Optimal. Leaf size=111 \[ -2 c \text{PolyLog}(2,c x)-c \text{PolyLog}(3,c x)-2 c \text{PolyLog}(3,1-c x)+c \log (1-c x) \text{PolyLog}(2,c x)-\frac{\log (1-c x) \text{PolyLog}(2,c x)}{x}+2 c \log (1-c x) \text{PolyLog}(2,1-c x)+\frac{(1-c x) \log ^2(1-c x)}{x}+c \log (c x) \log ^2(1-c x) \]
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Rubi [A] time = 0.163861, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 13, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.812, Rules used = {6591, 2395, 36, 29, 31, 6603, 2397, 2391, 6589, 6596, 2396, 2433, 2374} \[ -2 c \text{PolyLog}(2,c x)-c \text{PolyLog}(3,c x)-2 c \text{PolyLog}(3,1-c x)+c \log (1-c x) \text{PolyLog}(2,c x)-\frac{\log (1-c x) \text{PolyLog}(2,c x)}{x}+2 c \log (1-c x) \text{PolyLog}(2,1-c x)+\frac{(1-c x) \log ^2(1-c x)}{x}+c \log (c x) \log ^2(1-c x) \]
Antiderivative was successfully verified.
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Rule 6591
Rule 2395
Rule 36
Rule 29
Rule 31
Rule 6603
Rule 2397
Rule 2391
Rule 6589
Rule 6596
Rule 2396
Rule 2433
Rule 2374
Rubi steps
\begin{align*} \int \frac{\log (1-c x) \text{Li}_2(c x)}{x^2} \, dx &=-\frac{\log (1-c x) \text{Li}_2(c x)}{x}-c \int \left (\frac{\text{Li}_2(c x)}{x}-\frac{c \text{Li}_2(c x)}{-1+c x}\right ) \, dx-\int \frac{\log ^2(1-c x)}{x^2} \, dx\\ &=\frac{(1-c x) \log ^2(1-c x)}{x}-\frac{\log (1-c x) \text{Li}_2(c x)}{x}-c \int \frac{\text{Li}_2(c x)}{x} \, dx+(2 c) \int \frac{\log (1-c x)}{x} \, dx+c^2 \int \frac{\text{Li}_2(c x)}{-1+c x} \, dx\\ &=\frac{(1-c x) \log ^2(1-c x)}{x}-2 c \text{Li}_2(c x)+c \log (1-c x) \text{Li}_2(c x)-\frac{\log (1-c x) \text{Li}_2(c x)}{x}-c \text{Li}_3(c x)+c \int \frac{\log ^2(1-c x)}{x} \, dx\\ &=\frac{(1-c x) \log ^2(1-c x)}{x}+c \log (c x) \log ^2(1-c x)-2 c \text{Li}_2(c x)+c \log (1-c x) \text{Li}_2(c x)-\frac{\log (1-c x) \text{Li}_2(c x)}{x}-c \text{Li}_3(c x)+\left (2 c^2\right ) \int \frac{\log (c x) \log (1-c x)}{1-c x} \, dx\\ &=\frac{(1-c x) \log ^2(1-c x)}{x}+c \log (c x) \log ^2(1-c x)-2 c \text{Li}_2(c x)+c \log (1-c x) \text{Li}_2(c x)-\frac{\log (1-c x) \text{Li}_2(c x)}{x}-c \text{Li}_3(c x)-(2 c) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (c \left (\frac{1}{c}-\frac{x}{c}\right )\right )}{x} \, dx,x,1-c x\right )\\ &=\frac{(1-c x) \log ^2(1-c x)}{x}+c \log (c x) \log ^2(1-c x)-2 c \text{Li}_2(c x)+c \log (1-c x) \text{Li}_2(c x)-\frac{\log (1-c x) \text{Li}_2(c x)}{x}+2 c \log (1-c x) \text{Li}_2(1-c x)-c \text{Li}_3(c x)-(2 c) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,1-c x\right )\\ &=\frac{(1-c x) \log ^2(1-c x)}{x}+c \log (c x) \log ^2(1-c x)-2 c \text{Li}_2(c x)+c \log (1-c x) \text{Li}_2(c x)-\frac{\log (1-c x) \text{Li}_2(c x)}{x}+2 c \log (1-c x) \text{Li}_2(1-c x)-c \text{Li}_3(c x)-2 c \text{Li}_3(1-c x)\\ \end{align*}
Mathematica [A] time = 0.135536, size = 115, normalized size = 1.04 \[ -c \text{PolyLog}(3,c x)-2 c \text{PolyLog}(3,1-c x)+\frac{(c x-1) \log (1-c x) \text{PolyLog}(2,c x)}{x}+2 c (\log (1-c x)+1) \text{PolyLog}(2,1-c x)-c \log ^2(1-c x)+c \log (c x) \log ^2(1-c x)+\frac{\log ^2(1-c x)}{x}+2 c \log (c x) \log (1-c x) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.049, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( -cx+1 \right ){\it polylog} \left ( 2,cx \right ) }{{x}^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16597, size = 153, normalized size = 1.38 \begin{align*}{\left (\log \left (c x\right ) \log \left (-c x + 1\right )^{2} + 2 \,{\rm Li}_2\left (-c x + 1\right ) \log \left (-c x + 1\right ) - 2 \,{\rm Li}_{3}(-c x + 1)\right )} c + 2 \,{\left (\log \left (c x\right ) \log \left (-c x + 1\right ) +{\rm Li}_2\left (-c x + 1\right )\right )} c - c{\rm Li}_{3}(c x) + \frac{{\left (c x - 1\right )}{\rm Li}_2\left (c x\right ) \log \left (-c x + 1\right ) -{\left (c x - 1\right )} \log \left (-c x + 1\right )^{2}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\rm Li}_2\left (c x\right ) \log \left (-c x + 1\right )}{x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log{\left (- c x + 1 \right )} \operatorname{Li}_{2}\left (c x\right )}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Li}_2\left (c x\right ) \log \left (-c x + 1\right )}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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