Optimal. Leaf size=20 \[ \log (x) \text {Li}_{1+n}(a x)-\text {Li}_{2+n}(a x) \]
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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 = {2430, 6724}
\begin {gather*} \log (x) \text {PolyLog}(n+1,a x)-\text {PolyLog}(n+2,a x) \end {gather*}
Antiderivative was successfully verified.
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Rule 2430
Rule 6724
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 \begin {gather*} \log (x) \text {Li}_{1+n}(a x)-\text {Li}_{2+n}(a x) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (x \right ) \polylog \left (n , a x \right )}{x}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.14, size = 15, normalized size = 0.75 \begin {gather*} \log {\left (x \right )} \operatorname {Li}_{n + 1}\left (a x\right ) - \operatorname {Li}_{n + 2}\left (a x\right ) \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\ln \left (x\right )\,\mathrm {polylog}\left (n,a\,x\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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