Optimal. Leaf size=33 \[ 2 \text{PolyLog}(n+3,a x)+\log ^2(x) \text{PolyLog}(n+1,a x)-2 \log (x) \text{PolyLog}(n+2,a x) \]
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Rubi [A] time = 0.0443116, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2383, 6589} \[ 2 \text{PolyLog}(n+3,a x)+\log ^2(x) \text{PolyLog}(n+1,a x)-2 \log (x) \text{PolyLog}(n+2,a x) \]
Antiderivative was successfully verified.
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Rule 2383
Rule 6589
Rubi steps
\begin{align*} \int \frac{\log ^2(x) \text{Li}_n(a x)}{x} \, dx &=\log ^2(x) \text{Li}_{1+n}(a x)-2 \int \frac{\log (x) \text{Li}_{1+n}(a x)}{x} \, dx\\ &=\log ^2(x) \text{Li}_{1+n}(a x)-2 \log (x) \text{Li}_{2+n}(a x)+2 \int \frac{\text{Li}_{2+n}(a x)}{x} \, dx\\ &=\log ^2(x) \text{Li}_{1+n}(a x)-2 \log (x) \text{Li}_{2+n}(a x)+2 \text{Li}_{3+n}(a x)\\ \end{align*}
Mathematica [A] time = 0.0027551, size = 33, normalized size = 1. \[ 2 \text{PolyLog}(n+3,a x)+\log ^2(x) \text{PolyLog}(n+1,a x)-2 \log (x) \text{PolyLog}(n+2,a x) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.018, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \ln \left ( x \right ) \right ) ^{2}{\it polylog} \left ( n,ax \right ) }{x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (x\right )^{2}{\rm Li}_{n}(a x)}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left (x\right )^{2}{\rm polylog}\left (n, a x\right )}{x}, 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{\log{\left (x \right )}^{2} \operatorname{Li}_{n}\left (a x\right )}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (x\right )^{2}{\rm Li}_{n}(a x)}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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