Optimal. Leaf size=26 \[ (-\log (x))^{-m p} \log ^m(x)^p \Gamma (m p+1,-\log (x)) \]
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Rubi [A] time = 0.03, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 2181
Rule 2299
Rule 6720
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*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 1.00 \[ (-\log (x))^{-m p} \log ^m(x)^p \Gamma (m p+1,-\log (x)) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 16, normalized size = 0.62 \[ \cos \left (\pi m p\right ) \Gamma \left (m p + 1, -\log \relax (x)\right ) \]
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 \[ \int {\left (\log \relax (x)^{m}\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.28, size = 0, normalized size = 0.00 \[ \int \left (\ln \relax (x )^{m}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (\log \relax (x)^{m}\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int {\left ({\ln \relax (x)}^m\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (\log {\relax (x )}^{m}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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