Optimal. Leaf size=27 \[ \frac{\log \left (a x^n\right ) \log ^2\left (a x^n\right )^p}{n (2 p+1)} \]
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Rubi [A] time = 0.026034, 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} \[ \frac{\log \left (a x^n\right ) \log ^2\left (a x^n\right )^p}{n (2 p+1)} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin{align*} \int \frac{\log ^2\left (a x^n\right )^p}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \left (x^2\right )^p \, dx,x,\log \left (a x^n\right )\right )}{n}\\ &=\frac{\left (\log ^{-2 p}\left (a x^n\right ) \log ^2\left (a x^n\right )^p\right ) \operatorname{Subst}\left (\int x^{2 p} \, dx,x,\log \left (a x^n\right )\right )}{n}\\ &=\frac{\log \left (a x^n\right ) \log ^2\left (a x^n\right )^p}{n (1+2 p)}\\ \end{align*}
Mathematica [A] time = 0.01064, size = 27, normalized size = 1. \[ \frac{\log \left (a x^n\right ) \log ^2\left (a x^n\right )^p}{n (2 p+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.181, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \left ( \ln \left ( a{x}^{n} \right ) \right ) ^{2} \right ) ^{p}}{x}}\, dx \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 = 2.13831, size = 109, normalized size = 4.04 \begin{align*} \frac{{\left (n \log \left (x\right ) + \log \left (a\right )\right )}{\left (n^{2} \log \left (x\right )^{2} + 2 \, n \log \left (a\right ) \log \left (x\right ) + \log \left (a\right )^{2}\right )}^{p}}{2 \, 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 )}^{2}\right )^{p}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.59817, size = 92, normalized size = 3.41 \begin{align*} \frac{{\left (n \log \left (x\right ) \mathrm{sgn}\left (\log \left (a x^{n}\right )\right ) + \log \left (a\right ) \mathrm{sgn}\left (\log \left (a x^{n}\right )\right )\right )}{\left (n \log \left (x\right ) \mathrm{sgn}\left (\log \left (a x^{n}\right )\right ) + \log \left (a\right ) \mathrm{sgn}\left (\log \left (a x^{n}\right )\right )\right )}^{2 \, p}}{n{\left (2 \, p + 1\right )} \mathrm{sgn}\left (\log \left (a x^{n}\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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