Optimal. Leaf size=76 \[ -\frac {\log (1-a x)}{16 a^4}-\frac {x}{16 a^3}-\frac {x^2}{32 a^2}+\frac {1}{4} x^4 \text {Li}_2(a x)+\frac {1}{16} x^4 \log (1-a x)-\frac {x^3}{48 a}-\frac {x^4}{64} \]
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Rubi [A] time = 0.04, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6591, 2395, 43} \[ \frac {1}{4} x^4 \text {PolyLog}(2,a x)-\frac {x^2}{32 a^2}-\frac {x}{16 a^3}-\frac {\log (1-a x)}{16 a^4}-\frac {x^3}{48 a}+\frac {1}{16} x^4 \log (1-a x)-\frac {x^4}{64} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2395
Rule 6591
Rubi steps
\begin {align*} \int x^3 \text {Li}_2(a x) \, dx &=\frac {1}{4} x^4 \text {Li}_2(a x)+\frac {1}{4} \int x^3 \log (1-a x) \, dx\\ &=\frac {1}{16} x^4 \log (1-a x)+\frac {1}{4} x^4 \text {Li}_2(a x)+\frac {1}{16} a \int \frac {x^4}{1-a x} \, dx\\ &=\frac {1}{16} x^4 \log (1-a x)+\frac {1}{4} x^4 \text {Li}_2(a x)+\frac {1}{16} a \int \left (-\frac {1}{a^4}-\frac {x}{a^3}-\frac {x^2}{a^2}-\frac {x^3}{a}-\frac {1}{a^4 (-1+a x)}\right ) \, dx\\ &=-\frac {x}{16 a^3}-\frac {x^2}{32 a^2}-\frac {x^3}{48 a}-\frac {x^4}{64}-\frac {\log (1-a x)}{16 a^4}+\frac {1}{16} x^4 \log (1-a x)+\frac {1}{4} x^4 \text {Li}_2(a x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 65, normalized size = 0.86 \[ \frac {48 a^4 x^4 \text {Li}_2(a x)+12 \left (a^4 x^4-1\right ) \log (1-a x)-a x \left (3 a^3 x^3+4 a^2 x^2+6 a x+12\right )}{192 a^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.52, size = 64, normalized size = 0.84 \[ \frac {48 \, a^{4} x^{4} {\rm Li}_2\left (a x\right ) - 3 \, a^{4} x^{4} - 4 \, a^{3} x^{3} - 6 \, a^{2} x^{2} - 12 \, a x + 12 \, {\left (a^{4} x^{4} - 1\right )} \log \left (-a x + 1\right )}{192 \, a^{4}} \]
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 \[ \int x^{3} {\rm Li}_2\left (a x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 68, normalized size = 0.89 \[ \frac {x^{4} \polylog \left (2, a x \right )}{4}+\frac {x^{4} \ln \left (-a x +1\right )}{16}-\frac {\ln \left (-a x +1\right )}{16 a^{4}}-\frac {x^{4}}{64}-\frac {x^{3}}{48 a}-\frac {x^{2}}{32 a^{2}}-\frac {x}{16 a^{3}}+\frac {25}{192 a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 64, normalized size = 0.84 \[ \frac {48 \, a^{4} x^{4} {\rm Li}_2\left (a x\right ) - 3 \, a^{4} x^{4} - 4 \, a^{3} x^{3} - 6 \, a^{2} x^{2} - 12 \, a x + 12 \, {\left (a^{4} x^{4} - 1\right )} \log \left (-a x + 1\right )}{192 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.25, size = 61, normalized size = 0.80 \[ \frac {x^4\,\ln \left (1-a\,x\right )}{16}-\frac {\ln \left (a\,x-1\right )}{16\,a^4}-\frac {x}{16\,a^3}-\frac {x^4}{64}+\frac {x^4\,\mathrm {polylog}\left (2,a\,x\right )}{4}-\frac {x^3}{48\,a}-\frac {x^2}{32\,a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.61, size = 58, normalized size = 0.76 \[ \begin {cases} - \frac {x^{4} \operatorname {Li}_{1}\left (a x\right )}{16} + \frac {x^{4} \operatorname {Li}_{2}\left (a x\right )}{4} - \frac {x^{4}}{64} - \frac {x^{3}}{48 a} - \frac {x^{2}}{32 a^{2}} - \frac {x}{16 a^{3}} + \frac {\operatorname {Li}_{1}\left (a x\right )}{16 a^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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