Optimal. Leaf size=60 \[ x \text {Li}_2(c (a+b x))+\frac {a \text {Li}_2(c (a+b x))}{b}-\frac {(-a c-b c x+1) \log (-a c-b c x+1)}{b c}-x \]
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Rubi [A] time = 0.05, antiderivative size = 60, normalized size of antiderivative = 1.00, 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 2295
Rule 2389
Rule 2391
Rule 2393
Rule 2421
Rule 2444
Rule 6595
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*}
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Mathematica [A] time = 0.01, size = 53, normalized size = 0.88 \[ \frac {c (a+b x) \text {Li}_2(c (a+b x))-c (a+b x)+(c (a+b x)-1) \log (1-c (a+b x))}{b c} \]
Antiderivative was successfully verified.
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fricas [A] time = 3.95, size = 55, normalized size = 0.92 \[ -\frac {b c x - {\left (b c x + a c\right )} {\rm Li}_2\left (b c x + a c\right ) - {\left (b c x + a c - 1\right )} \log \left (-b c x - a c + 1\right )}{b c} \]
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 \[ \int {\rm Li}_2\left ({\left (b x + a\right )} c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 96, normalized size = 1.60 \[ \polylog \left (2, b c x +a c \right ) x +\ln \left (-b c x -a c +1\right ) x +\frac {\polylog \left (2, b c x +a c \right ) a}{b}+\frac {\ln \left (-b c x -a c +1\right ) a}{b}-x -\frac {a}{b}-\frac {\ln \left (-b c x -a c +1\right )}{b c}+\frac {1}{b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 90, normalized size = 1.50 \[ -\frac {{\left (\log \left (b c x + a c\right ) \log \left (-b c x - a c + 1\right ) + {\rm Li}_2\left (-b c x - a c + 1\right )\right )} a}{b} + \frac {b c x {\rm Li}_2\left (b c x + a c\right ) - b c x + {\left (b c x + a c - 1\right )} \log \left (-b c x - a c + 1\right )}{b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.00, size = 61, normalized size = 1.02 \[ \frac {\mathrm {polylog}\left (2,c\,\left (a+b\,x\right )\right )\,\left (a+b\,x\right )}{b}-x-\frac {\ln \left (1-c\,\left (a+b\,x\right )\right )}{b\,c}+\frac {\ln \left (1-c\,\left (a+b\,x\right )\right )\,\left (a+b\,x\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.07, size = 73, normalized size = 1.22 \[ \begin {cases} 0 & \text {for}\: b = 0 \wedge c = 0 \\x \operatorname {Li}_{2}\left (a c\right ) & \text {for}\: b = 0 \\0 & \text {for}\: c = 0 \\- \frac {a \operatorname {Li}_{1}\left (a c + b c x\right )}{b} + \frac {a \operatorname {Li}_{2}\left (a c + b c x\right )}{b} - x \operatorname {Li}_{1}\left (a c + b c x\right ) + x \operatorname {Li}_{2}\left (a c + b c x\right ) - x + \frac {\operatorname {Li}_{1}\left (a c + b c x\right )}{b c} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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