Optimal. Leaf size=24 \[ (-\log (\log (x)))^{-n} \log ^n(\log (x)) \text{Gamma}(n+1,-\log (\log (x))) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.032174, 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, Rules used = {2299, 2181} \[ (-\log (\log (x)))^{-n} \log ^n(\log (x)) \text{Gamma}(n+1,-\log (\log (x))) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2299
Rule 2181
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*}
Mathematica [A] time = 0.0171608, 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]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.023, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \ln \left ( \ln \left ( x \right ) \right ) \right ) ^{n}}{x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.05007, size = 39, normalized size = 1.62 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.77683, size = 51, normalized size = 2.12 \begin{align*} \cos \left (\pi n\right ) \Gamma \left (n + 1, -\log \left (\log \left (x\right )\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 2.76667, size = 24, normalized size = 1. \begin{align*} \left (- \log{\left (\log{\left (x \right )} \right )}\right )^{- n} \log{\left (\log{\left (x \right )} \right )}^{n} \Gamma \left (n + 1, - \log{\left (\log{\left (x \right )} \right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (\log \left (x\right )\right )^{n}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]