Optimal. Leaf size=74 \[ -\frac {\log \left (1-a x^2\right )}{18 a^3}-\frac {x^2}{18 a^2}+\frac {1}{6} x^6 \text {Li}_2\left (a x^2\right )-\frac {x^4}{36 a}+\frac {1}{18} x^6 \log \left (1-a x^2\right )-\frac {x^6}{54} \]
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Rubi [A] time = 0.06, antiderivative size = 74, 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} \[ \frac {1}{6} x^6 \text {PolyLog}\left (2,a x^2\right )-\frac {x^2}{18 a^2}-\frac {\log \left (1-a x^2\right )}{18 a^3}-\frac {x^4}{36 a}+\frac {1}{18} x^6 \log \left (1-a x^2\right )-\frac {x^6}{54} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2395
Rule 2454
Rule 6591
Rubi steps
\begin {align*} \int x^5 \text {Li}_2\left (a x^2\right ) \, dx &=\frac {1}{6} x^6 \text {Li}_2\left (a x^2\right )+\frac {1}{3} \int x^5 \log \left (1-a x^2\right ) \, dx\\ &=\frac {1}{6} x^6 \text {Li}_2\left (a x^2\right )+\frac {1}{6} \operatorname {Subst}\left (\int x^2 \log (1-a x) \, dx,x,x^2\right )\\ &=\frac {1}{18} x^6 \log \left (1-a x^2\right )+\frac {1}{6} x^6 \text {Li}_2\left (a x^2\right )+\frac {1}{18} a \operatorname {Subst}\left (\int \frac {x^3}{1-a x} \, dx,x,x^2\right )\\ &=\frac {1}{18} x^6 \log \left (1-a x^2\right )+\frac {1}{6} x^6 \text {Li}_2\left (a x^2\right )+\frac {1}{18} a \operatorname {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}{18 a^2}-\frac {x^4}{36 a}-\frac {x^6}{54}-\frac {\log \left (1-a x^2\right )}{18 a^3}+\frac {1}{18} x^6 \log \left (1-a x^2\right )+\frac {1}{6} x^6 \text {Li}_2\left (a x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 65, normalized size = 0.88 \[ \frac {18 a^3 x^6 \text {Li}_2\left (a x^2\right )+6 \left (a^3 x^6-1\right ) \log \left (1-a x^2\right )-a x^2 \left (2 a^2 x^4+3 a x^2+6\right )}{108 a^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 62, normalized size = 0.84 \[ \frac {18 \, a^{3} x^{6} {\rm Li}_2\left (a x^{2}\right ) - 2 \, a^{3} x^{6} - 3 \, a^{2} x^{4} - 6 \, a x^{2} + 6 \, {\left (a^{3} x^{6} - 1\right )} \log \left (-a x^{2} + 1\right )}{108 \, a^{3}} \]
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 \[ \int x^{5} {\rm Li}_2\left (a x^{2}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 62, normalized size = 0.84 \[ \frac {x^{6} \polylog \left (2, a \,x^{2}\right )}{6}+\frac {x^{6} \ln \left (-a \,x^{2}+1\right )}{18}-\frac {x^{6}}{54}-\frac {x^{4}}{36 a}-\frac {x^{2}}{18 a^{2}}-\frac {\ln \left (a \,x^{2}-1\right )}{18 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 62, normalized size = 0.84 \[ \frac {18 \, a^{3} x^{6} {\rm Li}_2\left (a x^{2}\right ) - 2 \, a^{3} x^{6} - 3 \, a^{2} x^{4} - 6 \, a x^{2} + 6 \, {\left (a^{3} x^{6} - 1\right )} \log \left (-a x^{2} + 1\right )}{108 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 61, normalized size = 0.82 \[ \frac {x^6\,\mathrm {polylog}\left (2,a\,x^2\right )}{6}-\frac {\ln \left (a\,x^2-1\right )}{18\,a^3}+\frac {x^6\,\ln \left (1-a\,x^2\right )}{18}-\frac {x^6}{54}-\frac {x^2}{18\,a^2}-\frac {x^4}{36\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 19.53, size = 56, normalized size = 0.76 \[ \begin {cases} - \frac {x^{6} \operatorname {Li}_{1}\left (a x^{2}\right )}{18} + \frac {x^{6} \operatorname {Li}_{2}\left (a x^{2}\right )}{6} - \frac {x^{6}}{54} - \frac {x^{4}}{36 a} - \frac {x^{2}}{18 a^{2}} + \frac {\operatorname {Li}_{1}\left (a x^{2}\right )}{18 a^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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