Optimal. Leaf size=486 \[ -\frac {2 b \text {Li}_3(c (a+b x))}{a}+\frac {\left (b-\frac {a}{x}\right ) \text {Li}_3(c (a+b x))}{a}+\frac {b \text {Li}_3\left (-\frac {b x}{a (1-c (a+b x))}\right )}{a}-\frac {b \text {Li}_3\left (-\frac {b c x}{1-c (a+b x)}\right )}{a}-\frac {b \text {Li}_3(1-c (a+b x))}{a}+\frac {b \text {Li}_2\left (-\frac {b x}{a (1-c (a+b x))}\right ) \log \left (-\frac {a (1-c (a+b x))}{b x}\right )}{a}-\frac {b \text {Li}_2\left (-\frac {b c x}{1-c (a+b x)}\right ) \log \left (-\frac {a (1-c (a+b x))}{b x}\right )}{a}+\frac {b \text {Li}_2\left (-\frac {b x}{a}\right ) \left (\log (1-c (a+b x))-\log \left (-\frac {a (1-c (a+b x))}{b x}\right )\right )}{a}+\frac {b \log (x) \text {Li}_2(c (a+b x))}{a}+\frac {b \text {Li}_2(1-c (a+b x)) \left (\log \left (-\frac {a (1-c (a+b x))}{b x}\right )+\log (x)\right )}{a}+\frac {b \left (\log \left (\frac {1-a c}{1-c (a+b x)}\right )-\log \left (\frac {(1-a c) (a+b x)}{a (1-c (a+b x))}\right )+\log \left (\frac {b x}{a}+1\right )\right ) \log ^2\left (-\frac {a (1-c (a+b x))}{b x}\right )}{2 a}+\frac {b \left (\log (c (a+b x))-\log \left (\frac {b x}{a}+1\right )\right ) \left (\log \left (-\frac {a (1-c (a+b x))}{b x}\right )+\log (x)\right )^2}{2 a}+\frac {b \log (x) \log \left (\frac {b x}{a}+1\right ) \log (1-c (a+b x))}{a}-\frac {b \text {Li}_3\left (-\frac {b x}{a}\right )}{a} \]
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Rubi [A] time = 0.56, antiderivative size = 486, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {6599, 6597, 2440, 2435, 6589} \[ -\frac {2 b \text {PolyLog}(3,c (a+b x))}{a}+\frac {\left (b-\frac {a}{x}\right ) \text {PolyLog}(3,c (a+b x))}{a}+\frac {b \text {PolyLog}\left (3,-\frac {b x}{a (1-c (a+b x))}\right )}{a}-\frac {b \text {PolyLog}\left (3,-\frac {b c x}{1-c (a+b x)}\right )}{a}-\frac {b \text {PolyLog}(3,1-c (a+b x))}{a}+\frac {b \log \left (-\frac {a (1-c (a+b x))}{b x}\right ) \text {PolyLog}\left (2,-\frac {b x}{a (1-c (a+b x))}\right )}{a}-\frac {b \log \left (-\frac {a (1-c (a+b x))}{b x}\right ) \text {PolyLog}\left (2,-\frac {b c x}{1-c (a+b x)}\right )}{a}+\frac {b \text {PolyLog}\left (2,-\frac {b x}{a}\right ) \left (\log (1-c (a+b x))-\log \left (-\frac {a (1-c (a+b x))}{b x}\right )\right )}{a}+\frac {b \log (x) \text {PolyLog}(2,c (a+b x))}{a}+\frac {b \left (\log \left (-\frac {a (1-c (a+b x))}{b x}\right )+\log (x)\right ) \text {PolyLog}(2,1-c (a+b x))}{a}-\frac {b \text {PolyLog}\left (3,-\frac {b x}{a}\right )}{a}+\frac {b \left (\log \left (\frac {1-a c}{1-c (a+b x)}\right )-\log \left (\frac {(1-a c) (a+b x)}{a (1-c (a+b x))}\right )+\log \left (\frac {b x}{a}+1\right )\right ) \log ^2\left (-\frac {a (1-c (a+b x))}{b x}\right )}{2 a}+\frac {b \left (\log (c (a+b x))-\log \left (\frac {b x}{a}+1\right )\right ) \left (\log \left (-\frac {a (1-c (a+b x))}{b x}\right )+\log (x)\right )^2}{2 a}+\frac {b \log (x) \log \left (\frac {b x}{a}+1\right ) \log (1-c (a+b x))}{a} \]
Antiderivative was successfully verified.
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Rule 2435
Rule 2440
Rule 6589
Rule 6597
Rule 6599
Rubi steps
\begin {align*} \int \frac {\text {Li}_3(c (a+b x))}{x^2} \, dx &=\frac {\left (b-\frac {a}{x}\right ) \text {Li}_3(c (a+b x))}{a}-b^2 \int \left (-\frac {\text {Li}_2(c (a+b x))}{a b x}+\frac {2 \text {Li}_2(c (a+b x))}{a (a+b x)}\right ) \, dx\\ &=\frac {\left (b-\frac {a}{x}\right ) \text {Li}_3(c (a+b x))}{a}+\frac {b \int \frac {\text {Li}_2(c (a+b x))}{x} \, dx}{a}-\frac {\left (2 b^2\right ) \int \frac {\text {Li}_2(c (a+b x))}{a+b x} \, dx}{a}\\ &=\frac {b \log (x) \text {Li}_2(c (a+b x))}{a}-\frac {2 b \text {Li}_3(c (a+b x))}{a}+\frac {\left (b-\frac {a}{x}\right ) \text {Li}_3(c (a+b x))}{a}+\frac {b^2 \int \frac {\log (x) \log (1-a c-b c x)}{a+b x} \, dx}{a}\\ &=\frac {b \log (x) \text {Li}_2(c (a+b x))}{a}-\frac {2 b \text {Li}_3(c (a+b x))}{a}+\frac {\left (b-\frac {a}{x}\right ) \text {Li}_3(c (a+b x))}{a}+\frac {b \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 )}{a}\\ &=\frac {b \log (x) \log \left (1+\frac {b x}{a}\right ) \log (1-c (a+b x))}{a}+\frac {b \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 )}{2 a}+\frac {b \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}{2 a}+\frac {b \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 )}{a}+\frac {b \log (x) \text {Li}_2(c (a+b x))}{a}+\frac {b \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 )}{a}-\frac {b \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 )}{a}+\frac {b \left (\log (x)+\log \left (-\frac {a (1-c (a+b x))}{b x}\right )\right ) \text {Li}_2(1-c (a+b x))}{a}-\frac {b \text {Li}_3\left (-\frac {b x}{a}\right )}{a}-\frac {2 b \text {Li}_3(c (a+b x))}{a}+\frac {\left (b-\frac {a}{x}\right ) \text {Li}_3(c (a+b x))}{a}+\frac {b \text {Li}_3\left (-\frac {b x}{a (1-c (a+b x))}\right )}{a}-\frac {b \text {Li}_3\left (-\frac {b c x}{1-c (a+b x)}\right )}{a}-\frac {b \text {Li}_3(1-c (a+b x))}{a}\\ \end {align*}
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Mathematica [A] time = 0.70, size = 477, normalized size = 0.98 \[ \frac {b \left (-\text {Li}_3(c (a+b x))-\text {Li}_3(-a c-b x c+1)+\text {Li}_3\left (\frac {a (a c+b x c-1)}{b x}\right )-\text {Li}_3\left (\frac {a c+b x c-1}{b c x}\right )+\left (\text {Li}_2\left (\frac {a c+b x c-1}{b c x}\right )-\text {Li}_2\left (\frac {a (a c+b x c-1)}{b x}\right )\right ) \log \left (\frac {a (a c+b c x-1)}{b x}\right )+\text {Li}_2\left (-\frac {b x}{a}\right ) \left (\log (-a c-b c x+1)-\log \left (\frac {a (a c+b c x-1)}{b x}\right )\right )+(\log (x)-\log (a+b x)) \text {Li}_2(c (a+b x))+\log (a+b x) \text {Li}_2(c (a+b x))+\text {Li}_2(-a c-b x c+1) \left (\log \left (\frac {a (a c+b c x-1)}{b x}\right )+\log (x)\right )+\frac {1}{2} \left (\log \left (\frac {1-a c}{b c x}\right )-\log \left (\frac {(1-a c) (a+b x)}{b x}\right )+\log \left (\frac {b x}{a}+1\right )\right ) \log ^2\left (\frac {a (a c+b c x-1)}{b x}\right )+\left (\log (c (a+b x))-\log \left (\frac {b x}{a}+1\right )\right ) \log (-a c-b c x+1) \log \left (\frac {a (a c+b c x-1)}{b x}\right )+\log (x) \log \left (\frac {b x}{a}+1\right ) \log (-a c-b c x+1)+\frac {1}{2} \left (\log \left (\frac {b x}{a}+1\right )-\log (c (a+b x))\right ) \log (-a c-b c x+1) (\log (-a c-b c x+1)-2 \log (x))-\text {Li}_3\left (-\frac {b x}{a}\right )\right )}{a}-\frac {\text {Li}_3(c (a+b x))}{x} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\rm polylog}\left (3, b c x + a c\right )}{x^{2}}, x\right ) \]
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 \[ \int \frac {{\rm Li}_{3}({\left (b x + a\right )} c)}{x^{2}}\,{d x} \]
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 \[ \int \frac {\polylog \left (3, c \left (b x +a \right )\right )}{x^{2}}\, dx \]
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 \[ b \int \frac {{\rm Li}_2\left (b c x + a c\right )}{b x^{2} + a x}\,{d x} - \frac {{\rm Li}_{3}(b c x + a c)}{x} \]
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 \[ \int \frac {\mathrm {polylog}\left (3,c\,\left (a+b\,x\right )\right )}{x^2} \,d x \]
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 \[ \int \frac {\operatorname {Li}_{3}\left (a c + b c x\right )}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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