\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) ={\frac {F \left ( - \left ( x-y \left ( x \right ) \right ) \left ( y \left ( x \right ) +x \right ) \right ) x}{y \left ( x \right ) }}=0} \]
Mathematica: cpu = 32.152083 (sec), leaf count = 179 \[ \text {Solve}\left [\int _1^{y(x)} \left (\frac {K[2]}{F((K[2]-x) (K[2]+x))-1}-\int _1^x \left (\frac {2 K[1] K[2] F((K[2]-K[1]) (K[1]+K[2])) F'((K[2]-K[1]) (K[1]+K[2]))}{(F((K[2]-K[1]) (K[1]+K[2]))-1)^2}-\frac {2 K[1] K[2] F'((K[2]-K[1]) (K[1]+K[2]))}{F((K[2]-K[1]) (K[1]+K[2]))-1}\right ) \, dK[1]\right ) \, dK[2]+\int _1^x -\frac {K[1] F((y(x)-K[1]) (K[1]+y(x)))}{F((y(x)-K[1]) (K[1]+y(x)))-1} \, dK[1]=c_1,y(x)\right ] \]
Maple: cpu = 0.125 (sec), leaf count = 61 \[ \left \{ y \left ( x \right ) =\sqrt {{x}^{2}+{\it RootOf} \left ( -{x}^{ 2}+\int ^{{\it \_Z}}\! \left ( F \left ( {\it \_a} \right ) -1 \right ) ^{ -1}{d{\it \_a}}+2\,{\it \_C1} \right ) },y \left ( x \right ) =-\sqrt {{x }^{2}+{\it RootOf} \left ( -{x}^{2}+\int ^{{\it \_Z}}\! \left ( F \left ( {\it \_a} \right ) -1 \right ) ^{-1}{d{\it \_a}}+2\,{\it \_C1} \right ) } \right \} \]