Optimal. Leaf size=46 \[ -2 \text {Li}_3(1-x)+2 \text {Li}_2(1-x) \log (1-x)+\text {Li}_2(x) \log (1-x)+\log (x) \log ^2(1-x) \]
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Rubi [A] time = 0.07, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {6596, 2396, 2433, 2374, 6589} \[ -2 \text {PolyLog}(3,1-x)+2 \log (1-x) \text {PolyLog}(2,1-x)+\log (1-x) \text {PolyLog}(2,x)+\log (x) \log ^2(1-x) \]
Antiderivative was successfully verified.
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Rule 2374
Rule 2396
Rule 2433
Rule 6589
Rule 6596
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 \operatorname {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 \operatorname {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 = 0.04, size = 46, normalized size = 1.00 \[ -2 \text {Li}_3(1-x)+2 \text {Li}_2(1-x) \log (1-x)+\text {Li}_2(x) \log (1-x)+\log (x) \log ^2(1-x) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\rm Li}_2\relax (x)}{x - 1}, x\right ) \]
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 \[ \int \frac {{\rm Li}_2\relax (x)}{x - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.11, size = 0, normalized size = 0.00 \[ \int \frac {\polylog \left (2, x\right )}{-1+x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 44, normalized size = 0.96 \[ \log \relax (x) \log \left (-x + 1\right )^{2} + {\rm Li}_2\relax (x) \log \left (-x + 1\right ) + 2 \, {\rm Li}_2\left (-x + 1\right ) \log \left (-x + 1\right ) - 2 \, {\rm Li}_{3}(-x + 1) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 46, normalized size = 1.00 \[ {\ln \left (1-x\right )}^2\,\ln \relax (x)-2\,\mathrm {polylog}\left (3,1-x\right )+2\,\ln \left (1-x\right )\,\mathrm {polylog}\left (2,1-x\right )+\ln \left (1-x\right )\,\mathrm {polylog}\left (2,x\right ) \]
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 \[ \int \frac {\operatorname {Li}_{2}\relax (x)}{x - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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