Optimal. Leaf size=20 \[ \log (x) \text{PolyLog}(n+1,a x)-\text{PolyLog}(n+2,a x) \]
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Rubi [A] time = 0.0238397, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2383, 6589} \[ \log (x) \text{PolyLog}(n+1,a x)-\text{PolyLog}(n+2,a x) \]
Antiderivative was successfully verified.
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Rule 2383
Rule 6589
Rubi steps
\begin{align*} \int \frac{\log (x) \text{Li}_n(a x)}{x} \, dx &=\log (x) \text{Li}_{1+n}(a x)-\int \frac{\text{Li}_{1+n}(a x)}{x} \, dx\\ &=\log (x) \text{Li}_{1+n}(a x)-\text{Li}_{2+n}(a x)\\ \end{align*}
Mathematica [A] time = 0.002117, size = 20, normalized size = 1. \[ \log (x) \text{PolyLog}(n+1,a x)-\text{PolyLog}(n+2,a x) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.007, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( x \right ){\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 ){\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 ){\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 [A] time = 2.21308, size = 15, normalized size = 0.75 \begin{align*} \log{\left (x \right )} \operatorname{Li}_{n + 1}\left (a x\right ) - \operatorname{Li}_{n + 2}\left (a x\right ) \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 ){\rm Li}_{n}(a x)}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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