\[ y'(x)=\text {$\_$F1}(y(x)-\log (\sin (x))+\log (\cos (x)+1))+\csc (x) \] ✓ Mathematica : cpu = 0.252632 (sec), leaf count = 114
\[\text {Solve}\left [c_1=\int _1^{y(x)} \left (\frac {1}{\text {$\_$F1}(K[2]-\log (\sin (x))+\log (\cos (x)+1))}-\int _1^x \frac {\csc (K[1]) \text {$\_$F1}'(K[2]-\log (\sin (K[1]))+\log (\cos (K[1])+1))}{(\text {$\_$F1}(K[2]-\log (\sin (K[1]))+\log (\cos (K[1])+1))){}^2} \, dK[1]\right ) \, dK[2]+\int _1^x \left (-\frac {\csc (K[1])}{\text {$\_$F1}(-\log (\sin (K[1]))+\log (\cos (K[1])+1)+y(x))}-1\right ) \, dK[1],y(x)\right ]\]
✓ Maple : cpu = 1.39 (sec), leaf count = 32
\[ \left \{ \int _{{\it \_b}}^{y \left ( x \right ) }\! \left ( {\it \_F1} \left ( {\it \_a}-\ln \left ( \sin \left ( x \right ) \right ) +\ln \left ( \cos \left ( x \right ) +1 \right ) \right ) \right ) ^{-1}\,{\rm d}{\it \_a}-x-{\it \_C1}=0 \right \} \]