Optimal. Leaf size=51 \[ \log ^2(1-x) \log (x)+2 \log (1-x) \text {PolyLog}(2,1-x)+\log (1-x) \text {PolyLog}(2,x)-2 \text {PolyLog}(3,1-x)-\text {PolyLog}(3,x) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.09, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6874, 6731,
2443, 2481, 2421, 6724} \begin {gather*} -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) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2421
Rule 2443
Rule 2481
Rule 6724
Rule 6731
Rule 6874
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 \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)-\text {Li}_3(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)-\text {Li}_3(x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 51, 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)-\text {PolyLog}(3,x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.72, size = 82, normalized size = 1.61
method | result | size |
default | \(\ln \left (-1+x \right ) \polylog \left (2, x\right )+\ln \left (-1+x \right )^{2} \ln \left (x \right )+2 \ln \left (-1+x \right ) \polylog \left (2, 1-x \right )-2 \polylog \left (3, 1-x \right )+\ln \left (x \right ) \ln \left (-1+x \right ) \left (\ln \left (1-x \right )-\ln \left (-1+x \right )\right )+\dilog \left (x \right ) \left (\ln \left (1-x \right )-\ln \left (-1+x \right )\right )-\polylog \left (3, x\right )\) | \(82\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.26, size = 49, 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 ) - {\rm Li}_{3}(x) - 2 \, {\rm Li}_{3}(-x + 1) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {Li}_{2}\left (x\right )}{x \left (x - 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\mathrm {polylog}\left (2,x\right )}{x\,\left (x-1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________