\right ) \sin (K[1]) \text {\_F1}'(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))}{-\cot ^2(K[1])+\text {\_F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1]))) \cot (K[1])+\csc ^2(K[1])+\csc (K[1]) \text {\_F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))-1}\right )dK[1] \csc (x)+\cot (x) \text {\_F1}(K[2]+\log (\cos (x)+1)-\log (\sin (x))) \int _1^x\left (\frac {\left (\cot ^2(K[1])+\csc (K[1]) \cot (K[1])+1\right ) \sin (K[1]) (\csc (K[1])+\text {\_F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))) \left (\cot (K[1]) \text {\_F1}'(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))+\csc (K[1]) \text {\_F1}'(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))\right )}{\left (-\cot ^2(K[1])+\text {\_F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1]))) \cot (K[1])+\csc ^2(K[1])+\csc (K[1]) \text {\_F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))-1\right ){}^2}-\frac {\left (\cot ^2(K[1])+\csc (K[1]) \cot (K[1])+1\right ) \sin (K[1]) \text {\_F1}'(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))}{-\cot ^2(K[1])+\text {\_F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1]))) \cot (K[1])+\csc ^2(K[1])+\csc (K[1]) \text {\_F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))-1}\right )dK[1] \csc (x)-\int _1^x\left (\frac {\left (\cot ^2(K[1])+\csc (K[1]) \cot (K[1])+1\right ) \sin (K[1]) (\csc (K[1])+\text {\_F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))) \left (\cot (K[1]) \text {\_F1}'(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))+\csc (K[1]) \text {\_F1}'(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))\right )}{\left (-\cot ^2(K[1])+\text {\_F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1]))) \cot (K[1])+\csc ^2(K[1])+\csc (K[1]) \text {\_F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))-1\right ){}^2}-\frac {\left (\cot ^2(K[1])+\csc (K[1]) \cot (K[1])+1\right ) \sin (K[1]) \text {\_F1}'(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))}{-\cot ^2(K[1])+\text {\_F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1]))) \cot (K[1])+\csc ^2(K[1])+\csc (K[1]) \text {\_F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))-1}\right )dK[1] \csc (x)-\cot ^2(x)-1\right )}{-\cot ^2(x)+\text {\_F1}(K[2]+\log (\cos (x)+1)-\log (\sin (x))) \cot (x)+\csc ^2(x)+\csc (x) \text {\_F1}(K[2]+\log (\cos (x)+1)-\log (\sin (x)))-1}dK[2]=c_1,y(x)\right ]$ Maple : cpu = 0.783 (sec), leaf count = 32

$\left \{ \int _{{\it \_b}}^{y \left ( x \right ) }\! \left ( {\it \_F1} \left ( {\it \_a}-\ln \left ( \sin \left ( x \right ) \right ) +\ln \left ( \cos \left ( x \right ) +1 \right ) \right ) \right ) ^{-1}\,{\rm d}{\it \_a}-x-{\it \_C1}=0 \right \}$