Optimal. Leaf size=60 \[ x \text{PolyLog}(2,c (a+b x))+\frac{a \text{PolyLog}(2,c (a+b x))}{b}-\frac{(-a c-b c x+1) \log (-a c-b c x+1)}{b c}-x \]
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Rubi [A] time = 0.0486846, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.778, Rules used = {6595, 2444, 2389, 2295, 2421, 2393, 2391} \[ x \text{PolyLog}(2,c (a+b x))+\frac{a \text{PolyLog}(2,c (a+b x))}{b}-\frac{(-a c-b c x+1) \log (-a c-b c x+1)}{b c}-x \]
Antiderivative was successfully verified.
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Rule 6595
Rule 2444
Rule 2389
Rule 2295
Rule 2421
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \text{Li}_2(c (a+b x)) \, dx &=x \text{Li}_2(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))-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 \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))\\ \end{align*}
Mathematica [A] time = 0.0177979, size = 53, normalized size = 0.88 \[ \frac{c (a+b x) \text{PolyLog}(2,c (a+b x))-c (a+b x)+(c (a+b x)-1) \log (1-c (a+b x))}{b c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 96, normalized size = 1.6 \begin{align*} \ln \left ( -xbc-ac+1 \right ) x+{\it polylog} \left ( 2,xbc+ac \right ) x+{\frac{\ln \left ( -xbc-ac+1 \right ) a}{b}}+{\frac{{\it polylog} \left ( 2,xbc+ac \right ) a}{b}}-x-{\frac{a}{b}}-{\frac{\ln \left ( -xbc-ac+1 \right ) }{bc}}+{\frac{1}{bc}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.982883, size = 122, normalized size = 2.03 \begin{align*} -\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{b c x{\rm Li}_2\left (b c x + a c\right ) - b c x +{\left (b c x + a c - 1\right )} \log \left (-b c x - a c + 1\right )}{b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.41585, size = 126, normalized size = 2.1 \begin{align*} -\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 )}{b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\rm Li}_2\left ({\left (b x + a\right )} c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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