\[ y'(x)=\frac {x^2 \log (2 x)+2 x y(x) \log (2 x)+y(x)^2 \log (2 x)-\log (x)+\log (2 x)}{\log (x)} \] ✓ Mathematica : cpu = 0.275193 (sec), leaf count = 19
\[\left \{\left \{y(x)\to \tan \left (c_1+\log (2) \text {li}(x)+x\right )-x\right \}\right \}\]
✓ Maple : cpu = 0.976 (sec), leaf count = 25
\[ \left \{ y \left ( x \right ) =-x-\tan \left ( \ln \left ( 2 \right ) {\it Ei} \left ( 1,-\ln \left ( x \right ) \right ) +{\it \_C1}-x \right ) \right \} \]