Optimal. Leaf size=34 \[ x (-\text{PolyLog}(2,a x))+x \text{PolyLog}(3,a x)+\frac{(1-a x) \log (1-a x)}{a}+x \]
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Rubi [A] time = 0.0108174, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 5, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {6586, 2389, 2295} \[ x (-\text{PolyLog}(2,a x))+x \text{PolyLog}(3,a x)+\frac{(1-a x) \log (1-a x)}{a}+x \]
Antiderivative was successfully verified.
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Rule 6586
Rule 2389
Rule 2295
Rubi steps
\begin{align*} \int \text{Li}_3(a x) \, dx &=x \text{Li}_3(a x)-\int \text{Li}_2(a x) \, dx\\ &=-x \text{Li}_2(a x)+x \text{Li}_3(a x)-\int \log (1-a x) \, dx\\ &=-x \text{Li}_2(a x)+x \text{Li}_3(a x)+\frac{\operatorname{Subst}(\int \log (x) \, dx,x,1-a x)}{a}\\ &=x+\frac{(1-a x) \log (1-a x)}{a}-x \text{Li}_2(a x)+x \text{Li}_3(a x)\\ \end{align*}
Mathematica [A] time = 0.0116823, size = 39, normalized size = 1.15 \[ x \left (-\text{PolyLog}(2,a x)+\text{PolyLog}(3,a x)+\frac{\log (1-a x)}{a x}-\log (1-a x)+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.092, size = 41, normalized size = 1.2 \begin{align*}{\frac{1}{a} \left ( ax+{\frac{ \left ( -2\,ax+2 \right ) \ln \left ( -ax+1 \right ) }{2}}-ax{\it polylog} \left ( 2,ax \right ) +ax{\it polylog} \left ( 3,ax \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.990846, size = 53, normalized size = 1.56 \begin{align*} -\frac{a x{\rm Li}_2\left (a x\right ) - a x{\rm Li}_{3}(a x) - a x +{\left (a x - 1\right )} \log \left (-a x + 1\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.5983, size = 151, normalized size = 4.44 \begin{align*} -\frac{a x{\rm \%iint}\left (a, x, -\frac{\log \left (-a x + 1\right )}{a}, -\frac{\log \left (-a x + 1\right )}{x}\right ) - a x{\rm polylog}\left (3, a x\right ) - a x +{\left (a x - 1\right )} \log \left (-a x + 1\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{Li}_{3}\left (a x\right )\, dx \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}_{3}(a x)\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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