Optimal. Leaf size=84 \[ x (-\text {Li}_2(c (a+b x)))+x \text {Li}_3(c (a+b x))-\frac {a \text {Li}_2(c (a+b x))}{b}+\frac {a \text {Li}_3(c (a+b x))}{b}+\frac {(-a c-b c x+1) \log (-a c-b c x+1)}{b c}+x \]
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Rubi [A] time = 0.07, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.889, Rules used = {6595, 2444, 2389, 2295, 2421, 2393, 2391, 6589} \[ x (-\text {PolyLog}(2,c (a+b x)))+x \text {PolyLog}(3,c (a+b x))-\frac {a \text {PolyLog}(2,c (a+b x))}{b}+\frac {a \text {PolyLog}(3,c (a+b x))}{b}+\frac {(-a c-b c x+1) \log (-a c-b c x+1)}{b c}+x \]
Antiderivative was successfully verified.
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Rule 2295
Rule 2389
Rule 2391
Rule 2393
Rule 2421
Rule 2444
Rule 6589
Rule 6595
Rubi steps
\begin {align*} \int \text {Li}_3(c (a+b x)) \, dx &=x \text {Li}_3(c (a+b x))+a \int \frac {\text {Li}_2(c (a+b x))}{a+b x} \, dx-\int \text {Li}_2(c (a+b x)) \, dx\\ &=-x \text {Li}_2(c (a+b x))+\frac {a \text {Li}_3(c (a+b x))}{b}+x \text {Li}_3(c (a+b x))+a \int \frac {\log (1-c (a+b x))}{a+b x} \, dx-\int \log (1-c (a+b x)) \, dx\\ &=-x \text {Li}_2(c (a+b x))+\frac {a \text {Li}_3(c (a+b x))}{b}+x \text {Li}_3(c (a+b x))+a \int \frac {\log (1-a c-b c x)}{a+b x} \, dx-\int \log (1-a c-b c x) \, dx\\ &=-x \text {Li}_2(c (a+b x))+\frac {a \text {Li}_3(c (a+b x))}{b}+x \text {Li}_3(c (a+b x))+\frac {a \operatorname {Subst}\left (\int \frac {\log (1-c x)}{x} \, dx,x,a+b x\right )}{b}+\frac {\operatorname {Subst}(\int \log (x) \, dx,x,1-a c-b c x)}{b c}\\ &=x+\frac {(1-a c-b c x) \log (1-a c-b c x)}{b c}-\frac {a \text {Li}_2(c (a+b x))}{b}-x \text {Li}_2(c (a+b x))+\frac {a \text {Li}_3(c (a+b x))}{b}+x \text {Li}_3(c (a+b x))\\ \end {align*}
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Mathematica [A] time = 0.02, size = 66, normalized size = 0.79 \[ \frac {(a+b x) \left (-\text {Li}_2(c (a+b x))+\text {Li}_3(c (a+b x))+\frac {\log (1-c (a+b x))}{c (a+b x)}-\log (1-c (a+b x))+1\right )}{b} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.98, size = 73, normalized size = 0.87 \[ \frac {b c x - {\left (b c x + a c\right )} {\rm Li}_2\left (b c x + a c\right ) - {\left (b c x + a c - 1\right )} \log \left (-b c x - a c + 1\right ) + {\left (b c x + a c\right )} {\rm polylog}\left (3, b c x + a c\right )}{b c} \]
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 \[ \int {\rm Li}_{3}({\left (b x + a\right )} c)\,{d x} \]
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 \[ \int \polylog \left (3, c \left (b x +a \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 120, normalized size = 1.43 \[ \frac {{\left (\log \left (b c x + a c\right ) \log \left (-b c x - a c + 1\right ) + {\rm Li}_2\left (-b c x - a c + 1\right )\right )} a}{b} + \frac {a {\rm Li}_{3}(b c x + a c)}{b} - \frac {b c x {\rm Li}_2\left (b c x + a c\right ) - b c x {\rm Li}_{3}(b c x + a c) - b c x + {\left (b c x + a c - 1\right )} \log \left (-b c x - a c + 1\right )}{b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.18, size = 77, normalized size = 0.92 \[ x-\frac {\mathrm {polylog}\left (2,c\,\left (a+b\,x\right )\right )\,\left (a+b\,x\right )}{b}+\frac {\mathrm {polylog}\left (3,c\,\left (a+b\,x\right )\right )\,\left (a+b\,x\right )}{b}+\frac {\ln \left (c\,\left (a+b\,x\right )-1\right )}{b\,c}-\frac {\ln \left (1-c\,\left (a+b\,x\right )\right )\,\left (a+b\,x\right )}{b} \]
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 \[ \int \operatorname {Li}_{3}\left (c \left (a + b x\right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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