\[ 2 y(x) y''(x)+y'(x)^2+1=0 \] ✓ Mathematica : cpu = 0.449197 (sec), leaf count = 166
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {\text {$\#$1}} \left (\text {$\#$1}-e^{2 c_1}\right )+e^{3 c_1} \sqrt {1-\text {$\#$1} e^{-2 c_1}} \sin ^{-1}\left (\sqrt {\text {$\#$1}} e^{-c_1}\right )}{\sqrt {-\text {$\#$1}+e^{2 c_1}}}\& \right ][x+c_2]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {\sqrt {\text {$\#$1}} \left (\text {$\#$1}-e^{2 c_1}\right )+e^{3 c_1} \sqrt {1-\text {$\#$1} e^{-2 c_1}} \sin ^{-1}\left (\sqrt {\text {$\#$1}} e^{-c_1}\right )}{\sqrt {-\text {$\#$1}+e^{2 c_1}}}\& \right ][x+c_2]\right \}\right \}\] ✓ Maple : cpu = 1.708 (sec), leaf count = 823
\[\left \{y \left (x \right ) = \frac {c_{1}}{2}+\frac {\left (-c_{1} \RootOf \left (c_{1}^{2} \textit {\_Z}^{2} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )-4 c_{1} c_{2} \textit {\_Z} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )-4 c_{1} \textit {\_Z} x \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+c_{1}^{2} \textit {\_Z}^{2}+4 c_{2}^{2} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+8 c_{2} x \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+4 x^{2} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )-4 c_{1} c_{2} \textit {\_Z} -4 c_{1} \textit {\_Z} x -c_{1}^{2}+4 c_{2}^{2}+8 c_{2} x +4 x^{2}\right )+2 c_{2}+2 x \right ) \tan \left (\RootOf \left (c_{1}^{2} \textit {\_Z}^{2} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )-4 c_{1} c_{2} \textit {\_Z} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )-4 c_{1} \textit {\_Z} x \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+c_{1}^{2} \textit {\_Z}^{2}+4 c_{2}^{2} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+8 c_{2} x \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+4 x^{2} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )-4 c_{1} c_{2} \textit {\_Z} -4 c_{1} \textit {\_Z} x -c_{1}^{2}+4 c_{2}^{2}+8 c_{2} x +4 x^{2}\right )\right )}{2}, y \left (x \right ) = \frac {c_{1}}{2}+\frac {\left (c_{1} \RootOf \left (c_{1}^{2} \textit {\_Z}^{2} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )-4 c_{1} c_{2} \textit {\_Z} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )-4 c_{1} \textit {\_Z} x \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+c_{1}^{2} \textit {\_Z}^{2}+4 c_{2}^{2} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+8 c_{2} x \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+4 x^{2} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )-4 c_{1} c_{2} \textit {\_Z} -4 c_{1} \textit {\_Z} x -c_{1}^{2}+4 c_{2}^{2}+8 c_{2} x +4 x^{2}\right )-2 c_{2}-2 x \right ) \tan \left (\RootOf \left (c_{1}^{2} \textit {\_Z}^{2} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )-4 c_{1} c_{2} \textit {\_Z} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )-4 c_{1} \textit {\_Z} x \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+c_{1}^{2} \textit {\_Z}^{2}+4 c_{2}^{2} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+8 c_{2} x \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+4 x^{2} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )-4 c_{1} c_{2} \textit {\_Z} -4 c_{1} \textit {\_Z} x -c_{1}^{2}+4 c_{2}^{2}+8 c_{2} x +4 x^{2}\right )\right )}{2}, y \left (x \right ) = \frac {c_{1}}{2}+\frac {\left (-c_{1} \RootOf \left (c_{1}^{2} \textit {\_Z}^{2} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+4 c_{1} c_{2} \textit {\_Z} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+4 c_{1} \textit {\_Z} x \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+c_{1}^{2} \textit {\_Z}^{2}+4 c_{2}^{2} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+8 c_{2} x \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+4 x^{2} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+4 c_{1} c_{2} \textit {\_Z} +4 c_{1} \textit {\_Z} x -c_{1}^{2}+4 c_{2}^{2}+8 c_{2} x +4 x^{2}\right )-2 c_{2}-2 x \right ) \tan \left (\RootOf \left (c_{1}^{2} \textit {\_Z}^{2} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+4 c_{1} c_{2} \textit {\_Z} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+4 c_{1} \textit {\_Z} x \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+c_{1}^{2} \textit {\_Z}^{2}+4 c_{2}^{2} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+8 c_{2} x \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+4 x^{2} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+4 c_{1} c_{2} \textit {\_Z} +4 c_{1} \textit {\_Z} x -c_{1}^{2}+4 c_{2}^{2}+8 c_{2} x +4 x^{2}\right )\right )}{2}, y \left (x \right ) = \frac {c_{1}}{2}+\frac {\left (c_{1} \RootOf \left (c_{1}^{2} \textit {\_Z}^{2} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+4 c_{1} c_{2} \textit {\_Z} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+4 c_{1} \textit {\_Z} x \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+c_{1}^{2} \textit {\_Z}^{2}+4 c_{2}^{2} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+8 c_{2} x \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+4 x^{2} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+4 c_{1} c_{2} \textit {\_Z} +4 c_{1} \textit {\_Z} x -c_{1}^{2}+4 c_{2}^{2}+8 c_{2} x +4 x^{2}\right )+2 c_{2}+2 x \right ) \tan \left (\RootOf \left (c_{1}^{2} \textit {\_Z}^{2} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+4 c_{1} c_{2} \textit {\_Z} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+4 c_{1} \textit {\_Z} x \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+c_{1}^{2} \textit {\_Z}^{2}+4 c_{2}^{2} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+8 c_{2} x \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+4 x^{2} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+4 c_{1} c_{2} \textit {\_Z} +4 c_{1} \textit {\_Z} x -c_{1}^{2}+4 c_{2}^{2}+8 c_{2} x +4 x^{2}\right )\right )}{2}\right \}\]