\[ y'(x)=\frac {x F\left (\frac {y(x)}{\sqrt {x^2+1}}\right )}{\sqrt {x^2+1}} \] ✓ Mathematica : cpu = 185.373 (sec), leaf count = 398
\[\text {Solve}\left [c_1=\int _1^{y(x)} \frac {-\left (\left (x^2+1\right ) F\left (\frac {K[2]}{\sqrt {x^2+1}}\right )^2-K[2]^2\right ) \left (\int _1^x -\frac {K[1] \left (-\left (K[1]^2+1\right )^{3/2} K[2] F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )^2 \left (F'\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )-2\right )-\left (K[1]^2+1\right ) K[2]^2 F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right ) \left (2 F'\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )-1\right )-\sqrt {K[1]^2+1} K[2]^3 F'\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )+\left (K[1]^2+1\right )^2 F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )^3\right )}{\left (K[1]^2+1\right )^{3/2} \left (K[2]^2-\left (K[1]^2+1\right ) F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )^2\right )^2} \, dK[1]\right )+\sqrt {x^2+1} F\left (\frac {K[2]}{\sqrt {x^2+1}}\right )+K[2]}{\left (x^2+1\right ) F\left (\frac {K[2]}{\sqrt {x^2+1}}\right )^2-K[2]^2} \, dK[2]+\int _1^x -\frac {K[1] F\left (\frac {y(x)}{\sqrt {K[1]^2+1}}\right ) \left (\sqrt {K[1]^2+1} F\left (\frac {y(x)}{\sqrt {K[1]^2+1}}\right )+y(x)\right )}{\sqrt {K[1]^2+1} \left (\left (K[1]^2+1\right ) F\left (\frac {y(x)}{\sqrt {K[1]^2+1}}\right )^2-y(x)^2\right )} \, dK[1],y(x)\right ]\]
✓ Maple : cpu = 1.078 (sec), leaf count = 39
\[ \left \{ y \left ( x \right ) ={\it RootOf} \left ( -\ln \left ( {x}^{2}+1 \right ) +2\,\int ^{{\it \_Z}}\! \left ( F \left ( {\it \_a} \right ) -{\it \_a} \right ) ^{-1}{d{\it \_a}}+2\,{\it \_C1} \right ) \sqrt {{x}^{2}+1} \right \} \]