3.850   ODE No. 850

$$\boxed{{\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}y\left(x\right)={(\sin \left(x\right))}^{-1}+\mathit{\_}\mathit{F}\mathit{1}(y\left(x\right)-\ln (\sin \left(x\right))+\ln (\cos \left(x\right)+1))=0}}$$

1. Problem in Latex
2. Mathematica input
3. Maple input

Mathematica: cpu = 0.197025 (sec), leaf count = 1478 $$\text{Solve}[{\int }_1^{y(x)}-\frac{\sin (x)(({\int }_1^x(\frac{({\cot }^2(K[1])+\csc (K[1])\cot (K[1])+1)\sin (K[1])(\csc (K[1])+\mathrm{\_}\text{F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1]))))(\cot (K[1]){\mathrm{\_}\text{F1}}^{\prime }(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))+\csc (K[1]){\mathrm{\_}\text{F1}}^{\prime }(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1]))))}{(-{\cot }^2(K[1])+\mathrm{\_}\text{F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))\cot (K[1])+{\csc }^2(K[1])+\csc (K[1])\mathrm{\_}\text{F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))-1){}^2}-\frac{({\cot }^2(K[1])+\csc (K[1])\cot (K[1])+1)\sin (K[1]){\mathrm{\_}\text{F1}}^{\prime }(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))}{-{\cot }^2(K[1])+\mathrm{\_}\text{F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))\cot (K[1])+{\csc }^2(K[1])+\csc (K[1])\mathrm{\_}\text{F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))-1})dK[1]){\csc }^3(x)+({\int }_1^x(\frac{({\cot }^2(K[1])+\csc (K[1])\cot (K[1])+1)\sin (K[1])(\csc (K[1])+\mathrm{\_}\text{F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1]))))(\cot (K[1]){\mathrm{\_}\text{F1}}^{\prime }(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))+\csc (K[1]){\mathrm{\_}\text{F1}}^{\prime }(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1]))))}{(-{\cot }^2(K[1])+\mathrm{\_}\text{F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))\cot (K[1])+{\csc }^2(K[1])+\csc (K[1])\mathrm{\_}\text{F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))-1){}^2}-\frac{({\cot }^2(K[1])+\csc (K[1])\cot (K[1])+1)\sin (K[1]){\mathrm{\_}\text{F1}}^{\prime }(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))}{-{\cot }^2(K[1])+\mathrm{\_}\text{F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))\cot (K[1])+{\csc }^2(K[1])+\csc (K[1])\mathrm{\_}\text{F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))-1})dK[1])\mathrm{\_}\text{F1}(K[2]+\log (\cos (x)+1)-\log (\sin (x))){\csc }^2(x)-\cot (x)\csc (x)-{\cot }^2(x)({\int }_1^x(\frac{({\cot }^2(K[1])+\csc (K[1])\cot (K[1])+1)\sin (K[1])(\csc (K[1])+\mathrm{\_}\text{F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1]))))(\cot (K[1]){\mathrm{\_}\text{F1}}^{\prime }(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))+\csc (K[1]){\mathrm{\_}\text{F1}}^{\prime }(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1]))))}{(-{\cot }^2(K[1])+\mathrm{\_}\text{F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))\cot (K[1])+{\csc }^2(K[1])+\csc (K[1])\mathrm{\_}\text{F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))-1){}^2}-\frac{({\cot }^2(K[1])+\csc (K[1])\cot (K[1])+1)\sin (K[1]){\mathrm{\_}\text{F1}}^{\prime }(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))}{-{\cot }^2(K[1])+\mathrm{\_}\text{F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))\cot (K[1])+{\csc }^2(K[1])+\csc (K[1])\mathrm{\_}\text{F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))-1})dK[1])\csc (x)-({\int }_1^x(\frac{({\cot }^2(K[1])+\csc (K[1])\cot (K[1])+1)\sin (K[1])(\csc (K[1])+\mathrm{\_}\text{F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1]))))(\cot (K[1]){\mathrm{\_}\text{F1}}^{\prime }(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))+\csc (K[1]){\mathrm{\_}\text{F1}}^{\prime }(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1]))))}{(-{\cot }^2(K[1])+\mathrm{\_}\text{F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))\cot (K[1])+{\csc }^2(K[1])+\csc (K[1])\mathrm{\_}\text{F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))-1){}^2}-\frac{({\cot }^2(K[1])+\csc (K[1])\cot (K[1])+1)\sin (K[1]){\mathrm{\_}\text{F1}}^{\prime }(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))}{-{\cot }^2(K[1])+\mathrm{\_}\text{F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))\cot (K[1])+{\csc }^2(K[1])+\csc (K[1])\mathrm{\_}\text{F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))-1})dK[1])\csc (x)+\cot (x)({\int }_1^x(\frac{({\cot }^2(K[1])+\csc (K[1])\cot (K[1])+1)\sin (K[1])(\csc (K[1])+\mathrm{\_}\text{F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1]))))(\cot (K[1]){\mathrm{\_}\text{F1}}^{\prime }(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))+\csc (K[1]){\mathrm{\_}\text{F1}}^{\prime }(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1]))))}{(-{\cot }^2(K[1])+\mathrm{\_}\text{F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))\cot (K[1])+{\csc }^2(K[1])+\csc (K[1])\mathrm{\_}\text{F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))-1){}^2}-\frac{({\cot }^2(K[1])+\csc (K[1])\cot (K[1])+1)\sin (K[1]){\mathrm{\_}\text{F1}}^{\prime }(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))}{-{\cot }^2(K[1])+\mathrm{\_}\text{F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))\cot (K[1])+{\csc }^2(K[1])+\csc (K[1])\mathrm{\_}\text{F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))-1})dK[1])\mathrm{\_}\text{F1}(K[2]+\log (\cos (x)+1)-\log (\sin (x)))\csc (x)-{\cot }^2(x)-1)}{-{\cot }^2(x)+\mathrm{\_}\text{F1}(K[2]+\log (\cos (x)+1)-\log (\sin (x)))\cot (x)+{\csc }^2(x)+\csc (x)\mathrm{\_}\text{F1}(K[2]+\log (\cos (x)+1)-\log (\sin (x)))-1}dK[2]+{\int }_1^x-\frac{({\cot }^2(K[1])+\csc (K[1])\cot (K[1])+1)\sin (K[1])(\csc (K[1])+\mathrm{\_}\text{F1}(\log (\cos (K[1])+1)-\log (\sin (K[1]))+y(x)))}{-{\cot }^2(K[1])+\mathrm{\_}\text{F1}(\log (\cos (K[1])+1)-\log (\sin (K[1]))+y(x))\cot (K[1])+{\csc }^2(K[1])+\csc (K[1])\mathrm{\_}\text{F1}(\log (\cos (K[1])+1)-\log (\sin (K[1]))+y(x))-1}dK[1]=c_1,y(x)]$$

Maple: cpu = 0.795 (sec), leaf count = 32 $$\{{\int }_{\mathit{\_}\mathit{b}}^{y\left(x\right)}{\left(\mathit{\_}\mathit{F}\mathit{1}(\mathit{\_}\mathit{a}-\ln (\sin \left(x\right))+\ln (\cos \left(x\right)+1))\right)}^{-1}\mathrm{d}\mathit{\_}\mathit{a}-x-\mathit{\_}\mathit{C}\mathit{1}=0\}$$