Optimal. Leaf size=66 \[ -\frac {x}{9 a^2}-\frac {x^2}{18 a}-\frac {x^3}{27}-\frac {\log (1-a x)}{9 a^3}+\frac {1}{9} x^3 \log (1-a x)+\frac {1}{3} x^3 \text {PolyLog}(2,a x) \]
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Rubi [A]
time = 0.03, antiderivative size = 66, 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 = {6726, 2442, 45}
\begin {gather*} -\frac {\log (1-a x)}{9 a^3}-\frac {x}{9 a^2}+\frac {1}{3} x^3 \text {Li}_2(a x)+\frac {1}{9} x^3 \log (1-a x)-\frac {x^2}{18 a}-\frac {x^3}{27} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2442
Rule 6726
Rubi steps
\begin {align*} \int x^2 \text {Li}_2(a x) \, dx &=\frac {1}{3} x^3 \text {Li}_2(a x)+\frac {1}{3} \int x^2 \log (1-a x) \, dx\\ &=\frac {1}{9} x^3 \log (1-a x)+\frac {1}{3} x^3 \text {Li}_2(a x)+\frac {1}{9} a \int \frac {x^3}{1-a x} \, dx\\ &=\frac {1}{9} x^3 \log (1-a x)+\frac {1}{3} x^3 \text {Li}_2(a x)+\frac {1}{9} a \int \left (-\frac {1}{a^3}-\frac {x}{a^2}-\frac {x^2}{a}-\frac {1}{a^3 (-1+a x)}\right ) \, dx\\ &=-\frac {x}{9 a^2}-\frac {x^2}{18 a}-\frac {x^3}{27}-\frac {\log (1-a x)}{9 a^3}+\frac {1}{9} x^3 \log (1-a x)+\frac {1}{3} x^3 \text {Li}_2(a x)\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 57, normalized size = 0.86 \begin {gather*} \frac {-a x \left (6+3 a x+2 a^2 x^2\right )+6 \left (-1+a^3 x^3\right ) \log (1-a x)+18 a^3 x^3 \text {PolyLog}(2,a x)}{54 a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 93, normalized size = 1.41
method | result | size |
meijerg | \(\frac {-\frac {a x \left (4 a^{2} x^{2}+6 a x +12\right )}{108}-\frac {\left (-4 a^{3} x^{3}+4\right ) \ln \left (-a x +1\right )}{36}+\frac {a^{3} x^{3} \polylog \left (2, a x \right )}{3}}{a^{3}}\) | \(56\) |
derivativedivides | \(\frac {\frac {a^{3} x^{3} \polylog \left (2, a x \right )}{3}-\frac {\ln \left (-a x +1\right ) \left (-a x +1\right )^{3}}{9}+\frac {\left (-a x +1\right )^{3}}{27}+\frac {\ln \left (-a x +1\right ) \left (-a x +1\right )^{2}}{3}-\frac {\left (-a x +1\right )^{2}}{6}-\frac {\ln \left (-a x +1\right ) \left (-a x +1\right )}{3}+\frac {1}{3}-\frac {a x}{3}}{a^{3}}\) | \(93\) |
default | \(\frac {\frac {a^{3} x^{3} \polylog \left (2, a x \right )}{3}-\frac {\ln \left (-a x +1\right ) \left (-a x +1\right )^{3}}{9}+\frac {\left (-a x +1\right )^{3}}{27}+\frac {\ln \left (-a x +1\right ) \left (-a x +1\right )^{2}}{3}-\frac {\left (-a x +1\right )^{2}}{6}-\frac {\ln \left (-a x +1\right ) \left (-a x +1\right )}{3}+\frac {1}{3}-\frac {a x}{3}}{a^{3}}\) | \(93\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 56, normalized size = 0.85 \begin {gather*} \frac {18 \, a^{3} x^{3} {\rm Li}_2\left (a x\right ) - 2 \, a^{3} x^{3} - 3 \, a^{2} x^{2} - 6 \, a x + 6 \, {\left (a^{3} x^{3} - 1\right )} \log \left (-a x + 1\right )}{54 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 56, normalized size = 0.85 \begin {gather*} \frac {18 \, a^{3} x^{3} {\rm Li}_2\left (a x\right ) - 2 \, a^{3} x^{3} - 3 \, a^{2} x^{2} - 6 \, a x + 6 \, {\left (a^{3} x^{3} - 1\right )} \log \left (-a x + 1\right )}{54 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.89, size = 49, normalized size = 0.74 \begin {gather*} \begin {cases} - \frac {x^{3} \operatorname {Li}_{1}\left (a x\right )}{9} + \frac {x^{3} \operatorname {Li}_{2}\left (a x\right )}{3} - \frac {x^{3}}{27} - \frac {x^{2}}{18 a} - \frac {x}{9 a^{2}} + \frac {\operatorname {Li}_{1}\left (a x\right )}{9 a^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.23, size = 53, normalized size = 0.80 \begin {gather*} \frac {x^3\,\ln \left (1-a\,x\right )}{9}-\frac {\ln \left (a\,x-1\right )}{9\,a^3}-\frac {x}{9\,a^2}-\frac {x^3}{27}+\frac {x^3\,\mathrm {polylog}\left (2,a\,x\right )}{3}-\frac {x^2}{18\,a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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