Optimal. Leaf size=51 \[ -2 \text{PolyLog}(3,1-x)-\text{PolyLog}(3,x)+2 \log (1-x) \text{PolyLog}(2,1-x)+\log (1-x) \text{PolyLog}(2,x)+\log (x) \log ^2(1-x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.134478, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, 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.
[In]
[Out]
Rule 6742
Rule 6596
Rule 2396
Rule 2433
Rule 2374
Rule 6589
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*}
Mathematica [A] time = 0.0149876, size = 51, normalized size = 1. \[ -2 \text{PolyLog}(3,1-x)-\text{PolyLog}(3,x)+2 \log (1-x) \text{PolyLog}(2,1-x)+\log (1-x) \text{PolyLog}(2,x)+\log (x) \log ^2(1-x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.282, size = 0, normalized size = 0. \begin{align*} \int -{\frac{{\it polylog} \left ( 2,x \right ) }{ \left ( 1-x \right ) x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.977682, size = 66, normalized size = 1.29 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\rm Li}_2\left (x\right )}{x^{2} - x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{Li}_{2}\left (x\right )}{x \left (x - 1\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Li}_2\left (x\right )}{{\left (x - 1\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]