Optimal. Leaf size=51 \[ -2 \text {Li}_3(1-x)-\text {Li}_3(x)+2 \text {Li}_2(1-x) \log (1-x)+\text {Li}_2(x) \log (1-x)+\log (x) \log ^2(1-x) \]
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Rubi [A] time = 0.13, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6742, 6596, 2396, 2433, 2374, 6589} \[ -2 \text {PolyLog}(3,1-x)-\text {PolyLog}(3,x)+2 \log (1-x) \text {PolyLog}(2,1-x)+\log (1-x) \text {PolyLog}(2,x)+\log (x) \log ^2(1-x) \]
Antiderivative was successfully verified.
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Rule 2374
Rule 2396
Rule 2433
Rule 6589
Rule 6596
Rule 6742
Rubi steps
\begin {align*} \int \frac {\text {Li}_2(x)}{(-1+x) x} \, dx &=\int \left (\frac {\text {Li}_2(x)}{-1+x}-\frac {\text {Li}_2(x)}{x}\right ) \, dx\\ &=\int \frac {\text {Li}_2(x)}{-1+x} \, dx-\int \frac {\text {Li}_2(x)}{x} \, dx\\ &=\log (1-x) \text {Li}_2(x)-\text {Li}_3(x)+\int \frac {\log ^2(1-x)}{x} \, dx\\ &=\log ^2(1-x) \log (x)+\log (1-x) \text {Li}_2(x)-\text {Li}_3(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)-\text {Li}_3(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)-\text {Li}_3(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)-\text {Li}_3(x)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 51, normalized size = 1.00 \[ -2 \text {Li}_3(1-x)-\text {Li}_3(x)+2 \text {Li}_2(1-x) \log (1-x)+\text {Li}_2(x) \log (1-x)+\log (x) \log ^2(1-x) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\rm Li}_2\relax (x)}{x^{2} - x}, x\right ) \]
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 \[ \int \frac {{\rm Li}_2\relax (x)}{{\left (x - 1\right )} x}\,{d x} \]
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 \[ \int \frac {\polylog \left (2, x\right )}{\left (-1+x \right ) x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 49, normalized size = 0.96 \[ \log \relax (x) \log \left (-x + 1\right )^{2} + {\rm Li}_2\relax (x) \log \left (-x + 1\right ) + 2 \, {\rm Li}_2\left (-x + 1\right ) \log \left (-x + 1\right ) - {\rm Li}_{3}(x) - 2 \, {\rm Li}_{3}(-x + 1) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {polylog}\left (2,x\right )}{x\,\left (x-1\right )} \,d x \]
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 \[ \int \frac {\operatorname {Li}_{2}\relax (x)}{x \left (x - 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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