Optimal. Leaf size=645 \[ \frac {\text {Li}_3(1-d x) (3 d (2 a d+b)+2 c)}{3 d^3}-\frac {\text {Li}_2(d x) \log (1-d x) (3 d (2 a d+b)+2 c)}{6 d^3}-\frac {\text {Li}_2(1-d x) \log (1-d x) (3 d (2 a d+b)+2 c)}{3 d^3}-\frac {\log (d x) \log ^2(1-d x) (3 d (2 a d+b)+2 c)}{6 d^3}+\frac {(1-d x) \log (1-d x) (3 d (2 a d+b)+2 c)}{6 d^3}-\frac {x \text {Li}_2(d x) (3 d (2 a d+b)+2 c)}{6 d^2}+\frac {x (3 d (2 a d+b)+2 c)}{6 d^2}+\frac {1}{6} \text {Li}_2(d x) \log (1-d x) \left (6 a x+3 b x^2+2 c x^3\right )-\frac {a (1-d x) \log ^2(1-d x)}{d}+\frac {2 a (1-d x) \log (1-d x)}{d}+2 a x+\frac {(3 b d+2 c) \log (1-d x)}{24 d^3}+\frac {x (3 b d+2 c)}{24 d^2}-\frac {x^2 (3 b d+2 c) \text {Li}_2(d x)}{12 d}+\frac {x^2 (3 b d+2 c)}{48 d}-\frac {x^2 (3 b d+2 c) \log (1-d x)}{24 d}+\frac {b (1-d x)^2}{8 d^2}+\frac {b (1-d x)^2 \log ^2(1-d x)}{4 d^2}-\frac {b (1-d x) \log ^2(1-d x)}{2 d^2}-\frac {b (1-d x)^2 \log (1-d x)}{4 d^2}+\frac {b (1-d x) \log (1-d x)}{d^2}+\frac {b x}{d}-\frac {c \log ^2(1-d x)}{9 d^3}+\frac {2 c (1-d x) \log (1-d x)}{9 d^3}+\frac {2 c \log (1-d x)}{9 d^3}+\frac {4 c x}{9 d^2}-\frac {1}{9} c x^3 \text {Li}_2(d x)+\frac {1}{9} c x^3 \log ^2(1-d x)-\frac {1}{9} c x^3 \log (1-d x)+\frac {c x^2}{9 d}-\frac {c x^2 \log (1-d x)}{9 d}+\frac {c x^3}{27} \]
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Rubi [A] time = 0.82, antiderivative size = 645, normalized size of antiderivative = 1.00, number of steps used = 43, number of rules used = 21, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.913, Rules used = {6742, 6586, 2389, 2295, 6591, 2395, 43, 6604, 2296, 2401, 2390, 2305, 2304, 2398, 2410, 2301, 6596, 2396, 2433, 2374, 6589} \[ -\frac {x \text {PolyLog}(2,d x) (3 d (2 a d+b)+2 c)}{6 d^2}+\frac {\text {PolyLog}(3,1-d x) (3 d (2 a d+b)+2 c)}{3 d^3}-\frac {\log (1-d x) \text {PolyLog}(2,d x) (3 d (2 a d+b)+2 c)}{6 d^3}-\frac {\log (1-d x) \text {PolyLog}(2,1-d x) (3 d (2 a d+b)+2 c)}{3 d^3}+\frac {1}{6} \log (1-d x) \text {PolyLog}(2,d x) \left (6 a x+3 b x^2+2 c x^3\right )-\frac {x^2 (3 b d+2 c) \text {PolyLog}(2,d x)}{12 d}-\frac {1}{9} c x^3 \text {PolyLog}(2,d x)+\frac {x (3 d (2 a d+b)+2 c)}{6 d^2}-\frac {\log (d x) \log ^2(1-d x) (3 d (2 a d+b)+2 c)}{6 d^3}+\frac {(1-d x) \log (1-d x) (3 d (2 a d+b)+2 c)}{6 d^3}-\frac {a (1-d x) \log ^2(1-d x)}{d}+\frac {2 a (1-d x) \log (1-d x)}{d}+2 a x+\frac {x (3 b d+2 c)}{24 d^2}+\frac {(3 b d+2 c) \log (1-d x)}{24 d^3}+\frac {x^2 (3 b d+2 c)}{48 d}-\frac {x^2 (3 b d+2 c) \log (1-d x)}{24 d}+\frac {b (1-d x)^2}{8 d^2}+\frac {b (1-d x)^2 \log ^2(1-d x)}{4 d^2}-\frac {b (1-d x) \log ^2(1-d x)}{2 d^2}-\frac {b (1-d x)^2 \log (1-d x)}{4 d^2}+\frac {b (1-d x) \log (1-d x)}{d^2}+\frac {b x}{d}+\frac {4 c x}{9 d^2}-\frac {c \log ^2(1-d x)}{9 d^3}+\frac {2 c (1-d x) \log (1-d x)}{9 d^3}+\frac {2 c \log (1-d x)}{9 d^3}+\frac {c x^2}{9 d}+\frac {1}{9} c x^3 \log ^2(1-d x)-\frac {1}{9} c x^3 \log (1-d x)-\frac {c x^2 \log (1-d x)}{9 d}+\frac {c x^3}{27} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2295
Rule 2296
Rule 2301
Rule 2304
Rule 2305
Rule 2374
Rule 2389
Rule 2390
Rule 2395
Rule 2396
Rule 2398
Rule 2401
Rule 2410
Rule 2433
Rule 6586
Rule 6589
Rule 6591
Rule 6596
Rule 6604
Rule 6742
Rubi steps
\begin {align*} \int \left (a+b x+c x^2\right ) \log (1-d x) \text {Li}_2(d x) \, dx &=\frac {1}{6} \left (6 a x+3 b x^2+2 c x^3\right ) \log (1-d x) \text {Li}_2(d x)+d \int \left (\frac {(-2 c-3 d (b+2 a d)) \text {Li}_2(d x)}{6 d^3}-\frac {(2 c+3 b d) x \text {Li}_2(d x)}{6 d^2}-\frac {c x^2 \text {Li}_2(d x)}{3 d}+\frac {(2 c+3 d (b+2 a d)) \text {Li}_2(d x)}{6 d^3 (1-d x)}\right ) \, dx+\int \left (a \log ^2(1-d x)+\frac {1}{2} b x \log ^2(1-d x)+\frac {1}{3} c x^2 \log ^2(1-d x)\right ) \, dx\\ &=\frac {1}{6} \left (6 a x+3 b x^2+2 c x^3\right ) \log (1-d x) \text {Li}_2(d x)+a \int \log ^2(1-d x) \, dx+\frac {1}{2} b \int x \log ^2(1-d x) \, dx+\frac {1}{3} c \int x^2 \log ^2(1-d x) \, dx-\frac {1}{3} c \int x^2 \text {Li}_2(d x) \, dx-\frac {(2 c+3 b d) \int x \text {Li}_2(d x) \, dx}{6 d}-\frac {(2 c+3 d (b+2 a d)) \int \text {Li}_2(d x) \, dx}{6 d^2}+\frac {(2 c+3 d (b+2 a d)) \int \frac {\text {Li}_2(d x)}{1-d x} \, dx}{6 d^2}\\ &=\frac {1}{9} c x^3 \log ^2(1-d x)-\frac {(2 c+3 d (b+2 a d)) x \text {Li}_2(d x)}{6 d^2}-\frac {(2 c+3 b d) x^2 \text {Li}_2(d x)}{12 d}-\frac {1}{9} c x^3 \text {Li}_2(d x)-\frac {(2 c+3 d (b+2 a d)) \log (1-d x) \text {Li}_2(d x)}{6 d^3}+\frac {1}{6} \left (6 a x+3 b x^2+2 c x^3\right ) \log (1-d x) \text {Li}_2(d x)+\frac {1}{2} b \int \left (\frac {\log ^2(1-d x)}{d}-\frac {(1-d x) \log ^2(1-d x)}{d}\right ) \, dx-\frac {1}{9} c \int x^2 \log (1-d x) \, dx-\frac {a \operatorname {Subst}\left (\int \log ^2(x) \, dx,x,1-d x\right )}{d}+\frac {1}{9} (2 c d) \int \frac {x^3 \log (1-d x)}{1-d x} \, dx-\frac {(2 c+3 b d) \int x \log (1-d x) \, dx}{12 d}-\frac {(2 c+3 d (b+2 a d)) \int \frac {\log ^2(1-d x)}{x} \, dx}{6 d^3}-\frac {(2 c+3 d (b+2 a d)) \int \log (1-d x) \, dx}{6 d^2}\\ &=-\frac {(2 c+3 b d) x^2 \log (1-d x)}{24 d}-\frac {1}{27} c x^3 \log (1-d x)+\frac {1}{9} c x^3 \log ^2(1-d x)-\frac {a (1-d x) \log ^2(1-d x)}{d}-\frac {(2 c+3 d (b+2 a d)) \log (d x) \log ^2(1-d x)}{6 d^3}-\frac {(2 c+3 d (b+2 a d)) x \text {Li}_2(d x)}{6 d^2}-\frac {(2 c+3 b d) x^2 \text {Li}_2(d x)}{12 d}-\frac {1}{9} c x^3 \text {Li}_2(d x)-\frac {(2 c+3 d (b+2 a d)) \log (1-d x) \text {Li}_2(d x)}{6 d^3}+\frac {1}{6} \left (6 a x+3 b x^2+2 c x^3\right ) \log (1-d x) \text {Li}_2(d x)+\frac {(2 a) \operatorname {Subst}(\int \log (x) \, dx,x,1-d x)}{d}+\frac {b \int \log ^2(1-d x) \, dx}{2 d}-\frac {b \int (1-d x) \log ^2(1-d x) \, dx}{2 d}-\frac {1}{27} (c d) \int \frac {x^3}{1-d x} \, dx+\frac {1}{9} (2 c d) \int \left (-\frac {\log (1-d x)}{d^3}-\frac {x \log (1-d x)}{d^2}-\frac {x^2 \log (1-d x)}{d}-\frac {\log (1-d x)}{d^3 (-1+d x)}\right ) \, dx-\frac {1}{24} (2 c+3 b d) \int \frac {x^2}{1-d x} \, dx+\frac {(2 c+3 d (b+2 a d)) \operatorname {Subst}(\int \log (x) \, dx,x,1-d x)}{6 d^3}-\frac {(2 c+3 d (b+2 a d)) \int \frac {\log (d x) \log (1-d x)}{1-d x} \, dx}{3 d^2}\\ &=2 a x+\frac {(2 c+3 d (b+2 a d)) x}{6 d^2}-\frac {(2 c+3 b d) x^2 \log (1-d x)}{24 d}-\frac {1}{27} c x^3 \log (1-d x)+\frac {2 a (1-d x) \log (1-d x)}{d}+\frac {(2 c+3 d (b+2 a d)) (1-d x) \log (1-d x)}{6 d^3}+\frac {1}{9} c x^3 \log ^2(1-d x)-\frac {a (1-d x) \log ^2(1-d x)}{d}-\frac {(2 c+3 d (b+2 a d)) \log (d x) \log ^2(1-d x)}{6 d^3}-\frac {(2 c+3 d (b+2 a d)) x \text {Li}_2(d x)}{6 d^2}-\frac {(2 c+3 b d) x^2 \text {Li}_2(d x)}{12 d}-\frac {1}{9} c x^3 \text {Li}_2(d x)-\frac {(2 c+3 d (b+2 a d)) \log (1-d x) \text {Li}_2(d x)}{6 d^3}+\frac {1}{6} \left (6 a x+3 b x^2+2 c x^3\right ) \log (1-d x) \text {Li}_2(d x)-\frac {1}{9} (2 c) \int x^2 \log (1-d x) \, dx-\frac {b \operatorname {Subst}\left (\int \log ^2(x) \, dx,x,1-d x\right )}{2 d^2}+\frac {b \operatorname {Subst}\left (\int x \log ^2(x) \, dx,x,1-d x\right )}{2 d^2}-\frac {(2 c) \int \log (1-d x) \, dx}{9 d^2}-\frac {(2 c) \int \frac {\log (1-d x)}{-1+d x} \, dx}{9 d^2}-\frac {(2 c) \int x \log (1-d x) \, dx}{9 d}-\frac {1}{27} (c d) \int \left (-\frac {1}{d^3}-\frac {x}{d^2}-\frac {x^2}{d}-\frac {1}{d^3 (-1+d x)}\right ) \, dx-\frac {1}{24} (2 c+3 b d) \int \left (-\frac {1}{d^2}-\frac {x}{d}-\frac {1}{d^2 (-1+d x)}\right ) \, dx+\frac {(2 c+3 d (b+2 a d)) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (d \left (\frac {1}{d}-\frac {x}{d}\right )\right )}{x} \, dx,x,1-d x\right )}{3 d^3}\\ &=2 a x+\frac {c x}{27 d^2}+\frac {(2 c+3 b d) x}{24 d^2}+\frac {(2 c+3 d (b+2 a d)) x}{6 d^2}+\frac {c x^2}{54 d}+\frac {(2 c+3 b d) x^2}{48 d}+\frac {c x^3}{81}+\frac {c \log (1-d x)}{27 d^3}+\frac {(2 c+3 b d) \log (1-d x)}{24 d^3}-\frac {c x^2 \log (1-d x)}{9 d}-\frac {(2 c+3 b d) x^2 \log (1-d x)}{24 d}-\frac {1}{9} c x^3 \log (1-d x)+\frac {2 a (1-d x) \log (1-d x)}{d}+\frac {(2 c+3 d (b+2 a d)) (1-d x) \log (1-d x)}{6 d^3}+\frac {1}{9} c x^3 \log ^2(1-d x)-\frac {b (1-d x) \log ^2(1-d x)}{2 d^2}-\frac {a (1-d x) \log ^2(1-d x)}{d}+\frac {b (1-d x)^2 \log ^2(1-d x)}{4 d^2}-\frac {(2 c+3 d (b+2 a d)) \log (d x) \log ^2(1-d x)}{6 d^3}-\frac {(2 c+3 d (b+2 a d)) x \text {Li}_2(d x)}{6 d^2}-\frac {(2 c+3 b d) x^2 \text {Li}_2(d x)}{12 d}-\frac {1}{9} c x^3 \text {Li}_2(d x)-\frac {(2 c+3 d (b+2 a d)) \log (1-d x) \text {Li}_2(d x)}{6 d^3}+\frac {1}{6} \left (6 a x+3 b x^2+2 c x^3\right ) \log (1-d x) \text {Li}_2(d x)-\frac {(2 c+3 d (b+2 a d)) \log (1-d x) \text {Li}_2(1-d x)}{3 d^3}-\frac {1}{9} c \int \frac {x^2}{1-d x} \, dx+\frac {(2 c) \operatorname {Subst}(\int \log (x) \, dx,x,1-d x)}{9 d^3}-\frac {(2 c) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,1-d x\right )}{9 d^3}-\frac {b \operatorname {Subst}(\int x \log (x) \, dx,x,1-d x)}{2 d^2}+\frac {b \operatorname {Subst}(\int \log (x) \, dx,x,1-d x)}{d^2}-\frac {1}{27} (2 c d) \int \frac {x^3}{1-d x} \, dx+\frac {(2 c+3 d (b+2 a d)) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,1-d x\right )}{3 d^3}\\ &=2 a x+\frac {7 c x}{27 d^2}+\frac {b x}{d}+\frac {(2 c+3 b d) x}{24 d^2}+\frac {(2 c+3 d (b+2 a d)) x}{6 d^2}+\frac {c x^2}{54 d}+\frac {(2 c+3 b d) x^2}{48 d}+\frac {c x^3}{81}+\frac {b (1-d x)^2}{8 d^2}+\frac {c \log (1-d x)}{27 d^3}+\frac {(2 c+3 b d) \log (1-d x)}{24 d^3}-\frac {c x^2 \log (1-d x)}{9 d}-\frac {(2 c+3 b d) x^2 \log (1-d x)}{24 d}-\frac {1}{9} c x^3 \log (1-d x)+\frac {2 c (1-d x) \log (1-d x)}{9 d^3}+\frac {b (1-d x) \log (1-d x)}{d^2}+\frac {2 a (1-d x) \log (1-d x)}{d}+\frac {(2 c+3 d (b+2 a d)) (1-d x) \log (1-d x)}{6 d^3}-\frac {b (1-d x)^2 \log (1-d x)}{4 d^2}-\frac {c \log ^2(1-d x)}{9 d^3}+\frac {1}{9} c x^3 \log ^2(1-d x)-\frac {b (1-d x) \log ^2(1-d x)}{2 d^2}-\frac {a (1-d x) \log ^2(1-d x)}{d}+\frac {b (1-d x)^2 \log ^2(1-d x)}{4 d^2}-\frac {(2 c+3 d (b+2 a d)) \log (d x) \log ^2(1-d x)}{6 d^3}-\frac {(2 c+3 d (b+2 a d)) x \text {Li}_2(d x)}{6 d^2}-\frac {(2 c+3 b d) x^2 \text {Li}_2(d x)}{12 d}-\frac {1}{9} c x^3 \text {Li}_2(d x)-\frac {(2 c+3 d (b+2 a d)) \log (1-d x) \text {Li}_2(d x)}{6 d^3}+\frac {1}{6} \left (6 a x+3 b x^2+2 c x^3\right ) \log (1-d x) \text {Li}_2(d x)-\frac {(2 c+3 d (b+2 a d)) \log (1-d x) \text {Li}_2(1-d x)}{3 d^3}+\frac {(2 c+3 d (b+2 a d)) \text {Li}_3(1-d x)}{3 d^3}-\frac {1}{9} c \int \left (-\frac {1}{d^2}-\frac {x}{d}-\frac {1}{d^2 (-1+d x)}\right ) \, dx-\frac {1}{27} (2 c d) \int \left (-\frac {1}{d^3}-\frac {x}{d^2}-\frac {x^2}{d}-\frac {1}{d^3 (-1+d x)}\right ) \, dx\\ &=2 a x+\frac {4 c x}{9 d^2}+\frac {b x}{d}+\frac {(2 c+3 b d) x}{24 d^2}+\frac {(2 c+3 d (b+2 a d)) x}{6 d^2}+\frac {c x^2}{9 d}+\frac {(2 c+3 b d) x^2}{48 d}+\frac {c x^3}{27}+\frac {b (1-d x)^2}{8 d^2}+\frac {2 c \log (1-d x)}{9 d^3}+\frac {(2 c+3 b d) \log (1-d x)}{24 d^3}-\frac {c x^2 \log (1-d x)}{9 d}-\frac {(2 c+3 b d) x^2 \log (1-d x)}{24 d}-\frac {1}{9} c x^3 \log (1-d x)+\frac {2 c (1-d x) \log (1-d x)}{9 d^3}+\frac {b (1-d x) \log (1-d x)}{d^2}+\frac {2 a (1-d x) \log (1-d x)}{d}+\frac {(2 c+3 d (b+2 a d)) (1-d x) \log (1-d x)}{6 d^3}-\frac {b (1-d x)^2 \log (1-d x)}{4 d^2}-\frac {c \log ^2(1-d x)}{9 d^3}+\frac {1}{9} c x^3 \log ^2(1-d x)-\frac {b (1-d x) \log ^2(1-d x)}{2 d^2}-\frac {a (1-d x) \log ^2(1-d x)}{d}+\frac {b (1-d x)^2 \log ^2(1-d x)}{4 d^2}-\frac {(2 c+3 d (b+2 a d)) \log (d x) \log ^2(1-d x)}{6 d^3}-\frac {(2 c+3 d (b+2 a d)) x \text {Li}_2(d x)}{6 d^2}-\frac {(2 c+3 b d) x^2 \text {Li}_2(d x)}{12 d}-\frac {1}{9} c x^3 \text {Li}_2(d x)-\frac {(2 c+3 d (b+2 a d)) \log (1-d x) \text {Li}_2(d x)}{6 d^3}+\frac {1}{6} \left (6 a x+3 b x^2+2 c x^3\right ) \log (1-d x) \text {Li}_2(d x)-\frac {(2 c+3 d (b+2 a d)) \log (1-d x) \text {Li}_2(1-d x)}{3 d^3}+\frac {(2 c+3 d (b+2 a d)) \text {Li}_3(1-d x)}{3 d^3}\\ \end {align*}
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Mathematica [A] time = 0.98, size = 472, normalized size = 0.73 \[ \frac {\text {Li}_2(d x) \left (6 (d x-1) \log (1-d x) \left (3 d (2 a d+b d x+b)+2 c \left (d^2 x^2+d x+1\right )\right )-d x \left (9 d (4 a d+b d x+2 b)+2 c \left (2 d^2 x^2+3 d x+6\right )\right )\right )+12 \text {Li}_3(1-d x) (3 d (2 a d+b)+2 c)-12 \text {Li}_2(1-d x) \log (1-d x) (3 d (2 a d+b)+2 c)+108 a d^3 x+36 a d^3 x \log ^2(1-d x)-108 a d^3 x \log (1-d x)-36 a d^2 \log ^2(1-d x)-36 a d^2 \log (d x) \log ^2(1-d x)+108 a d^2 \log (1-d x)+\frac {27}{4} b d^3 x^2+9 b d^3 x^2 \log ^2(1-d x)-\frac {27}{2} b d^3 x^2 \log (1-d x)+\frac {99}{2} b d^2 x-36 b d^2 x \log (1-d x)-9 b d \log ^2(1-d x)-18 b d \log (d x) \log ^2(1-d x)+\frac {99}{2} b d \log (1-d x)+\frac {4}{3} c d^3 x^3+4 c d^3 x^3 \log ^2(1-d x)-4 c d^3 x^3 \log (1-d x)+\frac {11}{2} c d^2 x^2-7 c d^2 x^2 \log (1-d x)+31 c d x-4 c \log ^2(1-d x)-12 c \log (d x) \log ^2(1-d x)-20 c d x \log (1-d x)+31 c \log (1-d x)}{36 d^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 2.00, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c x^{2} + b x + a\right )} {\rm Li}_2\left (d x\right ) \log \left (-d x + 1\right ), x\right ) \]
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 \[ \int {\left (c x^{2} + b x + a\right )} {\rm Li}_2\left (d x\right ) \log \left (-d x + 1\right )\,{d x} \]
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 \[ \int \left (c \,x^{2}+b x +a \right ) \ln \left (-d x +1\right ) \polylog \left (2, d x \right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 412, normalized size = 0.64 \[ -\frac {1}{432} \, d {\left (\frac {72 \, {\left (6 \, a d^{2} + 3 \, b d + 2 \, c\right )} {\left (\log \left (d x\right ) \log \left (-d x + 1\right )^{2} + 2 \, {\rm Li}_2\left (-d x + 1\right ) \log \left (-d x + 1\right ) - 2 \, {\rm Li}_{3}(-d x + 1)\right )}}{d^{4}} - \frac {16 \, c d^{3} x^{3} + 3 \, {\left (27 \, b d^{3} + 22 \, c d^{2}\right )} x^{2} + 6 \, {\left (216 \, a d^{3} + 99 \, b d^{2} + 62 \, c d\right )} x - 12 \, {\left (4 \, c d^{3} x^{3} + 3 \, {\left (3 \, b d^{3} + 2 \, c d^{2}\right )} x^{2} + 6 \, {\left (6 \, a d^{3} + 3 \, b d^{2} + 2 \, c d\right )} x + 6 \, {\left (6 \, a d^{2} + 3 \, b d + 2 \, c\right )} \log \left (-d x + 1\right )\right )} {\rm Li}_2\left (d x\right ) - 2 \, {\left (16 \, c d^{3} x^{3} - 648 \, a d^{2} + 6 \, {\left (9 \, b d^{3} + 5 \, c d^{2}\right )} x^{2} - 297 \, b d + 6 \, {\left (72 \, a d^{3} + 27 \, b d^{2} + 16 \, c d\right )} x - 186 \, c\right )} \log \left (-d x + 1\right )}{d^{4}}\right )} + \frac {1}{216} \, {\left (\frac {216 \, {\left (d x {\rm Li}_2\left (d x\right ) - d x + {\left (d x - 1\right )} \log \left (-d x + 1\right )\right )} a}{d} + \frac {27 \, {\left (4 \, d^{2} x^{2} {\rm Li}_2\left (d x\right ) - d^{2} x^{2} - 2 \, d x + 2 \, {\left (d^{2} x^{2} - 1\right )} \log \left (-d x + 1\right )\right )} b}{d^{2}} + \frac {4 \, {\left (18 \, d^{3} x^{3} {\rm Li}_2\left (d x\right ) - 2 \, d^{3} x^{3} - 3 \, d^{2} x^{2} - 6 \, d x + 6 \, {\left (d^{3} x^{3} - 1\right )} \log \left (-d x + 1\right )\right )} c}{d^{3}}\right )} \log \left (-d x + 1\right ) \]
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 \[ \int \ln \left (1-d\,x\right )\,\mathrm {polylog}\left (2,d\,x\right )\,\left (c\,x^2+b\,x+a\right ) \,d x \]
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 \[ \int \left (a + b x + c x^{2}\right ) \log {\left (- d x + 1 \right )} \operatorname {Li}_{2}\left (d x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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