\[ \int \frac {x \log (4)+\log ^2(4)+e^x \left (-x^2 \log (4)-x \log ^2(4)\right )+\left (-x \log (4)+\log ^2(4)+e^x \left (x^3 \log (4)+x^2 \log ^2(4)\right )\right ) \log \left (\frac {x}{4}\right )+\left (x \log (4)+\log ^2(4)+\left (x \log (4)+\log ^2(4)\right ) \log \left (\frac {x}{4}\right )\right ) \log \left (x^2+2 x \log (4)+\log ^2(4)\right )}{\left (x^3+x^2 \log (4)\right ) \log ^2\left (\frac {x}{4}\right )} \, dx \]
Optimal antiderivative \[ \frac {2 \ln \! \left (2\right ) \left ({\mathrm e}^{x}-\frac {\ln \left (\left (x +2 \ln \left (2\right )\right )^{2}\right )+1}{x}\right )}{\ln \! \left (\frac {x}{4}\right )} \]
command
integrate((((4*ln(2)**2+2*x*ln(2))*ln(1/4*x)+4*ln(2)**2+2*x*ln(2))*ln(4*ln(2)**2+4*x*ln(2)+x**2)+((4*x**2*ln(2)**2+2*x**3*ln(2))*exp(x)+4*ln(2)**2-2*x*ln(2))*ln(1/4*x)+(-4*x*ln(2)**2-2*x**2*ln(2))*exp(x)+4*ln(2)**2+2*x*ln(2))/(2*x**2*ln(2)+x**3)/ln(1/4*x)**2,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Exception raised: TypeError} \]
Sympy 1.8 under Python 3.8.8 output
\[ \frac {2 e^{x} \log {\left (2 \right )}}{\log {\left (\frac {x}{4} \right )}} - \frac {2 \log {\left (2 \right )} \log {\left (x^{2} + 4 x \log {\left (2 \right )} + 4 \log {\left (2 \right )}^{2} \right )}}{x \log {\left (\frac {x}{4} \right )}} - \frac {2 \log {\left (2 \right )}}{x \log {\left (\frac {x}{4} \right )}} \]