22.10 Problem number 578

$$\int \frac{((4e{x}^3-8x^4)\log (3)+(4ex-8x^2){\log }^2(3)){\log }^3(x)+(2e{x}^3-4x^4){\log }^2(-e+2x)+{\log }^2(x)(-4x^5-4x^3\log (3)+(-2e{x}^4+4x^5+(-6e{x}^2+12x^3)\log (3))\log (-e+2x))+\log (x)((4x^4-2e{x}^4+4x^5+(-2e{x}^2+4x^3)\log (3))\log (-e+2x)+(2e{x}^3-4x^4){\log }^2(-e+2x))}{(e-2x){\log }^3(3){\log }^3(x)+(-3ex+6x^2){\log }^2(3){\log }^2(x)\log (-e+2x)+(3e{x}^2-6x^3)\log (3)\log (x){\log }^2(-e+2x)+(-e{x}^3+2x^4){\log }^3(-e+2x)}dx$$

Optimal antiderivative $${(1-\frac{x}{\frac{\ln (-\mathrm{e}+2x)}{\ln \left(x\right)}-\frac{\ln \left(3\right)}{x}})}^2$$

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

$$\int \frac{((4e{x}^3-8x^4)\log (3)+(4ex-8x^2){\log }^2(3)){\log }^3(x)+(2e{x}^3-4x^4){\log }^2(-e+2x)+{\log }^2(x)(-4x^5-4x^3\log (3)+(-2e{x}^4+4x^5+(-6e{x}^2+12x^3)\log (3))\log (-e+2x))+\log (x)((4x^4-2e{x}^4+4x^5+(-2e{x}^2+4x^3)\log (3))\log (-e+2x)+(2e{x}^3-4x^4){\log }^2(-e+2x))}{(e-2x){\log }^3(3){\log }^3(x)+(-3ex+6x^2){\log }^2(3){\log }^2(x)\log (-e+2x)+(3e{x}^2-6x^3)\log (3)\log (x){\log }^2(-e+2x)+(-e{x}^3+2x^4){\log }^3(-e+2x)}dx$$

Mathematica 12.3 output

$$\frac{x^2\log (x)(x^2\log (x)-\frac{(2{(2x^2+e\log (3)-x\log (9))}^3+(-e^3\log (3)(2{\log }^2(3)+{\log }^2(9))+4x^3(x-\log (3))(-4{\log }^2(3)-\log (9)\log (81)+x\log (531441))-2e{x}^2(-x(16{\log }^2(3)+{\log }^2(81)+\log (9)\log (6561))+x^2(-36{\log }^2(3)+\log (9)\log (729)+\log (81)\log (729)+\log (531441))+{\log }^2(3)\log (150094635296999121))+e^{2}x(-x(8{\log }^2(3)+\log (9)\log (6561))+x^2(-36{\log }^2(3)+\log (27)\log (81)+\log (9)\log (531441))+{\log }^2(3)\log (150094635296999121)))\log (x)+(4x^3(-4{\log }^2(3)\log (27)+x(-4{\log }^2(3)+{\log }^2(81)))+e^3{\log }^2(3)\log (729)+e^{2}x(x\log (27)\log (81)+x^2(-{\log }^2(9)+\log (3)\log (81))-4{\log }^2(3)\log (19683))+2e{x}^2(x(8{\log }^2(3)-8\log (3)\log (9)-{\log }^2(81))+4{\log }^2(3)\log (19683))){\log }^2(x)+(8x^3{\log }^2(3)\log (9)+e^3(-2{\log }^3(3)+x^2(-4{\log }^2(3)+{\log }^2(9)))+e^{2}x\log (3)(-12{\log }^2(3)+8\log (3)\log (9)+\log (9)\log (81))+e(-8x^2{\log }^2(3)\log (27)+x^4(-48{\log }^2(3)+10\log (3)\log (81)+\log (9)\log (81)))){\log }^3(x))(-\log (3)\log (x)+x\log (-e+2x))}{{(2x^2+e\log (3)-x\log (9)+(-e\log (3)+x\log (9))\log (x))}^3})}{(\log (3)\log (x)-x\log (-e+2x){)}^2}$$