Optimal. Leaf size=60 \[ -\frac{1}{2} x^2 \text{PolyLog}\left (2,a x^2\right )+\frac{1}{2} x^2 \text{PolyLog}\left (3,a x^2\right )+\frac{\left (1-a x^2\right ) \log \left (1-a x^2\right )}{2 a}+\frac{x^2}{2} \]
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Rubi [A] time = 0.029869, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {6591, 2454, 2389, 2295} \[ -\frac{1}{2} x^2 \text{PolyLog}\left (2,a x^2\right )+\frac{1}{2} x^2 \text{PolyLog}\left (3,a x^2\right )+\frac{\left (1-a x^2\right ) \log \left (1-a x^2\right )}{2 a}+\frac{x^2}{2} \]
Antiderivative was successfully verified.
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Rule 6591
Rule 2454
Rule 2389
Rule 2295
Rubi steps
\begin{align*} \int x \text{Li}_3\left (a x^2\right ) \, dx &=\frac{1}{2} x^2 \text{Li}_3\left (a x^2\right )-\int x \text{Li}_2\left (a x^2\right ) \, dx\\ &=-\frac{1}{2} x^2 \text{Li}_2\left (a x^2\right )+\frac{1}{2} x^2 \text{Li}_3\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} x^2 \text{Li}_3\left (a x^2\right )-\frac{1}{2} \operatorname{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{1}{2} x^2 \text{Li}_3\left (a x^2\right )+\frac{\operatorname{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 )+\frac{1}{2} x^2 \text{Li}_3\left (a x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0112108, size = 52, normalized size = 0.87 \[ \frac{1}{2} x^2 \left (-\text{PolyLog}\left (2,a x^2\right )+\text{PolyLog}\left (3,a x^2\right )+\frac{\log \left (1-a x^2\right )}{a x^2}-\log \left (1-a x^2\right )+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.055, size = 56, normalized size = 0.9 \begin{align*}{\frac{1}{2\,a} \left ( a{x}^{2}+{\frac{ \left ( -2\,a{x}^{2}+2 \right ) \ln \left ( -a{x}^{2}+1 \right ) }{2}}-a{x}^{2}{\it polylog} \left ( 2,a{x}^{2} \right ) +a{x}^{2}{\it polylog} \left ( 3,a{x}^{2} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00071, size = 72, normalized size = 1.2 \begin{align*} -\frac{a x^{2}{\rm Li}_2\left (a x^{2}\right ) - a x^{2}{\rm Li}_{3}(a x^{2}) - a x^{2} +{\left (a x^{2} - 1\right )} \log \left (-a x^{2} + 1\right )}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.66414, size = 181, normalized size = 3.02 \begin{align*} -\frac{a x^{2}{\rm \%iint}\left (a, x, -\frac{\log \left (-a x^{2} + 1\right )}{a}, -\frac{2 \, \log \left (-a x^{2} + 1\right )}{x}\right ) - a x^{2}{\rm polylog}\left (3, a x^{2}\right ) - a x^{2} +{\left (a x^{2} - 1\right )} \log \left (-a x^{2} + 1\right )}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \operatorname{Li}_{3}\left (a x^{2}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x{\rm Li}_{3}(a x^{2})\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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