Optimal. Leaf size=68 \[ \frac {\log (1-a x)}{8 a^2}-\frac {1}{4} x^2 \text {Li}_2(a x)+\frac {1}{2} x^2 \text {Li}_3(a x)-\frac {1}{8} x^2 \log (1-a x)+\frac {x}{8 a}+\frac {x^2}{16} \]
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Rubi [A] time = 0.04, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6591, 2395, 43} \[ -\frac {1}{4} x^2 \text {PolyLog}(2,a x)+\frac {1}{2} x^2 \text {PolyLog}(3,a x)+\frac {\log (1-a x)}{8 a^2}-\frac {1}{8} x^2 \log (1-a x)+\frac {x}{8 a}+\frac {x^2}{16} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2395
Rule 6591
Rubi steps
\begin {align*} \int x \text {Li}_3(a x) \, dx &=\frac {1}{2} x^2 \text {Li}_3(a x)-\frac {1}{2} \int x \text {Li}_2(a x) \, dx\\ &=-\frac {1}{4} x^2 \text {Li}_2(a x)+\frac {1}{2} x^2 \text {Li}_3(a x)-\frac {1}{4} \int x \log (1-a x) \, dx\\ &=-\frac {1}{8} x^2 \log (1-a x)-\frac {1}{4} x^2 \text {Li}_2(a x)+\frac {1}{2} x^2 \text {Li}_3(a x)-\frac {1}{8} a \int \frac {x^2}{1-a x} \, dx\\ &=-\frac {1}{8} x^2 \log (1-a x)-\frac {1}{4} x^2 \text {Li}_2(a x)+\frac {1}{2} x^2 \text {Li}_3(a x)-\frac {1}{8} a \int \left (-\frac {1}{a^2}-\frac {x}{a}-\frac {1}{a^2 (-1+a x)}\right ) \, dx\\ &=\frac {x}{8 a}+\frac {x^2}{16}+\frac {\log (1-a x)}{8 a^2}-\frac {1}{8} x^2 \log (1-a x)-\frac {1}{4} x^2 \text {Li}_2(a x)+\frac {1}{2} x^2 \text {Li}_3(a x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 69, normalized size = 1.01 \[ \frac {-4 a^2 x^2 \text {Li}_2(a x)+8 a^2 x^2 \text {Li}_3(a x)+a^2 x^2-2 a^2 x^2 \log (1-a x)+2 a x+2 \log (1-a x)}{16 a^2} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.77, size = 61, normalized size = 0.90 \[ -\frac {4 \, a^{2} x^{2} {\rm Li}_2\left (a x\right ) - 8 \, a^{2} x^{2} {\rm polylog}\left (3, a x\right ) - a^{2} x^{2} - 2 \, a x + 2 \, {\left (a^{2} x^{2} - 1\right )} \log \left (-a x + 1\right )}{16 \, a^{2}} \]
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 \[ \int x {\rm Li}_{3}(a x)\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 62, normalized size = 0.91 \[ -\frac {-\frac {a x \left (3 a x +6\right )}{48}-\frac {\left (-3 a^{2} x^{2}+3\right ) \ln \left (-a x +1\right )}{24}+\frac {a^{2} x^{2} \polylog \left (2, a x \right )}{4}-\frac {a^{2} x^{2} \polylog \left (3, a x \right )}{2}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 61, normalized size = 0.90 \[ -\frac {4 \, a^{2} x^{2} {\rm Li}_2\left (a x\right ) - 8 \, a^{2} x^{2} {\rm Li}_{3}(a x) - a^{2} x^{2} - 2 \, a x + 2 \, {\left (a^{2} x^{2} - 1\right )} \log \left (-a x + 1\right )}{16 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.90, size = 55, normalized size = 0.81 \[ \frac {\ln \left (a\,x-1\right )}{8\,a^2}-\frac {x^2\,\ln \left (1-a\,x\right )}{8}+\frac {x}{8\,a}+\frac {x^2}{16}-\frac {x^2\,\mathrm {polylog}\left (2,a\,x\right )}{4}+\frac {x^2\,\mathrm {polylog}\left (3,a\,x\right )}{2} \]
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 \[ \int x \operatorname {Li}_{3}\left (a x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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