3.360   ODE No. 360

$$\boxed{{\displaystyle \left(\frac{\mathrm{d}}{\mathrm{d}x}y\left(x\right)\right)\cos \left(ay\left(x\right)\right)-b(1-c\cos \left(ay\left(x\right)\right))\sqrt{{(\cos \left(ay\left(x\right)\right))}^2-1+c\cos \left(ay\left(x\right)\right)}=0}}$$

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

Mathematica: cpu = 4.990134 (sec), leaf count = 369 $$\left\{\{y(x)\to \text{InverseFunction}[-\frac{i(\cos (\mathrm{\#}\text{1}a)+1)\sqrt{\frac{2c\cos (\mathrm{\#}\text{1}a)+\cos (2\mathrm{\#}\text{1}a)-1}{(\cos (\mathrm{\#}\text{1}a)+1{)}^2}}\sqrt{\frac{c{\tan }^2\left(\frac{\mathrm{\#}\text{1}a}{2}\right)+\sqrt{c^2+4}+2}{\sqrt{c^2+4}+2}}\sqrt{1-\frac{c{\tan }^2\left(\frac{\mathrm{\#}\text{1}a}{2}\right)}{\sqrt{c^2+4}-2}}((c-1)F(i{\sinh }^{-1}(\sqrt{-\frac{c}{\sqrt{c^2+4}-2}}\tan \left(\frac{a\mathrm{\#}\text{1}}{2}\right))|\frac{2-\sqrt{c^2+4}}{\sqrt{c^2+4}+2})+2\mathrm{\Pi }(\frac{(c+1)(\sqrt{c^2+4}-2)}{(c-1)c};i{\sinh }^{-1}(\sqrt{-\frac{c}{\sqrt{c^2+4}-2}}\tan \left(\frac{a\mathrm{\#}\text{1}}{2}\right))|\frac{2-\sqrt{c^2+4}}{\sqrt{c^2+4}+2}))}{a(c^2-1)\sqrt{\frac{c}{4-2\sqrt{c^2+4}}}\sqrt{2c\cos (\mathrm{\#}\text{1}a)+\cos (2\mathrm{\#}\text{1}a)-1}\sqrt{-c{\tan }^4\left(\frac{\mathrm{\#}\text{1}a}{2}\right)-4{\tan }^2\left(\frac{\mathrm{\#}\text{1}a}{2}\right)+c}}\mathrm{\&}][c_1-\frac{bx}{\sqrt{2}}]\}\right\}$$

Maple: cpu = 0.124 (sec), leaf count = 48 $$\{x+{\int }^{y\left(x\right)}2\frac{\cos \left(\mathit{\_}\mathit{a}a\right)}{b(c\cos \left(\mathit{\_}\mathit{a}a\right)-1)\sqrt{2\cos \left(2\mathit{\_}\mathit{a}a\right)-2+4c\cos \left(\mathit{\_}\mathit{a}a\right)}}d\mathit{\_}\mathit{a}+\mathit{\_}\mathit{C}\mathit{1}=0\}$$