\[ y'(x)=\frac {e^{-\frac {2}{x^2-y(x)^2-1}}+x^2+2 x y(x)+y(x)^2}{-e^{-\frac {2}{x^2-y(x)^2-1}}+x^2+2 x y(x)+y(x)^2} \] ✓ Mathematica : cpu = 2.08742 (sec), leaf count = 1283
DSolve[Derivative[1][y][x] == (E^(-2/(-1 + x^2 - y[x]^2)) + x^2 + 2*x*y[x] + y[x]^2)/(-E^(-2/(-1 + x^2 - y[x]^2)) + x^2 + 2*x*y[x] + y[x]^2),y[x],x]
\[\text {Solve}\left [\int _1^x\left (-e^{\int _1^{(K[2]-y(x)) (K[2]+y(x))}\frac {2 ((K[1]-3) K[1]+1)}{\left (e^{-\frac {2}{K[1]-1}}-K[1]\right ) (K[1]-1)^2}dK[1]+\frac {2}{K[2]^2-y(x)^2-1}} K[2]^2-2 e^{\int _1^{(K[2]-y(x)) (K[2]+y(x))}\frac {2 ((K[1]-3) K[1]+1)}{\left (e^{-\frac {2}{K[1]-1}}-K[1]\right ) (K[1]-1)^2}dK[1]+\frac {2}{K[2]^2-y(x)^2-1}} y(x) K[2]-e^{\int _1^{(K[2]-y(x)) (K[2]+y(x))}\frac {2 ((K[1]-3) K[1]+1)}{\left (e^{-\frac {2}{K[1]-1}}-K[1]\right ) (K[1]-1)^2}dK[1]} \left (e^{\frac {2}{K[2]^2-y(x)^2-1}} y(x)^2+1\right )\right )dK[2]+\int _1^{y(x)}\left (e^{\int _1^{(x-K[3]) (x+K[3])}\frac {2 ((K[1]-3) K[1]+1)}{\left (e^{-\frac {2}{K[1]-1}}-K[1]\right ) (K[1]-1)^2}dK[1]+\frac {2}{x^2-K[3]^2-1}} x^2+2 e^{\int _1^{(x-K[3]) (x+K[3])}\frac {2 ((K[1]-3) K[1]+1)}{\left (e^{-\frac {2}{K[1]-1}}-K[1]\right ) (K[1]-1)^2}dK[1]+\frac {2}{x^2-K[3]^2-1}} K[3] x-e^{\int _1^{(x-K[3]) (x+K[3])}\frac {2 ((K[1]-3) K[1]+1)}{\left (e^{-\frac {2}{K[1]-1}}-K[1]\right ) (K[1]-1)^2}dK[1]}+e^{\int _1^{(x-K[3]) (x+K[3])}\frac {2 ((K[1]-3) K[1]+1)}{\left (e^{-\frac {2}{K[1]-1}}-K[1]\right ) (K[1]-1)^2}dK[1]+\frac {2}{x^2-K[3]^2-1}} K[3]^2-\int _1^x\left (-e^{\int _1^{(K[2]-K[3]) (K[2]+K[3])}\frac {2 ((K[1]-3) K[1]+1)}{\left (e^{-\frac {2}{K[1]-1}}-K[1]\right ) (K[1]-1)^2}dK[1]+\frac {2}{K[2]^2-K[3]^2-1}} \left (\frac {4 K[3]}{\left (K[2]^2-K[3]^2-1\right )^2}-\frac {4 K[3] ((K[2]-K[3]) (K[2]+K[3]) ((K[2]-K[3]) (K[2]+K[3])-3)+1)}{\left (e^{-\frac {2}{(K[2]-K[3]) (K[2]+K[3])-1}}-(K[2]-K[3]) (K[2]+K[3])\right ) ((K[2]-K[3]) (K[2]+K[3])-1)^2}\right ) K[2]^2-2 e^{\int _1^{(K[2]-K[3]) (K[2]+K[3])}\frac {2 ((K[1]-3) K[1]+1)}{\left (e^{-\frac {2}{K[1]-1}}-K[1]\right ) (K[1]-1)^2}dK[1]+\frac {2}{K[2]^2-K[3]^2-1}} K[2]-2 e^{\int _1^{(K[2]-K[3]) (K[2]+K[3])}\frac {2 ((K[1]-3) K[1]+1)}{\left (e^{-\frac {2}{K[1]-1}}-K[1]\right ) (K[1]-1)^2}dK[1]+\frac {2}{K[2]^2-K[3]^2-1}} K[3] \left (\frac {4 K[3]}{\left (K[2]^2-K[3]^2-1\right )^2}-\frac {4 K[3] ((K[2]-K[3]) (K[2]+K[3]) ((K[2]-K[3]) (K[2]+K[3])-3)+1)}{\left (e^{-\frac {2}{(K[2]-K[3]) (K[2]+K[3])-1}}-(K[2]-K[3]) (K[2]+K[3])\right ) ((K[2]-K[3]) (K[2]+K[3])-1)^2}\right ) K[2]-e^{\int _1^{(K[2]-K[3]) (K[2]+K[3])}\frac {2 ((K[1]-3) K[1]+1)}{\left (e^{-\frac {2}{K[1]-1}}-K[1]\right ) (K[1]-1)^2}dK[1]} \left (\frac {4 e^{\frac {2}{K[2]^2-K[3]^2-1}} K[3]^3}{\left (K[2]^2-K[3]^2-1\right )^2}+2 e^{\frac {2}{K[2]^2-K[3]^2-1}} K[3]\right )+\frac {4 e^{\int _1^{(K[2]-K[3]) (K[2]+K[3])}\frac {2 ((K[1]-3) K[1]+1)}{\left (e^{-\frac {2}{K[1]-1}}-K[1]\right ) (K[1]-1)^2}dK[1]} K[3] \left (e^{\frac {2}{K[2]^2-K[3]^2-1}} K[3]^2+1\right ) ((K[2]-K[3]) (K[2]+K[3]) ((K[2]-K[3]) (K[2]+K[3])-3)+1)}{\left (e^{-\frac {2}{(K[2]-K[3]) (K[2]+K[3])-1}}-(K[2]-K[3]) (K[2]+K[3])\right ) ((K[2]-K[3]) (K[2]+K[3])-1)^2}\right )dK[2]\right )dK[3]=c_1,y(x)\right ]\] ✓ Maple : cpu = 0.582 (sec), leaf count = 40
dsolve(diff(y(x),x) = (y(x)^2+2*x*y(x)+x^2+exp(-2/(-y(x)^2+x^2-1)))/(y(x)^2+2*x*y(x)+x^2-exp(-2/(-y(x)^2+x^2-1))),y(x))
\[y \left (x \right ) = {\mathrm e}^{\RootOf \left (-\textit {\_Z} +\int _{}^{{\mathrm e}^{2 \textit {\_Z}}-2 \,{\mathrm e}^{\textit {\_Z}} x}\frac {1}{{\mathrm e}^{\frac {2}{1+\textit {\_a}}}+\textit {\_a}}d \textit {\_a} +c_{1}\right )}-x\]