Optimal. Leaf size=64 \[ -\frac {\log \left (1-a x^2\right )}{8 a^2}+\frac {1}{4} x^4 \text {Li}_2\left (a x^2\right )-\frac {x^2}{8 a}+\frac {1}{8} x^4 \log \left (1-a x^2\right )-\frac {x^4}{16} \]
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Rubi [A] time = 0.05, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6591, 2454, 2395, 43} \[ \frac {1}{4} x^4 \text {PolyLog}\left (2,a x^2\right )-\frac {\log \left (1-a x^2\right )}{8 a^2}-\frac {x^2}{8 a}+\frac {1}{8} x^4 \log \left (1-a x^2\right )-\frac {x^4}{16} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2395
Rule 2454
Rule 6591
Rubi steps
\begin {align*} \int x^3 \text {Li}_2\left (a x^2\right ) \, dx &=\frac {1}{4} x^4 \text {Li}_2\left (a x^2\right )+\frac {1}{2} \int x^3 \log \left (1-a x^2\right ) \, dx\\ &=\frac {1}{4} x^4 \text {Li}_2\left (a x^2\right )+\frac {1}{4} \operatorname {Subst}\left (\int x \log (1-a x) \, dx,x,x^2\right )\\ &=\frac {1}{8} x^4 \log \left (1-a x^2\right )+\frac {1}{4} x^4 \text {Li}_2\left (a x^2\right )+\frac {1}{8} a \operatorname {Subst}\left (\int \frac {x^2}{1-a x} \, dx,x,x^2\right )\\ &=\frac {1}{8} x^4 \log \left (1-a x^2\right )+\frac {1}{4} x^4 \text {Li}_2\left (a x^2\right )+\frac {1}{8} a \operatorname {Subst}\left (\int \left (-\frac {1}{a^2}-\frac {x}{a}-\frac {1}{a^2 (-1+a x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {x^2}{8 a}-\frac {x^4}{16}-\frac {\log \left (1-a x^2\right )}{8 a^2}+\frac {1}{8} x^4 \log \left (1-a x^2\right )+\frac {1}{4} x^4 \text {Li}_2\left (a x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 56, normalized size = 0.88 \[ \frac {4 a^2 x^4 \text {Li}_2\left (a x^2\right )+2 \left (a^2 x^4-1\right ) \log \left (1-a x^2\right )-a x^2 \left (a x^2+2\right )}{16 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 54, normalized size = 0.84 \[ \frac {4 \, a^{2} x^{4} {\rm Li}_2\left (a x^{2}\right ) - a^{2} x^{4} - 2 \, a x^{2} + 2 \, {\left (a^{2} x^{4} - 1\right )} \log \left (-a x^{2} + 1\right )}{16 \, a^{2}} \]
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 \[ \int x^{3} {\rm Li}_2\left (a x^{2}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 54, normalized size = 0.84 \[ \frac {x^{4} \polylog \left (2, a \,x^{2}\right )}{4}+\frac {x^{4} \ln \left (-a \,x^{2}+1\right )}{8}-\frac {x^{4}}{16}-\frac {x^{2}}{8 a}-\frac {\ln \left (a \,x^{2}-1\right )}{8 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 54, normalized size = 0.84 \[ \frac {4 \, a^{2} x^{4} {\rm Li}_2\left (a x^{2}\right ) - a^{2} x^{4} - 2 \, a x^{2} + 2 \, {\left (a^{2} x^{4} - 1\right )} \log \left (-a x^{2} + 1\right )}{16 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.27, size = 53, normalized size = 0.83 \[ \frac {x^4\,\mathrm {polylog}\left (2,a\,x^2\right )}{4}-\frac {\ln \left (a\,x^2-1\right )}{8\,a^2}+\frac {x^4\,\ln \left (1-a\,x^2\right )}{8}-\frac {x^4}{16}-\frac {x^2}{8\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.48, size = 48, normalized size = 0.75 \[ \begin {cases} - \frac {x^{4} \operatorname {Li}_{1}\left (a x^{2}\right )}{8} + \frac {x^{4} \operatorname {Li}_{2}\left (a x^{2}\right )}{4} - \frac {x^{4}}{16} - \frac {x^{2}}{8 a} + \frac {\operatorname {Li}_{1}\left (a x^{2}\right )}{8 a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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