Optimal. Leaf size=56 \[ -\frac {x}{4 a}-\frac {x^2}{8}-\frac {\log (1-a x)}{4 a^2}+\frac {1}{4} x^2 \log (1-a x)+\frac {1}{2} x^2 \text {PolyLog}(2,a x) \]
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Rubi [A]
time = 0.02, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6726, 2442, 45}
\begin {gather*} -\frac {\log (1-a x)}{4 a^2}+\frac {1}{2} x^2 \text {Li}_2(a x)+\frac {1}{4} x^2 \log (1-a x)-\frac {x}{4 a}-\frac {x^2}{8} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2442
Rule 6726
Rubi steps
\begin {align*} \int x \text {Li}_2(a x) \, dx &=\frac {1}{2} x^2 \text {Li}_2(a x)+\frac {1}{2} \int x \log (1-a x) \, dx\\ &=\frac {1}{4} x^2 \log (1-a x)+\frac {1}{2} x^2 \text {Li}_2(a x)+\frac {1}{4} a \int \frac {x^2}{1-a x} \, dx\\ &=\frac {1}{4} x^2 \log (1-a x)+\frac {1}{2} x^2 \text {Li}_2(a x)+\frac {1}{4} a \int \left (-\frac {1}{a^2}-\frac {x}{a}-\frac {1}{a^2 (-1+a x)}\right ) \, dx\\ &=-\frac {x}{4 a}-\frac {x^2}{8}-\frac {\log (1-a x)}{4 a^2}+\frac {1}{4} x^2 \log (1-a x)+\frac {1}{2} x^2 \text {Li}_2(a x)\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 48, normalized size = 0.86 \begin {gather*} \frac {-a x (2+a x)+2 \left (-1+a^2 x^2\right ) \log (1-a x)+4 a^2 x^2 \text {PolyLog}(2,a x)}{8 a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.35, size = 66, normalized size = 1.18
method | result | size |
meijerg | \(-\frac {\frac {a x \left (3 a x +6\right )}{24}+\frac {\left (-3 a^{2} x^{2}+3\right ) \ln \left (-a x +1\right )}{12}-\frac {a^{2} x^{2} \polylog \left (2, a x \right )}{2}}{a^{2}}\) | \(49\) |
derivativedivides | \(\frac {\frac {a^{2} x^{2} \polylog \left (2, a x \right )}{2}+\frac {\ln \left (-a x +1\right ) \left (-a x +1\right )^{2}}{4}-\frac {\left (-a x +1\right )^{2}}{8}-\frac {\ln \left (-a x +1\right ) \left (-a x +1\right )}{2}+\frac {1}{2}-\frac {a x}{2}}{a^{2}}\) | \(66\) |
default | \(\frac {\frac {a^{2} x^{2} \polylog \left (2, a x \right )}{2}+\frac {\ln \left (-a x +1\right ) \left (-a x +1\right )^{2}}{4}-\frac {\left (-a x +1\right )^{2}}{8}-\frac {\ln \left (-a x +1\right ) \left (-a x +1\right )}{2}+\frac {1}{2}-\frac {a x}{2}}{a^{2}}\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 48, normalized size = 0.86 \begin {gather*} \frac {4 \, a^{2} x^{2} {\rm Li}_2\left (a x\right ) - a^{2} x^{2} - 2 \, a x + 2 \, {\left (a^{2} x^{2} - 1\right )} \log \left (-a x + 1\right )}{8 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 48, normalized size = 0.86 \begin {gather*} \frac {4 \, a^{2} x^{2} {\rm Li}_2\left (a x\right ) - a^{2} x^{2} - 2 \, a x + 2 \, {\left (a^{2} x^{2} - 1\right )} \log \left (-a x + 1\right )}{8 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.48, size = 41, normalized size = 0.73 \begin {gather*} \begin {cases} - \frac {x^{2} \operatorname {Li}_{1}\left (a x\right )}{4} + \frac {x^{2} \operatorname {Li}_{2}\left (a x\right )}{2} - \frac {x^{2}}{8} - \frac {x}{4 a} + \frac {\operatorname {Li}_{1}\left (a x\right )}{4 a^{2}} & \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 = 0.34, size = 46, normalized size = 0.82 \begin {gather*} \frac {x^2\,\ln \left (1-a\,x\right )}{4}-\frac {\ln \left (1-a\,x\right )}{4\,a^2}-\frac {x}{4\,a}-\frac {x^2}{8}+\frac {x^2\,\mathrm {polylog}\left (2,a\,x\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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