\[ y'(x)=\frac {x^2 \log (2 x)+2 x y(x) \log (2 x)+y(x)^2 \log (2 x)-\log (x)+\log (2 x)}{\log (x)} \] ✓ Mathematica : cpu = 0.360593 (sec), leaf count = 19
\[\{\{y(x)\to -x+\tan (\log (2) \text {li}(x)+x+c_1)\}\}\] ✓ Maple : cpu = 0.942 (sec), leaf count = 25
\[\{y \left (x \right ) = -x -\tan \left (c_{1}-x +\ln \left (2\right ) \Ei \left (1, -\ln \left (x \right )\right )\right )\}\]