\[ \int \frac {e^{-x} \left (-1+(1-x) \log \left (-\frac {x}{2}\right )-\log ^2\left (-\frac {x}{2}\right )\right )}{\log ^2\left (-\frac {x}{2}\right )} \, dx \]
Optimal antiderivative \[ \frac {\left (x +\frac {x^{2}}{\ln \left (-\frac {x}{2}\right )}\right ) {\mathrm e}^{-x}}{x} \]
command
integrate((-log(-1/2*x)^2+(1-x)*log(-1/2*x)-1)/exp(x)/log(-1/2*x)^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {could not integrate} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \frac {x e^{\left (-x\right )} + e^{\left (-x\right )} \log \left (-\frac {1}{2} \, x\right )}{\log \left (-\frac {1}{2} \, x\right )} \]