\[ \int \frac {4 x^3+4 x^2 \log (5)+\left (x-8 x^4+\left (1-8 x^2-8 x^3\right ) \log (5)\right ) \log (x) \log (\log (x))+\left (4 x^2+4 x \log (5)+\left (-12 x^3+\left (-8 x-12 x^2\right ) \log (5)\right ) \log (x) \log (\log (x))\right ) \log \left (\frac {x^2}{\left (x^2+2 x \log (5)+\log ^2(5)\right ) \log (\log (x))}\right )+\left (-4 x^2-4 x \log (5)\right ) \log (x) \log (\log (x)) \log ^2\left (\frac {x^2}{\left (x^2+2 x \log (5)+\log ^2(5)\right ) \log (\log (x))}\right )}{(2 x+2 \log (5)) \log (x) \log (\log (x))} \, dx \]
Optimal antiderivative \[ \frac {x}{2}-\left (\ln \! \left (\frac {x^{2}}{\left (\ln \! \left (5\right )+x \right )^{2} \ln \! \left (\ln \! \left (x \right )\right )}\right )+x \right )^{2} x^{2} \]
command
integrate(((-4*x*ln(5)-4*x**2)*ln(x)*ln(ln(x))*ln(x**2/(ln(5)**2+2*x*ln(5)+x**2)/ln(ln(x)))**2+(((-12*x**2-8*x)*ln(5)-12*x**3)*ln(x)*ln(ln(x))+4*x*ln(5)+4*x**2)*ln(x**2/(ln(5)**2+2*x*ln(5)+x**2)/ln(ln(x)))+((-8*x**3-8*x**2+1)*ln(5)-8*x**4+x)*ln(x)*ln(ln(x))+4*x**2*ln(5)+4*x**3)/(2*ln(5)+2*x)/ln(x)/ln(ln(x)),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Exception raised: TypeError} \]
Sympy 1.8 under Python 3.8.8 output
\[ - x^{4} - 2 x^{3} \log {\left (\frac {x^{2}}{\left (x^{2} + 2 x \log {\left (5 \right )} + \log {\left (5 \right )}^{2}\right ) \log {\left (\log {\left (x \right )} \right )}} \right )} - x^{2} \log {\left (\frac {x^{2}}{\left (x^{2} + 2 x \log {\left (5 \right )} + \log {\left (5 \right )}^{2}\right ) \log {\left (\log {\left (x \right )} \right )}} \right )}^{2} + \frac {x}{2} \]