\[ -\left (y'(x)-1\right )^2+e^{-2 x} y'(x)^2+e^{-2 y(x)}=0 \] ✓ Mathematica : cpu = 2.75519 (sec), leaf count = 271
\[\left \{\left \{y(x)\to \log \left (-\frac {e^{-c_1} \left (e^x+1\right ) \left (-e^x+e^{x+2 c_1}-1-e^{2 c_1}\right )}{\sqrt {8 e^x+4 e^{2 x}+4}}\right )\right \},\left \{y(x)\to \log \left (\frac {e^{-c_1} \left (e^x+1\right ) \left (-e^x+e^{x+2 c_1}-1-e^{2 c_1}\right )}{\sqrt {8 e^x+4 e^{2 x}+4}}\right )\right \},\text {Solve}\left [-\frac {1}{2} \log \left (1-e^{2 y(x)}\right )+\frac {1}{2} \log \left (\sqrt {e^{2 y(x)+2 x} \left (e^{2 y(x)}+e^{2 x}-1\right )}+e^{2 y(x)+x}-e^x-e^{2 x}\right )+\frac {1}{2} \log \left (\sqrt {e^{2 y(x)+2 x} \left (e^{2 y(x)}+e^{2 x}-1\right )}+e^{2 y(x)+x}-e^x+e^{2 x}\right )-x-\frac {1}{2} \log \left (1-e^x\right )-\frac {1}{2} \log \left (e^x-1\right )=c_1,y(x)\right ]\right \}\] ✓ Maple : cpu = 0.726 (sec), leaf count = 65
\[y \relax (x ) = x +\ln \left (\left (-1-\sqrt {{\mathrm e}^{2 x}-{\mathrm e}^{-2 c_{1}} {\mathrm e}^{2 x}}\right ) {\mathrm e}^{-x}\right )+c_{1}\]