Optimal. Leaf size=34 \[ x (-\text {Li}_2(a x))+x \text {Li}_3(a x)+\frac {(1-a x) \log (1-a x)}{a}+x \]
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Rubi [A] time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, 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 2295
Rule 2389
Rule 6586
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*}
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Mathematica [A] time = 0.01, size = 39, normalized size = 1.15 \[ x \left (-\text {Li}_2(a x)+\text {Li}_3(a x)+\frac {\log (1-a x)}{a x}-\log (1-a x)+1\right ) \]
Antiderivative was successfully verified.
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fricas [C] time = 0.49, size = 39, normalized size = 1.15 \[ -\frac {a x {\rm Li}_2\left (a x\right ) - a x {\rm polylog}\left (3, a x\right ) - a x + {\left (a x - 1\right )} \log \left (-a x + 1\right )}{a} \]
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 \[ \int {\rm Li}_{3}(a x)\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 41, normalized size = 1.21 \[ \frac {a x +\frac {\left (-2 a x +2\right ) \ln \left (-a x +1\right )}{2}-a x \polylog \left (2, a x \right )+a x \polylog \left (3, a x \right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 39, normalized size = 1.15 \[ -\frac {a x {\rm Li}_2\left (a x\right ) - a x {\rm Li}_{3}(a x) - a x + {\left (a x - 1\right )} \log \left (-a x + 1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.84, size = 37, normalized size = 1.09 \[ x+\frac {\ln \left (a\,x-1\right )}{a}-x\,\mathrm {polylog}\left (2,a\,x\right )+x\,\mathrm {polylog}\left (3,a\,x\right )-x\,\ln \left (1-a\,x\right ) \]
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 \[ \int \operatorname {Li}_{3}\left (a x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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