Optimal. Leaf size=343 \[ \frac {b^2 \text {Li}_3(1-d x)}{a}-\frac {b^2 \text {Li}_2(1-d x) \log (1-d x)}{a}-\frac {b^2 \log (d x) \log ^2(1-d x)}{2 a}-\frac {(a+b x)^2 \text {Li}_2(d x) \log (1-d x)}{2 a x^2}-\frac {1}{2} d (a d+2 b) \text {Li}_3(d x)-\frac {(a d+b)^2 \text {Li}_3(1-d x)}{a}+\frac {(a d+b)^2 \text {Li}_2(d x) \log (1-d x)}{2 a}+\frac {(a d+b)^2 \text {Li}_2(1-d x) \log (1-d x)}{a}+\frac {(a d+b)^2 \log (d x) \log ^2(1-d x)}{2 a}-\frac {1}{2} a d^2 \text {Li}_2(d x)-\frac {1}{4} a d^2 \log ^2(1-d x)-a d^2 \log (x)+a d^2 \log (1-d x)+\frac {a d \text {Li}_2(d x)}{2 x}+\frac {a \log ^2(1-d x)}{4 x^2}-\frac {a d \log (1-d x)}{x}-2 b d \text {Li}_2(d x)+\frac {b (1-d x) \log ^2(1-d x)}{x}-\frac {1}{2} c \text {Li}_2(d x){}^2 \]
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Rubi [A] time = 0.74, antiderivative size = 343, normalized size of antiderivative = 1.00, number of steps used = 32, number of rules used = 22, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.846, Rules used = {6742, 6591, 2395, 44, 36, 29, 31, 6589, 6605, 6601, 37, 6606, 2398, 2410, 2391, 2390, 2301, 2397, 2396, 2433, 2374, 6596} \[ \frac {b^2 \text {PolyLog}(3,1-d x)}{a}-\frac {b^2 \log (1-d x) \text {PolyLog}(2,1-d x)}{a}-\frac {(a+b x)^2 \log (1-d x) \text {PolyLog}(2,d x)}{2 a x^2}-\frac {1}{2} d (a d+2 b) \text {PolyLog}(3,d x)-\frac {(a d+b)^2 \text {PolyLog}(3,1-d x)}{a}+\frac {(a d+b)^2 \log (1-d x) \text {PolyLog}(2,d x)}{2 a}+\frac {(a d+b)^2 \log (1-d x) \text {PolyLog}(2,1-d x)}{a}-\frac {1}{2} a d^2 \text {PolyLog}(2,d x)+\frac {a d \text {PolyLog}(2,d x)}{2 x}-2 b d \text {PolyLog}(2,d x)-\frac {1}{2} c \text {PolyLog}(2,d x)^2-\frac {b^2 \log (d x) \log ^2(1-d x)}{2 a}+\frac {(a d+b)^2 \log (d x) \log ^2(1-d x)}{2 a}-\frac {1}{4} a d^2 \log ^2(1-d x)-a d^2 \log (x)+a d^2 \log (1-d x)+\frac {a \log ^2(1-d x)}{4 x^2}-\frac {a d \log (1-d x)}{x}+\frac {b (1-d x) \log ^2(1-d x)}{x} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 37
Rule 44
Rule 2301
Rule 2374
Rule 2390
Rule 2391
Rule 2395
Rule 2396
Rule 2397
Rule 2398
Rule 2410
Rule 2433
Rule 6589
Rule 6591
Rule 6596
Rule 6601
Rule 6605
Rule 6606
Rule 6742
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right ) \log (1-d x) \text {Li}_2(d x)}{x^3} \, dx &=c \int \frac {\log (1-d x) \text {Li}_2(d x)}{x} \, dx+\int \frac {(a+b x) \log (1-d x) \text {Li}_2(d x)}{x^3} \, dx\\ &=-\frac {(a+b x)^2 \log (1-d x) \text {Li}_2(d x)}{2 a x^2}-\frac {1}{2} c \text {Li}_2(d x){}^2+d \int \left (-\frac {a \text {Li}_2(d x)}{2 x^2}+\frac {(-2 b-a d) \text {Li}_2(d x)}{2 x}+\frac {(b+a d)^2 \text {Li}_2(d x)}{2 a (-1+d x)}\right ) \, dx+\int \left (-\frac {a \log ^2(1-d x)}{2 x^3}-\frac {b \log ^2(1-d x)}{x^2}-\frac {b^2 \log ^2(1-d x)}{2 a x}\right ) \, dx\\ &=-\frac {(a+b x)^2 \log (1-d x) \text {Li}_2(d x)}{2 a x^2}-\frac {1}{2} c \text {Li}_2(d x){}^2-\frac {1}{2} a \int \frac {\log ^2(1-d x)}{x^3} \, dx-b \int \frac {\log ^2(1-d x)}{x^2} \, dx-\frac {b^2 \int \frac {\log ^2(1-d x)}{x} \, dx}{2 a}-\frac {1}{2} (a d) \int \frac {\text {Li}_2(d x)}{x^2} \, dx+\frac {\left (d (b+a d)^2\right ) \int \frac {\text {Li}_2(d x)}{-1+d x} \, dx}{2 a}-\frac {1}{2} (d (2 b+a d)) \int \frac {\text {Li}_2(d x)}{x} \, dx\\ &=\frac {a \log ^2(1-d x)}{4 x^2}+\frac {b (1-d x) \log ^2(1-d x)}{x}-\frac {b^2 \log (d x) \log ^2(1-d x)}{2 a}+\frac {a d \text {Li}_2(d x)}{2 x}+\frac {(b+a d)^2 \log (1-d x) \text {Li}_2(d x)}{2 a}-\frac {(a+b x)^2 \log (1-d x) \text {Li}_2(d x)}{2 a x^2}-\frac {1}{2} c \text {Li}_2(d x){}^2-\frac {1}{2} d (2 b+a d) \text {Li}_3(d x)+\frac {1}{2} (a d) \int \frac {\log (1-d x)}{x^2} \, dx+\frac {1}{2} (a d) \int \frac {\log (1-d x)}{x^2 (1-d x)} \, dx+(2 b d) \int \frac {\log (1-d x)}{x} \, dx-\frac {\left (b^2 d\right ) \int \frac {\log (d x) \log (1-d x)}{1-d x} \, dx}{a}+\frac {(b+a d)^2 \int \frac {\log ^2(1-d x)}{x} \, dx}{2 a}\\ &=-\frac {a d \log (1-d x)}{2 x}+\frac {a \log ^2(1-d x)}{4 x^2}+\frac {b (1-d x) \log ^2(1-d x)}{x}-\frac {b^2 \log (d x) \log ^2(1-d x)}{2 a}+\frac {(b+a d)^2 \log (d x) \log ^2(1-d x)}{2 a}-2 b d \text {Li}_2(d x)+\frac {a d \text {Li}_2(d x)}{2 x}+\frac {(b+a d)^2 \log (1-d x) \text {Li}_2(d x)}{2 a}-\frac {(a+b x)^2 \log (1-d x) \text {Li}_2(d x)}{2 a x^2}-\frac {1}{2} c \text {Li}_2(d x){}^2-\frac {1}{2} d (2 b+a d) \text {Li}_3(d x)+\frac {b^2 \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 )}{a}+\frac {1}{2} (a d) \int \left (\frac {\log (1-d x)}{x^2}+\frac {d \log (1-d x)}{x}-\frac {d^2 \log (1-d x)}{-1+d x}\right ) \, dx-\frac {1}{2} \left (a d^2\right ) \int \frac {1}{x (1-d x)} \, dx+\frac {\left (d (b+a d)^2\right ) \int \frac {\log (d x) \log (1-d x)}{1-d x} \, dx}{a}\\ &=-\frac {a d \log (1-d x)}{2 x}+\frac {a \log ^2(1-d x)}{4 x^2}+\frac {b (1-d x) \log ^2(1-d x)}{x}-\frac {b^2 \log (d x) \log ^2(1-d x)}{2 a}+\frac {(b+a d)^2 \log (d x) \log ^2(1-d x)}{2 a}-2 b d \text {Li}_2(d x)+\frac {a d \text {Li}_2(d x)}{2 x}+\frac {(b+a d)^2 \log (1-d x) \text {Li}_2(d x)}{2 a}-\frac {(a+b x)^2 \log (1-d x) \text {Li}_2(d x)}{2 a x^2}-\frac {1}{2} c \text {Li}_2(d x){}^2-\frac {b^2 \log (1-d x) \text {Li}_2(1-d x)}{a}-\frac {1}{2} d (2 b+a d) \text {Li}_3(d x)+\frac {b^2 \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,1-d x\right )}{a}+\frac {1}{2} (a d) \int \frac {\log (1-d x)}{x^2} \, dx-\frac {1}{2} \left (a d^2\right ) \int \frac {1}{x} \, dx+\frac {1}{2} \left (a d^2\right ) \int \frac {\log (1-d x)}{x} \, dx-\frac {1}{2} \left (a d^3\right ) \int \frac {1}{1-d x} \, dx-\frac {1}{2} \left (a d^3\right ) \int \frac {\log (1-d x)}{-1+d x} \, dx-\frac {(b+a d)^2 \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 )}{a}\\ &=-\frac {1}{2} a d^2 \log (x)+\frac {1}{2} a d^2 \log (1-d x)-\frac {a d \log (1-d x)}{x}+\frac {a \log ^2(1-d x)}{4 x^2}+\frac {b (1-d x) \log ^2(1-d x)}{x}-\frac {b^2 \log (d x) \log ^2(1-d x)}{2 a}+\frac {(b+a d)^2 \log (d x) \log ^2(1-d x)}{2 a}-2 b d \text {Li}_2(d x)-\frac {1}{2} a d^2 \text {Li}_2(d x)+\frac {a d \text {Li}_2(d x)}{2 x}+\frac {(b+a d)^2 \log (1-d x) \text {Li}_2(d x)}{2 a}-\frac {(a+b x)^2 \log (1-d x) \text {Li}_2(d x)}{2 a x^2}-\frac {1}{2} c \text {Li}_2(d x){}^2-\frac {b^2 \log (1-d x) \text {Li}_2(1-d x)}{a}+\frac {(b+a d)^2 \log (1-d x) \text {Li}_2(1-d x)}{a}-\frac {1}{2} d (2 b+a d) \text {Li}_3(d x)+\frac {b^2 \text {Li}_3(1-d x)}{a}-\frac {1}{2} \left (a d^2\right ) \int \frac {1}{x (1-d x)} \, dx-\frac {1}{2} \left (a d^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,1-d x\right )-\frac {(b+a d)^2 \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,1-d x\right )}{a}\\ &=-\frac {1}{2} a d^2 \log (x)+\frac {1}{2} a d^2 \log (1-d x)-\frac {a d \log (1-d x)}{x}-\frac {1}{4} a d^2 \log ^2(1-d x)+\frac {a \log ^2(1-d x)}{4 x^2}+\frac {b (1-d x) \log ^2(1-d x)}{x}-\frac {b^2 \log (d x) \log ^2(1-d x)}{2 a}+\frac {(b+a d)^2 \log (d x) \log ^2(1-d x)}{2 a}-2 b d \text {Li}_2(d x)-\frac {1}{2} a d^2 \text {Li}_2(d x)+\frac {a d \text {Li}_2(d x)}{2 x}+\frac {(b+a d)^2 \log (1-d x) \text {Li}_2(d x)}{2 a}-\frac {(a+b x)^2 \log (1-d x) \text {Li}_2(d x)}{2 a x^2}-\frac {1}{2} c \text {Li}_2(d x){}^2-\frac {b^2 \log (1-d x) \text {Li}_2(1-d x)}{a}+\frac {(b+a d)^2 \log (1-d x) \text {Li}_2(1-d x)}{a}-\frac {1}{2} d (2 b+a d) \text {Li}_3(d x)+\frac {b^2 \text {Li}_3(1-d x)}{a}-\frac {(b+a d)^2 \text {Li}_3(1-d x)}{a}-\frac {1}{2} \left (a d^2\right ) \int \frac {1}{x} \, dx-\frac {1}{2} \left (a d^3\right ) \int \frac {1}{1-d x} \, dx\\ &=-a d^2 \log (x)+a d^2 \log (1-d x)-\frac {a d \log (1-d x)}{x}-\frac {1}{4} a d^2 \log ^2(1-d x)+\frac {a \log ^2(1-d x)}{4 x^2}+\frac {b (1-d x) \log ^2(1-d x)}{x}-\frac {b^2 \log (d x) \log ^2(1-d x)}{2 a}+\frac {(b+a d)^2 \log (d x) \log ^2(1-d x)}{2 a}-2 b d \text {Li}_2(d x)-\frac {1}{2} a d^2 \text {Li}_2(d x)+\frac {a d \text {Li}_2(d x)}{2 x}+\frac {(b+a d)^2 \log (1-d x) \text {Li}_2(d x)}{2 a}-\frac {(a+b x)^2 \log (1-d x) \text {Li}_2(d x)}{2 a x^2}-\frac {1}{2} c \text {Li}_2(d x){}^2-\frac {b^2 \log (1-d x) \text {Li}_2(1-d x)}{a}+\frac {(b+a d)^2 \log (1-d x) \text {Li}_2(1-d x)}{a}-\frac {1}{2} d (2 b+a d) \text {Li}_3(d x)+\frac {b^2 \text {Li}_3(1-d x)}{a}-\frac {(b+a d)^2 \text {Li}_3(1-d x)}{a}\\ \end {align*}
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Mathematica [F] time = 1.68, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b x+c x^2\right ) \log (1-d x) \text {Li}_2(d x)}{x^3} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 3.32, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c x^{2} + b x + a\right )} {\rm Li}_2\left (d x\right ) \log \left (-d x + 1\right )}{x^{3}}, x\right ) \]
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 \[ \int \frac {{\left (c x^{2} + b x + a\right )} {\rm Li}_2\left (d x\right ) \log \left (-d x + 1\right )}{x^{3}}\,{d x} \]
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 \[ \int \frac {\left (c \,x^{2}+b x +a \right ) \ln \left (-d x +1\right ) \polylog \left (2, d x \right )}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c x^{2} + b x + a\right )} {\rm Li}_2\left (d x\right ) \log \left (-d x + 1\right )}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\ln \left (1-d\,x\right )\,\mathrm {polylog}\left (2,d\,x\right )\,\left (c\,x^2+b\,x+a\right )}{x^3} \,d x \]
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 \[ \int \frac {\left (a + b x + c x^{2}\right ) \log {\left (- d x + 1 \right )} \operatorname {Li}_{2}\left (d x\right )}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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