Optimal. Leaf size=629 \[ \frac {\left (b^2-\frac {a^2}{x^2}\right ) \text {Li}_3(c (a+b x))}{2 a^2}-\frac {b^2 \text {Li}_2(c (a+b x))}{2 a^2}-\frac {b^2 \text {Li}_2\left (1-\frac {b c x}{1-a c}\right )}{2 a^2}-\frac {b^2 \text {Li}_3\left (-\frac {b x}{a (1-c (a+b x))}\right )}{2 a^2}+\frac {b^2 \text {Li}_3\left (-\frac {b c x}{1-c (a+b x)}\right )}{2 a^2}+\frac {b^2 \text {Li}_3(1-c (a+b x))}{2 a^2}-\frac {b^2 \text {Li}_2\left (-\frac {b x}{a}\right ) \left (\log (1-c (a+b x))-\log \left (-\frac {a (1-c (a+b x))}{b x}\right )\right )}{2 a^2}-\frac {b^2 \log (x) \text {Li}_2(c (a+b x))}{2 a^2}-\frac {b^2 \text {Li}_2\left (-\frac {b x}{a (1-c (a+b x))}\right ) \log \left (-\frac {a (1-c (a+b x))}{b x}\right )}{2 a^2}+\frac {b^2 \text {Li}_2\left (-\frac {b c x}{1-c (a+b x)}\right ) \log \left (-\frac {a (1-c (a+b x))}{b x}\right )}{2 a^2}-\frac {b^2 \text {Li}_2(1-c (a+b x)) \left (\log \left (-\frac {a (1-c (a+b x))}{b x}\right )+\log (x)\right )}{2 a^2}-\frac {b^2 \left (\log \left (\frac {1-a c}{1-c (a+b x)}\right )-\log \left (\frac {(1-a c) (a+b x)}{a (1-c (a+b x))}\right )+\log \left (\frac {b x}{a}+1\right )\right ) \log ^2\left (-\frac {a (1-c (a+b x))}{b x}\right )}{4 a^2}-\frac {b^2 \left (\log (c (a+b x))-\log \left (\frac {b x}{a}+1\right )\right ) \left (\log \left (-\frac {a (1-c (a+b x))}{b x}\right )+\log (x)\right )^2}{4 a^2}-\frac {b^2 \log \left (\frac {b c x}{1-a c}\right ) \log (-a c-b c x+1)}{2 a^2}-\frac {b^2 \log (x) \log \left (\frac {b x}{a}+1\right ) \log (1-c (a+b x))}{2 a^2}+\frac {b^2 \text {Li}_3\left (-\frac {b x}{a}\right )}{2 a^2}-\frac {b \text {Li}_2(c (a+b x))}{2 a x} \]
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Rubi [A] time = 0.66, antiderivative size = 629, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 13, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6599, 6598, 36, 29, 31, 2416, 2394, 2315, 2393, 2391, 6597, 2440, 2435} \[ \frac {\left (b^2-\frac {a^2}{x^2}\right ) \text {PolyLog}(3,c (a+b x))}{2 a^2}-\frac {b^2 \text {PolyLog}(2,c (a+b x))}{2 a^2}-\frac {b^2 \text {PolyLog}\left (2,1-\frac {b c x}{1-a c}\right )}{2 a^2}-\frac {b^2 \text {PolyLog}\left (3,-\frac {b x}{a (1-c (a+b x))}\right )}{2 a^2}+\frac {b^2 \text {PolyLog}\left (3,-\frac {b c x}{1-c (a+b x)}\right )}{2 a^2}+\frac {b^2 \text {PolyLog}(3,1-c (a+b x))}{2 a^2}-\frac {b^2 \text {PolyLog}\left (2,-\frac {b x}{a}\right ) \left (\log (1-c (a+b x))-\log \left (-\frac {a (1-c (a+b x))}{b x}\right )\right )}{2 a^2}-\frac {b^2 \log (x) \text {PolyLog}(2,c (a+b x))}{2 a^2}-\frac {b^2 \log \left (-\frac {a (1-c (a+b x))}{b x}\right ) \text {PolyLog}\left (2,-\frac {b x}{a (1-c (a+b x))}\right )}{2 a^2}+\frac {b^2 \log \left (-\frac {a (1-c (a+b x))}{b x}\right ) \text {PolyLog}\left (2,-\frac {b c x}{1-c (a+b x)}\right )}{2 a^2}-\frac {b^2 \left (\log \left (-\frac {a (1-c (a+b x))}{b x}\right )+\log (x)\right ) \text {PolyLog}(2,1-c (a+b x))}{2 a^2}+\frac {b^2 \text {PolyLog}\left (3,-\frac {b x}{a}\right )}{2 a^2}-\frac {b \text {PolyLog}(2,c (a+b x))}{2 a x}-\frac {b^2 \left (\log \left (\frac {1-a c}{1-c (a+b x)}\right )-\log \left (\frac {(1-a c) (a+b x)}{a (1-c (a+b x))}\right )+\log \left (\frac {b x}{a}+1\right )\right ) \log ^2\left (-\frac {a (1-c (a+b x))}{b x}\right )}{4 a^2}-\frac {b^2 \left (\log (c (a+b x))-\log \left (\frac {b x}{a}+1\right )\right ) \left (\log \left (-\frac {a (1-c (a+b x))}{b x}\right )+\log (x)\right )^2}{4 a^2}-\frac {b^2 \log \left (\frac {b c x}{1-a c}\right ) \log (-a c-b c x+1)}{2 a^2}-\frac {b^2 \log (x) \log \left (\frac {b x}{a}+1\right ) \log (1-c (a+b x))}{2 a^2} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 2315
Rule 2391
Rule 2393
Rule 2394
Rule 2416
Rule 2435
Rule 2440
Rule 6597
Rule 6598
Rule 6599
Rubi steps
\begin {align*} \int \frac {\text {Li}_3(c (a+b x))}{x^3} \, dx &=\frac {\left (b^2-\frac {a^2}{x^2}\right ) \text {Li}_3(c (a+b x))}{2 a^2}-\frac {1}{2} b^3 \int \left (-\frac {\text {Li}_2(c (a+b x))}{a b^2 x^2}+\frac {\text {Li}_2(c (a+b x))}{a^2 b x}\right ) \, dx\\ &=\frac {\left (b^2-\frac {a^2}{x^2}\right ) \text {Li}_3(c (a+b x))}{2 a^2}+\frac {b \int \frac {\text {Li}_2(c (a+b x))}{x^2} \, dx}{2 a}-\frac {b^2 \int \frac {\text {Li}_2(c (a+b x))}{x} \, dx}{2 a^2}\\ &=-\frac {b \text {Li}_2(c (a+b x))}{2 a x}-\frac {b^2 \log (x) \text {Li}_2(c (a+b x))}{2 a^2}+\frac {\left (b^2-\frac {a^2}{x^2}\right ) \text {Li}_3(c (a+b x))}{2 a^2}-\frac {b^2 \int \frac {\log (1-a c-b c x)}{x (a+b x)} \, dx}{2 a}-\frac {b^3 \int \frac {\log (x) \log (1-a c-b c x)}{a+b x} \, dx}{2 a^2}\\ &=-\frac {b \text {Li}_2(c (a+b x))}{2 a x}-\frac {b^2 \log (x) \text {Li}_2(c (a+b x))}{2 a^2}+\frac {\left (b^2-\frac {a^2}{x^2}\right ) \text {Li}_3(c (a+b x))}{2 a^2}-\frac {b^2 \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 )}{2 a^2}-\frac {b^2 \int \left (\frac {\log (1-a c-b c x)}{a x}-\frac {b \log (1-a c-b c x)}{a (a+b x)}\right ) \, dx}{2 a}\\ &=-\frac {b^2 \log (x) \log \left (1+\frac {b x}{a}\right ) \log (1-c (a+b x))}{2 a^2}-\frac {b^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 )}{4 a^2}-\frac {b^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}{4 a^2}-\frac {b^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 )}{2 a^2}-\frac {b \text {Li}_2(c (a+b x))}{2 a x}-\frac {b^2 \log (x) \text {Li}_2(c (a+b x))}{2 a^2}-\frac {b^2 \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 )}{2 a^2}+\frac {b^2 \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 )}{2 a^2}-\frac {b^2 \left (\log (x)+\log \left (-\frac {a (1-c (a+b x))}{b x}\right )\right ) \text {Li}_2(1-c (a+b x))}{2 a^2}+\frac {b^2 \text {Li}_3\left (-\frac {b x}{a}\right )}{2 a^2}+\frac {\left (b^2-\frac {a^2}{x^2}\right ) \text {Li}_3(c (a+b x))}{2 a^2}-\frac {b^2 \text {Li}_3\left (-\frac {b x}{a (1-c (a+b x))}\right )}{2 a^2}+\frac {b^2 \text {Li}_3\left (-\frac {b c x}{1-c (a+b x)}\right )}{2 a^2}+\frac {b^2 \text {Li}_3(1-c (a+b x))}{2 a^2}-\frac {b^2 \int \frac {\log (1-a c-b c x)}{x} \, dx}{2 a^2}+\frac {b^3 \int \frac {\log (1-a c-b c x)}{a+b x} \, dx}{2 a^2}\\ &=-\frac {b^2 \log \left (\frac {b c x}{1-a c}\right ) \log (1-a c-b c x)}{2 a^2}-\frac {b^2 \log (x) \log \left (1+\frac {b x}{a}\right ) \log (1-c (a+b x))}{2 a^2}-\frac {b^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 )}{4 a^2}-\frac {b^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}{4 a^2}-\frac {b^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 )}{2 a^2}-\frac {b \text {Li}_2(c (a+b x))}{2 a x}-\frac {b^2 \log (x) \text {Li}_2(c (a+b x))}{2 a^2}-\frac {b^2 \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 )}{2 a^2}+\frac {b^2 \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 )}{2 a^2}-\frac {b^2 \left (\log (x)+\log \left (-\frac {a (1-c (a+b x))}{b x}\right )\right ) \text {Li}_2(1-c (a+b x))}{2 a^2}+\frac {b^2 \text {Li}_3\left (-\frac {b x}{a}\right )}{2 a^2}+\frac {\left (b^2-\frac {a^2}{x^2}\right ) \text {Li}_3(c (a+b x))}{2 a^2}-\frac {b^2 \text {Li}_3\left (-\frac {b x}{a (1-c (a+b x))}\right )}{2 a^2}+\frac {b^2 \text {Li}_3\left (-\frac {b c x}{1-c (a+b x)}\right )}{2 a^2}+\frac {b^2 \text {Li}_3(1-c (a+b x))}{2 a^2}+\frac {b^2 \operatorname {Subst}\left (\int \frac {\log (1-c x)}{x} \, dx,x,a+b x\right )}{2 a^2}-\frac {\left (b^3 c\right ) \int \frac {\log \left (-\frac {b c x}{-1+a c}\right )}{1-a c-b c x} \, dx}{2 a^2}\\ &=-\frac {b^2 \log \left (\frac {b c x}{1-a c}\right ) \log (1-a c-b c x)}{2 a^2}-\frac {b^2 \log (x) \log \left (1+\frac {b x}{a}\right ) \log (1-c (a+b x))}{2 a^2}-\frac {b^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 )}{4 a^2}-\frac {b^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}{4 a^2}-\frac {b^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 )}{2 a^2}-\frac {b^2 \text {Li}_2(c (a+b x))}{2 a^2}-\frac {b \text {Li}_2(c (a+b x))}{2 a x}-\frac {b^2 \log (x) \text {Li}_2(c (a+b x))}{2 a^2}-\frac {b^2 \text {Li}_2\left (1-\frac {b c x}{1-a c}\right )}{2 a^2}-\frac {b^2 \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 )}{2 a^2}+\frac {b^2 \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 )}{2 a^2}-\frac {b^2 \left (\log (x)+\log \left (-\frac {a (1-c (a+b x))}{b x}\right )\right ) \text {Li}_2(1-c (a+b x))}{2 a^2}+\frac {b^2 \text {Li}_3\left (-\frac {b x}{a}\right )}{2 a^2}+\frac {\left (b^2-\frac {a^2}{x^2}\right ) \text {Li}_3(c (a+b x))}{2 a^2}-\frac {b^2 \text {Li}_3\left (-\frac {b x}{a (1-c (a+b x))}\right )}{2 a^2}+\frac {b^2 \text {Li}_3\left (-\frac {b c x}{1-c (a+b x)}\right )}{2 a^2}+\frac {b^2 \text {Li}_3(1-c (a+b x))}{2 a^2}\\ \end {align*}
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Mathematica [A] time = 1.73, size = 573, normalized size = 0.91 \[ \frac {\frac {b x \left (b x \left (\text {Li}_2\left (\frac {b c x}{1-a c}\right )+\text {Li}_2(-a c-b x c+1)+\text {Li}_3(c (a+b x))+\text {Li}_3(-a c-b x c+1)-\text {Li}_3\left (\frac {a (a c+b x c-1)}{b x}\right )+\text {Li}_3\left (\frac {a c+b x c-1}{b c x}\right )+\left (\text {Li}_2\left (\frac {a (a c+b x c-1)}{b x}\right )-\text {Li}_2\left (\frac {a c+b x c-1}{b c x}\right )\right ) \log \left (\frac {a (a c+b c x-1)}{b x}\right )-\text {Li}_2\left (-\frac {b x}{a}\right ) \left (\log (-a c-b c x+1)-\log \left (\frac {a (a c+b c x-1)}{b x}\right )\right )-\log (a+b x) \text {Li}_2(c (a+b x))-\text {Li}_2(-a c-b x c+1) \left (\log \left (\frac {a (a c+b c x-1)}{b x}\right )+\log (x)\right )-\frac {1}{2} \left (\log \left (\frac {1-a c}{b c x}\right )-\log \left (\frac {(1-a c) (a+b x)}{b x}\right )+\log \left (\frac {b x}{a}+1\right )\right ) \log ^2\left (\frac {a (a c+b c x-1)}{b x}\right )-\left (\log (c (a+b x))-\log \left (\frac {b x}{a}+1\right )\right ) \log (-a c-b c x+1) \log \left (\frac {a (a c+b c x-1)}{b x}\right )+\log (c (a+b x)) \log (-a c-b c x+1)-\log (x) \log \left (\frac {b x}{a}+1\right ) \log (-a c-b c x+1)+\frac {1}{2} \left (\log (c (a+b x))-\log \left (\frac {b x}{a}+1\right )\right ) \log (-a c-b c x+1) (\log (-a c-b c x+1)-2 \log (x))-\log (x) \left (\log (-a c-b c x+1)-\log \left (\frac {b c x}{a c-1}+1\right )\right )+\text {Li}_3\left (-\frac {b x}{a}\right )\right )-(-b x \log (a+b x)+a+b x \log (x)) \text {Li}_2(c (a+b x))\right )}{a^2}-\text {Li}_3(c (a+b x))}{2 x^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.25, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\rm polylog}\left (3, b c x + a c\right )}{x^{3}}, x\right ) \]
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 \[ \int \frac {{\rm Li}_{3}({\left (b x + a\right )} c)}{x^{3}}\,{d x} \]
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 \[ \int \frac {\polylog \left (3, c \left (b x +a \right )\right )}{x^{3}}\, dx \]
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 \[ b \int \frac {{\rm Li}_2\left (b c x + a c\right )}{2 \, {\left (b x^{3} + a x^{2}\right )}}\,{d x} - \frac {{\rm Li}_{3}(b c x + a c)}{2 \, x^{2}} \]
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 \[ \int \frac {\mathrm {polylog}\left (3,c\,\left (a+b\,x\right )\right )}{x^3} \,d x \]
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 \[ \int \frac {\operatorname {Li}_{3}\left (a c + b c x\right )}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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