Optimal. Leaf size=25 \[ \frac{\log \left (a x^n\right ) \sqrt{\log ^2\left (a x^n\right )}}{2 n} \]
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Rubi [A] time = 0.0200935, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {15, 30} \[ \frac{\log \left (a x^n\right ) \sqrt{\log ^2\left (a x^n\right )}}{2 n} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin{align*} \int \frac{\sqrt{\log ^2\left (a x^n\right )}}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \sqrt{x^2} \, dx,x,\log \left (a x^n\right )\right )}{n}\\ &=\frac{\sqrt{\log ^2\left (a x^n\right )} \operatorname{Subst}\left (\int x \, dx,x,\log \left (a x^n\right )\right )}{n \log \left (a x^n\right )}\\ &=\frac{\log \left (a x^n\right ) \sqrt{\log ^2\left (a x^n\right )}}{2 n}\\ \end{align*}
Mathematica [A] time = 0.0078259, size = 25, normalized size = 1. \[ \frac{\log \left (a x^n\right ) \sqrt{\log ^2\left (a x^n\right )}}{2 n} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.031, size = 21, normalized size = 0.8 \begin{align*}{\frac{{\it csgn} \left ( \ln \left ( a{x}^{n} \right ) \right ) \left ( \ln \left ( a{x}^{n} \right ) \right ) ^{2}}{2\,n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04749, size = 27, normalized size = 1.08 \begin{align*} -\frac{1}{2} \, n \log \left (x\right )^{2} + \log \left (a\right ) \log \left (x\right ) + \log \left (x\right ) \log \left (x^{n}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.09794, size = 43, normalized size = 1.72 \begin{align*} \frac{1}{2} \, n \log \left (x\right )^{2} + \log \left (a\right ) \log \left (x\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 \frac{\sqrt{\log{\left (a x^{n} \right )}^{2}}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.35093, size = 36, normalized size = 1.44 \begin{align*} \frac{1}{2} \, n \log \left (x\right )^{2} \mathrm{sgn}\left (\log \left (a x^{n}\right )\right ) + \log \left (a\right ) \log \left (x\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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