Optimal. Leaf size=46 \[ \log ^2(1-x) \log (x)+2 \log (1-x) \text {PolyLog}(2,1-x)+\log (1-x) \text {PolyLog}(2,x)-2 \text {PolyLog}(3,1-x) \]
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Rubi [A]
time = 0.04, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6731, 2443,
2481, 2421, 6724} \begin {gather*} -2 \text {Li}_3(1-x)+2 \text {Li}_2(1-x) \log (1-x)+\text {Li}_2(x) \log (1-x)+\log (x) \log ^2(1-x) \end {gather*}
Antiderivative was successfully verified.
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Rule 2421
Rule 2443
Rule 2481
Rule 6724
Rule 6731
Rubi steps
\begin {align*} \int -\frac {\text {Li}_2(x)}{1-x} \, dx &=\log (1-x) \text {Li}_2(x)+\int \frac {\log ^2(1-x)}{x} \, dx\\ &=\log ^2(1-x) \log (x)+\log (1-x) \text {Li}_2(x)+2 \int \frac {\log (1-x) \log (x)}{1-x} \, dx\\ &=\log ^2(1-x) \log (x)+\log (1-x) \text {Li}_2(x)-2 \text {Subst}\left (\int \frac {\log (1-x) \log (x)}{x} \, dx,x,1-x\right )\\ &=\log ^2(1-x) \log (x)+2 \log (1-x) \text {Li}_2(1-x)+\log (1-x) \text {Li}_2(x)-2 \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,1-x\right )\\ &=\log ^2(1-x) \log (x)+2 \log (1-x) \text {Li}_2(1-x)+\log (1-x) \text {Li}_2(x)-2 \text {Li}_3(1-x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 46, normalized size = 1.00 \begin {gather*} \log ^2(1-x) \log (x)+2 \log (1-x) \text {PolyLog}(2,1-x)+\log (1-x) \text {PolyLog}(2,x)-2 \text {PolyLog}(3,1-x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.53, size = 66, normalized size = 1.43
method | result | size |
default | \(\ln \left (-1+x \right ) \polylog \left (2, x\right )+\ln \left (1-x \right )^{2} \ln \left (x \right )+2 \ln \left (1-x \right ) \polylog \left (2, 1-x \right )-2 \polylog \left (3, 1-x \right )-\left (\ln \left (-1+x \right )-\ln \left (1-x \right )\right ) \dilog \left (1-x \right )\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 44, normalized size = 0.96 \begin {gather*} \log \left (x\right ) \log \left (-x + 1\right )^{2} + {\rm Li}_2\left (x\right ) \log \left (-x + 1\right ) + 2 \, {\rm Li}_2\left (-x + 1\right ) \log \left (-x + 1\right ) - 2 \, {\rm Li}_{3}(-x + 1) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 39, normalized size = 0.85 \begin {gather*} \log \left (x\right ) \log \left (-x + 1\right )^{2} + {\left ({\rm Li}_2\left (x\right ) + 2 \, {\rm Li}_2\left (-x + 1\right )\right )} \log \left (-x + 1\right ) - 2 \, {\rm polylog}\left (3, -x + 1\right ) \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 {\operatorname {Li}_{2}\left (x\right )}{x - 1}\, 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 = 0.00, size = 46, normalized size = 1.00 \begin {gather*} {\ln \left (1-x\right )}^2\,\ln \left (x\right )-2\,\mathrm {polylog}\left (3,1-x\right )+2\,\ln \left (1-x\right )\,\mathrm {polylog}\left (2,1-x\right )+\ln \left (1-x\right )\,\mathrm {polylog}\left (2,x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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