Optimal. Leaf size=111 \[ -2 c \text {Li}_2(c x)-c \text {Li}_3(c x)-2 c \text {Li}_3(1-c x)+c \text {Li}_2(c x) \log (1-c x)-\frac {\text {Li}_2(c x) \log (1-c x)}{x}+2 c \text {Li}_2(1-c x) \log (1-c x)+\frac {(1-c x) \log ^2(1-c x)}{x}+c \log (c x) \log ^2(1-c x) \]
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Rubi [A] time = 0.16, 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 = {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 29
Rule 31
Rule 36
Rule 2374
Rule 2391
Rule 2395
Rule 2396
Rule 2397
Rule 2433
Rule 6589
Rule 6591
Rule 6596
Rule 6603
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*}
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Mathematica [A] time = 0.13, size = 115, normalized size = 1.04 \[ -c \text {Li}_3(c x)-2 c \text {Li}_3(1-c x)+\frac {(c x-1) \text {Li}_2(c x) \log (1-c x)}{x}+2 c \text {Li}_2(1-c x) (\log (1-c x)+1)-c \log ^2(1-c x)+c \log (c x) \log ^2(1-c x)+\frac {\log ^2(1-c x)}{x}+2 c \log (c x) \log (1-c x) \]
Antiderivative was successfully verified.
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fricas [F] time = 1.13, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\rm Li}_2\left (c x\right ) \log \left (-c x + 1\right )}{x^{2}}, x\right ) \]
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 \[ \int \frac {{\rm Li}_2\left (c x\right ) \log \left (-c x + 1\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, 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 = 0.37, size = 113, normalized size = 1.02 \[ {\left (\log \left (c x\right ) \log \left (-c x + 1\right )^{2} + 2 \, {\rm Li}_2\left (-c x + 1\right ) \log \left (-c x + 1\right ) - 2 \, {\rm Li}_{3}(-c x + 1)\right )} c + 2 \, {\left (\log \left (c x\right ) \log \left (-c x + 1\right ) + {\rm Li}_2\left (-c x + 1\right )\right )} c - c {\rm Li}_{3}(c x) + \frac {{\left (c x - 1\right )} {\rm Li}_2\left (c x\right ) \log \left (-c x + 1\right ) - {\left (c x - 1\right )} \log \left (-c x + 1\right )^{2}}{x} \]
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 \[ \int \frac {\ln \left (1-c\,x\right )\,\mathrm {polylog}\left (2,c\,x\right )}{x^2} \,d x \]
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 \[ \int \frac {\log {\left (- c x + 1 \right )} \operatorname {Li}_{2}\left (c x\right )}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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