Optimal. Leaf size=26 \[ (-\log (x))^{-m p} \log ^m(x)^p \text{Gamma}(m p+1,-\log (x)) \]
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Rubi [A] time = 0.0252434, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6720, 2299, 2181} \[ (-\log (x))^{-m p} \log ^m(x)^p \text{Gamma}(m p+1,-\log (x)) \]
Antiderivative was successfully verified.
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Rule 6720
Rule 2299
Rule 2181
Rubi steps
\begin{align*} \int \log ^m(x)^p \, dx &=\left (\log ^{-m p}(x) \log ^m(x)^p\right ) \int \log ^{m p}(x) \, dx\\ &=\left (\log ^{-m p}(x) \log ^m(x)^p\right ) \operatorname{Subst}\left (\int e^x x^{m p} \, dx,x,\log (x)\right )\\ &=\Gamma (1+m p,-\log (x)) (-\log (x))^{-m p} \log ^m(x)^p\\ \end{align*}
Mathematica [A] time = 0.0137504, size = 26, normalized size = 1. \[ (-\log (x))^{-m p} \log ^m(x)^p \text{Gamma}(m p+1,-\log (x)) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.043, size = 0, normalized size = 0. \begin{align*} \int \left ( \left ( \ln \left ( x \right ) \right ) ^{m} \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (\log \left (x\right )^{m}\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.10844, size = 50, normalized size = 1.92 \begin{align*} \cos \left (\pi m p\right ) \Gamma \left (m p + 1, -\log \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (\log{\left (x \right )}^{m}\right )^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (\log \left (x\right )^{m}\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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