8.61   ODE No. 1651

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

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

Mathematica: cpu = 0.284036 (sec), leaf count = 414 $$\left\{\{y(x)\to \frac{a\text{InverseFunction}\left[\frac{\frac{b{\tan }^{-1}\left(\frac{\mathrm{\#}\text{1}b}{\sqrt{{\mathrm{\#}\text{1}}^2+1}\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}-\frac{b{\tan }^{-1}\left(\frac{\mathrm{\#}\text{1}a}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+{\sinh }^{-1}(\mathrm{\#}\text{1})}{a}\mathrm{\&}\right][c_1+x]{}^2-b\sqrt{\text{InverseFunction}\left[\frac{\frac{b{\tan }^{-1}\left(\frac{\mathrm{\#}\text{1}b}{\sqrt{{\mathrm{\#}\text{1}}^2+1}\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}-\frac{b{\tan }^{-1}\left(\frac{\mathrm{\#}\text{1}a}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+{\sinh }^{-1}(\mathrm{\#}\text{1})}{a}\mathrm{\&}\right][c_1+x]{}^2+1}\log (a\sqrt{\text{InverseFunction}\left[\frac{\frac{b{\tan }^{-1}\left(\frac{\mathrm{\#}\text{1}b}{\sqrt{{\mathrm{\#}\text{1}}^2+1}\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}-\frac{b{\tan }^{-1}\left(\frac{\mathrm{\#}\text{1}a}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+{\sinh }^{-1}(\mathrm{\#}\text{1})}{a}\mathrm{\&}\right][c_1+x]{}^2+1}+b)+a}{a^2\sqrt{\text{InverseFunction}\left[\frac{\frac{b{\tan }^{-1}\left(\frac{\mathrm{\#}\text{1}b}{\sqrt{{\mathrm{\#}\text{1}}^2+1}\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}-\frac{b{\tan }^{-1}\left(\frac{\mathrm{\#}\text{1}a}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+{\sinh }^{-1}(\mathrm{\#}\text{1})}{a}\mathrm{\&}\right][c_1+x]{}^2+1}}+c_2\}\right\}$$

Maple: cpu = 0.109 (sec), leaf count = 31 $$\{y\left(x\right)=\int \mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}(x-{\int }^{\mathit{\_}\mathit{Z}}{(a\sqrt{{\mathit{\_}\mathit{f}}^2+1}+b)}^{-1}d\mathit{\_}\mathit{f}+\mathit{\_}\mathit{C}\mathit{1})\mathrm{d}x+\mathit{\_}\mathit{C}\mathit{2}\}$$