2.514   ODE No. 514

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

$$y^{\prime }(x{)}^2(a\cos (y(x))+b)-c\cos (y(x))+d=0$$ ✓ Mathematica : cpu = 14.7912 (sec), leaf count = 605

$$\{\{y(x)\to \text{InverseFunction}\left[\frac{4{\sin }^2\left(\frac{\mathrm{\#}\text{1}}{2}\right)\csc (\mathrm{\#}\text{1})\sqrt{a\cos (\mathrm{\#}\text{1})+b}\sqrt{\frac{{\cot }^2\left(\frac{\mathrm{\#}\text{1}}{2}\right)(c-d)}{c+d}}\sqrt{\frac{{\csc }^2\left(\frac{\mathrm{\#}\text{1}}{2}\right)(a+b)(d-c\cos (\mathrm{\#}\text{1}))}{ad+bc}}(c(a+b)F({\sin }^{-1}\left(\frac{\sqrt{\frac{(a+b)(d-c\cos (\mathrm{\#}\text{1})){\csc }^2\left(\frac{\mathrm{\#}\text{1}}{2}\right)}{bc+ad}}}{\sqrt{2}}\right)|\frac{2(bc+ad)}{(a+b)(c+d)})+a(d-c)\mathrm{\Pi }(\frac{bc+ad}{ac+bc};{\sin }^{-1}\left(\frac{\sqrt{\frac{(a+b)(d-c\cos (\mathrm{\#}\text{1})){\csc }^2\left(\frac{\mathrm{\#}\text{1}}{2}\right)}{bc+ad}}}{\sqrt{2}}\right)|\frac{2(bc+ad)}{(a+b)(c+d)}))}{c(a+b)\sqrt{c\cos (\mathrm{\#}\text{1})-d}\sqrt{\frac{{\csc }^2\left(\frac{\mathrm{\#}\text{1}}{2}\right)(c-d)(a\cos (\mathrm{\#}\text{1})+b)}{ad+bc}}}\mathrm{\&}\right][c_1-x]\},\{y(x)\to \text{InverseFunction}\left[\frac{4{\sin }^2\left(\frac{\mathrm{\#}\text{1}}{2}\right)\csc (\mathrm{\#}\text{1})\sqrt{a\cos (\mathrm{\#}\text{1})+b}\sqrt{\frac{{\cot }^2\left(\frac{\mathrm{\#}\text{1}}{2}\right)(c-d)}{c+d}}\sqrt{\frac{{\csc }^2\left(\frac{\mathrm{\#}\text{1}}{2}\right)(a+b)(d-c\cos (\mathrm{\#}\text{1}))}{ad+bc}}(c(a+b)F({\sin }^{-1}\left(\frac{\sqrt{\frac{(a+b)(d-c\cos (\mathrm{\#}\text{1})){\csc }^2\left(\frac{\mathrm{\#}\text{1}}{2}\right)}{bc+ad}}}{\sqrt{2}}\right)|\frac{2(bc+ad)}{(a+b)(c+d)})+a(d-c)\mathrm{\Pi }(\frac{bc+ad}{ac+bc};{\sin }^{-1}\left(\frac{\sqrt{\frac{(a+b)(d-c\cos (\mathrm{\#}\text{1})){\csc }^2\left(\frac{\mathrm{\#}\text{1}}{2}\right)}{bc+ad}}}{\sqrt{2}}\right)|\frac{2(bc+ad)}{(a+b)(c+d)}))}{c(a+b)\sqrt{c\cos (\mathrm{\#}\text{1})-d}\sqrt{\frac{{\csc }^2\left(\frac{\mathrm{\#}\text{1}}{2}\right)(c-d)(a\cos (\mathrm{\#}\text{1})+b)}{ad+bc}}}\mathrm{\&}\right][c_1+x]\}\}$$

✓ Maple : cpu = 0.235 (sec), leaf count = 87

$$\{x-{\int }^{y\left(x\right)}(a\cos \left(\mathit{\_}\mathit{a}\right)+b)\frac{1}{\sqrt{(a\cos \left(\mathit{\_}\mathit{a}\right)+b)(c\cos \left(\mathit{\_}\mathit{a}\right)-d)}}d\mathit{\_}\mathit{a}-\mathit{\_}\mathit{C}\mathit{1}=0,x-{\int }^{y\left(x\right)}-(a\cos \left(\mathit{\_}\mathit{a}\right)+b)\frac{1}{\sqrt{(a\cos \left(\mathit{\_}\mathit{a}\right)+b)(c\cos \left(\mathit{\_}\mathit{a}\right)-d)}}d\mathit{\_}\mathit{a}-\mathit{\_}\mathit{C}\mathit{1}=0,y\left(x\right)=\arccos \left(\frac{d}{c}\right)\}$$