\[ \int \frac {(2+4 x) \log \left (\frac {1}{16} \left (x+x^2\right )\right )}{x+x^2} \, dx \]
Optimal antiderivative \[ \ln \left (\frac {\left (1+x \right ) x}{16}\right )^{2} \]
command
integrate((4*x+2)*log(1/16*x^2+1/16*x)/(x^2+x),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \log \left (\frac {1}{16} \, x^{2} + \frac {1}{16} \, x\right )^{2} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {2 \, {\left (2 \, x + 1\right )} \log \left (\frac {1}{16} \, x^{2} + \frac {1}{16} \, x\right )}{x^{2} + x}\,{d x} \]________________________________________________________________________________________