Optimal. Leaf size=26 \[ \left (a+b \log \left (c x^n\right )\right ) \text {Li}_4(e x)-b n \text {Li}_5(e x) \]
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Rubi [A]
time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2430, 6724}
\begin {gather*} \text {PolyLog}(4,e x) \left (a+b \log \left (c x^n\right )\right )-b n \text {PolyLog}(5,e x) \end {gather*}
Antiderivative was successfully verified.
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Rule 2430
Rule 6724
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(e x)}{x} \, dx &=\left (a+b \log \left (c x^n\right )\right ) \text {Li}_4(e x)-(b n) \int \frac {\text {Li}_4(e x)}{x} \, dx\\ &=\left (a+b \log \left (c x^n\right )\right ) \text {Li}_4(e x)-b n \text {Li}_5(e x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 30, normalized size = 1.15 \begin {gather*} a \text {Li}_4(e x)+b \log \left (c x^n\right ) \text {Li}_4(e x)-b n \text {Li}_5(e x) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right ) \polylog \left (3, e x \right )}{x}\, dx\]
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 = 5.11, size = 26, normalized size = 1.00 \begin {gather*} a \operatorname {Li}_{4}\left (e x\right ) + b \left (- n \operatorname {Li}_{5}\left (e x\right ) + \log {\left (c x^{n} \right )} \operatorname {Li}_{4}\left (e x\right )\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(-1)]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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