3.514   ODE No. 514

$$\boxed{{\displaystyle {\left(\frac{\mathrm{d}}{\mathrm{d}x}y\left(x\right)\right)}^2(a\cos \left(y\left(x\right)\right)+b)-c\cos \left(y\left(x\right)\right)+d=0}}$$

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

Mathematica: cpu = 15.435460 (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 = 0.140 (sec), leaf count = 87 $$\{x-{\int }^{y\left(x\right)}(a\cos \left(\mathit{\_}\mathit{a}\right)+b)\frac{1}{\sqrt{(a\cos \left(\mathit{\_}\mathit{a}\right)+b)(c\cos \left(\mathit{\_}\mathit{a}\right)-d)}}d\mathit{\_}\mathit{a}-\mathit{\_}\mathit{C}\mathit{1}=0,x-{\int }^{y\left(x\right)}-(a\cos \left(\mathit{\_}\mathit{a}\right)+b)\frac{1}{\sqrt{(a\cos \left(\mathit{\_}\mathit{a}\right)+b)(c\cos \left(\mathit{\_}\mathit{a}\right)-d)}}d\mathit{\_}\mathit{a}-\mathit{\_}\mathit{C}\mathit{1}=0,y\left(x\right)=\arccos \left(\frac{d}{c}\right)\}$$