3.632 \(\int \frac{\log ^n(\log (x))}{x} \, dx\)

Optimal. Leaf size=24 \[ (-\log (\log (x)))^{-n} \log ^n(\log (x)) \text{Gamma}(n+1,-\log (\log (x))) \]

[Out]

(Gamma[1 + n, -Log[Log[x]]]*Log[Log[x]]^n)/(-Log[Log[x]])^n

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Rubi [A]  time = 0.0497788, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ (-\log (\log (x)))^{-n} \log ^n(\log (x)) \text{Gamma}(n+1,-\log (\log (x))) \]

Antiderivative was successfully verified.

[In]  Int[Log[Log[x]]^n/x,x]

[Out]

(Gamma[1 + n, -Log[Log[x]]]*Log[Log[x]]^n)/(-Log[Log[x]])^n

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(log(log(x))**n/x, x)

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Mathematica [A]  time = 0.0177344, size = 24, normalized size = 1. \[ (-\log (\log (x)))^{-n} \log ^n(\log (x)) \text{Gamma}(n+1,-\log (\log (x))) \]

Antiderivative was successfully verified.

[In]  Integrate[Log[Log[x]]^n/x,x]

[Out]

(Gamma[1 + n, -Log[Log[x]]]*Log[Log[x]]^n)/(-Log[Log[x]])^n

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Maple [F]  time = 0.038, size = 0, normalized size = 0. \[ \int{\frac{ \left ( \ln \left ( \ln \left ( x \right ) \right ) \right ) ^{n}}{x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(ln(ln(x))^n/x,x)

[Out]

int(ln(ln(x))^n/x,x)

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Maxima [A]  time = 1.45075, size = 39, normalized size = 1.62 \[ -\left (-\log \left (\log \left (x\right )\right )\right )^{-n - 1} \log \left (\log \left (x\right )\right )^{n + 1} \Gamma \left (n + 1, -\log \left (\log \left (x\right )\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-log(log(x)))^(-n - 1)*log(log(x))^(n + 1)*gamma(n + 1, -log(log(x)))

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Fricas [A]  time = 0.276443, size = 19, normalized size = 0.79 \[ \cos \left (\pi n\right ) \Gamma \left (n + 1, -\log \left (\log \left (x\right )\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

cos(pi*n)*gamma(n + 1, -log(log(x)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(ln(ln(x))**n/x,x)

[Out]

Integral(log(log(x))**n/x, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\log \left (\log \left (x\right )\right )^{n}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(log(log(x))^n/x, x)