3.82.26 \(\int \frac {-32 x^4 \log (\frac {1}{3} x \log (3))-32 x^4 \log ^2(\frac {1}{3} x \log (3))+(-1792 x^3+256 e^2 x^3) \log ^3(\frac {1}{3} x \log (3))}{-x^3+(-84 x^2+12 e^2 x^2) \log (\frac {1}{3} x \log (3))+(-2352 x+672 e^2 x-48 e^4 x) \log ^2(\frac {1}{3} x \log (3))+(-21952+9408 e^2-1344 e^4+64 e^6) \log ^3(\frac {1}{3} x \log (3))} \, dx\)

Optimal. Leaf size=28 \[ \frac {x^4}{\left (7-e^2+\frac {x}{4 \log \left (\frac {1}{3} x \log (3)\right )}\right )^2} \]

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Rubi [F]  time = 90.15, 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 {-32 x^4 \log \left (\frac {1}{3} x \log (3)\right )-32 x^4 \log ^2\left (\frac {1}{3} x \log (3)\right )+\left (-1792 x^3+256 e^2 x^3\right ) \log ^3\left (\frac {1}{3} x \log (3)\right )}{-x^3+\left (-84 x^2+12 e^2 x^2\right ) \log \left (\frac {1}{3} x \log (3)\right )+\left (-2352 x+672 e^2 x-48 e^4 x\right ) \log ^2\left (\frac {1}{3} x \log (3)\right )+\left (-21952+9408 e^2-1344 e^4+64 e^6\right ) \log ^3\left (\frac {1}{3} x \log (3)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-32*x^4*Log[(x*Log[3])/3] - 32*x^4*Log[(x*Log[3])/3]^2 + (-1792*x^3 + 256*E^2*x^3)*Log[(x*Log[3])/3]^3)/(
-x^3 + (-84*x^2 + 12*E^2*x^2)*Log[(x*Log[3])/3] + (-2352*x + 672*E^2*x - 48*E^4*x)*Log[(x*Log[3])/3]^2 + (-219
52 + 9408*E^2 - 1344*E^4 + 64*E^6)*Log[(x*Log[3])/3]^3),x]

[Out]

x^4/(7 - E^2)^2 + (2*Defer[Int][x^6/(-x - 28*(1 - E^2/7)*Log[(x*Log[3])/3])^3, x])/(7 - E^2)^2 + (10*Defer[Int
][x^4/(-x - 28*(1 - E^2/7)*Log[(x*Log[3])/3]), x])/(7 - E^2)^2 - (8*Defer[Int][x^5/(x + 28*(1 - E^2/7)*Log[(x*
Log[3])/3])^3, x])/(7 - E^2) + (8*Defer[Int][x^4/(x + 28*(1 - E^2/7)*Log[(x*Log[3])/3])^2, x])/(7 - E^2) + (8*
Defer[Int][x^5/(x + 28*(1 - E^2/7)*Log[(x*Log[3])/3])^2, x])/(7 - E^2)^2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {32 x^3 \log \left (\frac {1}{3} x \log (3)\right ) \left (x+x \log \left (\frac {1}{3} x \log (3)\right )-8 \left (-7+e^2\right ) \log ^2\left (\frac {1}{3} x \log (3)\right )\right )}{\left (x-4 \left (-7+e^2\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3} \, dx\\ &=32 \int \frac {x^3 \log \left (\frac {1}{3} x \log (3)\right ) \left (x+x \log \left (\frac {1}{3} x \log (3)\right )-8 \left (-7+e^2\right ) \log ^2\left (\frac {1}{3} x \log (3)\right )\right )}{\left (x-4 \left (-7+e^2\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3} \, dx\\ &=32 \int \left (\frac {x^3}{8 \left (-7+e^2\right )^2}+\frac {5 x^4}{16 \left (7-e^2\right )^2 \left (-x-28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )}+\frac {\left (-28+4 e^2-x\right ) x^5}{16 \left (7-e^2\right )^2 \left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3}+\frac {x^4 \left (7-e^2+x\right )}{4 \left (7-e^2\right )^2 \left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^2}\right ) \, dx\\ &=\frac {x^4}{\left (7-e^2\right )^2}+\frac {2 \int \frac {\left (-28+4 e^2-x\right ) x^5}{\left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3} \, dx}{\left (7-e^2\right )^2}+\frac {8 \int \frac {x^4 \left (7-e^2+x\right )}{\left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^2} \, dx}{\left (7-e^2\right )^2}+\frac {10 \int \frac {x^4}{-x-28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )} \, dx}{\left (7-e^2\right )^2}\\ &=\frac {x^4}{\left (7-e^2\right )^2}+\frac {2 \int \left (\frac {x^6}{\left (-x-28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3}+\frac {4 \left (-7+e^2\right ) x^5}{\left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3}\right ) \, dx}{\left (7-e^2\right )^2}+\frac {8 \int \left (\frac {\left (7-e^2\right ) x^4}{\left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^2}+\frac {x^5}{\left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^2}\right ) \, dx}{\left (7-e^2\right )^2}+\frac {10 \int \frac {x^4}{-x-28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )} \, dx}{\left (7-e^2\right )^2}\\ &=\frac {x^4}{\left (7-e^2\right )^2}+\frac {2 \int \frac {x^6}{\left (-x-28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3} \, dx}{\left (7-e^2\right )^2}+\frac {8 \int \frac {x^5}{\left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^2} \, dx}{\left (7-e^2\right )^2}+\frac {10 \int \frac {x^4}{-x-28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )} \, dx}{\left (7-e^2\right )^2}-\frac {8 \int \frac {x^5}{\left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^3} \, dx}{7-e^2}+\frac {8 \int \frac {x^4}{\left (x+28 \left (1-\frac {e^2}{7}\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^2} \, dx}{7-e^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 5.39, size = 34, normalized size = 1.21 \begin {gather*} \frac {16 x^4 \log ^2\left (\frac {1}{3} x \log (3)\right )}{\left (x-4 \left (-7+e^2\right ) \log \left (\frac {1}{3} x \log (3)\right )\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-32*x^4*Log[(x*Log[3])/3] - 32*x^4*Log[(x*Log[3])/3]^2 + (-1792*x^3 + 256*E^2*x^3)*Log[(x*Log[3])/3
]^3)/(-x^3 + (-84*x^2 + 12*E^2*x^2)*Log[(x*Log[3])/3] + (-2352*x + 672*E^2*x - 48*E^4*x)*Log[(x*Log[3])/3]^2 +
(-21952 + 9408*E^2 - 1344*E^4 + 64*E^6)*Log[(x*Log[3])/3]^3),x]

[Out]

(16*x^4*Log[(x*Log[3])/3]^2)/(x - 4*(-7 + E^2)*Log[(x*Log[3])/3])^2

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fricas [B]  time = 0.57, size = 53, normalized size = 1.89 \begin {gather*} \frac {16 \, x^{4} \log \left (\frac {1}{3} \, x \log \relax (3)\right )^{2}}{16 \, {\left (e^{4} - 14 \, e^{2} + 49\right )} \log \left (\frac {1}{3} \, x \log \relax (3)\right )^{2} + x^{2} - 8 \, {\left (x e^{2} - 7 \, x\right )} \log \left (\frac {1}{3} \, x \log \relax (3)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((256*x^3*exp(2)-1792*x^3)*log(1/3*x*log(3))^3-32*x^4*log(1/3*x*log(3))^2-32*x^4*log(1/3*x*log(3)))/
((64*exp(2)^3-1344*exp(2)^2+9408*exp(2)-21952)*log(1/3*x*log(3))^3+(-48*x*exp(2)^2+672*exp(2)*x-2352*x)*log(1/
3*x*log(3))^2+(12*x^2*exp(2)-84*x^2)*log(1/3*x*log(3))-x^3),x, algorithm="fricas")

[Out]

16*x^4*log(1/3*x*log(3))^2/(16*(e^4 - 14*e^2 + 49)*log(1/3*x*log(3))^2 + x^2 - 8*(x*e^2 - 7*x)*log(1/3*x*log(3
)))

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giac [B]  time = 0.50, size = 154, normalized size = 5.50 \begin {gather*} \frac {16 \, {\left (x^{4} \log \relax (3)^{2} - 2 \, x^{4} \log \relax (3) \log \left (x \log \relax (3)\right ) + x^{4} \log \left (x \log \relax (3)\right )^{2}\right )}}{8 \, x e^{2} \log \relax (3) + 16 \, e^{4} \log \relax (3)^{2} - 224 \, e^{2} \log \relax (3)^{2} - 8 \, x e^{2} \log \left (x \log \relax (3)\right ) - 32 \, e^{4} \log \relax (3) \log \left (x \log \relax (3)\right ) + 448 \, e^{2} \log \relax (3) \log \left (x \log \relax (3)\right ) + 16 \, e^{4} \log \left (x \log \relax (3)\right )^{2} - 224 \, e^{2} \log \left (x \log \relax (3)\right )^{2} + x^{2} - 56 \, x \log \relax (3) + 784 \, \log \relax (3)^{2} + 56 \, x \log \left (x \log \relax (3)\right ) - 1568 \, \log \relax (3) \log \left (x \log \relax (3)\right ) + 784 \, \log \left (x \log \relax (3)\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((256*x^3*exp(2)-1792*x^3)*log(1/3*x*log(3))^3-32*x^4*log(1/3*x*log(3))^2-32*x^4*log(1/3*x*log(3)))/
((64*exp(2)^3-1344*exp(2)^2+9408*exp(2)-21952)*log(1/3*x*log(3))^3+(-48*x*exp(2)^2+672*exp(2)*x-2352*x)*log(1/
3*x*log(3))^2+(12*x^2*exp(2)-84*x^2)*log(1/3*x*log(3))-x^3),x, algorithm="giac")

[Out]

16*(x^4*log(3)^2 - 2*x^4*log(3)*log(x*log(3)) + x^4*log(x*log(3))^2)/(8*x*e^2*log(3) + 16*e^4*log(3)^2 - 224*e
^2*log(3)^2 - 8*x*e^2*log(x*log(3)) - 32*e^4*log(3)*log(x*log(3)) + 448*e^2*log(3)*log(x*log(3)) + 16*e^4*log(
x*log(3))^2 - 224*e^2*log(x*log(3))^2 + x^2 - 56*x*log(3) + 784*log(3)^2 + 56*x*log(x*log(3)) - 1568*log(3)*lo
g(x*log(3)) + 784*log(x*log(3))^2)

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maple [A]  time = 0.26, size = 38, normalized size = 1.36

method	result	size
norman	\(\frac {16 x^{4} \ln \left (\frac {x \ln \relax (3)}{3}\right )^{2}}{\left (4 \ln \left (\frac {x \ln \relax (3)}{3}\right ) {\mathrm e}^{2}-28 \ln \left (\frac {x \ln \relax (3)}{3}\right )-x \right )^{2}}\)	\(38\)
risch	\(\frac {x^{4}}{{\mathrm e}^{4}-14 \,{\mathrm e}^{2}+49}+\frac {x^{5} \left (8 \ln \left (\frac {x \ln \relax (3)}{3}\right ) {\mathrm e}^{2}-x -56 \ln \left (\frac {x \ln \relax (3)}{3}\right )\right )}{\left ({\mathrm e}^{4}-14 \,{\mathrm e}^{2}+49\right ) \left (4 \ln \left (\frac {x \ln \relax (3)}{3}\right ) {\mathrm e}^{2}-28 \ln \left (\frac {x \ln \relax (3)}{3}\right )-x \right )^{2}}\)	\(76\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((256*x^3*exp(2)-1792*x^3)*ln(1/3*x*ln(3))^3-32*x^4*ln(1/3*x*ln(3))^2-32*x^4*ln(1/3*x*ln(3)))/((64*exp(2)^
3-1344*exp(2)^2+9408*exp(2)-21952)*ln(1/3*x*ln(3))^3+(-48*x*exp(2)^2+672*exp(2)*x-2352*x)*ln(1/3*x*ln(3))^2+(1
2*x^2*exp(2)-84*x^2)*ln(1/3*x*ln(3))-x^3),x,method=_RETURNVERBOSE)

[Out]

16*x^4*ln(1/3*x*ln(3))^2/(4*ln(1/3*x*ln(3))*exp(2)-28*ln(1/3*x*ln(3))-x)^2

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maxima [B]  time = 0.60, size = 199, normalized size = 7.11 \begin {gather*} -\frac {16 \, {\left (2 \, x^{4} {\left (\log \relax (3) - \log \left (\log \relax (3)\right )\right )} \log \relax (x) - x^{4} \log \relax (x)^{2} - {\left (\log \relax (3)^{2} - 2 \, \log \relax (3) \log \left (\log \relax (3)\right ) + \log \left (\log \relax (3)\right )^{2}\right )} x^{4}\right )}}{16 \, {\left (e^{4} - 14 \, e^{2} + 49\right )} \log \relax (x)^{2} + 8 \, {\left ({\left (\log \relax (3) - \log \left (\log \relax (3)\right )\right )} e^{2} - 7 \, \log \relax (3) + 7 \, \log \left (\log \relax (3)\right )\right )} x + x^{2} + 16 \, {\left (\log \relax (3)^{2} - 2 \, \log \relax (3) \log \left (\log \relax (3)\right ) + \log \left (\log \relax (3)\right )^{2}\right )} e^{4} - 224 \, {\left (\log \relax (3)^{2} - 2 \, \log \relax (3) \log \left (\log \relax (3)\right ) + \log \left (\log \relax (3)\right )^{2}\right )} e^{2} + 784 \, \log \relax (3)^{2} - 8 \, {\left (x {\left (e^{2} - 7\right )} + 4 \, {\left (\log \relax (3) - \log \left (\log \relax (3)\right )\right )} e^{4} - 56 \, {\left (\log \relax (3) - \log \left (\log \relax (3)\right )\right )} e^{2} + 196 \, \log \relax (3) - 196 \, \log \left (\log \relax (3)\right )\right )} \log \relax (x) - 1568 \, \log \relax (3) \log \left (\log \relax (3)\right ) + 784 \, \log \left (\log \relax (3)\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((256*x^3*exp(2)-1792*x^3)*log(1/3*x*log(3))^3-32*x^4*log(1/3*x*log(3))^2-32*x^4*log(1/3*x*log(3)))/
((64*exp(2)^3-1344*exp(2)^2+9408*exp(2)-21952)*log(1/3*x*log(3))^3+(-48*x*exp(2)^2+672*exp(2)*x-2352*x)*log(1/
3*x*log(3))^2+(12*x^2*exp(2)-84*x^2)*log(1/3*x*log(3))-x^3),x, algorithm="maxima")

[Out]

-16*(2*x^4*(log(3) - log(log(3)))*log(x) - x^4*log(x)^2 - (log(3)^2 - 2*log(3)*log(log(3)) + log(log(3))^2)*x^
4)/(16*(e^4 - 14*e^2 + 49)*log(x)^2 + 8*((log(3) - log(log(3)))*e^2 - 7*log(3) + 7*log(log(3)))*x + x^2 + 16*(
log(3)^2 - 2*log(3)*log(log(3)) + log(log(3))^2)*e^4 - 224*(log(3)^2 - 2*log(3)*log(log(3)) + log(log(3))^2)*e
^2 + 784*log(3)^2 - 8*(x*(e^2 - 7) + 4*(log(3) - log(log(3)))*e^4 - 56*(log(3) - log(log(3)))*e^2 + 196*log(3)
- 196*log(log(3)))*log(x) - 1568*log(3)*log(log(3)) + 784*log(log(3))^2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} -\int \frac {32\,x^4\,{\ln \left (\frac {x\,\ln \relax (3)}{3}\right )}^2-{\ln \left (\frac {x\,\ln \relax (3)}{3}\right )}^3\,\left (256\,x^3\,{\mathrm {e}}^2-1792\,x^3\right )+32\,x^4\,\ln \left (\frac {x\,\ln \relax (3)}{3}\right )}{{\ln \left (\frac {x\,\ln \relax (3)}{3}\right )}^3\,\left (9408\,{\mathrm {e}}^2-1344\,{\mathrm {e}}^4+64\,{\mathrm {e}}^6-21952\right )-{\ln \left (\frac {x\,\ln \relax (3)}{3}\right )}^2\,\left (2352\,x-672\,x\,{\mathrm {e}}^2+48\,x\,{\mathrm {e}}^4\right )+\ln \left (\frac {x\,\ln \relax (3)}{3}\right )\,\left (12\,x^2\,{\mathrm {e}}^2-84\,x^2\right )-x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(32*x^4*log((x*log(3))/3)^2 - log((x*log(3))/3)^3*(256*x^3*exp(2) - 1792*x^3) + 32*x^4*log((x*log(3))/3))
/(log((x*log(3))/3)^3*(9408*exp(2) - 1344*exp(4) + 64*exp(6) - 21952) - log((x*log(3))/3)^2*(2352*x - 672*x*ex
p(2) + 48*x*exp(4)) + log((x*log(3))/3)*(12*x^2*exp(2) - 84*x^2) - x^3),x)

[Out]

-int((32*x^4*log((x*log(3))/3)^2 - log((x*log(3))/3)^3*(256*x^3*exp(2) - 1792*x^3) + 32*x^4*log((x*log(3))/3))
/(log((x*log(3))/3)^3*(9408*exp(2) - 1344*exp(4) + 64*exp(6) - 21952) - log((x*log(3))/3)^2*(2352*x - 672*x*ex
p(2) + 48*x*exp(4)) + log((x*log(3))/3)*(12*x^2*exp(2) - 84*x^2) - x^3), x)

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sympy [B]  time = 0.50, size = 122, normalized size = 4.36 \begin {gather*} \frac {x^{4}}{- 14 e^{2} + 49 + e^{4}} + \frac {- x^{6} + \left (- 56 x^{5} + 8 x^{5} e^{2}\right ) \log {\left (\frac {x \log {\relax (3 )}}{3} \right )}}{- 14 x^{2} e^{2} + 49 x^{2} + x^{2} e^{4} + \left (- 1176 x e^{2} - 8 x e^{6} + 2744 x + 168 x e^{4}\right ) \log {\left (\frac {x \log {\relax (3 )}}{3} \right )} + \left (- 448 e^{6} - 21952 e^{2} + 38416 + 16 e^{8} + 4704 e^{4}\right ) \log {\left (\frac {x \log {\relax (3 )}}{3} \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((256*x**3*exp(2)-1792*x**3)*ln(1/3*x*ln(3))**3-32*x**4*ln(1/3*x*ln(3))**2-32*x**4*ln(1/3*x*ln(3)))/
((64*exp(2)**3-1344*exp(2)**2+9408*exp(2)-21952)*ln(1/3*x*ln(3))**3+(-48*x*exp(2)**2+672*exp(2)*x-2352*x)*ln(1
/3*x*ln(3))**2+(12*x**2*exp(2)-84*x**2)*ln(1/3*x*ln(3))-x**3),x)

[Out]

x**4/(-14*exp(2) + 49 + exp(4)) + (-x**6 + (-56*x**5 + 8*x**5*exp(2))*log(x*log(3)/3))/(-14*x**2*exp(2) + 49*x
**2 + x**2*exp(4) + (-1176*x*exp(2) - 8*x*exp(6) + 2744*x + 168*x*exp(4))*log(x*log(3)/3) + (-448*exp(6) - 219
52*exp(2) + 38416 + 16*exp(8) + 4704*exp(4))*log(x*log(3)/3)**2)

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