Optimal. Leaf size=24 \[ (-\log (\log (x)))^{-n} \log ^n(\log (x)) \Gamma (n+1,-\log (\log (x))) \]
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Rubi [A] time = 0.03, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2299, 2181} \[ (-\log (\log (x)))^{-n} \log ^n(\log (x)) \text {Gamma}(n+1,-\log (\log (x))) \]
Antiderivative was successfully verified.
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Rule 2181
Rule 2299
Rubi steps
\begin {align*} \int \frac {\log ^n(\log (x))}{x} \, dx &=\operatorname {Subst}\left (\int \log ^n(x) \, dx,x,\log (x)\right )\\ &=\operatorname {Subst}\left (\int e^x x^n \, dx,x,\log (\log (x))\right )\\ &=\Gamma (1+n,-\log (\log (x))) (-\log (\log (x)))^{-n} \log ^n(\log (x))\\ \end {align*}
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Mathematica [A] time = 0.01, size = 24, normalized size = 1.00 \[ (-\log (\log (x)))^{-n} \log ^n(\log (x)) \Gamma (n+1,-\log (\log (x))) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log ^n(\log (x))}{x} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.05, size = 14, normalized size = 0.58 \[ \cos \left (\pi n\right ) \Gamma \left (n + 1, -\log \left (\log \relax (x)\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (\log \relax (x)\right )^{n}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (\ln \relax (x )\right )^{n}}{x}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 29, normalized size = 1.21 \[ -\left (-\log \left (\log \relax (x)\right )\right )^{-n - 1} \log \left (\log \relax (x)\right )^{n + 1} \Gamma \left (n + 1, -\log \left (\log \relax (x)\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.45, size = 24, normalized size = 1.00 \[ \frac {{\ln \left (\ln \relax (x)\right )}^n\,\Gamma \left (n+1,-\ln \left (\ln \relax (x)\right )\right )}{{\left (-\ln \left (\ln \relax (x)\right )\right )}^n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.85, size = 24, normalized size = 1.00 \[ \left (- \log {\left (\log {\relax (x )} \right )}\right )^{- n} \log {\left (\log {\relax (x )} \right )}^{n} \Gamma \left (n + 1, - \log {\left (\log {\relax (x )} \right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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