Optimal. Leaf size=401 \[ \text {Li}_3\left (-\frac {b x}{a (1-c (a+b x))}\right )-\text {Li}_3\left (-\frac {b c x}{1-c (a+b x)}\right )-\text {Li}_3(1-c (a+b x))+\text {Li}_2\left (-\frac {b x}{a (1-c (a+b x))}\right ) \log \left (-\frac {a (1-c (a+b x))}{b x}\right )-\text {Li}_2\left (-\frac {b c x}{1-c (a+b x)}\right ) \log \left (-\frac {a (1-c (a+b x))}{b x}\right )+\text {Li}_2\left (-\frac {b x}{a}\right ) \left (\log (1-c (a+b x))-\log \left (-\frac {a (1-c (a+b x))}{b x}\right )\right )+\log (x) \text {Li}_2(c (a+b x))+\text {Li}_2(1-c (a+b x)) \left (\log \left (-\frac {a (1-c (a+b x))}{b x}\right )+\log (x)\right )+\frac {1}{2} \left (\log \left (\frac {1-a c}{1-c (a+b x)}\right )-\log \left (\frac {(1-a c) (a+b x)}{a (1-c (a+b x))}\right )+\log \left (\frac {b x}{a}+1\right )\right ) \log ^2\left (-\frac {a (1-c (a+b x))}{b x}\right )+\frac {1}{2} \left (\log (c (a+b x))-\log \left (\frac {b x}{a}+1\right )\right ) \left (\log \left (-\frac {a (1-c (a+b x))}{b x}\right )+\log (x)\right )^2+\log (x) \log \left (\frac {b x}{a}+1\right ) \log (1-c (a+b x))-\text {Li}_3\left (-\frac {b x}{a}\right ) \]
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Rubi [A] time = 0.36, 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 = {6597, 2440, 2435} \[ \text {PolyLog}\left (3,-\frac {b x}{a (1-c (a+b x))}\right )-\text {PolyLog}\left (3,-\frac {b c x}{1-c (a+b x)}\right )-\text {PolyLog}(3,1-c (a+b x))+\log \left (-\frac {a (1-c (a+b x))}{b x}\right ) \text {PolyLog}\left (2,-\frac {b x}{a (1-c (a+b x))}\right )-\log \left (-\frac {a (1-c (a+b x))}{b x}\right ) \text {PolyLog}\left (2,-\frac {b c x}{1-c (a+b x)}\right )+\text {PolyLog}\left (2,-\frac {b x}{a}\right ) \left (\log (1-c (a+b x))-\log \left (-\frac {a (1-c (a+b x))}{b x}\right )\right )+\log (x) \text {PolyLog}(2,c (a+b x))+\left (\log \left (-\frac {a (1-c (a+b x))}{b x}\right )+\log (x)\right ) \text {PolyLog}(2,1-c (a+b x))-\text {PolyLog}\left (3,-\frac {b x}{a}\right )+\frac {1}{2} \left (\log \left (\frac {1-a c}{1-c (a+b x)}\right )-\log \left (\frac {(1-a c) (a+b x)}{a (1-c (a+b x))}\right )+\log \left (\frac {b x}{a}+1\right )\right ) \log ^2\left (-\frac {a (1-c (a+b x))}{b x}\right )+\frac {1}{2} \left (\log (c (a+b x))-\log \left (\frac {b x}{a}+1\right )\right ) \left (\log \left (-\frac {a (1-c (a+b x))}{b x}\right )+\log (x)\right )^2+\log (x) \log \left (\frac {b x}{a}+1\right ) \log (1-c (a+b x)) \]
Antiderivative was successfully verified.
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Rule 2435
Rule 2440
Rule 6597
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))+\operatorname {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 = 0.13, size = 422, normalized size = 1.05 \[ -\text {Li}_3(-a c-b x c+1)+\text {Li}_3\left (\frac {a (a c+b x c-1)}{b x}\right )-\text {Li}_3\left (\frac {a c+b x c-1}{b c x}\right )+\left (\text {Li}_2\left (\frac {a c+b x c-1}{b c x}\right )-\text {Li}_2\left (\frac {a (a c+b x c-1)}{b x}\right )\right ) \log \left (\frac {a (a c+b c x-1)}{b x}\right )+\text {Li}_2\left (-\frac {b x}{a}\right ) \left (\log (-a c-b c x+1)-\log \left (\frac {a (a c+b c x-1)}{b x}\right )\right )+\text {Li}_2(-a c-b x c+1) \left (\log \left (\frac {a (a c+b c x-1)}{b x}\right )+\log (x)\right )+\log (x) \text {Li}_2(a c+b x c)+\frac {1}{2} \left (\log \left (\frac {1-a c}{b c x}\right )-\log \left (\frac {(1-a c) (a+b x)}{b x}\right )+\log \left (\frac {b x}{a}+1\right )\right ) \log ^2\left (\frac {a (a c+b c x-1)}{b x}\right )+\left (\log (c (a+b x))-\log \left (\frac {b x}{a}+1\right )\right ) \log (-a c-b c x+1) \log \left (\frac {a (a c+b c x-1)}{b x}\right )+\log (x) \log \left (\frac {b x}{a}+1\right ) \log (-a c-b c x+1)+\frac {1}{2} \left (\log \left (\frac {b x}{a}+1\right )-\log (c (a+b x))\right ) \log (-a c-b c x+1) (\log (-a c-b c x+1)-2 \log (x))-\text {Li}_3\left (-\frac {b x}{a}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\rm Li}_2\left (b c x + a c\right )}{x}, x\right ) \]
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 \[ \int \frac {{\rm Li}_2\left ({\left (b x + a\right )} c\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, 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 \[ \int \frac {{\rm Li}_2\left ({\left (b x + a\right )} c\right )}{x}\,{d x} \]
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 \[ \int \frac {\mathrm {polylog}\left (2,c\,\left (a+b\,x\right )\right )}{x} \,d x \]
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 \[ \int \frac {\operatorname {Li}_{2}\left (a c + b c x\right )}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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