Optimal. Leaf size=287 \[ \frac {5 c^2}{144 x^2}+\frac {7 c^3}{36 x}-\frac {41}{72} c^4 \log (x)+\frac {41}{72} c^4 \log (1-c x)-\frac {5 c \log (1-c x)}{72 x^3}-\frac {c^2 \log (1-c x)}{8 x^2}-\frac {3 c^3 \log (1-c x)}{8 x}-\frac {1}{16} c^4 \log ^2(1-c x)+\frac {\log ^2(1-c x)}{16 x^4}+\frac {1}{4} c^4 \log (c x) \log ^2(1-c x)-\frac {1}{8} c^4 \text {PolyLog}(2,c x)+\frac {c \text {PolyLog}(2,c x)}{12 x^3}+\frac {c^2 \text {PolyLog}(2,c x)}{8 x^2}+\frac {c^3 \text {PolyLog}(2,c x)}{4 x}+\frac {1}{4} c^4 \log (1-c x) \text {PolyLog}(2,c x)-\frac {\log (1-c x) \text {PolyLog}(2,c x)}{4 x^4}+\frac {1}{2} c^4 \log (1-c x) \text {PolyLog}(2,1-c x)-\frac {1}{4} c^4 \text {PolyLog}(3,c x)-\frac {1}{2} c^4 \text {PolyLog}(3,1-c x) \]
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Rubi [A]
time = 0.31, antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps
used = 37, number of rules used = 17, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.062, Rules used = {6726,
2442, 46, 6738, 2445, 2457, 36, 29, 31, 2438, 2437, 2338, 6724, 6731, 2443, 2481, 2421}
\begin {gather*} -\frac {1}{8} c^4 \text {Li}_2(c x)-\frac {1}{4} c^4 \text {Li}_3(c x)-\frac {1}{2} c^4 \text {Li}_3(1-c x)+\frac {1}{4} c^4 \text {Li}_2(c x) \log (1-c x)+\frac {1}{2} c^4 \text {Li}_2(1-c x) \log (1-c x)+\frac {1}{4} c^4 \log (c x) \log ^2(1-c x)-\frac {1}{16} c^4 \log ^2(1-c x)-\frac {41}{72} c^4 \log (x)+\frac {41}{72} c^4 \log (1-c x)+\frac {c^3 \text {Li}_2(c x)}{4 x}+\frac {7 c^3}{36 x}-\frac {3 c^3 \log (1-c x)}{8 x}+\frac {c^2 \text {Li}_2(c x)}{8 x^2}+\frac {5 c^2}{144 x^2}-\frac {c^2 \log (1-c x)}{8 x^2}-\frac {\text {Li}_2(c x) \log (1-c x)}{4 x^4}+\frac {c \text {Li}_2(c x)}{12 x^3}+\frac {\log ^2(1-c x)}{16 x^4}-\frac {5 c \log (1-c x)}{72 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 46
Rule 2338
Rule 2421
Rule 2437
Rule 2438
Rule 2442
Rule 2443
Rule 2445
Rule 2457
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^5} \, dx &=-\frac {\log (1-c x) \text {Li}_2(c x)}{4 x^4}-\frac {1}{4} \int \frac {\log ^2(1-c x)}{x^5} \, dx-\frac {1}{4} c \int \left (\frac {\text {Li}_2(c x)}{x^4}+\frac {c \text {Li}_2(c x)}{x^3}+\frac {c^2 \text {Li}_2(c x)}{x^2}+\frac {c^3 \text {Li}_2(c x)}{x}-\frac {c^4 \text {Li}_2(c x)}{-1+c x}\right ) \, dx\\ &=\frac {\log ^2(1-c x)}{16 x^4}-\frac {\log (1-c x) \text {Li}_2(c x)}{4 x^4}+\frac {1}{8} c \int \frac {\log (1-c x)}{x^4 (1-c x)} \, dx-\frac {1}{4} c \int \frac {\text {Li}_2(c x)}{x^4} \, dx-\frac {1}{4} c^2 \int \frac {\text {Li}_2(c x)}{x^3} \, dx-\frac {1}{4} c^3 \int \frac {\text {Li}_2(c x)}{x^2} \, dx-\frac {1}{4} c^4 \int \frac {\text {Li}_2(c x)}{x} \, dx+\frac {1}{4} c^5 \int \frac {\text {Li}_2(c x)}{-1+c x} \, dx\\ &=\frac {\log ^2(1-c x)}{16 x^4}+\frac {c \text {Li}_2(c x)}{12 x^3}+\frac {c^2 \text {Li}_2(c x)}{8 x^2}+\frac {c^3 \text {Li}_2(c x)}{4 x}+\frac {1}{4} c^4 \log (1-c x) \text {Li}_2(c x)-\frac {\log (1-c x) \text {Li}_2(c x)}{4 x^4}-\frac {1}{4} c^4 \text {Li}_3(c x)+\frac {1}{12} c \int \frac {\log (1-c x)}{x^4} \, dx+\frac {1}{8} c \int \left (\frac {\log (1-c x)}{x^4}+\frac {c \log (1-c x)}{x^3}+\frac {c^2 \log (1-c x)}{x^2}+\frac {c^3 \log (1-c x)}{x}-\frac {c^4 \log (1-c x)}{-1+c x}\right ) \, dx+\frac {1}{8} c^2 \int \frac {\log (1-c x)}{x^3} \, dx+\frac {1}{4} c^3 \int \frac {\log (1-c x)}{x^2} \, dx+\frac {1}{4} c^4 \int \frac {\log ^2(1-c x)}{x} \, dx\\ &=-\frac {c \log (1-c x)}{36 x^3}-\frac {c^2 \log (1-c x)}{16 x^2}-\frac {c^3 \log (1-c x)}{4 x}+\frac {\log ^2(1-c x)}{16 x^4}+\frac {1}{4} c^4 \log (c x) \log ^2(1-c x)+\frac {c \text {Li}_2(c x)}{12 x^3}+\frac {c^2 \text {Li}_2(c x)}{8 x^2}+\frac {c^3 \text {Li}_2(c x)}{4 x}+\frac {1}{4} c^4 \log (1-c x) \text {Li}_2(c x)-\frac {\log (1-c x) \text {Li}_2(c x)}{4 x^4}-\frac {1}{4} c^4 \text {Li}_3(c x)+\frac {1}{8} c \int \frac {\log (1-c x)}{x^4} \, dx-\frac {1}{36} c^2 \int \frac {1}{x^3 (1-c x)} \, dx+\frac {1}{8} c^2 \int \frac {\log (1-c x)}{x^3} \, dx-\frac {1}{16} c^3 \int \frac {1}{x^2 (1-c x)} \, dx+\frac {1}{8} c^3 \int \frac {\log (1-c x)}{x^2} \, dx+\frac {1}{8} c^4 \int \frac {\log (1-c x)}{x} \, dx-\frac {1}{4} c^4 \int \frac {1}{x (1-c x)} \, dx-\frac {1}{8} c^5 \int \frac {\log (1-c x)}{-1+c x} \, dx+\frac {1}{2} c^5 \int \frac {\log (c x) \log (1-c x)}{1-c x} \, dx\\ &=-\frac {5 c \log (1-c x)}{72 x^3}-\frac {c^2 \log (1-c x)}{8 x^2}-\frac {3 c^3 \log (1-c x)}{8 x}+\frac {\log ^2(1-c x)}{16 x^4}+\frac {1}{4} c^4 \log (c x) \log ^2(1-c x)-\frac {1}{8} c^4 \text {Li}_2(c x)+\frac {c \text {Li}_2(c x)}{12 x^3}+\frac {c^2 \text {Li}_2(c x)}{8 x^2}+\frac {c^3 \text {Li}_2(c x)}{4 x}+\frac {1}{4} c^4 \log (1-c x) \text {Li}_2(c x)-\frac {\log (1-c x) \text {Li}_2(c x)}{4 x^4}-\frac {1}{4} c^4 \text {Li}_3(c x)-\frac {1}{36} c^2 \int \left (\frac {1}{x^3}+\frac {c}{x^2}+\frac {c^2}{x}-\frac {c^3}{-1+c x}\right ) \, dx-\frac {1}{24} c^2 \int \frac {1}{x^3 (1-c x)} \, dx-\frac {1}{16} c^3 \int \frac {1}{x^2 (1-c x)} \, dx-\frac {1}{16} c^3 \int \left (\frac {1}{x^2}+\frac {c}{x}-\frac {c^2}{-1+c x}\right ) \, dx-\frac {1}{8} c^4 \int \frac {1}{x (1-c x)} \, dx-\frac {1}{8} c^4 \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,1-c x\right )-\frac {1}{4} c^4 \int \frac {1}{x} \, dx-\frac {1}{2} c^4 \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}{4} c^5 \int \frac {1}{1-c x} \, dx\\ &=\frac {c^2}{72 x^2}+\frac {13 c^3}{144 x}-\frac {49}{144} c^4 \log (x)+\frac {49}{144} c^4 \log (1-c x)-\frac {5 c \log (1-c x)}{72 x^3}-\frac {c^2 \log (1-c x)}{8 x^2}-\frac {3 c^3 \log (1-c x)}{8 x}-\frac {1}{16} c^4 \log ^2(1-c x)+\frac {\log ^2(1-c x)}{16 x^4}+\frac {1}{4} c^4 \log (c x) \log ^2(1-c x)-\frac {1}{8} c^4 \text {Li}_2(c x)+\frac {c \text {Li}_2(c x)}{12 x^3}+\frac {c^2 \text {Li}_2(c x)}{8 x^2}+\frac {c^3 \text {Li}_2(c x)}{4 x}+\frac {1}{4} c^4 \log (1-c x) \text {Li}_2(c x)-\frac {\log (1-c x) \text {Li}_2(c x)}{4 x^4}+\frac {1}{2} c^4 \log (1-c x) \text {Li}_2(1-c x)-\frac {1}{4} c^4 \text {Li}_3(c x)-\frac {1}{24} c^2 \int \left (\frac {1}{x^3}+\frac {c}{x^2}+\frac {c^2}{x}-\frac {c^3}{-1+c x}\right ) \, dx-\frac {1}{16} c^3 \int \left (\frac {1}{x^2}+\frac {c}{x}-\frac {c^2}{-1+c x}\right ) \, dx-\frac {1}{8} c^4 \int \frac {1}{x} \, dx-\frac {1}{2} c^4 \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,1-c x\right )-\frac {1}{8} c^5 \int \frac {1}{1-c x} \, dx\\ &=\frac {5 c^2}{144 x^2}+\frac {7 c^3}{36 x}-\frac {41}{72} c^4 \log (x)+\frac {41}{72} c^4 \log (1-c x)-\frac {5 c \log (1-c x)}{72 x^3}-\frac {c^2 \log (1-c x)}{8 x^2}-\frac {3 c^3 \log (1-c x)}{8 x}-\frac {1}{16} c^4 \log ^2(1-c x)+\frac {\log ^2(1-c x)}{16 x^4}+\frac {1}{4} c^4 \log (c x) \log ^2(1-c x)-\frac {1}{8} c^4 \text {Li}_2(c x)+\frac {c \text {Li}_2(c x)}{12 x^3}+\frac {c^2 \text {Li}_2(c x)}{8 x^2}+\frac {c^3 \text {Li}_2(c x)}{4 x}+\frac {1}{4} c^4 \log (1-c x) \text {Li}_2(c x)-\frac {\log (1-c x) \text {Li}_2(c x)}{4 x^4}+\frac {1}{2} c^4 \log (1-c x) \text {Li}_2(1-c x)-\frac {1}{4} c^4 \text {Li}_3(c x)-\frac {1}{2} c^4 \text {Li}_3(1-c x)\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 277, normalized size = 0.97 \begin {gather*} \frac {5 c^2 x^2+28 c^3 x^3-18 c^4 x^4-49 c^4 x^4 \log (x)-33 c^4 x^4 \log (c x)-10 c x \log (1-c x)-18 c^2 x^2 \log (1-c x)-54 c^3 x^3 \log (1-c x)+82 c^4 x^4 \log (1-c x)+18 c^4 x^4 \log (c x) \log (1-c x)+9 \log ^2(1-c x)-9 c^4 x^4 \log ^2(1-c x)+36 c^4 x^4 \log (c x) \log ^2(1-c x)+6 \left (c x \left (2+3 c x+6 c^2 x^2\right )+6 \left (-1+c^4 x^4\right ) \log (1-c x)\right ) \text {PolyLog}(2,c x)+18 c^4 x^4 (1+4 \log (1-c x)) \text {PolyLog}(2,1-c x)-36 c^4 x^4 \text {PolyLog}(3,c x)-72 c^4 x^4 \text {PolyLog}(3,1-c x)}{144 x^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (-c x +1\right ) \polylog \left (2, c x \right )}{x^{5}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 214, normalized size = 0.75 \begin {gather*} \frac {1}{4} \, {\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^{4} + \frac {1}{8} \, {\left (\log \left (c x\right ) \log \left (-c x + 1\right ) + {\rm Li}_2\left (-c x + 1\right )\right )} c^{4} - \frac {41}{72} \, c^{4} \log \left (x\right ) - \frac {1}{4} \, c^{4} {\rm Li}_{3}(c x) + \frac {28 \, c^{3} x^{3} + 5 \, c^{2} x^{2} - 9 \, {\left (c^{4} x^{4} - 1\right )} \log \left (-c x + 1\right )^{2} + 6 \, {\left (6 \, c^{3} x^{3} + 3 \, c^{2} x^{2} + 2 \, c x + 6 \, {\left (c^{4} x^{4} - 1\right )} \log \left (-c x + 1\right )\right )} {\rm Li}_2\left (c x\right ) + 2 \, {\left (41 \, c^{4} x^{4} - 27 \, c^{3} x^{3} - 9 \, c^{2} x^{2} - 5 \, c x\right )} \log \left (-c x + 1\right )}{144 \, x^{4}} \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^{5}}\, 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 = -0.00 \begin {gather*} \int \frac {\ln \left (1-c\,x\right )\,\mathrm {polylog}\left (2,c\,x\right )}{x^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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