Optimal. Leaf size=33 \[ \log ^2(x) \text {Li}_{1+n}(a x)-2 \log (x) \text {Li}_{2+n}(a x)+2 \text {Li}_{3+n}(a x) \]
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Rubi [A]
time = 0.03, antiderivative size = 33, normalized size of antiderivative = 1.00, 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 = {2430, 6724}
\begin {gather*} 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) \end {gather*}
Antiderivative was successfully verified.
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Rule 2430
Rule 6724
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*}
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Mathematica [A]
time = 0.00, size = 33, normalized size = 1.00 \begin {gather*} \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 {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (x \right )^{2} \polylog \left (n , a x \right )}{x}\, dx\]
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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\log {\left (x \right )}^{2} \operatorname {Li}_{n}\left (a x\right )}{x}\, dx \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 = -0.03 \begin {gather*} \int \frac {{\ln \left (x\right )}^2\,\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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