Optimal. Leaf size=460 \[ -\frac {1}{6} c^2 (2 a c+3 b) \text {Li}_3(c x)-\frac {1}{3} c^2 (2 a c+3 b) \text {Li}_3(1-c x)+\frac {1}{6} c^2 (2 a c+3 b) \text {Li}_2(c x) \log (1-c x)+\frac {1}{3} c^2 (2 a c+3 b) \text {Li}_2(1-c x) \log (1-c x)+\frac {1}{6} c^2 (2 a c+3 b) \log (c x) \log ^2(1-c x)-\frac {1}{6} c^2 \log (x) (2 a c+3 b)+\frac {1}{6} c^2 (2 a c+3 b) \log (1-c x)-\frac {1}{6} \left (\frac {2 a}{x^3}+\frac {3 b}{x^2}\right ) \text {Li}_2(c x) \log (1-c x)+\frac {c (2 a c+3 b) \text {Li}_2(c x)}{6 x}-\frac {c (2 a c+3 b) \log (1-c x)}{6 x}-\frac {2}{9} a c^3 \text {Li}_2(c x)-\frac {1}{9} a c^3 \log ^2(1-c x)-\frac {5}{12} a c^3 \log (x)+\frac {5}{12} a c^3 \log (1-c x)+\frac {7 a c^2}{36 x}-\frac {2 a c^2 \log (1-c x)}{9 x}+\frac {a c \text {Li}_2(c x)}{6 x^2}+\frac {a \log ^2(1-c x)}{9 x^3}-\frac {7 a c \log (1-c x)}{36 x^2}-\frac {1}{2} b c^2 \text {Li}_2(c x)-\frac {1}{4} b c^2 \log ^2(1-c x)-\frac {1}{2} b c^2 \log (x)+\frac {1}{2} b c^2 \log (1-c x)+\frac {b \log ^2(1-c x)}{4 x^2}-\frac {b c \log (1-c x)}{2 x} \]
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Rubi [A] time = 0.67, antiderivative size = 460, normalized size of antiderivative = 1.00, number of steps used = 41, number of rules used = 19, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.905, Rules used = {6742, 6591, 2395, 44, 43, 6606, 2398, 2410, 36, 29, 31, 2391, 2390, 2301, 6589, 6596, 2396, 2433, 2374} \[ -\frac {1}{6} c^2 (2 a c+3 b) \text {PolyLog}(3,c x)-\frac {1}{3} c^2 (2 a c+3 b) \text {PolyLog}(3,1-c x)+\frac {1}{6} c^2 (2 a c+3 b) \log (1-c x) \text {PolyLog}(2,c x)+\frac {1}{3} c^2 (2 a c+3 b) \log (1-c x) \text {PolyLog}(2,1-c x)-\frac {1}{6} \left (\frac {2 a}{x^3}+\frac {3 b}{x^2}\right ) \log (1-c x) \text {PolyLog}(2,c x)+\frac {c (2 a c+3 b) \text {PolyLog}(2,c x)}{6 x}-\frac {2}{9} a c^3 \text {PolyLog}(2,c x)+\frac {a c \text {PolyLog}(2,c x)}{6 x^2}-\frac {1}{2} b c^2 \text {PolyLog}(2,c x)+\frac {1}{6} c^2 (2 a c+3 b) \log (c x) \log ^2(1-c x)-\frac {1}{6} c^2 \log (x) (2 a c+3 b)+\frac {1}{6} c^2 (2 a c+3 b) \log (1-c x)-\frac {c (2 a c+3 b) \log (1-c x)}{6 x}+\frac {7 a c^2}{36 x}-\frac {1}{9} a c^3 \log ^2(1-c x)-\frac {5}{12} a c^3 \log (x)+\frac {5}{12} a c^3 \log (1-c x)-\frac {2 a c^2 \log (1-c x)}{9 x}+\frac {a \log ^2(1-c x)}{9 x^3}-\frac {7 a c \log (1-c x)}{36 x^2}-\frac {1}{4} b c^2 \log ^2(1-c x)-\frac {1}{2} b c^2 \log (x)+\frac {1}{2} b c^2 \log (1-c x)+\frac {b \log ^2(1-c x)}{4 x^2}-\frac {b c \log (1-c x)}{2 x} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 43
Rule 44
Rule 2301
Rule 2374
Rule 2390
Rule 2391
Rule 2395
Rule 2396
Rule 2398
Rule 2410
Rule 2433
Rule 6589
Rule 6591
Rule 6596
Rule 6606
Rule 6742
Rubi steps
\begin {align*} \int \frac {(a+b x) \log (1-c x) \text {Li}_2(c x)}{x^4} \, dx &=-\frac {1}{6} \left (\frac {2 a}{x^3}+\frac {3 b}{x^2}\right ) \log (1-c x) \text {Li}_2(c x)+c \int \left (-\frac {a \text {Li}_2(c x)}{3 x^3}+\frac {(-3 b-2 a c) \text {Li}_2(c x)}{6 x^2}-\frac {c (3 b+2 a c) \text {Li}_2(c x)}{6 x}+\frac {c^2 (3 b+2 a c) \text {Li}_2(c x)}{6 (-1+c x)}\right ) \, dx+\int \left (-\frac {a \log ^2(1-c x)}{3 x^4}-\frac {b \log ^2(1-c x)}{2 x^3}\right ) \, dx\\ &=-\frac {1}{6} \left (\frac {2 a}{x^3}+\frac {3 b}{x^2}\right ) \log (1-c x) \text {Li}_2(c x)-\frac {1}{3} a \int \frac {\log ^2(1-c x)}{x^4} \, dx-\frac {1}{2} b \int \frac {\log ^2(1-c x)}{x^3} \, dx-\frac {1}{3} (a c) \int \frac {\text {Li}_2(c x)}{x^3} \, dx-\frac {1}{6} (c (3 b+2 a c)) \int \frac {\text {Li}_2(c x)}{x^2} \, dx-\frac {1}{6} \left (c^2 (3 b+2 a c)\right ) \int \frac {\text {Li}_2(c x)}{x} \, dx+\frac {1}{6} \left (c^3 (3 b+2 a c)\right ) \int \frac {\text {Li}_2(c x)}{-1+c x} \, dx\\ &=\frac {a \log ^2(1-c x)}{9 x^3}+\frac {b \log ^2(1-c x)}{4 x^2}+\frac {a c \text {Li}_2(c x)}{6 x^2}+\frac {c (3 b+2 a c) \text {Li}_2(c x)}{6 x}+\frac {1}{6} c^2 (3 b+2 a c) \log (1-c x) \text {Li}_2(c x)-\frac {1}{6} \left (\frac {2 a}{x^3}+\frac {3 b}{x^2}\right ) \log (1-c x) \text {Li}_2(c x)-\frac {1}{6} c^2 (3 b+2 a c) \text {Li}_3(c x)+\frac {1}{6} (a c) \int \frac {\log (1-c x)}{x^3} \, dx+\frac {1}{9} (2 a c) \int \frac {\log (1-c x)}{x^3 (1-c x)} \, dx+\frac {1}{2} (b c) \int \frac {\log (1-c x)}{x^2 (1-c x)} \, dx+\frac {1}{6} (c (3 b+2 a c)) \int \frac {\log (1-c x)}{x^2} \, dx+\frac {1}{6} \left (c^2 (3 b+2 a c)\right ) \int \frac {\log ^2(1-c x)}{x} \, dx\\ &=-\frac {a c \log (1-c x)}{12 x^2}-\frac {c (3 b+2 a c) \log (1-c x)}{6 x}+\frac {a \log ^2(1-c x)}{9 x^3}+\frac {b \log ^2(1-c x)}{4 x^2}+\frac {1}{6} c^2 (3 b+2 a c) \log (c x) \log ^2(1-c x)+\frac {a c \text {Li}_2(c x)}{6 x^2}+\frac {c (3 b+2 a c) \text {Li}_2(c x)}{6 x}+\frac {1}{6} c^2 (3 b+2 a c) \log (1-c x) \text {Li}_2(c x)-\frac {1}{6} \left (\frac {2 a}{x^3}+\frac {3 b}{x^2}\right ) \log (1-c x) \text {Li}_2(c x)-\frac {1}{6} c^2 (3 b+2 a c) \text {Li}_3(c x)+\frac {1}{9} (2 a c) \int \left (\frac {\log (1-c x)}{x^3}+\frac {c \log (1-c x)}{x^2}+\frac {c^2 \log (1-c x)}{x}-\frac {c^3 \log (1-c x)}{-1+c x}\right ) \, dx+\frac {1}{2} (b c) \int \left (\frac {\log (1-c x)}{x^2}+\frac {c \log (1-c x)}{x}-\frac {c^2 \log (1-c x)}{-1+c x}\right ) \, dx-\frac {1}{12} \left (a c^2\right ) \int \frac {1}{x^2 (1-c x)} \, dx-\frac {1}{6} \left (c^2 (3 b+2 a c)\right ) \int \frac {1}{x (1-c x)} \, dx+\frac {1}{3} \left (c^3 (3 b+2 a c)\right ) \int \frac {\log (c x) \log (1-c x)}{1-c x} \, dx\\ &=-\frac {a c \log (1-c x)}{12 x^2}-\frac {c (3 b+2 a c) \log (1-c x)}{6 x}+\frac {a \log ^2(1-c x)}{9 x^3}+\frac {b \log ^2(1-c x)}{4 x^2}+\frac {1}{6} c^2 (3 b+2 a c) \log (c x) \log ^2(1-c x)+\frac {a c \text {Li}_2(c x)}{6 x^2}+\frac {c (3 b+2 a c) \text {Li}_2(c x)}{6 x}+\frac {1}{6} c^2 (3 b+2 a c) \log (1-c x) \text {Li}_2(c x)-\frac {1}{6} \left (\frac {2 a}{x^3}+\frac {3 b}{x^2}\right ) \log (1-c x) \text {Li}_2(c x)-\frac {1}{6} c^2 (3 b+2 a c) \text {Li}_3(c x)+\frac {1}{9} (2 a c) \int \frac {\log (1-c x)}{x^3} \, dx+\frac {1}{2} (b c) \int \frac {\log (1-c x)}{x^2} \, dx-\frac {1}{12} \left (a c^2\right ) \int \left (\frac {1}{x^2}+\frac {c}{x}-\frac {c^2}{-1+c x}\right ) \, dx+\frac {1}{9} \left (2 a c^2\right ) \int \frac {\log (1-c x)}{x^2} \, dx+\frac {1}{2} \left (b c^2\right ) \int \frac {\log (1-c x)}{x} \, dx+\frac {1}{9} \left (2 a c^3\right ) \int \frac {\log (1-c x)}{x} \, dx-\frac {1}{2} \left (b c^3\right ) \int \frac {\log (1-c x)}{-1+c x} \, dx-\frac {1}{9} \left (2 a c^4\right ) \int \frac {\log (1-c x)}{-1+c x} \, dx-\frac {1}{6} \left (c^2 (3 b+2 a c)\right ) \int \frac {1}{x} \, dx-\frac {1}{3} \left (c^2 (3 b+2 a c)\right ) \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}{6} \left (c^3 (3 b+2 a c)\right ) \int \frac {1}{1-c x} \, dx\\ &=\frac {a c^2}{12 x}-\frac {1}{12} a c^3 \log (x)-\frac {1}{6} c^2 (3 b+2 a c) \log (x)+\frac {1}{12} a c^3 \log (1-c x)+\frac {1}{6} c^2 (3 b+2 a c) \log (1-c x)-\frac {7 a c \log (1-c x)}{36 x^2}-\frac {b c \log (1-c x)}{2 x}-\frac {2 a c^2 \log (1-c x)}{9 x}-\frac {c (3 b+2 a c) \log (1-c x)}{6 x}+\frac {a \log ^2(1-c x)}{9 x^3}+\frac {b \log ^2(1-c x)}{4 x^2}+\frac {1}{6} c^2 (3 b+2 a c) \log (c x) \log ^2(1-c x)-\frac {1}{2} b c^2 \text {Li}_2(c x)-\frac {2}{9} a c^3 \text {Li}_2(c x)+\frac {a c \text {Li}_2(c x)}{6 x^2}+\frac {c (3 b+2 a c) \text {Li}_2(c x)}{6 x}+\frac {1}{6} c^2 (3 b+2 a c) \log (1-c x) \text {Li}_2(c x)-\frac {1}{6} \left (\frac {2 a}{x^3}+\frac {3 b}{x^2}\right ) \log (1-c x) \text {Li}_2(c x)+\frac {1}{3} c^2 (3 b+2 a c) \log (1-c x) \text {Li}_2(1-c x)-\frac {1}{6} c^2 (3 b+2 a c) \text {Li}_3(c x)-\frac {1}{9} \left (a c^2\right ) \int \frac {1}{x^2 (1-c x)} \, dx-\frac {1}{2} \left (b c^2\right ) \int \frac {1}{x (1-c x)} \, dx-\frac {1}{2} \left (b c^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,1-c x\right )-\frac {1}{9} \left (2 a c^3\right ) \int \frac {1}{x (1-c x)} \, dx-\frac {1}{9} \left (2 a c^3\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,1-c x\right )-\frac {1}{3} \left (c^2 (3 b+2 a c)\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,1-c x\right )\\ &=\frac {a c^2}{12 x}-\frac {1}{12} a c^3 \log (x)-\frac {1}{6} c^2 (3 b+2 a c) \log (x)+\frac {1}{12} a c^3 \log (1-c x)+\frac {1}{6} c^2 (3 b+2 a c) \log (1-c x)-\frac {7 a c \log (1-c x)}{36 x^2}-\frac {b c \log (1-c x)}{2 x}-\frac {2 a c^2 \log (1-c x)}{9 x}-\frac {c (3 b+2 a c) \log (1-c x)}{6 x}-\frac {1}{4} b c^2 \log ^2(1-c x)-\frac {1}{9} a c^3 \log ^2(1-c x)+\frac {a \log ^2(1-c x)}{9 x^3}+\frac {b \log ^2(1-c x)}{4 x^2}+\frac {1}{6} c^2 (3 b+2 a c) \log (c x) \log ^2(1-c x)-\frac {1}{2} b c^2 \text {Li}_2(c x)-\frac {2}{9} a c^3 \text {Li}_2(c x)+\frac {a c \text {Li}_2(c x)}{6 x^2}+\frac {c (3 b+2 a c) \text {Li}_2(c x)}{6 x}+\frac {1}{6} c^2 (3 b+2 a c) \log (1-c x) \text {Li}_2(c x)-\frac {1}{6} \left (\frac {2 a}{x^3}+\frac {3 b}{x^2}\right ) \log (1-c x) \text {Li}_2(c x)+\frac {1}{3} c^2 (3 b+2 a c) \log (1-c x) \text {Li}_2(1-c x)-\frac {1}{6} c^2 (3 b+2 a c) \text {Li}_3(c x)-\frac {1}{3} c^2 (3 b+2 a c) \text {Li}_3(1-c x)-\frac {1}{9} \left (a c^2\right ) \int \left (\frac {1}{x^2}+\frac {c}{x}-\frac {c^2}{-1+c x}\right ) \, dx-\frac {1}{2} \left (b c^2\right ) \int \frac {1}{x} \, dx-\frac {1}{9} \left (2 a c^3\right ) \int \frac {1}{x} \, dx-\frac {1}{2} \left (b c^3\right ) \int \frac {1}{1-c x} \, dx-\frac {1}{9} \left (2 a c^4\right ) \int \frac {1}{1-c x} \, dx\\ &=\frac {7 a c^2}{36 x}-\frac {1}{2} b c^2 \log (x)-\frac {5}{12} a c^3 \log (x)-\frac {1}{6} c^2 (3 b+2 a c) \log (x)+\frac {1}{2} b c^2 \log (1-c x)+\frac {5}{12} a c^3 \log (1-c x)+\frac {1}{6} c^2 (3 b+2 a c) \log (1-c x)-\frac {7 a c \log (1-c x)}{36 x^2}-\frac {b c \log (1-c x)}{2 x}-\frac {2 a c^2 \log (1-c x)}{9 x}-\frac {c (3 b+2 a c) \log (1-c x)}{6 x}-\frac {1}{4} b c^2 \log ^2(1-c x)-\frac {1}{9} a c^3 \log ^2(1-c x)+\frac {a \log ^2(1-c x)}{9 x^3}+\frac {b \log ^2(1-c x)}{4 x^2}+\frac {1}{6} c^2 (3 b+2 a c) \log (c x) \log ^2(1-c x)-\frac {1}{2} b c^2 \text {Li}_2(c x)-\frac {2}{9} a c^3 \text {Li}_2(c x)+\frac {a c \text {Li}_2(c x)}{6 x^2}+\frac {c (3 b+2 a c) \text {Li}_2(c x)}{6 x}+\frac {1}{6} c^2 (3 b+2 a c) \log (1-c x) \text {Li}_2(c x)-\frac {1}{6} \left (\frac {2 a}{x^3}+\frac {3 b}{x^2}\right ) \log (1-c x) \text {Li}_2(c x)+\frac {1}{3} c^2 (3 b+2 a c) \log (1-c x) \text {Li}_2(1-c x)-\frac {1}{6} c^2 (3 b+2 a c) \text {Li}_3(c x)-\frac {1}{3} c^2 (3 b+2 a c) \text {Li}_3(1-c x)\\ \end {align*}
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Mathematica [A] time = 1.36, size = 389, normalized size = 0.85 \[ \frac {1}{36} \left (2 c^2 \text {Li}_2(1-c x) (6 (2 a c+3 b) \log (1-c x)+4 a c+9 b)+\frac {6 \text {Li}_2(c x) \left (\log (1-c x) \left (2 a c^3 x^3-2 a+3 b c^2 x^3-3 b x\right )+c x (2 a c x+a+3 b x)\right )}{x^3}-12 a c^3 \text {Li}_3(c x)-24 a c^3 \text {Li}_3(1-c x)-4 a c^3 \log ^2(1-c x)+12 a c^3 \log (c x) \log ^2(1-c x)-27 a c^3 \log (c x)+27 a c^3 \log (1-c x)+8 a c^3 \log (c x) \log (1-c x)-7 a c^3+\frac {7 a c^2}{x}-\frac {20 a c^2 \log (1-c x)}{x}+\frac {4 a \log ^2(1-c x)}{x^3}-\frac {7 a c \log (1-c x)}{x^2}-18 b c^2 \text {Li}_3(c x)-36 b c^2 \text {Li}_3(1-c x)-9 b c^2 \log ^2(1-c x)+18 b c^2 \log (c x) \log ^2(1-c x)-36 b c^2 \log (c x)+36 b c^2 \log (1-c x)+18 b c^2 \log (c x) \log (1-c x)+\frac {9 b \log ^2(1-c x)}{x^2}-\frac {36 b c \log (1-c x)}{x}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x + a\right )} {\rm Li}_2\left (c x\right ) \log \left (-c x + 1\right )}{x^{4}}, x\right ) \]
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 \[ \int \frac {{\left (b x + a\right )} {\rm Li}_2\left (c x\right ) \log \left (-c x + 1\right )}{x^{4}}\,{d x} \]
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 \[ \int \frac {\left (b x +a \right ) \ln \left (-c x +1\right ) \polylog \left (2, c x \right )}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 287, normalized size = 0.62 \[ \frac {1}{6} \, {\left (2 \, a c^{3} + 3 \, b c^{2}\right )} {\left (\log \left (c x\right ) \log \left (-c x + 1\right )^{2} + 2 \, {\rm Li}_2\left (-c x + 1\right ) \log \left (-c x + 1\right ) - 2 \, {\rm Li}_{3}(-c x + 1)\right )} + \frac {1}{18} \, {\left (4 \, a c^{3} + 9 \, b c^{2}\right )} {\left (\log \left (c x\right ) \log \left (-c x + 1\right ) + {\rm Li}_2\left (-c x + 1\right )\right )} - \frac {1}{4} \, {\left (3 \, a c^{3} + 4 \, b c^{2}\right )} \log \relax (x) - \frac {1}{6} \, {\left (2 \, a c^{3} + 3 \, b c^{2}\right )} {\rm Li}_{3}(c x) + \frac {7 \, a c^{2} x^{2} - {\left ({\left (4 \, a c^{3} + 9 \, b c^{2}\right )} x^{3} - 9 \, b x - 4 \, a\right )} \log \left (-c x + 1\right )^{2} + 6 \, {\left (a c x + {\left (2 \, a c^{2} + 3 \, b c\right )} x^{2} + {\left ({\left (2 \, a c^{3} + 3 \, b c^{2}\right )} x^{3} - 3 \, b x - 2 \, a\right )} \log \left (-c x + 1\right )\right )} {\rm Li}_2\left (c x\right ) + {\left (9 \, {\left (3 \, a c^{3} + 4 \, b c^{2}\right )} x^{3} - 7 \, a c x - 4 \, {\left (5 \, a c^{2} + 9 \, b c\right )} x^{2}\right )} \log \left (-c x + 1\right )}{36 \, x^{3}} \]
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 \[ \int \frac {\ln \left (1-c\,x\right )\,\mathrm {polylog}\left (2,c\,x\right )\,\left (a+b\,x\right )}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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