Optimal. Leaf size=63 \[ -\frac{\text{PolyLog}\left (2,a x^2\right )}{2 x^2}-\frac{\text{PolyLog}\left (3,a x^2\right )}{2 x^2}-\frac{1}{2} a \log \left (1-a x^2\right )+\frac{\log \left (1-a x^2\right )}{2 x^2}+a \log (x) \]
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Rubi [A] time = 0.047025, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546, Rules used = {6591, 2454, 2395, 36, 29, 31} \[ -\frac{\text{PolyLog}\left (2,a x^2\right )}{2 x^2}-\frac{\text{PolyLog}\left (3,a x^2\right )}{2 x^2}-\frac{1}{2} a \log \left (1-a x^2\right )+\frac{\log \left (1-a x^2\right )}{2 x^2}+a \log (x) \]
Antiderivative was successfully verified.
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Rule 6591
Rule 2454
Rule 2395
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{\text{Li}_3\left (a x^2\right )}{x^3} \, dx &=-\frac{\text{Li}_3\left (a x^2\right )}{2 x^2}+\int \frac{\text{Li}_2\left (a x^2\right )}{x^3} \, dx\\ &=-\frac{\text{Li}_2\left (a x^2\right )}{2 x^2}-\frac{\text{Li}_3\left (a x^2\right )}{2 x^2}-\int \frac{\log \left (1-a x^2\right )}{x^3} \, dx\\ &=-\frac{\text{Li}_2\left (a x^2\right )}{2 x^2}-\frac{\text{Li}_3\left (a x^2\right )}{2 x^2}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{\log (1-a x)}{x^2} \, dx,x,x^2\right )\\ &=\frac{\log \left (1-a x^2\right )}{2 x^2}-\frac{\text{Li}_2\left (a x^2\right )}{2 x^2}-\frac{\text{Li}_3\left (a x^2\right )}{2 x^2}+\frac{1}{2} a \operatorname{Subst}\left (\int \frac{1}{x (1-a x)} \, dx,x,x^2\right )\\ &=\frac{\log \left (1-a x^2\right )}{2 x^2}-\frac{\text{Li}_2\left (a x^2\right )}{2 x^2}-\frac{\text{Li}_3\left (a x^2\right )}{2 x^2}+\frac{1}{2} a \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )+\frac{1}{2} a^2 \operatorname{Subst}\left (\int \frac{1}{1-a x} \, dx,x,x^2\right )\\ &=a \log (x)-\frac{1}{2} a \log \left (1-a x^2\right )+\frac{\log \left (1-a x^2\right )}{2 x^2}-\frac{\text{Li}_2\left (a x^2\right )}{2 x^2}-\frac{\text{Li}_3\left (a x^2\right )}{2 x^2}\\ \end{align*}
Mathematica [A] time = 0.0269275, size = 60, normalized size = 0.95 \[ -\frac{\text{PolyLog}\left (2,a x^2\right )+\text{PolyLog}\left (3,a x^2\right )-a x^2 \log \left (-a x^2\right )+a x^2 \log \left (1-a x^2\right )-\log \left (1-a x^2\right )}{2 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.061, size = 68, normalized size = 1.1 \begin{align*}{\frac{a}{2} \left ( 2\,\ln \left ( x \right ) +\ln \left ( -a \right ) +{\frac{ \left ( -8\,a{x}^{2}+8 \right ) \ln \left ( -a{x}^{2}+1 \right ) }{8\,a{x}^{2}}}-{\frac{{\it polylog} \left ( 2,a{x}^{2} \right ) }{a{x}^{2}}}-{\frac{{\it polylog} \left ( 3,a{x}^{2} \right ) }{a{x}^{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.989681, size = 55, normalized size = 0.87 \begin{align*} a \log \left (x\right ) - \frac{{\left (a x^{2} - 1\right )} \log \left (-a x^{2} + 1\right ) +{\rm Li}_2\left (a x^{2}\right ) +{\rm Li}_{3}(a x^{2})}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.71896, size = 194, normalized size = 3.08 \begin{align*} -\frac{a x^{2} \log \left (a x^{2} - 1\right ) - 2 \, a x^{2} \log \left (x\right ) +{\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 ) - \log \left (-a x^{2} + 1\right ) +{\rm polylog}\left (3, a x^{2}\right )}{2 \, x^{2}} \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 \frac{\operatorname{Li}_{3}\left (a x^{2}\right )}{x^{3}}\, 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 \frac{{\rm Li}_{3}(a x^{2})}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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