Optimal. Leaf size=20 \[ \log (x) \text {Li}_{n+1}(a x)-\text {Li}_{n+2}(a x) \]
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Rubi [A] time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.00, size = 20, normalized size = 1.00 \[ \log (x) \text {Li}_{n+1}(a x)-\text {Li}_{n+2}(a x) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \relax (x) {\rm polylog}\left (n, a x\right )}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \relax (x) {\rm Li}_{n}(a x)}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {\polylog \left (n , a x \right ) \ln \relax (x )}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \relax (x) {\rm Li}_{n}(a x)}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {\ln \relax (x)\,\mathrm {polylog}\left (n,a\,x\right )}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.33, size = 15, normalized size = 0.75 \[ \log {\relax (x )} \operatorname {Li}_{n + 1}\left (a x\right ) - \operatorname {Li}_{n + 2}\left (a x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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