\[ y'(x)=\frac {4 (a-1) (a+1) x \left (a^2 x^2-x^2-y(x)^2-2\right )}{a^8 x^6-4 a^6 x^6-3 a^6 x^4 y(x)^2+6 a^4 x^6+9 a^4 x^4 y(x)^2+3 a^4 x^2 y(x)^4-4 a^2 x^6-9 a^2 x^4 y(x)^2-6 a^2 x^2 y(x)^4+4 a^2 x^2 y(x)-a^2 y(x)^6+x^6+3 x^4 y(x)^2+3 x^2 y(x)^4-4 x^2 y(x)+y(x)^6-4 y(x)^3-8 y(x)} \] ✓ Mathematica : cpu = 3.94572 (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 = 1.407 (sec), leaf count = 79
\[-\frac {y \relax (x )}{\left (a -1\right ) \left (a +1\right )}+\frac {2}{\left (a^{2}-1\right )^{2} \left (a^{2} x^{2}-x^{2}-y \relax (x )^{2}\right )^{2}}-\frac {2}{\left (a^{2}-1\right )^{2} \left (a^{2} x^{2}-x^{2}-y \relax (x )^{2}\right )}+c_{1} = 0\]