\[ \int \frac {\left (\left (4 e x^3-8 x^4\right ) \log (3)+\left (4 e x-8 x^2\right ) \log ^2(3)\right ) \log ^3(x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)+\log ^2(x) \left (-4 x^5-4 x^3 \log (3)+\left (-2 e x^4+4 x^5+\left (-6 e x^2+12 x^3\right ) \log (3)\right ) \log (-e+2 x)\right )+\log (x) \left (\left (4 x^4-2 e x^4+4 x^5+\left (-2 e x^2+4 x^3\right ) \log (3)\right ) \log (-e+2 x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)\right )}{(e-2 x) \log ^3(3) \log ^3(x)+\left (-3 e x+6 x^2\right ) \log ^2(3) \log ^2(x) \log (-e+2 x)+\left (3 e x^2-6 x^3\right ) \log (3) \log (x) \log ^2(-e+2 x)+\left (-e x^3+2 x^4\right ) \log ^3(-e+2 x)} \, dx \]
Optimal antiderivative \[ \left (1-\frac {x}{\frac {\ln \left (-{\mathrm e}+2 x \right )}{\ln \left (x \right )}-\frac {\ln \left (3\right )}{x}}\right )^{2} \]
command
Integrate[(((4*E*x^3 - 8*x^4)*Log[3] + (4*E*x - 8*x^2)*Log[3]^2)*Log[x]^3 + (2*E*x^3 - 4*x^4)*Log[-E + 2*x]^2 + Log[x]^2*(-4*x^5 - 4*x^3*Log[3] + (-2*E*x^4 + 4*x^5 + (-6*E*x^2 + 12*x^3)*Log[3])*Log[-E + 2*x]) + Log[x]*((4*x^4 - 2*E*x^4 + 4*x^5 + (-2*E*x^2 + 4*x^3)*Log[3])*Log[-E + 2*x] + (2*E*x^3 - 4*x^4)*Log[-E + 2*x]^2))/((E - 2*x)*Log[3]^3*Log[x]^3 + (-3*E*x + 6*x^2)*Log[3]^2*Log[x]^2*Log[-E + 2*x] + (3*E*x^2 - 6*x^3)*Log[3]*Log[x]*Log[-E + 2*x]^2 + (-(E*x^3) + 2*x^4)*Log[-E + 2*x]^3),x]
Mathematica 13.1 output
\[ \int \frac {\left (\left (4 e x^3-8 x^4\right ) \log (3)+\left (4 e x-8 x^2\right ) \log ^2(3)\right ) \log ^3(x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)+\log ^2(x) \left (-4 x^5-4 x^3 \log (3)+\left (-2 e x^4+4 x^5+\left (-6 e x^2+12 x^3\right ) \log (3)\right ) \log (-e+2 x)\right )+\log (x) \left (\left (4 x^4-2 e x^4+4 x^5+\left (-2 e x^2+4 x^3\right ) \log (3)\right ) \log (-e+2 x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)\right )}{(e-2 x) \log ^3(3) \log ^3(x)+\left (-3 e x+6 x^2\right ) \log ^2(3) \log ^2(x) \log (-e+2 x)+\left (3 e x^2-6 x^3\right ) \log (3) \log (x) \log ^2(-e+2 x)+\left (-e x^3+2 x^4\right ) \log ^3(-e+2 x)} \, dx \]
Mathematica 12.3 output
\[ \frac {x^2 \log (x) \left (x^2 \log (x)-\frac {\left (2 \left (2 x^2+e \log (3)-x \log (9)\right )^3+\left (-e^3 \log (3) \left (2 \log ^2(3)+\log ^2(9)\right )+4 x^3 (x-\log (3)) \left (-4 \log ^2(3)-\log (9) \log (81)+x \log (531441)\right )-2 e x^2 \left (-x \left (16 \log ^2(3)+\log ^2(81)+\log (9) \log (6561)\right )+x^2 \left (-36 \log ^2(3)+\log (9) \log (729)+\log (81) \log (729)+\log (531441)\right )+\log ^2(3) \log (150094635296999121)\right )+e^2 x \left (-x \left (8 \log ^2(3)+\log (9) \log (6561)\right )+x^2 \left (-36 \log ^2(3)+\log (27) \log (81)+\log (9) \log (531441)\right )+\log ^2(3) \log (150094635296999121)\right )\right ) \log (x)+\left (4 x^3 \left (-4 \log ^2(3) \log (27)+x \left (-4 \log ^2(3)+\log ^2(81)\right )\right )+e^3 \log ^2(3) \log (729)+e^2 x \left (x \log (27) \log (81)+x^2 \left (-\log ^2(9)+\log (3) \log (81)\right )-4 \log ^2(3) \log (19683)\right )+2 e x^2 \left (x \left (8 \log ^2(3)-8 \log (3) \log (9)-\log ^2(81)\right )+4 \log ^2(3) \log (19683)\right )\right ) \log ^2(x)+\left (8 x^3 \log ^2(3) \log (9)+e^3 \left (-2 \log ^3(3)+x^2 \left (-4 \log ^2(3)+\log ^2(9)\right )\right )+e^2 x \log (3) \left (-12 \log ^2(3)+8 \log (3) \log (9)+\log (9) \log (81)\right )+e \left (-8 x^2 \log ^2(3) \log (27)+x^4 \left (-48 \log ^2(3)+10 \log (3) \log (81)+\log (9) \log (81)\right )\right )\right ) \log ^3(x)\right ) (-\log (3) \log (x)+x \log (-e+2 x))}{\left (2 x^2+e \log (3)-x \log (9)+(-e \log (3)+x \log (9)) \log (x)\right )^3}\right )}{(\log (3) \log (x)-x \log (-e+2 x))^2} \]