\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) =4\,{\frac {x \left ( a-1 \right ) \left ( a+1 \right ) \left ( - \left ( y \left ( x \right ) \right ) ^{2}+{a}^{2}{x}^{2}-{x}^{2}-2 \right ) }{-4\, \left ( y \left ( x \right ) \right ) ^{3}+4\,{a}^{2}{x}^{2}y \left ( x \right ) -4\,{x}^{2}y \left ( x \right ) -8\,y \left ( x \right ) -{a}^{2} \left ( y \left ( x \right ) \right ) ^{6}+3\,{a}^{4} \left ( y \left ( x \right ) \right ) ^{4}{x}^{2}-6\, \left ( y \left ( x \right ) \right ) ^{4}{a}^{2}{x}^{2}-3\,{a}^{6} \left ( y \left ( x \right ) \right ) ^{2}{x}^{4}+9\, \left ( y \left ( x \right ) \right ) ^{2}{a}^{4}{x}^{4}-9\, \left ( y \left ( x \right ) \right ) ^{2}{a}^{2}{x}^{4}+{a}^{8}{x}^{6}-4\,{a}^{6}{x}^{6}+6\,{a}^{4}{x}^{6}-4\,{a}^{2}{x}^{6}+ \left ( y \left ( x \right ) \right ) ^{6}+3\,{x}^{2} \left ( y \left ( x \right ) \right ) ^{4}+3\,{x}^{4} \left ( y \left ( x \right ) \right ) ^{2}+{x}^{6}}}=0} \]
Mathematica: cpu = 5.355180 (sec), leaf count = 1191 \[ \left \{\left \{y(x)\to \text {Root}\left [2 x^4 a^8-8 x^4 a^6+e^{c_1} x^4 a^4+11 x^4 a^4-2 e^{c_1} x^4 a^2-6 x^4 a^2+4 x^2 a^2+\left (2 a^2-2\right ) \text {$\#$1}^5+e^{c_1} x^4+x^4+\left (2 a^4-4 a^2+e^{c_1}+1\right ) \text {$\#$1}^4+\left (-4 x^2 a^4+8 x^2 a^2-4 x^2\right ) \text {$\#$1}^3-4 x^2+\left (-4 x^2 a^6+12 x^2 a^4-2 e^{c_1} x^2 a^2-10 x^2 a^2+2 e^{c_1} x^2+2 x^2-4\right ) \text {$\#$1}^2+\left (2 x^4 a^6-6 x^4 a^4+6 x^4 a^2-2 x^4\right ) \text {$\#$1}-4\& ,1\right ]\right \},\left \{y(x)\to \text {Root}\left [2 x^4 a^8-8 x^4 a^6+e^{c_1} x^4 a^4+11 x^4 a^4-2 e^{c_1} x^4 a^2-6 x^4 a^2+4 x^2 a^2+\left (2 a^2-2\right ) \text {$\#$1}^5+e^{c_1} x^4+x^4+\left (2 a^4-4 a^2+e^{c_1}+1\right ) \text {$\#$1}^4+\left (-4 x^2 a^4+8 x^2 a^2-4 x^2\right ) \text {$\#$1}^3-4 x^2+\left (-4 x^2 a^6+12 x^2 a^4-2 e^{c_1} x^2 a^2-10 x^2 a^2+2 e^{c_1} x^2+2 x^2-4\right ) \text {$\#$1}^2+\left (2 x^4 a^6-6 x^4 a^4+6 x^4 a^2-2 x^4\right ) \text {$\#$1}-4\& ,2\right ]\right \},\left \{y(x)\to \text {Root}\left [2 x^4 a^8-8 x^4 a^6+e^{c_1} x^4 a^4+11 x^4 a^4-2 e^{c_1} x^4 a^2-6 x^4 a^2+4 x^2 a^2+\left (2 a^2-2\right ) \text {$\#$1}^5+e^{c_1} x^4+x^4+\left (2 a^4-4 a^2+e^{c_1}+1\right ) \text {$\#$1}^4+\left (-4 x^2 a^4+8 x^2 a^2-4 x^2\right ) \text {$\#$1}^3-4 x^2+\left (-4 x^2 a^6+12 x^2 a^4-2 e^{c_1} x^2 a^2-10 x^2 a^2+2 e^{c_1} x^2+2 x^2-4\right ) \text {$\#$1}^2+\left (2 x^4 a^6-6 x^4 a^4+6 x^4 a^2-2 x^4\right ) \text {$\#$1}-4\& ,3\right ]\right \},\left \{y(x)\to \text {Root}\left [2 x^4 a^8-8 x^4 a^6+e^{c_1} x^4 a^4+11 x^4 a^4-2 e^{c_1} x^4 a^2-6 x^4 a^2+4 x^2 a^2+\left (2 a^2-2\right ) \text {$\#$1}^5+e^{c_1} x^4+x^4+\left (2 a^4-4 a^2+e^{c_1}+1\right ) \text {$\#$1}^4+\left (-4 x^2 a^4+8 x^2 a^2-4 x^2\right ) \text {$\#$1}^3-4 x^2+\left (-4 x^2 a^6+12 x^2 a^4-2 e^{c_1} x^2 a^2-10 x^2 a^2+2 e^{c_1} x^2+2 x^2-4\right ) \text {$\#$1}^2+\left (2 x^4 a^6-6 x^4 a^4+6 x^4 a^2-2 x^4\right ) \text {$\#$1}-4\& ,4\right ]\right \},\left \{y(x)\to \text {Root}\left [2 x^4 a^8-8 x^4 a^6+e^{c_1} x^4 a^4+11 x^4 a^4-2 e^{c_1} x^4 a^2-6 x^4 a^2+4 x^2 a^2+\left (2 a^2-2\right ) \text {$\#$1}^5+e^{c_1} x^4+x^4+\left (2 a^4-4 a^2+e^{c_1}+1\right ) \text {$\#$1}^4+\left (-4 x^2 a^4+8 x^2 a^2-4 x^2\right ) \text {$\#$1}^3-4 x^2+\left (-4 x^2 a^6+12 x^2 a^4-2 e^{c_1} x^2 a^2-10 x^2 a^2+2 e^{c_1} x^2+2 x^2-4\right ) \text {$\#$1}^2+\left (2 x^4 a^6-6 x^4 a^4+6 x^4 a^2-2 x^4\right ) \text {$\#$1}-4\& ,5\right ]\right \}\right \} \]
Maple: cpu = 1.092 (sec), leaf count = 79 \[ \left \{ -{\frac {y \left ( x \right ) }{ \left ( a-1 \right ) \left ( a+1 \right ) }}+2\,{\frac {1}{ \left ( {a}^{2}-1 \right ) ^{2} \left ( {a}^{2 }{x}^{2}-{x}^{2}- \left ( y \left ( x \right ) \right ) ^{2} \right ) ^{2} }}-2\,{\frac {1}{ \left ( {a}^{2}-1 \right ) ^{2} \left ( {a}^{2}{x}^{2}- {x}^{2}- \left ( y \left ( x \right ) \right ) ^{2} \right ) }}+{\it \_C1} =0 \right \} \]