Optimal. Leaf size=46 \[ -\frac {x^2}{2}-\frac {\left (1-a x^2\right ) \log \left (1-a x^2\right )}{2 a}+\frac {1}{2} x^2 \text {PolyLog}\left (2,a x^2\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {6726, 2504,
2436, 2332} \begin {gather*} \frac {1}{2} x^2 \text {Li}_2\left (a x^2\right )-\frac {\left (1-a x^2\right ) \log \left (1-a x^2\right )}{2 a}-\frac {x^2}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2332
Rule 2436
Rule 2504
Rule 6726
Rubi steps
\begin {align*} \int x \text {Li}_2\left (a x^2\right ) \, dx &=\frac {1}{2} x^2 \text {Li}_2\left (a x^2\right )+\int x \log \left (1-a x^2\right ) \, dx\\ &=\frac {1}{2} x^2 \text {Li}_2\left (a x^2\right )+\frac {1}{2} \text {Subst}\left (\int \log (1-a x) \, dx,x,x^2\right )\\ &=\frac {1}{2} x^2 \text {Li}_2\left (a x^2\right )-\frac {\text {Subst}\left (\int \log (x) \, dx,x,1-a x^2\right )}{2 a}\\ &=-\frac {x^2}{2}-\frac {\left (1-a x^2\right ) \log \left (1-a x^2\right )}{2 a}+\frac {1}{2} x^2 \text {Li}_2\left (a x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 43, normalized size = 0.93 \begin {gather*} \frac {-a x^2+\left (-1+a x^2\right ) \log \left (1-a x^2\right )+a x^2 \text {PolyLog}\left (2,a x^2\right )}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.29, size = 45, normalized size = 0.98
method | result | size |
meijerg | \(\frac {-a \,x^{2}-\frac {\left (-2 a \,x^{2}+2\right ) \ln \left (-a \,x^{2}+1\right )}{2}+a \,x^{2} \polylog \left (2, a \,x^{2}\right )}{2 a}\) | \(44\) |
derivativedivides | \(\frac {a \,x^{2} \polylog \left (2, a \,x^{2}\right )-\ln \left (-a \,x^{2}+1\right ) \left (-a \,x^{2}+1\right )+1-a \,x^{2}}{2 a}\) | \(45\) |
default | \(\frac {a \,x^{2} \polylog \left (2, a \,x^{2}\right )-\ln \left (-a \,x^{2}+1\right ) \left (-a \,x^{2}+1\right )+1-a \,x^{2}}{2 a}\) | \(45\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 40, normalized size = 0.87 \begin {gather*} \frac {a x^{2} {\rm Li}_2\left (a x^{2}\right ) - a x^{2} + {\left (a x^{2} - 1\right )} \log \left (-a x^{2} + 1\right )}{2 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 40, normalized size = 0.87 \begin {gather*} \frac {a x^{2} {\rm Li}_2\left (a x^{2}\right ) - a x^{2} + {\left (a x^{2} - 1\right )} \log \left (-a x^{2} + 1\right )}{2 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.67, size = 39, normalized size = 0.85 \begin {gather*} \begin {cases} - \frac {x^{2} \operatorname {Li}_{1}\left (a x^{2}\right )}{2} + \frac {x^{2} \operatorname {Li}_{2}\left (a x^{2}\right )}{2} - \frac {x^{2}}{2} + \frac {\operatorname {Li}_{1}\left (a x^{2}\right )}{2 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 = 0.21, size = 45, normalized size = 0.98 \begin {gather*} \frac {x^2\,\mathrm {polylog}\left (2,a\,x^2\right )}{2}-\frac {\ln \left (a\,x^2-1\right )}{2\,a}+\frac {x^2\,\ln \left (1-a\,x^2\right )}{2}-\frac {x^2}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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