Optimal. Leaf size=401 \[ \log (x) \log \left (1+\frac {b x}{a}\right ) \log (1-c (a+b x))+\frac {1}{2} \left (\log \left (1+\frac {b x}{a}\right )+\log \left (\frac {1-a c}{1-c (a+b x)}\right )-\log \left (\frac {(1-a c) (a+b x)}{a (1-c (a+b x))}\right )\right ) \log ^2\left (-\frac {a (1-c (a+b x))}{b x}\right )+\frac {1}{2} \left (\log (c (a+b x))-\log \left (1+\frac {b x}{a}\right )\right ) \left (\log (x)+\log \left (-\frac {a (1-c (a+b x))}{b x}\right )\right )^2+\left (\log (1-c (a+b x))-\log \left (-\frac {a (1-c (a+b x))}{b x}\right )\right ) \text {PolyLog}\left (2,-\frac {b x}{a}\right )+\log (x) \text {PolyLog}(2,c (a+b x))+\log \left (-\frac {a (1-c (a+b x))}{b x}\right ) \text {PolyLog}\left (2,-\frac {b x}{a (1-c (a+b x))}\right )-\log \left (-\frac {a (1-c (a+b x))}{b x}\right ) \text {PolyLog}\left (2,-\frac {b c x}{1-c (a+b x)}\right )+\left (\log (x)+\log \left (-\frac {a (1-c (a+b x))}{b x}\right )\right ) \text {PolyLog}(2,1-c (a+b x))-\text {PolyLog}\left (3,-\frac {b x}{a}\right )+\text {PolyLog}\left (3,-\frac {b x}{a (1-c (a+b x))}\right )-\text {PolyLog}\left (3,-\frac {b c x}{1-c (a+b x)}\right )-\text {PolyLog}(3,1-c (a+b x)) \]
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Rubi [A]
time = 0.24, antiderivative size = 401, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {6732, 2490,
2485} \begin {gather*} \text {Li}_3\left (-\frac {b x}{a (1-c (a+b x))}\right )-\text {Li}_3\left (-\frac {b c x}{1-c (a+b x)}\right )-\text {Li}_3(1-c (a+b x))+\text {Li}_2\left (-\frac {b x}{a (1-c (a+b x))}\right ) \log \left (-\frac {a (1-c (a+b x))}{b x}\right )-\text {Li}_2\left (-\frac {b c x}{1-c (a+b x)}\right ) \log \left (-\frac {a (1-c (a+b x))}{b x}\right )+\text {Li}_2\left (-\frac {b x}{a}\right ) \left (\log (1-c (a+b x))-\log \left (-\frac {a (1-c (a+b x))}{b x}\right )\right )+\log (x) \text {Li}_2(c (a+b x))+\text {Li}_2(1-c (a+b x)) \left (\log \left (-\frac {a (1-c (a+b x))}{b x}\right )+\log (x)\right )+\frac {1}{2} \left (\log \left (\frac {1-a c}{1-c (a+b x)}\right )-\log \left (\frac {(1-a c) (a+b x)}{a (1-c (a+b x))}\right )+\log \left (\frac {b x}{a}+1\right )\right ) \log ^2\left (-\frac {a (1-c (a+b x))}{b x}\right )+\frac {1}{2} \left (\log (c (a+b x))-\log \left (\frac {b x}{a}+1\right )\right ) \left (\log \left (-\frac {a (1-c (a+b x))}{b x}\right )+\log (x)\right )^2+\log (x) \log \left (\frac {b x}{a}+1\right ) \log (1-c (a+b x))-\text {Li}_3\left (-\frac {b x}{a}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2485
Rule 2490
Rule 6732
Rubi steps
\begin {align*} \int \frac {\text {Li}_2(c (a+b x))}{x} \, dx &=\log (x) \text {Li}_2(c (a+b x))+b \int \frac {\log (x) \log (1-a c-b c x)}{a+b x} \, dx\\ &=\log (x) \text {Li}_2(c (a+b x))+\text {Subst}\left (\int \frac {\log \left (-\frac {a}{b}+\frac {x}{b}\right ) \log \left (-\frac {-a b c-b (1-a c)}{b}-c x\right )}{x} \, dx,x,a+b x\right )\\ &=\log (x) \log \left (1+\frac {b x}{a}\right ) \log (1-c (a+b x))+\frac {1}{2} \left (\log \left (1+\frac {b x}{a}\right )+\log \left (\frac {1-a c}{1-c (a+b x)}\right )-\log \left (\frac {(1-a c) (a+b x)}{a (1-c (a+b x))}\right )\right ) \log ^2\left (-\frac {a (1-c (a+b x))}{b x}\right )-\frac {1}{2} \left (-\log (c (a+b x))+\log \left (1+\frac {b x}{a}\right )\right ) \left (\log (x)+\log \left (-\frac {a (1-c (a+b x))}{b x}\right )\right )^2+\left (\log (1-c (a+b x))-\log \left (-\frac {a (1-c (a+b x))}{b x}\right )\right ) \text {Li}_2\left (-\frac {b x}{a}\right )+\log (x) \text {Li}_2(c (a+b x))+\log \left (-\frac {a (1-c (a+b x))}{b x}\right ) \text {Li}_2\left (-\frac {b x}{a (1-c (a+b x))}\right )-\log \left (-\frac {a (1-c (a+b x))}{b x}\right ) \text {Li}_2\left (-\frac {b c x}{1-c (a+b x)}\right )+\left (\log (x)+\log \left (-\frac {a (1-c (a+b x))}{b x}\right )\right ) \text {Li}_2(1-c (a+b x))-\text {Li}_3\left (-\frac {b x}{a}\right )+\text {Li}_3\left (-\frac {b x}{a (1-c (a+b x))}\right )-\text {Li}_3\left (-\frac {b c x}{1-c (a+b x)}\right )-\text {Li}_3(1-c (a+b x))\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 422, normalized size = 1.05 \begin {gather*} \log (x) \log \left (1+\frac {b x}{a}\right ) \log (1-a c-b c x)+\frac {1}{2} \left (-\log (c (a+b x))+\log \left (1+\frac {b x}{a}\right )\right ) \log (1-a c-b c x) (-2 \log (x)+\log (1-a c-b c x))+\left (\log (c (a+b x))-\log \left (1+\frac {b x}{a}\right )\right ) \log (1-a c-b c x) \log \left (\frac {a (-1+a c+b c x)}{b x}\right )+\frac {1}{2} \left (\log \left (\frac {1-a c}{b c x}\right )-\log \left (\frac {(1-a c) (a+b x)}{b x}\right )+\log \left (1+\frac {b x}{a}\right )\right ) \log ^2\left (\frac {a (-1+a c+b c x)}{b x}\right )+\left (\log (1-a c-b c x)-\log \left (\frac {a (-1+a c+b c x)}{b x}\right )\right ) \text {PolyLog}\left (2,-\frac {b x}{a}\right )+\left (\log (x)+\log \left (\frac {a (-1+a c+b c x)}{b x}\right )\right ) \text {PolyLog}(2,1-a c-b c x)+\log \left (\frac {a (-1+a c+b c x)}{b x}\right ) \left (-\text {PolyLog}\left (2,\frac {a (-1+a c+b c x)}{b x}\right )+\text {PolyLog}\left (2,\frac {-1+a c+b c x}{b c x}\right )\right )+\log (x) \text {PolyLog}(2,a c+b c x)-\text {PolyLog}\left (3,-\frac {b x}{a}\right )-\text {PolyLog}(3,1-a c-b c x)+\text {PolyLog}\left (3,\frac {a (-1+a c+b c x)}{b x}\right )-\text {PolyLog}\left (3,\frac {-1+a c+b c x}{b c x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.47, size = 0, normalized size = 0.00 \[\int \frac {\polylog \left (2, c \left (b x +a \right )\right )}{x}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \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 {\operatorname {Li}_{2}\left (a c + b c x\right )}{x}\, 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 {\mathrm {polylog}\left (2,c\,\left (a+b\,x\right )\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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