\(\int \frac {((4 e x^3-8 x^4) \log (3)+(4 e x-8 x^2) \log ^2(3)) \log ^3(x)+(2 e x^3-4 x^4) \log ^2(-e+2 x)+\log ^2(x) (-4 x^5-4 x^3 \log (3)+(-2 e x^4+4 x^5+(-6 e x^2+12 x^3) \log (3)) \log (-e+2 x))+\log (x) ((4 x^4-2 e x^4+4 x^5+(-2 e x^2+4 x^3) \log (3)) \log (-e+2 x)+(2 e x^3-4 x^4) \log ^2(-e+2 x))}{(e-2 x) \log ^3(3) \log ^3(x)+(-3 e x+6 x^2) \log ^2(3) \log ^2(x) \log (-e+2 x)+(3 e x^2-6 x^3) \log (3) \log (x) \log ^2(-e+2 x)+(-e x^3+2 x^4) \log ^3(-e+2 x)} \, dx\) [578]

Optimal result

Integrand size = 277, antiderivative size = 30 \[ \int \frac {\left (\left (4 e x^3-8 x^4\right ) \log (3)+\left (4 e x-8 x^2\right ) \log ^2(3)\right ) \log ^3(x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)+\log ^2(x) \left (-4 x^5-4 x^3 \log (3)+\left (-2 e x^4+4 x^5+\left (-6 e x^2+12 x^3\right ) \log (3)\right ) \log (-e+2 x)\right )+\log (x) \left (\left (4 x^4-2 e x^4+4 x^5+\left (-2 e x^2+4 x^3\right ) \log (3)\right ) \log (-e+2 x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)\right )}{(e-2 x) \log ^3(3) \log ^3(x)+\left (-3 e x+6 x^2\right ) \log ^2(3) \log ^2(x) \log (-e+2 x)+\left (3 e x^2-6 x^3\right ) \log (3) \log (x) \log ^2(-e+2 x)+\left (-e x^3+2 x^4\right ) \log ^3(-e+2 x)} \, dx=\left (1-\frac {x}{-\frac {\log (3)}{x}+\frac {\log (-e+2 x)}{\log (x)}}\right )^2 \]

[Out]

(1-x/(ln(-exp(1)+2*x)/ln(x)-ln(3)/x))^2

Rubi [F]

\[ \int \frac {\left (\left (4 e x^3-8 x^4\right ) \log (3)+\left (4 e x-8 x^2\right ) \log ^2(3)\right ) \log ^3(x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)+\log ^2(x) \left (-4 x^5-4 x^3 \log (3)+\left (-2 e x^4+4 x^5+\left (-6 e x^2+12 x^3\right ) \log (3)\right ) \log (-e+2 x)\right )+\log (x) \left (\left (4 x^4-2 e x^4+4 x^5+\left (-2 e x^2+4 x^3\right ) \log (3)\right ) \log (-e+2 x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)\right )}{(e-2 x) \log ^3(3) \log ^3(x)+\left (-3 e x+6 x^2\right ) \log ^2(3) \log ^2(x) \log (-e+2 x)+\left (3 e x^2-6 x^3\right ) \log (3) \log (x) \log ^2(-e+2 x)+\left (-e x^3+2 x^4\right ) \log ^3(-e+2 x)} \, dx=\int \frac {\left (\left (4 e x^3-8 x^4\right ) \log (3)+\left (4 e x-8 x^2\right ) \log ^2(3)\right ) \log ^3(x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)+\log ^2(x) \left (-4 x^5-4 x^3 \log (3)+\left (-2 e x^4+4 x^5+\left (-6 e x^2+12 x^3\right ) \log (3)\right ) \log (-e+2 x)\right )+\log (x) \left (\left (4 x^4-2 e x^4+4 x^5+\left (-2 e x^2+4 x^3\right ) \log (3)\right ) \log (-e+2 x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)\right )}{(e-2 x) \log ^3(3) \log ^3(x)+\left (-3 e x+6 x^2\right ) \log ^2(3) \log ^2(x) \log (-e+2 x)+\left (3 e x^2-6 x^3\right ) \log (3) \log (x) \log ^2(-e+2 x)+\left (-e x^3+2 x^4\right ) \log ^3(-e+2 x)} \, dx \]

[In]

Int[(((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]

[Out]

(E^4*Defer[Int][Log[x]^2/(Log[3]*Log[x] - x*Log[-E + 2*x])^3, x])/8 + (E^3*Log[3]*Defer[Int][Log[x]^2/(Log[3]*
Log[x] - x*Log[-E + 2*x])^3, x])/4 - (E^3*Log[9]*Defer[Int][Log[x]^2/(Log[3]*Log[x] - x*Log[-E + 2*x])^3, x])/
8 + (E^3*Defer[Int][(x*Log[x]^2)/(Log[3]*Log[x] - x*Log[-E + 2*x])^3, x])/4 + (E^2*Log[3]*Defer[Int][(x*Log[x]
^2)/(Log[3]*Log[x] - x*Log[-E + 2*x])^3, x])/2 - (E^2*Log[9]*Defer[Int][(x*Log[x]^2)/(Log[3]*Log[x] - x*Log[-E
+ 2*x])^3, x])/4 + (E^2*Defer[Int][(x^2*Log[x]^2)/(Log[3]*Log[x] - x*Log[-E + 2*x])^3, x])/2 + E*Log[3]*Defer
[Int][(x^2*Log[x]^2)/(Log[3]*Log[x] - x*Log[-E + 2*x])^3, x] - (E*Log[9]*Defer[Int][(x^2*Log[x]^2)/(Log[3]*Log
[x] - x*Log[-E + 2*x])^3, x])/2 + E*Defer[Int][(x^3*Log[x]^2)/(Log[3]*Log[x] - x*Log[-E + 2*x])^3, x] - Log[9]
*Defer[Int][(x^3*Log[x]^2)/(Log[3]*Log[x] - x*Log[-E + 2*x])^3, x] + 2*Defer[Int][(x^4*Log[x]^2)/(Log[3]*Log[x
] - x*Log[-E + 2*x])^3, x] - (E^3*Log[3]*Defer[Int][Log[x]^3/(Log[3]*Log[x] - x*Log[-E + 2*x])^3, x])/4 + (E^3
*Log[9]*Defer[Int][Log[x]^3/(Log[3]*Log[x] - x*Log[-E + 2*x])^3, x])/8 - (E^2*Log[3]*Defer[Int][(x*Log[x]^3)/(
Log[3]*Log[x] - x*Log[-E + 2*x])^3, x])/2 + (E^2*Log[9]*Defer[Int][(x*Log[x]^3)/(Log[3]*Log[x] - x*Log[-E + 2*
x])^3, x])/4 - E*Log[3]*Defer[Int][(x^2*Log[x]^3)/(Log[3]*Log[x] - x*Log[-E + 2*x])^3, x] + (E*Log[9]*Defer[In
t][(x^2*Log[x]^3)/(Log[3]*Log[x] - x*Log[-E + 2*x])^3, x])/2 + Log[9]*Defer[Int][(x^3*Log[x]^3)/(Log[3]*Log[x]
- x*Log[-E + 2*x])^3, x] + ((2 - E)*E^2*Defer[Int][Log[x]/(Log[3]*Log[x] - x*Log[-E + 2*x])^2, x])/4 + (E^3*D
efer[Int][Log[x]/(Log[3]*Log[x] - x*Log[-E + 2*x])^2, x])/4 + E*Log[3]*Defer[Int][Log[x]/(Log[3]*Log[x] - x*Lo
g[-E + 2*x])^2, x] - (E*Log[9]*Defer[Int][Log[x]/(Log[3]*Log[x] - x*Log[-E + 2*x])^2, x])/2 + ((2 - E)*E*Defer
[Int][(x*Log[x])/(Log[3]*Log[x] - x*Log[-E + 2*x])^2, x])/2 + (E^2*Defer[Int][(x*Log[x])/(Log[3]*Log[x] - x*Lo
g[-E + 2*x])^2, x])/2 - Log[9]*Defer[Int][(x*Log[x])/(Log[3]*Log[x] - x*Log[-E + 2*x])^2, x] + (2 - E)*Defer[I
nt][(x^2*Log[x])/(Log[3]*Log[x] - x*Log[-E + 2*x])^2, x] + E*Defer[Int][(x^2*Log[x])/(Log[3]*Log[x] - x*Log[-E
+ 2*x])^2, x] + 2*Defer[Int][(x^3*Log[x])/(Log[3]*Log[x] - x*Log[-E + 2*x])^2, x] - E*Log[3]*Defer[Int][Log[x
]^2/(Log[3]*Log[x] - x*Log[-E + 2*x])^2, x] + (E*Log[9]*Defer[Int][Log[x]^2/(Log[3]*Log[x] - x*Log[-E + 2*x])^
2, x])/2 + Log[9]*Defer[Int][(x*Log[x]^2)/(Log[3]*Log[x] - x*Log[-E + 2*x])^2, x] + 2*Defer[Int][(x^3*Log[x]^2
)/(Log[3]*Log[x] - x*Log[-E + 2*x])^2, x] + (E^5*Defer[Int][Log[x]^2/((E - 2*x)*(-(Log[3]*Log[x]) + x*Log[-E +
2*x])^3), x])/8 + (E^4*Log[3]*Defer[Int][Log[x]^2/((E - 2*x)*(-(Log[3]*Log[x]) + x*Log[-E + 2*x])^3), x])/4 -
(E^4*Log[9]*Defer[Int][Log[x]^2/((E - 2*x)*(-(Log[3]*Log[x]) + x*Log[-E + 2*x])^3), x])/8 - (E^4*Log[3]*Defer
[Int][Log[x]^3/((E - 2*x)*(-(Log[3]*Log[x]) + x*Log[-E + 2*x])^3), x])/4 + (E^4*Log[9]*Defer[Int][Log[x]^3/((E
- 2*x)*(-(Log[3]*Log[x]) + x*Log[-E + 2*x])^3), x])/8 - ((2 - E)*E^3*Defer[Int][Log[x]/((E - 2*x)*(-(Log[3]*L
og[x]) + x*Log[-E + 2*x])^2), x])/4 - (E^4*Defer[Int][Log[x]/((E - 2*x)*(-(Log[3]*Log[x]) + x*Log[-E + 2*x])^2
), x])/4 - E^2*Log[3]*Defer[Int][Log[x]/((E - 2*x)*(-(Log[3]*Log[x]) + x*Log[-E + 2*x])^2), x] + (E^2*Log[9]*D
efer[Int][Log[x]/((E - 2*x)*(-(Log[3]*Log[x]) + x*Log[-E + 2*x])^2), x])/2 + E^2*Log[3]*Defer[Int][Log[x]^2/((
E - 2*x)*(-(Log[3]*Log[x]) + x*Log[-E + 2*x])^2), x] - (E^2*Log[9]*Defer[Int][Log[x]^2/((E - 2*x)*(-(Log[3]*Lo
g[x]) + x*Log[-E + 2*x])^2), x])/2 - 2*Defer[Int][x/(-(Log[3]*Log[x]) + x*Log[-E + 2*x]), x] - 2*Defer[Int][(x
*Log[x])/(-(Log[3]*Log[x]) + x*Log[-E + 2*x]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {2 x \left (\left (x^2+\log (3)\right ) \log (x)-x \log (-e+2 x)\right ) \left (-\left ((-e \log (9)+x \log (81)) \log ^2(x)\right )-(e-2 x) x \log (-e+2 x)-x \log (x) (2 x+(e-2 x) \log (-e+2 x))\right )}{(e-2 x) (\log (3) \log (x)-x \log (-e+2 x))^3} \, dx \\ & = 2 \int \frac {x \left (\left (x^2+\log (3)\right ) \log (x)-x \log (-e+2 x)\right ) \left (-\left ((-e \log (9)+x \log (81)) \log ^2(x)\right )-(e-2 x) x \log (-e+2 x)-x \log (x) (2 x+(e-2 x) \log (-e+2 x))\right )}{(e-2 x) (\log (3) \log (x)-x \log (-e+2 x))^3} \, dx \\ & = 2 \int \left (\frac {x \log (x) \left (-2 \left (1-\frac {e}{2}\right ) x^2-2 x^3-e \log (3)+x \log (9)+e x^2 \log (x)-2 x^3 \log (x)+e \log (3) \log (x)-x \log (9) \log (x)\right )}{(e-2 x) (\log (3) \log (x)-x \log (-e+2 x))^2}-\frac {x^3 \log ^2(x) \left (-2 x^2-e \log (3)+x \log (9)+e \log (3) \log (x)-x \log (9) \log (x)\right )}{(e-2 x) (-\log (3) \log (x)+x \log (-e+2 x))^3}-\frac {x (1+\log (x))}{-\log (3) \log (x)+x \log (-e+2 x)}\right ) \, dx \\ & = 2 \int \frac {x \log (x) \left (-2 \left (1-\frac {e}{2}\right ) x^2-2 x^3-e \log (3)+x \log (9)+e x^2 \log (x)-2 x^3 \log (x)+e \log (3) \log (x)-x \log (9) \log (x)\right )}{(e-2 x) (\log (3) \log (x)-x \log (-e+2 x))^2} \, dx-2 \int \frac {x^3 \log ^2(x) \left (-2 x^2-e \log (3)+x \log (9)+e \log (3) \log (x)-x \log (9) \log (x)\right )}{(e-2 x) (-\log (3) \log (x)+x \log (-e+2 x))^3} \, dx-2 \int \frac {x (1+\log (x))}{-\log (3) \log (x)+x \log (-e+2 x)} \, dx \\ & = 2 \int \left (\frac {e \log (x) \left (-2 \left (1-\frac {e}{2}\right ) x^2-2 x^3-e \log (3)+x \log (9)+e x^2 \log (x)-2 x^3 \log (x)+e \log (3) \log (x)-x \log (9) \log (x)\right )}{2 (e-2 x) (\log (3) \log (x)-x \log (-e+2 x))^2}+\frac {\log (x) \left (2 \left (1-\frac {e}{2}\right ) x^2+2 x^3+e \log (3)-x \log (9)-e x^2 \log (x)+2 x^3 \log (x)-e \log (3) \log (x)+x \log (9) \log (x)\right )}{2 (\log (3) \log (x)-x \log (-e+2 x))^2}\right ) \, dx-2 \int \left (\frac {e^2 \log ^2(x) \left (-2 x^2-e \log (3)+x \log (9)+e \log (3) \log (x)-x \log (9) \log (x)\right )}{8 (\log (3) \log (x)-x \log (-e+2 x))^3}+\frac {e x \log ^2(x) \left (-2 x^2-e \log (3)+x \log (9)+e \log (3) \log (x)-x \log (9) \log (x)\right )}{4 (\log (3) \log (x)-x \log (-e+2 x))^3}-\frac {x^2 \log ^2(x) \left (2 x^2+e \log (3)-x \log (9)-e \log (3) \log (x)+x \log (9) \log (x)\right )}{2 (\log (3) \log (x)-x \log (-e+2 x))^3}+\frac {e^3 \log ^2(x) \left (-2 x^2-e \log (3)+x \log (9)+e \log (3) \log (x)-x \log (9) \log (x)\right )}{8 (e-2 x) (-\log (3) \log (x)+x \log (-e+2 x))^3}\right ) \, dx-2 \int \left (\frac {x}{-\log (3) \log (x)+x \log (-e+2 x)}+\frac {x \log (x)}{-\log (3) \log (x)+x \log (-e+2 x)}\right ) \, dx \\ & = -\left (2 \int \frac {x}{-\log (3) \log (x)+x \log (-e+2 x)} \, dx\right )-2 \int \frac {x \log (x)}{-\log (3) \log (x)+x \log (-e+2 x)} \, dx-\frac {1}{2} e \int \frac {x \log ^2(x) \left (-2 x^2-e \log (3)+x \log (9)+e \log (3) \log (x)-x \log (9) \log (x)\right )}{(\log (3) \log (x)-x \log (-e+2 x))^3} \, dx+e \int \frac {\log (x) \left (-2 \left (1-\frac {e}{2}\right ) x^2-2 x^3-e \log (3)+x \log (9)+e x^2 \log (x)-2 x^3 \log (x)+e \log (3) \log (x)-x \log (9) \log (x)\right )}{(e-2 x) (\log (3) \log (x)-x \log (-e+2 x))^2} \, dx-\frac {1}{4} e^2 \int \frac {\log ^2(x) \left (-2 x^2-e \log (3)+x \log (9)+e \log (3) \log (x)-x \log (9) \log (x)\right )}{(\log (3) \log (x)-x \log (-e+2 x))^3} \, dx-\frac {1}{4} e^3 \int \frac {\log ^2(x) \left (-2 x^2-e \log (3)+x \log (9)+e \log (3) \log (x)-x \log (9) \log (x)\right )}{(e-2 x) (-\log (3) \log (x)+x \log (-e+2 x))^3} \, dx+\int \frac {x^2 \log ^2(x) \left (2 x^2+e \log (3)-x \log (9)-e \log (3) \log (x)+x \log (9) \log (x)\right )}{(\log (3) \log (x)-x \log (-e+2 x))^3} \, dx+\int \frac {\log (x) \left (2 \left (1-\frac {e}{2}\right ) x^2+2 x^3+e \log (3)-x \log (9)-e x^2 \log (x)+2 x^3 \log (x)-e \log (3) \log (x)+x \log (9) \log (x)\right )}{(\log (3) \log (x)-x \log (-e+2 x))^2} \, dx \\ & = -\left (2 \int \frac {x}{-\log (3) \log (x)+x \log (-e+2 x)} \, dx\right )-2 \int \frac {x \log (x)}{-\log (3) \log (x)+x \log (-e+2 x)} \, dx-\frac {1}{2} e \int \left (-\frac {2 x^3 \log ^2(x)}{(\log (3) \log (x)-x \log (-e+2 x))^3}-\frac {e x \log (3) \log ^2(x)}{(\log (3) \log (x)-x \log (-e+2 x))^3}+\frac {x^2 \log (9) \log ^2(x)}{(\log (3) \log (x)-x \log (-e+2 x))^3}+\frac {e x \log (3) \log ^3(x)}{(\log (3) \log (x)-x \log (-e+2 x))^3}-\frac {x^2 \log (9) \log ^3(x)}{(\log (3) \log (x)-x \log (-e+2 x))^3}\right ) \, dx+e \int \left (\frac {(-2+e) x^2 \log (x)}{(e-2 x) (-\log (3) \log (x)+x \log (-e+2 x))^2}-\frac {2 x^3 \log (x)}{(e-2 x) (-\log (3) \log (x)+x \log (-e+2 x))^2}-\frac {e \log (3) \log (x)}{(e-2 x) (-\log (3) \log (x)+x \log (-e+2 x))^2}+\frac {x \log (9) \log (x)}{(e-2 x) (-\log (3) \log (x)+x \log (-e+2 x))^2}+\frac {e x^2 \log ^2(x)}{(e-2 x) (-\log (3) \log (x)+x \log (-e+2 x))^2}-\frac {2 x^3 \log ^2(x)}{(e-2 x) (-\log (3) \log (x)+x \log (-e+2 x))^2}+\frac {e \log (3) \log ^2(x)}{(e-2 x) (-\log (3) \log (x)+x \log (-e+2 x))^2}-\frac {x \log (9) \log ^2(x)}{(e-2 x) (-\log (3) \log (x)+x \log (-e+2 x))^2}\right ) \, dx-\frac {1}{4} e^2 \int \left (-\frac {2 x^2 \log ^2(x)}{(\log (3) \log (x)-x \log (-e+2 x))^3}-\frac {e \log (3) \log ^2(x)}{(\log (3) \log (x)-x \log (-e+2 x))^3}+\frac {x \log (9) \log ^2(x)}{(\log (3) \log (x)-x \log (-e+2 x))^3}+\frac {e \log (3) \log ^3(x)}{(\log (3) \log (x)-x \log (-e+2 x))^3}-\frac {x \log (9) \log ^3(x)}{(\log (3) \log (x)-x \log (-e+2 x))^3}\right ) \, dx-\frac {1}{4} e^3 \int \left (-\frac {2 x^2 \log ^2(x)}{(e-2 x) (-\log (3) \log (x)+x \log (-e+2 x))^3}-\frac {e \log (3) \log ^2(x)}{(e-2 x) (-\log (3) \log (x)+x \log (-e+2 x))^3}+\frac {x \log (9) \log ^2(x)}{(e-2 x) (-\log (3) \log (x)+x \log (-e+2 x))^3}+\frac {e \log (3) \log ^3(x)}{(e-2 x) (-\log (3) \log (x)+x \log (-e+2 x))^3}-\frac {x \log (9) \log ^3(x)}{(e-2 x) (-\log (3) \log (x)+x \log (-e+2 x))^3}\right ) \, dx+\int \left (\frac {2 x^4 \log ^2(x)}{(\log (3) \log (x)-x \log (-e+2 x))^3}+\frac {e x^2 \log (3) \log ^2(x)}{(\log (3) \log (x)-x \log (-e+2 x))^3}-\frac {x^3 \log (9) \log ^2(x)}{(\log (3) \log (x)-x \log (-e+2 x))^3}-\frac {e x^2 \log (3) \log ^3(x)}{(\log (3) \log (x)-x \log (-e+2 x))^3}+\frac {x^3 \log (9) \log ^3(x)}{(\log (3) \log (x)-x \log (-e+2 x))^3}\right ) \, dx+\int \left (-\frac {(-2+e) x^2 \log (x)}{(\log (3) \log (x)-x \log (-e+2 x))^2}+\frac {2 x^3 \log (x)}{(\log (3) \log (x)-x \log (-e+2 x))^2}+\frac {e \log (3) \log (x)}{(\log (3) \log (x)-x \log (-e+2 x))^2}-\frac {x \log (9) \log (x)}{(\log (3) \log (x)-x \log (-e+2 x))^2}-\frac {e x^2 \log ^2(x)}{(\log (3) \log (x)-x \log (-e+2 x))^2}+\frac {2 x^3 \log ^2(x)}{(\log (3) \log (x)-x \log (-e+2 x))^2}-\frac {e \log (3) \log ^2(x)}{(\log (3) \log (x)-x \log (-e+2 x))^2}+\frac {x \log (9) \log ^2(x)}{(\log (3) \log (x)-x \log (-e+2 x))^2}\right ) \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [F]

\[ \int \frac {\left (\left (4 e x^3-8 x^4\right ) \log (3)+\left (4 e x-8 x^2\right ) \log ^2(3)\right ) \log ^3(x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)+\log ^2(x) \left (-4 x^5-4 x^3 \log (3)+\left (-2 e x^4+4 x^5+\left (-6 e x^2+12 x^3\right ) \log (3)\right ) \log (-e+2 x)\right )+\log (x) \left (\left (4 x^4-2 e x^4+4 x^5+\left (-2 e x^2+4 x^3\right ) \log (3)\right ) \log (-e+2 x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)\right )}{(e-2 x) \log ^3(3) \log ^3(x)+\left (-3 e x+6 x^2\right ) \log ^2(3) \log ^2(x) \log (-e+2 x)+\left (3 e x^2-6 x^3\right ) \log (3) \log (x) \log ^2(-e+2 x)+\left (-e x^3+2 x^4\right ) \log ^3(-e+2 x)} \, dx=\int \frac {\left (\left (4 e x^3-8 x^4\right ) \log (3)+\left (4 e x-8 x^2\right ) \log ^2(3)\right ) \log ^3(x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)+\log ^2(x) \left (-4 x^5-4 x^3 \log (3)+\left (-2 e x^4+4 x^5+\left (-6 e x^2+12 x^3\right ) \log (3)\right ) \log (-e+2 x)\right )+\log (x) \left (\left (4 x^4-2 e x^4+4 x^5+\left (-2 e x^2+4 x^3\right ) \log (3)\right ) \log (-e+2 x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)\right )}{(e-2 x) \log ^3(3) \log ^3(x)+\left (-3 e x+6 x^2\right ) \log ^2(3) \log ^2(x) \log (-e+2 x)+\left (3 e x^2-6 x^3\right ) \log (3) \log (x) \log ^2(-e+2 x)+\left (-e x^3+2 x^4\right ) \log ^3(-e+2 x)} \, dx \]

[In]

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]*L
og[-E + 2*x]^2 + (-(E*x^3) + 2*x^4)*Log[-E + 2*x]^3),x]

[Out]

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]*L
og[-E + 2*x]^2 + (-(E*x^3) + 2*x^4)*Log[-E + 2*x]^3), x]

Maple [A] (verified)

Time = 3.41 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.73

method	result	size
default	\(\frac {\left (x^{2} \ln \left (x \right )+2 \ln \left (3\right ) \ln \left (x \right )-2 \ln \left (-{\mathrm e}+2 x \right ) x \right ) x^{2} \ln \left (x \right )}{\left (\ln \left (3\right ) \ln \left (x \right )-\ln \left (-{\mathrm e}+2 x \right ) x \right )^{2}}\)	\(52\)
risch	\(\frac {\left (x^{2} \ln \left (x \right )+2 \ln \left (3\right ) \ln \left (x \right )-2 \ln \left (-{\mathrm e}+2 x \right ) x \right ) x^{2} \ln \left (x \right )}{\left (\ln \left (3\right ) \ln \left (x \right )-\ln \left (-{\mathrm e}+2 x \right ) x \right )^{2}}\)	\(52\)
parallelrisch	\(-\frac {\left (-4 x^{4} \ln \left (x \right )^{2} {\mathrm e}-8 x^{2} \ln \left (x \right )^{2} \ln \left (3\right ) {\mathrm e}+8 \ln \left (x \right ) {\mathrm e} \ln \left (-{\mathrm e}+2 x \right ) x^{3}\right ) {\mathrm e}^{-1}}{4 \left (\ln \left (3\right )^{2} \ln \left (x \right )^{2}-2 \ln \left (3\right ) x \ln \left (x \right ) \ln \left (-{\mathrm e}+2 x \right )+\ln \left (-{\mathrm e}+2 x \right )^{2} x^{2}\right )}\)	\(93\)

[In]

int((((4*x*exp(1)-8*x^2)*ln(3)^2+(4*x^3*exp(1)-8*x^4)*ln(3))*ln(x)^3+(((-6*x^2*exp(1)+12*x^3)*ln(3)-2*x^4*exp(
1)+4*x^5)*ln(-exp(1)+2*x)-4*x^3*ln(3)-4*x^5)*ln(x)^2+((2*x^3*exp(1)-4*x^4)*ln(-exp(1)+2*x)^2+((-2*x^2*exp(1)+4
*x^3)*ln(3)-2*x^4*exp(1)+4*x^5+4*x^4)*ln(-exp(1)+2*x))*ln(x)+(2*x^3*exp(1)-4*x^4)*ln(-exp(1)+2*x)^2)/((exp(1)-
2*x)*ln(3)^3*ln(x)^3+(-3*x*exp(1)+6*x^2)*ln(3)^2*ln(-exp(1)+2*x)*ln(x)^2+(3*x^2*exp(1)-6*x^3)*ln(3)*ln(-exp(1)
+2*x)^2*ln(x)+(-x^3*exp(1)+2*x^4)*ln(-exp(1)+2*x)^3),x,method=_RETURNVERBOSE)

[Out]

(x^2*ln(x)+2*ln(3)*ln(x)-2*ln(-exp(1)+2*x)*x)*x^2*ln(x)/(ln(3)*ln(x)-ln(-exp(1)+2*x)*x)^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (30) = 60\).

Time = 0.28 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.63 \[ \int \frac {\left (\left (4 e x^3-8 x^4\right ) \log (3)+\left (4 e x-8 x^2\right ) \log ^2(3)\right ) \log ^3(x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)+\log ^2(x) \left (-4 x^5-4 x^3 \log (3)+\left (-2 e x^4+4 x^5+\left (-6 e x^2+12 x^3\right ) \log (3)\right ) \log (-e+2 x)\right )+\log (x) \left (\left (4 x^4-2 e x^4+4 x^5+\left (-2 e x^2+4 x^3\right ) \log (3)\right ) \log (-e+2 x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)\right )}{(e-2 x) \log ^3(3) \log ^3(x)+\left (-3 e x+6 x^2\right ) \log ^2(3) \log ^2(x) \log (-e+2 x)+\left (3 e x^2-6 x^3\right ) \log (3) \log (x) \log ^2(-e+2 x)+\left (-e x^3+2 x^4\right ) \log ^3(-e+2 x)} \, dx=-\frac {2 \, x^{3} \log \left (2 \, x - e\right ) \log \left (x\right ) - {\left (x^{4} + 2 \, x^{2} \log \left (3\right )\right )} \log \left (x\right )^{2}}{x^{2} \log \left (2 \, x - e\right )^{2} - 2 \, x \log \left (3\right ) \log \left (2 \, x - e\right ) \log \left (x\right ) + \log \left (3\right )^{2} \log \left (x\right )^{2}} \]

[In]

integrate((((4*x*exp(1)-8*x^2)*log(3)^2+(4*x^3*exp(1)-8*x^4)*log(3))*log(x)^3+(((-6*x^2*exp(1)+12*x^3)*log(3)-
2*x^4*exp(1)+4*x^5)*log(-exp(1)+2*x)-4*x^3*log(3)-4*x^5)*log(x)^2+((2*x^3*exp(1)-4*x^4)*log(-exp(1)+2*x)^2+((-
2*x^2*exp(1)+4*x^3)*log(3)-2*x^4*exp(1)+4*x^5+4*x^4)*log(-exp(1)+2*x))*log(x)+(2*x^3*exp(1)-4*x^4)*log(-exp(1)
+2*x)^2)/((exp(1)-2*x)*log(3)^3*log(x)^3+(-3*x*exp(1)+6*x^2)*log(3)^2*log(-exp(1)+2*x)*log(x)^2+(3*x^2*exp(1)-
6*x^3)*log(3)*log(-exp(1)+2*x)^2*log(x)+(-x^3*exp(1)+2*x^4)*log(-exp(1)+2*x)^3),x, algorithm="fricas")

[Out]

-(2*x^3*log(2*x - e)*log(x) - (x^4 + 2*x^2*log(3))*log(x)^2)/(x^2*log(2*x - e)^2 - 2*x*log(3)*log(2*x - e)*log
(x) + log(3)^2*log(x)^2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (20) = 40\).

Time = 0.18 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.67 \[ \int \frac {\left (\left (4 e x^3-8 x^4\right ) \log (3)+\left (4 e x-8 x^2\right ) \log ^2(3)\right ) \log ^3(x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)+\log ^2(x) \left (-4 x^5-4 x^3 \log (3)+\left (-2 e x^4+4 x^5+\left (-6 e x^2+12 x^3\right ) \log (3)\right ) \log (-e+2 x)\right )+\log (x) \left (\left (4 x^4-2 e x^4+4 x^5+\left (-2 e x^2+4 x^3\right ) \log (3)\right ) \log (-e+2 x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)\right )}{(e-2 x) \log ^3(3) \log ^3(x)+\left (-3 e x+6 x^2\right ) \log ^2(3) \log ^2(x) \log (-e+2 x)+\left (3 e x^2-6 x^3\right ) \log (3) \log (x) \log ^2(-e+2 x)+\left (-e x^3+2 x^4\right ) \log ^3(-e+2 x)} \, dx=\frac {x^{4} \log {\left (x \right )}^{2} - 2 x^{3} \log {\left (x \right )} \log {\left (2 x - e \right )} + 2 x^{2} \log {\left (3 \right )} \log {\left (x \right )}^{2}}{x^{2} \log {\left (2 x - e \right )}^{2} - 2 x \log {\left (3 \right )} \log {\left (x \right )} \log {\left (2 x - e \right )} + \log {\left (3 \right )}^{2} \log {\left (x \right )}^{2}} \]

[In]

integrate((((4*x*exp(1)-8*x**2)*ln(3)**2+(4*x**3*exp(1)-8*x**4)*ln(3))*ln(x)**3+(((-6*x**2*exp(1)+12*x**3)*ln(
3)-2*x**4*exp(1)+4*x**5)*ln(-exp(1)+2*x)-4*x**3*ln(3)-4*x**5)*ln(x)**2+((2*x**3*exp(1)-4*x**4)*ln(-exp(1)+2*x)
**2+((-2*x**2*exp(1)+4*x**3)*ln(3)-2*x**4*exp(1)+4*x**5+4*x**4)*ln(-exp(1)+2*x))*ln(x)+(2*x**3*exp(1)-4*x**4)*
ln(-exp(1)+2*x)**2)/((exp(1)-2*x)*ln(3)**3*ln(x)**3+(-3*x*exp(1)+6*x**2)*ln(3)**2*ln(-exp(1)+2*x)*ln(x)**2+(3*
x**2*exp(1)-6*x**3)*ln(3)*ln(-exp(1)+2*x)**2*ln(x)+(-x**3*exp(1)+2*x**4)*ln(-exp(1)+2*x)**3),x)

[Out]

(x**4*log(x)**2 - 2*x**3*log(x)*log(2*x - E) + 2*x**2*log(3)*log(x)**2)/(x**2*log(2*x - E)**2 - 2*x*log(3)*log
(x)*log(2*x - E) + log(3)**2*log(x)**2)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (30) = 60\).

Time = 0.56 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.63 \[ \int \frac {\left (\left (4 e x^3-8 x^4\right ) \log (3)+\left (4 e x-8 x^2\right ) \log ^2(3)\right ) \log ^3(x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)+\log ^2(x) \left (-4 x^5-4 x^3 \log (3)+\left (-2 e x^4+4 x^5+\left (-6 e x^2+12 x^3\right ) \log (3)\right ) \log (-e+2 x)\right )+\log (x) \left (\left (4 x^4-2 e x^4+4 x^5+\left (-2 e x^2+4 x^3\right ) \log (3)\right ) \log (-e+2 x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)\right )}{(e-2 x) \log ^3(3) \log ^3(x)+\left (-3 e x+6 x^2\right ) \log ^2(3) \log ^2(x) \log (-e+2 x)+\left (3 e x^2-6 x^3\right ) \log (3) \log (x) \log ^2(-e+2 x)+\left (-e x^3+2 x^4\right ) \log ^3(-e+2 x)} \, dx=-\frac {2 \, x^{3} \log \left (2 \, x - e\right ) \log \left (x\right ) - {\left (x^{4} + 2 \, x^{2} \log \left (3\right )\right )} \log \left (x\right )^{2}}{x^{2} \log \left (2 \, x - e\right )^{2} - 2 \, x \log \left (3\right ) \log \left (2 \, x - e\right ) \log \left (x\right ) + \log \left (3\right )^{2} \log \left (x\right )^{2}} \]

[In]

integrate((((4*x*exp(1)-8*x^2)*log(3)^2+(4*x^3*exp(1)-8*x^4)*log(3))*log(x)^3+(((-6*x^2*exp(1)+12*x^3)*log(3)-
2*x^4*exp(1)+4*x^5)*log(-exp(1)+2*x)-4*x^3*log(3)-4*x^5)*log(x)^2+((2*x^3*exp(1)-4*x^4)*log(-exp(1)+2*x)^2+((-
2*x^2*exp(1)+4*x^3)*log(3)-2*x^4*exp(1)+4*x^5+4*x^4)*log(-exp(1)+2*x))*log(x)+(2*x^3*exp(1)-4*x^4)*log(-exp(1)
+2*x)^2)/((exp(1)-2*x)*log(3)^3*log(x)^3+(-3*x*exp(1)+6*x^2)*log(3)^2*log(-exp(1)+2*x)*log(x)^2+(3*x^2*exp(1)-
6*x^3)*log(3)*log(-exp(1)+2*x)^2*log(x)+(-x^3*exp(1)+2*x^4)*log(-exp(1)+2*x)^3),x, algorithm="maxima")

[Out]

-(2*x^3*log(2*x - e)*log(x) - (x^4 + 2*x^2*log(3))*log(x)^2)/(x^2*log(2*x - e)^2 - 2*x*log(3)*log(2*x - e)*log
(x) + log(3)^2*log(x)^2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (30) = 60\).

Time = 0.57 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.67 \[ \int \frac {\left (\left (4 e x^3-8 x^4\right ) \log (3)+\left (4 e x-8 x^2\right ) \log ^2(3)\right ) \log ^3(x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)+\log ^2(x) \left (-4 x^5-4 x^3 \log (3)+\left (-2 e x^4+4 x^5+\left (-6 e x^2+12 x^3\right ) \log (3)\right ) \log (-e+2 x)\right )+\log (x) \left (\left (4 x^4-2 e x^4+4 x^5+\left (-2 e x^2+4 x^3\right ) \log (3)\right ) \log (-e+2 x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)\right )}{(e-2 x) \log ^3(3) \log ^3(x)+\left (-3 e x+6 x^2\right ) \log ^2(3) \log ^2(x) \log (-e+2 x)+\left (3 e x^2-6 x^3\right ) \log (3) \log (x) \log ^2(-e+2 x)+\left (-e x^3+2 x^4\right ) \log ^3(-e+2 x)} \, dx=\frac {x^{4} \log \left (x\right )^{2} - 2 \, x^{3} \log \left (2 \, x - e\right ) \log \left (x\right ) + 2 \, x^{2} \log \left (3\right ) \log \left (x\right )^{2}}{x^{2} \log \left (2 \, x - e\right )^{2} - 2 \, x \log \left (3\right ) \log \left (2 \, x - e\right ) \log \left (x\right ) + \log \left (3\right )^{2} \log \left (x\right )^{2}} \]

[In]

integrate((((4*x*exp(1)-8*x^2)*log(3)^2+(4*x^3*exp(1)-8*x^4)*log(3))*log(x)^3+(((-6*x^2*exp(1)+12*x^3)*log(3)-
2*x^4*exp(1)+4*x^5)*log(-exp(1)+2*x)-4*x^3*log(3)-4*x^5)*log(x)^2+((2*x^3*exp(1)-4*x^4)*log(-exp(1)+2*x)^2+((-
2*x^2*exp(1)+4*x^3)*log(3)-2*x^4*exp(1)+4*x^5+4*x^4)*log(-exp(1)+2*x))*log(x)+(2*x^3*exp(1)-4*x^4)*log(-exp(1)
+2*x)^2)/((exp(1)-2*x)*log(3)^3*log(x)^3+(-3*x*exp(1)+6*x^2)*log(3)^2*log(-exp(1)+2*x)*log(x)^2+(3*x^2*exp(1)-
6*x^3)*log(3)*log(-exp(1)+2*x)^2*log(x)+(-x^3*exp(1)+2*x^4)*log(-exp(1)+2*x)^3),x, algorithm="giac")

[Out]

(x^4*log(x)^2 - 2*x^3*log(2*x - e)*log(x) + 2*x^2*log(3)*log(x)^2)/(x^2*log(2*x - e)^2 - 2*x*log(3)*log(2*x -
e)*log(x) + log(3)^2*log(x)^2)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (\left (4 e x^3-8 x^4\right ) \log (3)+\left (4 e x-8 x^2\right ) \log ^2(3)\right ) \log ^3(x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)+\log ^2(x) \left (-4 x^5-4 x^3 \log (3)+\left (-2 e x^4+4 x^5+\left (-6 e x^2+12 x^3\right ) \log (3)\right ) \log (-e+2 x)\right )+\log (x) \left (\left (4 x^4-2 e x^4+4 x^5+\left (-2 e x^2+4 x^3\right ) \log (3)\right ) \log (-e+2 x)+\left (2 e x^3-4 x^4\right ) \log ^2(-e+2 x)\right )}{(e-2 x) \log ^3(3) \log ^3(x)+\left (-3 e x+6 x^2\right ) \log ^2(3) \log ^2(x) \log (-e+2 x)+\left (3 e x^2-6 x^3\right ) \log (3) \log (x) \log ^2(-e+2 x)+\left (-e x^3+2 x^4\right ) \log ^3(-e+2 x)} \, dx=\int \frac {\ln \left (x\right )\,\left (\ln \left (2\,x-\mathrm {e}\right )\,\left (2\,x^4\,\mathrm {e}-4\,x^4-4\,x^5+\ln \left (3\right )\,\left (2\,x^2\,\mathrm {e}-4\,x^3\right )\right )-{\ln \left (2\,x-\mathrm {e}\right )}^2\,\left (2\,x^3\,\mathrm {e}-4\,x^4\right )\right )-{\ln \left (2\,x-\mathrm {e}\right )}^2\,\left (2\,x^3\,\mathrm {e}-4\,x^4\right )+{\ln \left (x\right )}^2\,\left (4\,x^3\,\ln \left (3\right )+4\,x^5+\ln \left (2\,x-\mathrm {e}\right )\,\left (2\,x^4\,\mathrm {e}-4\,x^5+\ln \left (3\right )\,\left (6\,x^2\,\mathrm {e}-12\,x^3\right )\right )\right )-{\ln \left (x\right )}^3\,\left ({\ln \left (3\right )}^2\,\left (4\,x\,\mathrm {e}-8\,x^2\right )+\ln \left (3\right )\,\left (4\,x^3\,\mathrm {e}-8\,x^4\right )\right )}{\left (x^3\,\mathrm {e}-2\,x^4\right )\,{\ln \left (2\,x-\mathrm {e}\right )}^3-\ln \left (3\right )\,\left (3\,x^2\,\mathrm {e}-6\,x^3\right )\,{\ln \left (2\,x-\mathrm {e}\right )}^2\,\ln \left (x\right )+{\ln \left (3\right )}^2\,\left (3\,x\,\mathrm {e}-6\,x^2\right )\,\ln \left (2\,x-\mathrm {e}\right )\,{\ln \left (x\right )}^2+{\ln \left (3\right )}^3\,\left (2\,x-\mathrm {e}\right )\,{\ln \left (x\right )}^3} \,d x \]

[In]

int((log(x)*(log(2*x - exp(1))*(2*x^4*exp(1) - 4*x^4 - 4*x^5 + log(3)*(2*x^2*exp(1) - 4*x^3)) - log(2*x - exp(
1))^2*(2*x^3*exp(1) - 4*x^4)) - log(2*x - exp(1))^2*(2*x^3*exp(1) - 4*x^4) + log(x)^2*(4*x^3*log(3) + 4*x^5 +
log(2*x - exp(1))*(2*x^4*exp(1) - 4*x^5 + log(3)*(6*x^2*exp(1) - 12*x^3))) - log(x)^3*(log(3)^2*(4*x*exp(1) -
8*x^2) + log(3)*(4*x^3*exp(1) - 8*x^4)))/(log(2*x - exp(1))^3*(x^3*exp(1) - 2*x^4) + log(3)^3*log(x)^3*(2*x -
exp(1)) + log(2*x - exp(1))*log(3)^2*log(x)^2*(3*x*exp(1) - 6*x^2) - log(2*x - exp(1))^2*log(3)*log(x)*(3*x^2*
exp(1) - 6*x^3)),x)

[Out]

int((log(x)*(log(2*x - exp(1))*(2*x^4*exp(1) - 4*x^4 - 4*x^5 + log(3)*(2*x^2*exp(1) - 4*x^3)) - log(2*x - exp(
1))^2*(2*x^3*exp(1) - 4*x^4)) - log(2*x - exp(1))^2*(2*x^3*exp(1) - 4*x^4) + log(x)^2*(4*x^3*log(3) + 4*x^5 +
log(2*x - exp(1))*(2*x^4*exp(1) - 4*x^5 + log(3)*(6*x^2*exp(1) - 12*x^3))) - log(x)^3*(log(3)^2*(4*x*exp(1) -
8*x^2) + log(3)*(4*x^3*exp(1) - 8*x^4)))/(log(2*x - exp(1))^3*(x^3*exp(1) - 2*x^4) + log(3)^3*log(x)^3*(2*x -
exp(1)) + log(2*x - exp(1))*log(3)^2*log(x)^2*(3*x*exp(1) - 6*x^2) - log(2*x - exp(1))^2*log(3)*log(x)*(3*x^2*
exp(1) - 6*x^3)), x)