Optimal. Leaf size=111 \[ \frac {(1-c x) \log ^2(1-c x)}{x}+c \log (c x) \log ^2(1-c x)-2 c \text {PolyLog}(2,c x)+c \log (1-c x) \text {PolyLog}(2,c x)-\frac {\log (1-c x) \text {PolyLog}(2,c x)}{x}+2 c \log (1-c x) \text {PolyLog}(2,1-c x)-c \text {PolyLog}(3,c x)-2 c \text {PolyLog}(3,1-c x) \]
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Rubi [A]
time = 0.11, antiderivative size = 111, normalized size of antiderivative = 1.00, 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 = {6726, 2442,
36, 29, 31, 6738, 2444, 2438, 6724, 6731, 2443, 2481, 2421} \begin {gather*} -2 c \text {Li}_2(c x)-c \text {Li}_3(c x)-2 c \text {Li}_3(1-c x)+c \text {Li}_2(c x) \log (1-c x)-\frac {\text {Li}_2(c x) \log (1-c x)}{x}+2 c \text {Li}_2(1-c x) \log (1-c x)+\frac {(1-c x) \log ^2(1-c x)}{x}+c \log (c x) \log ^2(1-c x) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 2421
Rule 2438
Rule 2442
Rule 2443
Rule 2444
Rule 2481
Rule 6724
Rule 6726
Rule 6731
Rule 6738
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) \text {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) \text {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*}
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Mathematica [A]
time = 0.10, size = 115, normalized size = 1.04 \begin {gather*} 2 c \log (c x) \log (1-c x)-c \log ^2(1-c x)+\frac {\log ^2(1-c x)}{x}+c \log (c x) \log ^2(1-c x)+\frac {(-1+c x) \log (1-c x) \text {PolyLog}(2,c x)}{x}+2 c (1+\log (1-c x)) \text {PolyLog}(2,1-c x)-c \text {PolyLog}(3,c x)-2 c \text {PolyLog}(3,1-c x) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (-c x +1\right ) \polylog \left (2, c x \right )}{x^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 113, normalized size = 1.02 \begin {gather*} {\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\log {\left (- c x + 1 \right )} \operatorname {Li}_{2}\left (c x\right )}{x^{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\ln \left (1-c\,x\right )\,\mathrm {polylog}\left (2,c\,x\right )}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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