\[ y'(x)=\text {$\_$F1}(y(x)-\log (\sin (x))+\log (\cos (x)+1))+\csc (x) \] ✓ Mathematica : cpu = 0.203701 (sec), leaf count = 1485
DSolve[Derivative[1][y][x] == Csc[x] + _F1[Log[1 + Cos[x]] - Log[Sin[x]] + y[x]],y[x],x]
\[\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 = 0.982 (sec), leaf count = 27
dsolve(diff(y(x),x) = 1/sin(x)+_F1(y(x)-ln(sin(x))+ln(cos(x)+1)),y(x))
\[y \left (x \right ) = -\ln \left (\csc \left (x \right )+\cot \left (x \right )\right )+\operatorname {RootOf}\left (-x +\int _{}^{\textit {\_Z}}\frac {1}{\textit {\_F1} \left (\textit {\_a} \right )}d \textit {\_a} +c_{1}\right )\]