Optimal. Leaf size=20 \[ -\frac {1}{2} h \text {PolyLog}(2,c x)^2+g \text {PolyLog}(3,c x) \]
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Rubi [A]
time = 0.04, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6737, 6724,
6736} \begin {gather*} g \text {Li}_3(c x)-\frac {1}{2} h \text {Li}_2(c x){}^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 6724
Rule 6736
Rule 6737
Rubi steps
\begin {align*} \int \frac {(g+h \log (1-c x)) \text {Li}_2(c x)}{x} \, dx &=g \int \frac {\text {Li}_2(c x)}{x} \, dx+h \int \frac {\log (1-c x) \text {Li}_2(c x)}{x} \, dx\\ &=-\frac {1}{2} h \text {Li}_2(c x){}^2+g \text {Li}_3(c x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 20, normalized size = 1.00 \begin {gather*} -\frac {1}{2} h \text {PolyLog}(2,c x)^2+g \text {PolyLog}(3,c x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 19, normalized size = 0.95
method | result | size |
default | \(-\frac {h \polylog \left (2, c x \right )^{2}}{2}+g \polylog \left (3, c x \right )\) | \(19\) |
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 {\left (g + h \log {\left (- c x + 1 \right )}\right ) \operatorname {Li}_{2}\left (c 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 [B]
time = 0.25, size = 18, normalized size = 0.90 \begin {gather*} g\,\mathrm {polylog}\left (3,c\,x\right )-\frac {h\,{\mathrm {polylog}\left (2,c\,x\right )}^2}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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