Optimal. Leaf size=26 \[ \Gamma (1+m p,-\log (x)) (-\log (x))^{-m p} \log ^m(x)^p \]
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Rubi [A]
time = 0.02, 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 = {6852, 2336,
2212} \begin {gather*} (-\log (x))^{-m p} \log ^m(x)^p \text {Gamma}(m p+1,-\log (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 2212
Rule 2336
Rule 6852
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 ) \text {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 = 0.02, size = 26, normalized size = 1.00 \begin {gather*} \Gamma (1+m p,-\log (x)) (-\log (x))^{-m p} \log ^m(x)^p \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (\ln \left (x \right )^{m}\right )^{p}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.10, size = 16, normalized size = 0.62 \begin {gather*} \cos \left (\pi m p\right ) \Gamma \left (m p + 1, -\log \left (x\right )\right ) \end {gather*}
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 \begin {gather*} \int \left (\log {\left (x \right )}^{m}\right )^{p}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\left ({\ln \left (x\right )}^m\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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