\[ \boxed { {{\rm e}^{-2\,x}} \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}- \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) -1 \right ) ^{2}+{{\rm e}^{-2\,y \left ( x \right ) }}=0} \]
Mathematica: cpu = 2.602831 (sec), leaf count = 271 \[ \left \{\left \{y(x)\to \log \left (-\frac {e^{-c_1} \left (e^x+1\right ) \left (e^{2 c_1+x}-e^{2 c_1}-e^x-1\right )}{\sqrt {8 e^x+4 e^{2 x}+4}}\right )\right \},\left \{y(x)\to \log \left (\frac {e^{-c_1} \left (e^x+1\right ) \left (e^{2 c_1+x}-e^{2 c_1}-e^x-1\right )}{\sqrt {8 e^x+4 e^{2 x}+4}}\right )\right \},\text {Solve}\left [-\frac {1}{2} \log \left (1-e^{2 y(x)}\right )+\frac {1}{2} \log \left (\sqrt {e^{2 y(x)+2 x} \left (e^{2 y(x)}+e^{2 x}-1\right )}+e^{2 y(x)+x}-e^x-e^{2 x}\right )+\frac {1}{2} \log \left (\sqrt {e^{2 y(x)+2 x} \left (e^{2 y(x)}+e^{2 x}-1\right )}+e^{2 y(x)+x}-e^x+e^{2 x}\right )-x-\frac {1}{2} \log \left (1-e^x\right )-\frac {1}{2} \log \left (e^x-1\right )=c_1,y(x)\right ]\right \} \]
Maple: cpu = 0.422 (sec), leaf count = 65 \[ \left \{ y \left ( x \right ) =x+\ln \left ( {\frac {1}{{{\rm e}^{x}}} \left ( -1-\sqrt { \left ( {{\rm e}^{x}} \right ) ^{2}-{\frac { \left ( { {\rm e}^{x}} \right ) ^{2}}{ \left ( {{\rm e}^{{\it \_C1}}} \right ) ^{2} }}} \right ) } \right ) +{\it \_C1},y \left ( x \right ) =x+\ln \left ( { \frac {1}{{{\rm e}^{x}}} \left ( -1+\sqrt { \left ( {{\rm e}^{x}} \right ) ^{2}-{\frac { \left ( {{\rm e}^{x}} \right ) ^{2}}{ \left ( { {\rm e}^{{\it \_C1}}} \right ) ^{2}}}} \right ) } \right ) +{\it \_C1} \right \} \]