\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) ={\frac {x \left ( a-1 \right ) \left ( a+1 \right ) }{y \left ( x \right ) +F \left ( 1/2\, \left ( y \left ( x \right ) \right ) ^{2}-1/2\,{a}^{2}{x}^{2}+1/2\,{x}^{2} \right ) {a}^{2}-F \left ( 1/2\, \left ( y \left ( x \right ) \right ) ^{2}-1/2\,{a}^{2}{x}^{2}+1/2\,{x}^{2} \right ) }}=0} \]
Mathematica: cpu = 72.937762 (sec), leaf count = 171 \[ \text {Solve}\left [\int _1^{y(x)} \left (-\int _1^x \frac {K[1] K[2] F'\left (-\frac {1}{2} a^2 K[1]^2+\frac {K[1]^2}{2}+\frac {K[2]^2}{2}\right )}{F\left (-\frac {1}{2} a^2 K[1]^2+\frac {K[1]^2}{2}+\frac {K[2]^2}{2}\right )^2} \, dK[1]+\frac {K[2]}{\left (a^2-1\right ) F\left (\frac {K[2]^2}{2}-\frac {1}{2} a^2 x^2+\frac {x^2}{2}\right )}+1\right ) \, dK[2]+\int _1^x -\frac {K[1]}{F\left (-\frac {1}{2} a^2 K[1]^2+\frac {K[1]^2}{2}+\frac {y(x)^2}{2}\right )} \, dK[1]=c_1,y(x)\right ] \]
Maple: cpu = 0.328 (sec), leaf count = 59 \[ \left \{ {\frac {y \left ( x \right ) }{ \left ( a-1 \right ) \left ( a+1 \right ) }}+{\frac {1}{2\,{a}^{4}-4\,{a}^{2}+2}\int ^{-{a}^{2}{x}^{2}+ {x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}}\! \left ( F \left ( { \frac {{\it \_a}}{2}} \right ) \right ) ^{-1}{d{\it \_a}}}-{\it \_C1}=0 \right \} \]