8.56   ODE No. 1646

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

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

Mathematica: cpu = 10.814373 (sec), leaf count = 262 $$\{\{y(x)\to \text{InverseFunction}[-\frac{\sqrt{\frac{{\mathrm{\#}\text{1}}^2(-a)+2c_1+1}{2c_1+1}}\sqrt{2{\mathrm{\#}\text{1}}^{2}a-4c_1}E({\sin }^{-1}\left(\sqrt{\frac{a}{2c_1+1}}\mathrm{\#}\text{1}\right)|1+\frac{1}{2c_1})}{\sqrt{\frac{a}{2c_1+1}}\sqrt{{\mathrm{\#}\text{1}}^2(-a)+2c_1+1}\sqrt{2-\frac{{\mathrm{\#}\text{1}}^{2}a}{c_1}}}\mathrm{\&}][c_2+x]\},\{y(x)\to \text{InverseFunction}\left[\frac{\sqrt{\frac{{\mathrm{\#}\text{1}}^2(-a)+2c_1+1}{2c_1+1}}\sqrt{2{\mathrm{\#}\text{1}}^{2}a-4c_1}E({\sin }^{-1}\left(\sqrt{\frac{a}{2c_1+1}}\mathrm{\#}\text{1}\right)|1+\frac{1}{2c_1})}{\sqrt{\frac{a}{2c_1+1}}\sqrt{{\mathrm{\#}\text{1}}^2(-a)+2c_1+1}\sqrt{2-\frac{{\mathrm{\#}\text{1}}^{2}a}{c_1}}}\mathrm{\&}\right][c_2+x]\}\}$$

Maple: cpu = 0.187 (sec), leaf count = 94 $$\{{\int }^{y\left(x\right)}a({\mathit{\_}\mathit{a}}^2+2\mathit{\_}\mathit{C}\mathit{1})\frac{1}{\sqrt{-a({\mathit{\_}\mathit{a}}^2+2\mathit{\_}\mathit{C}\mathit{1})({\mathit{\_}\mathit{a}}^{2}a+2\mathit{\_}\mathit{C}\mathit{1}a-1)}}d\mathit{\_}\mathit{a}-x-\mathit{\_}\mathit{C}\mathit{2}=0,{\int }^{y\left(x\right)}-a({\mathit{\_}\mathit{a}}^2+2\mathit{\_}\mathit{C}\mathit{1})\frac{1}{\sqrt{-a({\mathit{\_}\mathit{a}}^2+2\mathit{\_}\mathit{C}\mathit{1})({\mathit{\_}\mathit{a}}^{2}a+2\mathit{\_}\mathit{C}\mathit{1}a-1)}}d\mathit{\_}\mathit{a}-x-\mathit{\_}\mathit{C}\mathit{2}=0\}$$