\[ \int \frac {\left (-8 x^2+2 x^3-2 x^4\right ) \log (5) \log \left (\frac {1}{4} \left (-4 x+x^2-x^3\right )\right )+\left (16-8 x+12 x^2+\left (-4 x^2+2 x^3-3 x^4\right ) \log (5)\right ) \log \left (-4+x^2 \log (5)\right )+\left (8 x^2-2 x^3+2 x^4\right ) \log (5) \log \left (\frac {1}{4} \left (-4 x+x^2-x^3\right )\right ) \log \left (\log \left (\frac {1}{4} \left (-4 x+x^2-x^3\right )\right )\right )}{\left (-16 x+4 x^2-4 x^3+\left (4 x^3-x^4+x^5\right ) \log (5)\right ) \log \left (\frac {1}{4} \left (-4 x+x^2-x^3\right )\right ) \log ^2\left (-4+x^2 \log (5)\right )} \, dx \]
Optimal antiderivative \[ \frac {1-\ln \! \left (\ln \! \left (\frac {x \left (-x^{2}+x \right )}{4}-x \right )\right )}{\ln \! \left (x^{2} \ln \! \left (5\right )-4\right )} \]
command
integrate(((2*x**4-2*x**3+8*x**2)*ln(5)*ln(-1/4*x**3+1/4*x**2-x)*ln(ln(-1/4*x**3+1/4*x**2-x))+((-3*x**4+2*x**3-4*x**2)*ln(5)+12*x**2-8*x+16)*ln(x**2*ln(5)-4)+(-2*x**4+2*x**3-8*x**2)*ln(5)*ln(-1/4*x**3+1/4*x**2-x))/((x**5-x**4+4*x**3)*ln(5)-4*x**3+4*x**2-16*x)/ln(-1/4*x**3+1/4*x**2-x)/ln(x**2*ln(5)-4)**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
\[ - \frac {\log {\left (\log {\left (- \frac {x^{3}}{4} + \frac {x^{2}}{4} - x \right )} \right )}}{\log {\left (x^{2} \log {\left (5 \right )} - 4 \right )}} + \frac {1}{\log {\left (x^{2} \log {\left (5 \right )} - 4 \right )}} \]