3.57.3 \(\int \frac {-4 x^2+6 x^3+(-2 x^2+9 x^3) \log (x)+(2 x^2-6 x^3) \log ^2(x)+((-14 x^2+21 x^3) \log (x)+(6 x^2-9 x^3) \log ^2(x)) \log ((-4 x+6 x^2) \log (x))}{(-8+12 x) \log (x)+(8-12 x) \log ^2(x)+(-2+3 x) \log ^3(x)} \, dx\)

Optimal. Leaf size=23 \[ \frac {x^3 \log (2 x (-2+3 x) \log (x))}{2-\log (x)} \]

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Rubi [F]  time = 3.62, 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 {-4 x^2+6 x^3+\left (-2 x^2+9 x^3\right ) \log (x)+\left (2 x^2-6 x^3\right ) \log ^2(x)+\left (\left (-14 x^2+21 x^3\right ) \log (x)+\left (6 x^2-9 x^3\right ) \log ^2(x)\right ) \log \left (\left (-4 x+6 x^2\right ) \log (x)\right )}{(-8+12 x) \log (x)+(8-12 x) \log ^2(x)+(-2+3 x) \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-4*x^2 + 6*x^3 + (-2*x^2 + 9*x^3)*Log[x] + (2*x^2 - 6*x^3)*Log[x]^2 + ((-14*x^2 + 21*x^3)*Log[x] + (6*x^2
- 9*x^3)*Log[x]^2)*Log[(-4*x + 6*x^2)*Log[x]])/((-8 + 12*x)*Log[x] + (8 - 12*x)*Log[x]^2 + (-2 + 3*x)*Log[x]^
3),x]

[Out]

3*Defer[Int][x^2/((-2 + 3*x)*(-2 + Log[x])), x] - (15*Defer[Int][x^3/((-2 + 3*x)*(-2 + Log[x])), x])/2 - Defer
[Int][x^2/((-2 + 3*x)*Log[x]), x] + (3*Defer[Int][x^3/((-2 + 3*x)*Log[x]), x])/2 + 7*Defer[Int][(x^2*Log[2*x*(
-2 + 3*x)*Log[x]])/(-2 + Log[x])^2, x] - 3*Defer[Int][(x^2*Log[x]*Log[2*x*(-2 + 3*x)*Log[x]])/(-2 + Log[x])^2,
x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^2 \left (4-6 x-\log ^2(x) (2-6 x+(6-9 x) \log (2 x (-2+3 x) \log (x)))-\log (x) (-2+9 x+7 (-2+3 x) \log (2 x (-2+3 x) \log (x)))\right )}{(2-3 x) (2-\log (x))^2 \log (x)} \, dx\\ &=\int \left (-\frac {2 x^2}{(-2+3 x) (-2+\log (x))^2}+\frac {9 x^3}{(-2+3 x) (-2+\log (x))^2}-\frac {4 x^2}{(-2+3 x) (-2+\log (x))^2 \log (x)}+\frac {6 x^3}{(-2+3 x) (-2+\log (x))^2 \log (x)}+\frac {2 x^2 \log (x)}{(-2+3 x) (-2+\log (x))^2}-\frac {6 x^3 \log (x)}{(-2+3 x) (-2+\log (x))^2}-\frac {x^2 (-7+3 \log (x)) \log (2 x (-2+3 x) \log (x))}{(-2+\log (x))^2}\right ) \, dx\\ &=-\left (2 \int \frac {x^2}{(-2+3 x) (-2+\log (x))^2} \, dx\right )+2 \int \frac {x^2 \log (x)}{(-2+3 x) (-2+\log (x))^2} \, dx-4 \int \frac {x^2}{(-2+3 x) (-2+\log (x))^2 \log (x)} \, dx+6 \int \frac {x^3}{(-2+3 x) (-2+\log (x))^2 \log (x)} \, dx-6 \int \frac {x^3 \log (x)}{(-2+3 x) (-2+\log (x))^2} \, dx+9 \int \frac {x^3}{(-2+3 x) (-2+\log (x))^2} \, dx-\int \frac {x^2 (-7+3 \log (x)) \log (2 x (-2+3 x) \log (x))}{(-2+\log (x))^2} \, dx\\ &=2 \int \left (\frac {2 x^2}{(-2+3 x) (-2+\log (x))^2}+\frac {x^2}{(-2+3 x) (-2+\log (x))}\right ) \, dx-2 \int \frac {x^2}{(-2+3 x) (-2+\log (x))^2} \, dx-4 \int \left (\frac {x^2}{2 (-2+3 x) (-2+\log (x))^2}-\frac {x^2}{4 (-2+3 x) (-2+\log (x))}+\frac {x^2}{4 (-2+3 x) \log (x)}\right ) \, dx-6 \int \left (\frac {2 x^3}{(-2+3 x) (-2+\log (x))^2}+\frac {x^3}{(-2+3 x) (-2+\log (x))}\right ) \, dx+6 \int \left (\frac {x^3}{2 (-2+3 x) (-2+\log (x))^2}-\frac {x^3}{4 (-2+3 x) (-2+\log (x))}+\frac {x^3}{4 (-2+3 x) \log (x)}\right ) \, dx+9 \int \frac {x^3}{(-2+3 x) (-2+\log (x))^2} \, dx-\int \left (-\frac {7 x^2 \log (2 x (-2+3 x) \log (x))}{(-2+\log (x))^2}+\frac {3 x^2 \log (x) \log (2 x (-2+3 x) \log (x))}{(-2+\log (x))^2}\right ) \, dx\\ &=-\left (\frac {3}{2} \int \frac {x^3}{(-2+3 x) (-2+\log (x))} \, dx\right )+\frac {3}{2} \int \frac {x^3}{(-2+3 x) \log (x)} \, dx-2 \left (2 \int \frac {x^2}{(-2+3 x) (-2+\log (x))^2} \, dx\right )+2 \int \frac {x^2}{(-2+3 x) (-2+\log (x))} \, dx+3 \int \frac {x^3}{(-2+3 x) (-2+\log (x))^2} \, dx-3 \int \frac {x^2 \log (x) \log (2 x (-2+3 x) \log (x))}{(-2+\log (x))^2} \, dx+4 \int \frac {x^2}{(-2+3 x) (-2+\log (x))^2} \, dx-6 \int \frac {x^3}{(-2+3 x) (-2+\log (x))} \, dx+7 \int \frac {x^2 \log (2 x (-2+3 x) \log (x))}{(-2+\log (x))^2} \, dx+9 \int \frac {x^3}{(-2+3 x) (-2+\log (x))^2} \, dx-12 \int \frac {x^3}{(-2+3 x) (-2+\log (x))^2} \, dx+\int \frac {x^2}{(-2+3 x) (-2+\log (x))} \, dx-\int \frac {x^2}{(-2+3 x) \log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.30, size = 22, normalized size = 0.96 \begin {gather*} -\frac {x^3 \log (2 x (-2+3 x) \log (x))}{-2+\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4*x^2 + 6*x^3 + (-2*x^2 + 9*x^3)*Log[x] + (2*x^2 - 6*x^3)*Log[x]^2 + ((-14*x^2 + 21*x^3)*Log[x] +
(6*x^2 - 9*x^3)*Log[x]^2)*Log[(-4*x + 6*x^2)*Log[x]])/((-8 + 12*x)*Log[x] + (8 - 12*x)*Log[x]^2 + (-2 + 3*x)*L
og[x]^3),x]

[Out]

-((x^3*Log[2*x*(-2 + 3*x)*Log[x]])/(-2 + Log[x]))

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fricas [A]  time = 0.59, size = 25, normalized size = 1.09 \begin {gather*} -\frac {x^{3} \log \left (2 \, {\left (3 \, x^{2} - 2 \, x\right )} \log \relax (x)\right )}{\log \relax (x) - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-9*x^3+6*x^2)*log(x)^2+(21*x^3-14*x^2)*log(x))*log((6*x^2-4*x)*log(x))+(-6*x^3+2*x^2)*log(x)^2+(9
*x^3-2*x^2)*log(x)+6*x^3-4*x^2)/((3*x-2)*log(x)^3+(-12*x+8)*log(x)^2+(12*x-8)*log(x)),x, algorithm="fricas")

[Out]

-x^3*log(2*(3*x^2 - 2*x)*log(x))/(log(x) - 2)

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giac [A]  time = 0.18, size = 39, normalized size = 1.70 \begin {gather*} -x^{3} - \frac {x^{3} \log \left (6 \, x \log \relax (x) - 4 \, \log \relax (x)\right )}{\log \relax (x) - 2} - \frac {2 \, x^{3}}{\log \relax (x) - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-9*x^3+6*x^2)*log(x)^2+(21*x^3-14*x^2)*log(x))*log((6*x^2-4*x)*log(x))+(-6*x^3+2*x^2)*log(x)^2+(9
*x^3-2*x^2)*log(x)+6*x^3-4*x^2)/((3*x-2)*log(x)^3+(-12*x+8)*log(x)^2+(12*x-8)*log(x)),x, algorithm="giac")

[Out]

-x^3 - x^3*log(6*x*log(x) - 4*log(x))/(log(x) - 2) - 2*x^3/(log(x) - 2)

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maple [C]  time = 0.32, size = 220, normalized size = 9.57

method	result	size
risch	\(-\frac {x^{3} \ln \left (x -\frac {2}{3}\right )}{\ln \relax (x )-2}-\frac {x^{3} \left (i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \relax (x ) \left (x -\frac {2}{3}\right )\right )^{2}+i \pi \,\mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i \ln \relax (x ) \left (x -\frac {2}{3}\right )\right )^{2}-i \pi \mathrm {csgn}\left (i x \ln \relax (x ) \left (x -\frac {2}{3}\right )\right )^{3}-i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x ) \left (x -\frac {2}{3}\right )\right ) \mathrm {csgn}\left (i x \ln \relax (x ) \left (x -\frac {2}{3}\right )\right )+i \pi \,\mathrm {csgn}\left (i \left (x -\frac {2}{3}\right )\right ) \mathrm {csgn}\left (i \ln \relax (x ) \left (x -\frac {2}{3}\right )\right )^{2}-i \pi \,\mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i \left (x -\frac {2}{3}\right )\right ) \mathrm {csgn}\left (i \ln \relax (x ) \left (x -\frac {2}{3}\right )\right )+i \pi \,\mathrm {csgn}\left (i \ln \relax (x ) \left (x -\frac {2}{3}\right )\right ) \mathrm {csgn}\left (i x \ln \relax (x ) \left (x -\frac {2}{3}\right )\right )^{2}-i \pi \mathrm {csgn}\left (i \ln \relax (x ) \left (x -\frac {2}{3}\right )\right )^{3}+2 \ln \relax (3)+2 \ln \relax (2)+2 \ln \relax (x )+2 \ln \left (\ln \relax (x )\right )\right )}{2 \left (\ln \relax (x )-2\right )}\)	\(220\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-9*x^3+6*x^2)*ln(x)^2+(21*x^3-14*x^2)*ln(x))*ln((6*x^2-4*x)*ln(x))+(-6*x^3+2*x^2)*ln(x)^2+(9*x^3-2*x^2)
*ln(x)+6*x^3-4*x^2)/((3*x-2)*ln(x)^3+(-12*x+8)*ln(x)^2+(12*x-8)*ln(x)),x,method=_RETURNVERBOSE)

[Out]

-x^3/(ln(x)-2)*ln(x-2/3)-1/2*x^3*(I*Pi*csgn(I*x)*csgn(I*x*ln(x)*(x-2/3))^2+I*Pi*csgn(I*ln(x))*csgn(I*ln(x)*(x-
2/3))^2-I*Pi*csgn(I*x*ln(x)*(x-2/3))^3-I*Pi*csgn(I*x)*csgn(I*ln(x)*(x-2/3))*csgn(I*x*ln(x)*(x-2/3))+I*Pi*csgn(
I*(x-2/3))*csgn(I*ln(x)*(x-2/3))^2-I*Pi*csgn(I*ln(x))*csgn(I*(x-2/3))*csgn(I*ln(x)*(x-2/3))+I*Pi*csgn(I*ln(x)*
(x-2/3))*csgn(I*x*ln(x)*(x-2/3))^2-I*Pi*csgn(I*ln(x)*(x-2/3))^3+2*ln(3)+2*ln(2)+2*ln(x)+2*ln(ln(x)))/(ln(x)-2)

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maxima [A]  time = 0.53, size = 38, normalized size = 1.65 \begin {gather*} -\frac {x^{3} \log \relax (2) + x^{3} \log \left (3 \, x - 2\right ) + x^{3} \log \relax (x) + x^{3} \log \left (\log \relax (x)\right )}{\log \relax (x) - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-9*x^3+6*x^2)*log(x)^2+(21*x^3-14*x^2)*log(x))*log((6*x^2-4*x)*log(x))+(-6*x^3+2*x^2)*log(x)^2+(9
*x^3-2*x^2)*log(x)+6*x^3-4*x^2)/((3*x-2)*log(x)^3+(-12*x+8)*log(x)^2+(12*x-8)*log(x)),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 4.06, size = 25, normalized size = 1.09 \begin {gather*} -\frac {x^3\,\ln \left (-2\,\ln \relax (x)\,\left (2\,x-3\,x^2\right )\right )}{\ln \relax (x)-2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x)*(2*x^2 - 9*x^3) + log(-log(x)*(4*x - 6*x^2))*(log(x)*(14*x^2 - 21*x^3) - log(x)^2*(6*x^2 - 9*x^3)
) - log(x)^2*(2*x^2 - 6*x^3) + 4*x^2 - 6*x^3)/(log(x)*(12*x - 8) + log(x)^3*(3*x - 2) - log(x)^2*(12*x - 8)),x
)

[Out]

-(x^3*log(-2*log(x)*(2*x - 3*x^2)))/(log(x) - 2)

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sympy [A]  time = 0.51, size = 22, normalized size = 0.96 \begin {gather*} - \frac {x^{3} \log {\left (\left (6 x^{2} - 4 x\right ) \log {\relax (x )} \right )}}{\log {\relax (x )} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-9*x**3+6*x**2)*ln(x)**2+(21*x**3-14*x**2)*ln(x))*ln((6*x**2-4*x)*ln(x))+(-6*x**3+2*x**2)*ln(x)**
2+(9*x**3-2*x**2)*ln(x)+6*x**3-4*x**2)/((3*x-2)*ln(x)**3+(-12*x+8)*ln(x)**2+(12*x-8)*ln(x)),x)

[Out]

-x**3*log((6*x**2 - 4*x)*log(x))/(log(x) - 2)

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