Optimal. Leaf size=460 \[ -\frac{1}{6} c^2 (2 a c+3 b) \text{PolyLog}(3,c x)-\frac{1}{3} c^2 (2 a c+3 b) \text{PolyLog}(3,1-c x)+\frac{1}{6} c^2 (2 a c+3 b) \log (1-c x) \text{PolyLog}(2,c x)+\frac{1}{3} c^2 (2 a c+3 b) \log (1-c x) \text{PolyLog}(2,1-c x)-\frac{1}{6} \left (\frac{2 a}{x^3}+\frac{3 b}{x^2}\right ) \log (1-c x) \text{PolyLog}(2,c x)+\frac{c (2 a c+3 b) \text{PolyLog}(2,c x)}{6 x}-\frac{2}{9} a c^3 \text{PolyLog}(2,c x)+\frac{a c \text{PolyLog}(2,c x)}{6 x^2}-\frac{1}{2} b c^2 \text{PolyLog}(2,c x)+\frac{1}{6} c^2 (2 a c+3 b) \log (c x) \log ^2(1-c x)-\frac{1}{6} c^2 \log (x) (2 a c+3 b)+\frac{1}{6} c^2 (2 a c+3 b) \log (1-c x)-\frac{c (2 a c+3 b) \log (1-c x)}{6 x}+\frac{7 a c^2}{36 x}-\frac{1}{9} a c^3 \log ^2(1-c x)-\frac{5}{12} a c^3 \log (x)+\frac{5}{12} a c^3 \log (1-c x)-\frac{2 a c^2 \log (1-c x)}{9 x}+\frac{a \log ^2(1-c x)}{9 x^3}-\frac{7 a c \log (1-c x)}{36 x^2}-\frac{1}{4} b c^2 \log ^2(1-c x)-\frac{1}{2} b c^2 \log (x)+\frac{1}{2} b c^2 \log (1-c x)+\frac{b \log ^2(1-c x)}{4 x^2}-\frac{b c \log (1-c x)}{2 x} \]
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Rubi [A] time = 0.671801, antiderivative size = 460, normalized size of antiderivative = 1., 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 6742
Rule 6591
Rule 2395
Rule 44
Rule 43
Rule 6606
Rule 2398
Rule 2410
Rule 36
Rule 29
Rule 31
Rule 2391
Rule 2390
Rule 2301
Rule 6589
Rule 6596
Rule 2396
Rule 2433
Rule 2374
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*}
Mathematica [A] time = 1.30438, size = 389, normalized size = 0.85 \[ \frac{1}{36} \left (\frac{6 \text{PolyLog}(2,c x) \left (\log (1-c x) \left (2 a c^3 x^3-2 a+3 b c^2 x^3-3 b x\right )+c x (2 a c x+a+3 b x)\right )}{x^3}+2 c^2 \text{PolyLog}(2,1-c x) (6 (2 a c+3 b) \log (1-c x)+4 a c+9 b)-12 a c^3 \text{PolyLog}(3,c x)-24 a c^3 \text{PolyLog}(3,1-c x)-18 b c^2 \text{PolyLog}(3,c x)-36 b c^2 \text{PolyLog}(3,1-c x)+\frac{7 a c^2}{x}-4 a c^3 \log ^2(1-c x)+12 a c^3 \log (c x) \log ^2(1-c x)-27 a c^3 \log (c x)+27 a c^3 \log (1-c x)+8 a c^3 \log (c x) \log (1-c x)-\frac{20 a c^2 \log (1-c x)}{x}-7 a c^3+\frac{4 a \log ^2(1-c x)}{x^3}-\frac{7 a c \log (1-c x)}{x^2}-9 b c^2 \log ^2(1-c x)+18 b c^2 \log (c x) \log ^2(1-c x)-36 b c^2 \log (c x)+36 b c^2 \log (1-c x)+18 b c^2 \log (c x) \log (1-c x)+\frac{9 b \log ^2(1-c x)}{x^2}-\frac{36 b c \log (1-c x)}{x}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.006, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) \ln \left ( -cx+1 \right ){\it polylog} \left ( 2,cx \right ) }{{x}^{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.20069, size = 387, normalized size = 0.84 \begin{align*} \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 \left (x\right ) - \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}} \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{{\left (b x + a\right )}{\rm Li}_2\left (c x\right ) \log \left (-c x + 1\right )}{x^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{{\left (b x + a\right )}{\rm Li}_2\left (c x\right ) \log \left (-c x + 1\right )}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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