\[ y'(x)=\text {$\_$F1}(y(x)-\log (\sin (x))+\log (\cos (x)+1))+\csc (x) \] ✓ Mathematica : cpu = 0.253636 (sec), leaf count = 1478
\[\text {Solve}\left [\int _1^{y(x)} -\frac {\sin (x) \left (\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]\right ) \csc ^3(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]\right ) \text {$\_$F1}(K[2]+\log (\cos (x)+1)-\log (\sin (x))) \csc ^2(x)-\cot (x) \csc (x)-\cot ^2(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]\right ) \csc (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]\right ) \csc (x)+\cot (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]\right ) \text {$\_$F1}(K[2]+\log (\cos (x)+1)-\log (\sin (x))) \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]+\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]=c_1,y(x)\right ]\] ✓ Maple : cpu = 1.171 (sec), leaf count = 32
\[ \left \{ \int _{{\it \_b}}^{y \left ( x \right ) }\! \left ( {\it \_F1} \left ( {\it \_a}-\ln \left ( \sin \left ( x \right ) \right ) +\ln \left ( \cos \left ( x \right ) +1 \right ) \right ) \right ) ^{-1}\,{\rm d}{\it \_a}-x-{\it \_C1}=0 \right \} \]