Optimal. Leaf size=27 \[ \frac{\log \left (a x^n\right ) \log ^m\left (a x^n\right )^p}{n (m p+1)} \]
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Rubi [A] time = 0.0347011, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {15, 30} \[ \frac{\log \left (a x^n\right ) \log ^m\left (a x^n\right )^p}{n (m p+1)} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin{align*} \int \frac{\log ^m\left (a x^n\right )^p}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \left (x^m\right )^p \, dx,x,\log \left (a x^n\right )\right )}{n}\\ &=\frac{\left (\log ^{-m p}\left (a x^n\right ) \log ^m\left (a x^n\right )^p\right ) \operatorname{Subst}\left (\int x^{m p} \, dx,x,\log \left (a x^n\right )\right )}{n}\\ &=\frac{\log \left (a x^n\right ) \log ^m\left (a x^n\right )^p}{n (1+m p)}\\ \end{align*}
Mathematica [A] time = 0.0059178, size = 27, normalized size = 1. \[ \frac{\log \left (a x^n\right ) \log ^m\left (a x^n\right )^p}{n (m p+1)} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.245, size = 71, normalized size = 2.6 \begin{align*}{\frac{ \left ( \ln \left ( a \right ) +\ln \left ({x}^{n} \right ) -{\frac{i}{2}}\pi \,{\it csgn} \left ( ia{x}^{n} \right ) \left ( -{\it csgn} \left ( ia{x}^{n} \right ) +{\it csgn} \left ( ia \right ) \right ) \left ( -{\it csgn} \left ( ia{x}^{n} \right ) +{\it csgn} \left ( i{x}^{n} \right ) \right ) \right ) ^{mp+1}}{n \left ( mp+1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.17485, size = 80, normalized size = 2.96 \begin{align*} \frac{{\left (n \log \left (x\right ) + \log \left (a\right )\right )}{\left (n \log \left (x\right ) + \log \left (a\right )\right )}^{m p}}{m n p + n} \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 \frac{\left (\log{\left (a x^{n} \right )}^{m}\right )^{p}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.36615, size = 32, normalized size = 1.19 \begin{align*} \frac{{\left (n \log \left (x\right ) + \log \left (a\right )\right )}^{m p + 1}}{{\left (m p + 1\right )} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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