\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) ={\frac {{{\rm e}^{x}}}{y \left ( x \right ) {{\rm e}^{-x}}+1}}=0} \]
Mathematica: cpu = 0.168521 (sec), leaf count = 65 \[ \text {Solve}\left [\frac {1}{2} \log \left (-e^{-2 x} y(x)^2-e^{-x} y(x)+1\right )+x=c_1+\frac {\tanh ^{-1}\left (\frac {y(x)+3 e^x}{\sqrt {5} \left (y(x)+e^x\right )}\right )}{\sqrt {5}},y(x)\right ] \]
Maple: cpu = 0.203 (sec), leaf count = 52 \[ \left \{ x+{\frac {\ln \left ( \left ( y \left ( x \right ) \right ) ^{2 } \left ( {{\rm e}^{-x}} \right ) ^{2}+y \left ( x \right ) {{\rm e}^{-x}} -1 \right ) }{2}}-{\frac {\sqrt {5}}{5}{\it Artanh} \left ( {\frac { \left ( 1+2\,y \left ( x \right ) {{\rm e}^{-x}} \right ) \sqrt {5}}{5}} \right ) }-{\it \_C1}=0 \right \} \]