\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) ={\frac { \left ( y \left ( x \right ) \right ) ^{3}{{\rm e}^{-2\,x}}}{y \left ( x \right ) {{\rm e}^{-x}}+1}}=0} \]
Mathematica: cpu = 0.536568 (sec), leaf count = 78 \[ \text {Solve}\left [\log (y(x))+y(x)^2 \left (\frac {x}{y(x)^2}-\frac {\log \left (-y(x)^2+e^x y(x)+e^{2 x}\right )}{2 y(x)^2}+\frac {\tanh ^{-1}\left (\frac {y(x)+2 e^x}{\sqrt {5} y(x)}\right )}{\sqrt {5} y(x)^2}\right )=c_1,y(x)\right ] \]
Maple: cpu = 0.546 (sec), leaf count = 58 \[ \left \{ y \left ( x \right ) ={{\rm e}^{{\it RootOf} \left ( 2\,\sqrt {5 }{\it Artanh} \left ( 1/5\,{\frac { \left ( {{\rm e}^{x}}-2\,{{\rm e}^{{ \it \_Z}}} \right ) \sqrt {5}}{{{\rm e}^{x}}}} \right ) +5\,\ln \left ( \left ( {{\rm e}^{{\it \_Z}}} \right ) ^{2}-{{\rm e}^{x+{\it \_Z}}}- \left ( {{\rm e}^{x}} \right ) ^{2} \right ) +10\,{\it \_C1}-10\,{\it \_Z}-10\,x \right ) }} \right \} \]