Optimal. Leaf size=63 \[ -\frac {4 x}{9 a}-\frac {4 x^3}{27}+\frac {4 \tanh ^{-1}\left (\sqrt {a} x\right )}{9 a^{3/2}}+\frac {2}{9} x^3 \log \left (1-a x^2\right )+\frac {1}{3} x^3 \text {PolyLog}\left (2,a x^2\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 63, 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 = {6726, 2505,
308, 212} \begin {gather*} \frac {4 \tanh ^{-1}\left (\sqrt {a} x\right )}{9 a^{3/2}}+\frac {1}{3} x^3 \text {Li}_2\left (a x^2\right )+\frac {2}{9} x^3 \log \left (1-a x^2\right )-\frac {4 x}{9 a}-\frac {4 x^3}{27} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 308
Rule 2505
Rule 6726
Rubi steps
\begin {align*} \int x^2 \text {Li}_2\left (a x^2\right ) \, dx &=\frac {1}{3} x^3 \text {Li}_2\left (a x^2\right )+\frac {2}{3} \int x^2 \log \left (1-a x^2\right ) \, dx\\ &=\frac {2}{9} x^3 \log \left (1-a x^2\right )+\frac {1}{3} x^3 \text {Li}_2\left (a x^2\right )+\frac {1}{9} (4 a) \int \frac {x^4}{1-a x^2} \, dx\\ &=\frac {2}{9} x^3 \log \left (1-a x^2\right )+\frac {1}{3} x^3 \text {Li}_2\left (a x^2\right )+\frac {1}{9} (4 a) \int \left (-\frac {1}{a^2}-\frac {x^2}{a}+\frac {1}{a^2 \left (1-a x^2\right )}\right ) \, dx\\ &=-\frac {4 x}{9 a}-\frac {4 x^3}{27}+\frac {2}{9} x^3 \log \left (1-a x^2\right )+\frac {1}{3} x^3 \text {Li}_2\left (a x^2\right )+\frac {4 \int \frac {1}{1-a x^2} \, dx}{9 a}\\ &=-\frac {4 x}{9 a}-\frac {4 x^3}{27}+\frac {4 \tanh ^{-1}\left (\sqrt {a} x\right )}{9 a^{3/2}}+\frac {2}{9} x^3 \log \left (1-a x^2\right )+\frac {1}{3} x^3 \text {Li}_2\left (a x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 57, normalized size = 0.90 \begin {gather*} \frac {1}{27} \left (-\frac {12 x}{a}-4 x^3+\frac {12 \tanh ^{-1}\left (\sqrt {a} x\right )}{a^{3/2}}+6 x^3 \log \left (1-a x^2\right )+9 x^3 \text {PolyLog}\left (2,a x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 55, normalized size = 0.87
method | result | size |
default | \(\frac {x^{3} \polylog \left (2, a \,x^{2}\right )}{3}+\frac {2 x^{3} \ln \left (-a \,x^{2}+1\right )}{9}+\frac {4 a \left (-\frac {\frac {1}{3} a \,x^{3}+x}{a^{2}}+\frac {\arctanh \left (x \sqrt {a}\right )}{a^{\frac {5}{2}}}\right )}{9}\) | \(55\) |
meijerg | \(\frac {-\frac {2 x \left (-a \right )^{\frac {5}{2}} \left (20 a \,x^{2}+60\right )}{135 a^{2}}-\frac {4 x \left (-a \right )^{\frac {5}{2}} \left (\ln \left (1-\sqrt {a \,x^{2}}\right )-\ln \left (1+\sqrt {a \,x^{2}}\right )\right )}{9 a^{2} \sqrt {a \,x^{2}}}+\frac {4 x^{3} \left (-a \right )^{\frac {5}{2}} \ln \left (-a \,x^{2}+1\right )}{9 a}+\frac {2 x^{3} \left (-a \right )^{\frac {5}{2}} \polylog \left (2, a \,x^{2}\right )}{3 a}}{2 a \sqrt {-a}}\) | \(116\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 68, normalized size = 1.08 \begin {gather*} \frac {9 \, a x^{3} {\rm Li}_2\left (a x^{2}\right ) + 6 \, a x^{3} \log \left (-a x^{2} + 1\right ) - 4 \, a x^{3} - 12 \, x}{27 \, a} - \frac {2 \, \log \left (\frac {a x - \sqrt {a}}{a x + \sqrt {a}}\right )}{9 \, a^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.54, size = 143, normalized size = 2.27 \begin {gather*} \left [\frac {9 \, a^{2} x^{3} {\rm Li}_2\left (a x^{2}\right ) + 6 \, a^{2} x^{3} \log \left (-a x^{2} + 1\right ) - 4 \, a^{2} x^{3} - 12 \, a x + 6 \, \sqrt {a} \log \left (\frac {a x^{2} + 2 \, \sqrt {a} x + 1}{a x^{2} - 1}\right )}{27 \, a^{2}}, \frac {9 \, a^{2} x^{3} {\rm Li}_2\left (a x^{2}\right ) + 6 \, a^{2} x^{3} \log \left (-a x^{2} + 1\right ) - 4 \, a^{2} x^{3} - 12 \, a x - 12 \, \sqrt {-a} \arctan \left (\sqrt {-a} x\right )}{27 \, a^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 13.13, size = 83, normalized size = 1.32 \begin {gather*} \begin {cases} - \frac {2 x^{3} \operatorname {Li}_{1}\left (a x^{2}\right )}{9} + \frac {x^{3} \operatorname {Li}_{2}\left (a x^{2}\right )}{3} - \frac {4 x^{3}}{27} - \frac {4 x}{9 a} - \frac {4 \log {\left (x - \sqrt {\frac {1}{a}} \right )}}{9 a^{2} \sqrt {\frac {1}{a}}} - \frac {2 \operatorname {Li}_{1}\left (a x^{2}\right )}{9 a^{2} \sqrt {\frac {1}{a}}} & \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 = 0.28, size = 52, normalized size = 0.83 \begin {gather*} \frac {x^3\,\mathrm {polylog}\left (2,a\,x^2\right )}{3}-\frac {4\,x}{9\,a}+\frac {2\,x^3\,\ln \left (1-a\,x^2\right )}{9}-\frac {4\,x^3}{27}-\frac {\mathrm {atan}\left (\sqrt {a}\,x\,1{}\mathrm {i}\right )\,4{}\mathrm {i}}{9\,a^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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