Optimal. Leaf size=78 \[ \frac {x^2}{16 a}+\frac {x^4}{32}+\frac {\log \left (1-a x^2\right )}{16 a^2}-\frac {1}{16} x^4 \log \left (1-a x^2\right )-\frac {1}{8} x^4 \text {PolyLog}\left (2,a x^2\right )+\frac {1}{4} x^4 \text {PolyLog}\left (3,a x^2\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6726, 2504,
2442, 45} \begin {gather*} \frac {\log \left (1-a x^2\right )}{16 a^2}-\frac {1}{8} x^4 \text {Li}_2\left (a x^2\right )+\frac {1}{4} x^4 \text {Li}_3\left (a x^2\right )+\frac {x^2}{16 a}-\frac {1}{16} x^4 \log \left (1-a x^2\right )+\frac {x^4}{32} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2442
Rule 2504
Rule 6726
Rubi steps
\begin {align*} \int x^3 \text {Li}_3\left (a x^2\right ) \, dx &=\frac {1}{4} x^4 \text {Li}_3\left (a x^2\right )-\frac {1}{2} \int x^3 \text {Li}_2\left (a x^2\right ) \, dx\\ &=-\frac {1}{8} x^4 \text {Li}_2\left (a x^2\right )+\frac {1}{4} x^4 \text {Li}_3\left (a x^2\right )-\frac {1}{4} \int x^3 \log \left (1-a x^2\right ) \, dx\\ &=-\frac {1}{8} x^4 \text {Li}_2\left (a x^2\right )+\frac {1}{4} x^4 \text {Li}_3\left (a x^2\right )-\frac {1}{8} \text {Subst}\left (\int x \log (1-a x) \, dx,x,x^2\right )\\ &=-\frac {1}{16} x^4 \log \left (1-a x^2\right )-\frac {1}{8} x^4 \text {Li}_2\left (a x^2\right )+\frac {1}{4} x^4 \text {Li}_3\left (a x^2\right )-\frac {1}{16} a \text {Subst}\left (\int \frac {x^2}{1-a x} \, dx,x,x^2\right )\\ &=-\frac {1}{16} x^4 \log \left (1-a x^2\right )-\frac {1}{8} x^4 \text {Li}_2\left (a x^2\right )+\frac {1}{4} x^4 \text {Li}_3\left (a x^2\right )-\frac {1}{16} a \text {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}{16 a}+\frac {x^4}{32}+\frac {\log \left (1-a x^2\right )}{16 a^2}-\frac {1}{16} x^4 \log \left (1-a x^2\right )-\frac {1}{8} x^4 \text {Li}_2\left (a x^2\right )+\frac {1}{4} x^4 \text {Li}_3\left (a x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 79, normalized size = 1.01 \begin {gather*} \frac {2 a x^2+a^2 x^4+2 \log \left (1-a x^2\right )-2 a^2 x^4 \log \left (1-a x^2\right )-4 a^2 x^4 \text {PolyLog}\left (2,a x^2\right )+8 a^2 x^4 \text {PolyLog}\left (3,a x^2\right )}{32 a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 72, normalized size = 0.92
method | result | size |
meijerg | \(-\frac {-\frac {a \,x^{2} \left (3 a \,x^{2}+6\right )}{48}-\frac {\left (-3 a^{2} x^{4}+3\right ) \ln \left (-a \,x^{2}+1\right )}{24}+\frac {a^{2} x^{4} \polylog \left (2, a \,x^{2}\right )}{4}-\frac {a^{2} x^{4} \polylog \left (3, a \,x^{2}\right )}{2}}{2 a^{2}}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 69, normalized size = 0.88 \begin {gather*} -\frac {4 \, a^{2} x^{4} {\rm Li}_2\left (a x^{2}\right ) - 8 \, a^{2} x^{4} {\rm Li}_{3}(a x^{2}) - a^{2} x^{4} - 2 \, a x^{2} + 2 \, {\left (a^{2} x^{4} - 1\right )} \log \left (-a x^{2} + 1\right )}{32 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.45, size = 69, normalized size = 0.88 \begin {gather*} -\frac {4 \, a^{2} x^{4} {\rm Li}_2\left (a x^{2}\right ) - 8 \, a^{2} x^{4} {\rm polylog}\left (3, 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 )}{32 \, a^{2}} \end {gather*}
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 \begin {gather*} \int x^{3} \operatorname {Li}_{3}\left (a x^{2}\right )\, dx \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 = 0.30, size = 65, normalized size = 0.83 \begin {gather*} \frac {x^4\,\mathrm {polylog}\left (3,a\,x^2\right )}{4}-\frac {x^4\,\mathrm {polylog}\left (2,a\,x^2\right )}{8}+\frac {\ln \left (a\,x^2-1\right )}{16\,a^2}-\frac {x^4\,\ln \left (1-a\,x^2\right )}{16}+\frac {x^4}{32}+\frac {x^2}{16\,a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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