3.630 \(\int \frac{\log ^3(\log (x))}{x} \, dx\)

Optimal. Leaf size=29 \[ \log (x) \log ^3(\log (x))-3 \log (x) \log ^2(\log (x))+6 \log (x) \log (\log (x))-6 \log (x) \]

[Out]

-6*Log[x] + 6*Log[x]*Log[Log[x]] - 3*Log[x]*Log[Log[x]]^2 + Log[x]*Log[Log[x]]^3

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Rubi [A]  time = 0.0344065, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \log (x) \log ^3(\log (x))-3 \log (x) \log ^2(\log (x))+6 \log (x) \log (\log (x))-6 \log (x) \]

Antiderivative was successfully verified.

[In]  Int[Log[Log[x]]^3/x,x]

[Out]

-6*Log[x] + 6*Log[x]*Log[Log[x]] - 3*Log[x]*Log[Log[x]]^2 + Log[x]*Log[Log[x]]^3

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\log{\left (\log{\left (x \right )} \right )}^{3}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(ln(ln(x))**3/x,x)

[Out]

Integral(log(log(x))**3/x, x)

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Mathematica [A]  time = 0.00241203, size = 29, normalized size = 1. \[ \log (x) \log ^3(\log (x))-3 \log (x) \log ^2(\log (x))+6 \log (x) \log (\log (x))-6 \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[Log[Log[x]]^3/x,x]

[Out]

-6*Log[x] + 6*Log[x]*Log[Log[x]] - 3*Log[x]*Log[Log[x]]^2 + Log[x]*Log[Log[x]]^3

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Maple [A]  time = 0.003, size = 30, normalized size = 1. \[ -6\,\ln \left ( x \right ) +6\,\ln \left ( x \right ) \ln \left ( \ln \left ( x \right ) \right ) -3\,\ln \left ( x \right ) \left ( \ln \left ( \ln \left ( x \right ) \right ) \right ) ^{2}+\ln \left ( x \right ) \left ( \ln \left ( \ln \left ( x \right ) \right ) \right ) ^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(ln(ln(x))^3/x,x)

[Out]

-6*ln(x)+6*ln(x)*ln(ln(x))-3*ln(x)*ln(ln(x))^2+ln(x)*ln(ln(x))^3

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Maxima [A]  time = 1.33847, size = 30, normalized size = 1.03 \[{\left (\log \left (\log \left (x\right )\right )^{3} - 3 \, \log \left (\log \left (x\right )\right )^{2} + 6 \, \log \left (\log \left (x\right )\right ) - 6\right )} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(log(log(x))^3/x,x, algorithm="maxima")

[Out]

(log(log(x))^3 - 3*log(log(x))^2 + 6*log(log(x)) - 6)*log(x)

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Fricas [A]  time = 0.213367, size = 39, normalized size = 1.34 \[ \log \left (x\right ) \log \left (\log \left (x\right )\right )^{3} - 3 \, \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} + 6 \, \log \left (x\right ) \log \left (\log \left (x\right )\right ) - 6 \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(log(log(x))^3/x,x, algorithm="fricas")

[Out]

log(x)*log(log(x))^3 - 3*log(x)*log(log(x))^2 + 6*log(x)*log(log(x)) - 6*log(x)

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Sympy [A]  time = 0.688063, size = 36, normalized size = 1.24 \[ \log{\left (x \right )} \log{\left (\log{\left (x \right )} \right )}^{3} - 3 \log{\left (x \right )} \log{\left (\log{\left (x \right )} \right )}^{2} + 6 \log{\left (x \right )} \log{\left (\log{\left (x \right )} \right )} - 6 \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(ln(ln(x))**3/x,x)

[Out]

log(x)*log(log(x))**3 - 3*log(x)*log(log(x))**2 + 6*log(x)*log(log(x)) - 6*log(x
)

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GIAC/XCAS [A]  time = 0.223345, size = 39, normalized size = 1.34 \[{\rm ln}\left (x\right ){\rm ln}\left ({\rm ln}\left (x\right )\right )^{3} - 3 \,{\rm ln}\left (x\right ){\rm ln}\left ({\rm ln}\left (x\right )\right )^{2} + 6 \,{\rm ln}\left (x\right ){\rm ln}\left ({\rm ln}\left (x\right )\right ) - 6 \,{\rm ln}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(log(log(x))^3/x,x, algorithm="giac")

[Out]

ln(x)*ln(ln(x))^3 - 3*ln(x)*ln(ln(x))^2 + 6*ln(x)*ln(ln(x)) - 6*ln(x)