Optimal. Leaf size=88 \[ \frac {x^2}{54 a^2}+\frac {x^4}{108 a}+\frac {x^6}{162}+\frac {\log \left (1-a x^2\right )}{54 a^3}-\frac {1}{54} x^6 \log \left (1-a x^2\right )-\frac {1}{18} x^6 \text {PolyLog}\left (2,a x^2\right )+\frac {1}{6} x^6 \text {PolyLog}\left (3,a x^2\right ) \]
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Rubi [A]
time = 0.05, antiderivative size = 88, 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 )}{54 a^3}+\frac {x^2}{54 a^2}-\frac {1}{18} x^6 \text {Li}_2\left (a x^2\right )+\frac {1}{6} x^6 \text {Li}_3\left (a x^2\right )+\frac {x^4}{108 a}-\frac {1}{54} x^6 \log \left (1-a x^2\right )+\frac {x^6}{162} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2442
Rule 2504
Rule 6726
Rubi steps
\begin {align*} \int x^5 \text {Li}_3\left (a x^2\right ) \, dx &=\frac {1}{6} x^6 \text {Li}_3\left (a x^2\right )-\frac {1}{3} \int x^5 \text {Li}_2\left (a x^2\right ) \, dx\\ &=-\frac {1}{18} x^6 \text {Li}_2\left (a x^2\right )+\frac {1}{6} x^6 \text {Li}_3\left (a x^2\right )-\frac {1}{9} \int x^5 \log \left (1-a x^2\right ) \, dx\\ &=-\frac {1}{18} x^6 \text {Li}_2\left (a x^2\right )+\frac {1}{6} x^6 \text {Li}_3\left (a x^2\right )-\frac {1}{18} \text {Subst}\left (\int x^2 \log (1-a x) \, dx,x,x^2\right )\\ &=-\frac {1}{54} x^6 \log \left (1-a x^2\right )-\frac {1}{18} x^6 \text {Li}_2\left (a x^2\right )+\frac {1}{6} x^6 \text {Li}_3\left (a x^2\right )-\frac {1}{54} a \text {Subst}\left (\int \frac {x^3}{1-a x} \, dx,x,x^2\right )\\ &=-\frac {1}{54} x^6 \log \left (1-a x^2\right )-\frac {1}{18} x^6 \text {Li}_2\left (a x^2\right )+\frac {1}{6} x^6 \text {Li}_3\left (a x^2\right )-\frac {1}{54} a \text {Subst}\left (\int \left (-\frac {1}{a^3}-\frac {x}{a^2}-\frac {x^2}{a}-\frac {1}{a^3 (-1+a x)}\right ) \, dx,x,x^2\right )\\ &=\frac {x^2}{54 a^2}+\frac {x^4}{108 a}+\frac {x^6}{162}+\frac {\log \left (1-a x^2\right )}{54 a^3}-\frac {1}{54} x^6 \log \left (1-a x^2\right )-\frac {1}{18} x^6 \text {Li}_2\left (a x^2\right )+\frac {1}{6} x^6 \text {Li}_3\left (a x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 88, normalized size = 1.00 \begin {gather*} \frac {6 a x^2+3 a^2 x^4+2 a^3 x^6+6 \log \left (1-a x^2\right )-6 a^3 x^6 \log \left (1-a x^2\right )-18 a^3 x^6 \text {PolyLog}\left (2,a x^2\right )+54 a^3 x^6 \text {PolyLog}\left (3,a x^2\right )}{324 a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 80, normalized size = 0.91
method | result | size |
meijerg | \(\frac {\frac {x^{2} a \left (4 a^{2} x^{4}+6 a \,x^{2}+12\right )}{324}+\frac {\left (-4 a^{3} x^{6}+4\right ) \ln \left (-a \,x^{2}+1\right )}{108}-\frac {x^{6} a^{3} \polylog \left (2, a \,x^{2}\right )}{9}+\frac {x^{6} a^{3} \polylog \left (3, a \,x^{2}\right )}{3}}{2 a^{3}}\) | \(80\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 77, normalized size = 0.88 \begin {gather*} -\frac {18 \, a^{3} x^{6} {\rm Li}_2\left (a x^{2}\right ) - 54 \, a^{3} x^{6} {\rm Li}_{3}(a x^{2}) - 2 \, a^{3} x^{6} - 3 \, a^{2} x^{4} - 6 \, a x^{2} + 6 \, {\left (a^{3} x^{6} - 1\right )} \log \left (-a x^{2} + 1\right )}{324 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 77, normalized size = 0.88 \begin {gather*} -\frac {18 \, a^{3} x^{6} {\rm Li}_2\left (a x^{2}\right ) - 54 \, a^{3} x^{6} {\rm polylog}\left (3, a x^{2}\right ) - 2 \, a^{3} x^{6} - 3 \, a^{2} x^{4} - 6 \, a x^{2} + 6 \, {\left (a^{3} x^{6} - 1\right )} \log \left (-a x^{2} + 1\right )}{324 \, a^{3}} \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^{5} \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 = 0.34, size = 73, normalized size = 0.83 \begin {gather*} \frac {x^6\,\mathrm {polylog}\left (3,a\,x^2\right )}{6}-\frac {x^6\,\mathrm {polylog}\left (2,a\,x^2\right )}{18}+\frac {\ln \left (a\,x^2-1\right )}{54\,a^3}-\frac {x^6\,\ln \left (1-a\,x^2\right )}{54}+\frac {x^6}{162}+\frac {x^2}{54\,a^2}+\frac {x^4}{108\,a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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