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

Optimal. Leaf size=30 \[ \left (1-\frac {x}{-\frac {\log (3)}{x}+\frac {\log (-e+2 x)}{\log (x)}}\right )^2 \]

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Rubi [F]  time = 33.42, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \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\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [B]  time = 1.08, size = 475, normalized size = 15.83 \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[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]

(x^2*Log[x]*(x^2*Log[x] - ((2*(2*x^2 + E*Log[3] - x*Log[9])^3 + (-(E^3*Log[3]*(2*Log[3]^2 + Log[9]^2)) + 4*x^3
*(x - Log[3])*(-4*Log[3]^2 - Log[9]*Log[81] + x*Log[531441]) - 2*E*x^2*(-(x*(16*Log[3]^2 + Log[81]^2 + Log[9]*
Log[6561])) + x^2*(-36*Log[3]^2 + Log[9]*Log[729] + Log[81]*Log[729] + Log[531441]) + Log[3]^2*Log[15009463529
6999121]) + E^2*x*(-(x*(8*Log[3]^2 + Log[9]*Log[6561])) + x^2*(-36*Log[3]^2 + Log[27]*Log[81] + Log[9]*Log[531
441]) + Log[3]^2*Log[150094635296999121]))*Log[x] + (4*x^3*(-4*Log[3]^2*Log[27] + x*(-4*Log[3]^2 + Log[81]^2))
+ E^3*Log[3]^2*Log[729] + E^2*x*(x*Log[27]*Log[81] + x^2*(-Log[9]^2 + Log[3]*Log[81]) - 4*Log[3]^2*Log[19683]
) + 2*E*x^2*(x*(8*Log[3]^2 - 8*Log[3]*Log[9] - Log[81]^2) + 4*Log[3]^2*Log[19683]))*Log[x]^2 + (8*x^3*Log[3]^2
*Log[9] + E^3*(-2*Log[3]^3 + x^2*(-4*Log[3]^2 + Log[9]^2)) + E^2*x*Log[3]*(-12*Log[3]^2 + 8*Log[3]*Log[9] + Lo
g[9]*Log[81]) + E*(-8*x^2*Log[3]^2*Log[27] + x^4*(-48*Log[3]^2 + 10*Log[3]*Log[81] + Log[9]*Log[81])))*Log[x]^
3)*(-(Log[3]*Log[x]) + x*Log[-E + 2*x]))/(2*x^2 + E*Log[3] - x*Log[9] + (-(E*Log[3]) + x*Log[9])*Log[x])^3))/(
Log[3]*Log[x] - x*Log[-E + 2*x])^2

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fricas [B]  time = 0.60, size = 79, normalized size = 2.63 \begin {gather*} -\frac {2 \, x^{3} \log \left (2 \, x - e\right ) \log \relax (x) - {\left (x^{4} + 2 \, x^{2} \log \relax (3)\right )} \log \relax (x)^{2}}{x^{2} \log \left (2 \, x - e\right )^{2} - 2 \, x \log \relax (3) \log \left (2 \, x - e\right ) \log \relax (x) + \log \relax (3)^{2} \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [B]  time = 3.11, size = 80, normalized size = 2.67 \begin {gather*} \frac {x^{4} \log \relax (x)^{2} - 2 \, x^{3} \log \left (2 \, x - e\right ) \log \relax (x) + 2 \, x^{2} \log \relax (3) \log \relax (x)^{2}}{x^{2} \log \left (2 \, x - e\right )^{2} - 2 \, x \log \relax (3) \log \left (2 \, x - e\right ) \log \relax (x) + \log \relax (3)^{2} \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.15, size = 52, normalized size = 1.73

method	result	size
risch	\(\frac {\left (x^{2} \ln \relax (x )+2 \ln \relax (3) \ln \relax (x )-2 \ln \left (-{\mathrm e}+2 x \right ) x \right ) x^{2} \ln \relax (x )}{\left (\ln \relax (3) \ln \relax (x )-\ln \left (-{\mathrm e}+2 x \right ) x \right )^{2}}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[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

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maxima [B]  time = 1.00, size = 79, normalized size = 2.63 \begin {gather*} -\frac {2 \, x^{3} \log \left (2 \, x - e\right ) \log \relax (x) - {\left (x^{4} + 2 \, x^{2} \log \relax (3)\right )} \log \relax (x)^{2}}{x^{2} \log \left (2 \, x - e\right )^{2} - 2 \, x \log \relax (3) \log \left (2 \, x - e\right ) \log \relax (x) + \log \relax (3)^{2} \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\ln \relax (x)\,\left (\ln \left (2\,x-\mathrm {e}\right )\,\left (2\,x^4\,\mathrm {e}-4\,x^4-4\,x^5+\ln \relax (3)\,\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 \relax (x)}^2\,\left (4\,x^3\,\ln \relax (3)+4\,x^5+\ln \left (2\,x-\mathrm {e}\right )\,\left (2\,x^4\,\mathrm {e}-4\,x^5+\ln \relax (3)\,\left (6\,x^2\,\mathrm {e}-12\,x^3\right )\right )\right )-{\ln \relax (x)}^3\,\left ({\ln \relax (3)}^2\,\left (4\,x\,\mathrm {e}-8\,x^2\right )+\ln \relax (3)\,\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 \relax (3)\,\left (3\,x^2\,\mathrm {e}-6\,x^3\right )\,{\ln \left (2\,x-\mathrm {e}\right )}^2\,\ln \relax (x)+{\ln \relax (3)}^2\,\left (3\,x\,\mathrm {e}-6\,x^2\right )\,\ln \left (2\,x-\mathrm {e}\right )\,{\ln \relax (x)}^2+{\ln \relax (3)}^3\,\left (2\,x-\mathrm {e}\right )\,{\ln \relax (x)}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [B]  time = 0.50, size = 80, normalized size = 2.67 \begin {gather*} \frac {x^{4} \log {\relax (x )}^{2} - 2 x^{3} \log {\relax (x )} \log {\left (2 x - e \right )} + 2 x^{2} \log {\relax (3 )} \log {\relax (x )}^{2}}{x^{2} \log {\left (2 x - e \right )}^{2} - 2 x \log {\relax (3 )} \log {\relax (x )} \log {\left (2 x - e \right )} + \log {\relax (3 )}^{2} \log {\relax (x )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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