Optimal. Leaf size=56 \[ -\frac{\text{PolyLog}\left (2,a x^2\right )}{3 x^3}+\frac{4}{9} a^{3/2} \tanh ^{-1}\left (\sqrt{a} x\right )+\frac{2 \log \left (1-a x^2\right )}{9 x^3}-\frac{4 a}{9 x} \]
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Rubi [A] time = 0.0321837, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {6591, 2455, 325, 206} \[ -\frac{\text{PolyLog}\left (2,a x^2\right )}{3 x^3}+\frac{4}{9} a^{3/2} \tanh ^{-1}\left (\sqrt{a} x\right )+\frac{2 \log \left (1-a x^2\right )}{9 x^3}-\frac{4 a}{9 x} \]
Antiderivative was successfully verified.
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Rule 6591
Rule 2455
Rule 325
Rule 206
Rubi steps
\begin{align*} \int \frac{\text{Li}_2\left (a x^2\right )}{x^4} \, dx &=-\frac{\text{Li}_2\left (a x^2\right )}{3 x^3}-\frac{2}{3} \int \frac{\log \left (1-a x^2\right )}{x^4} \, dx\\ &=\frac{2 \log \left (1-a x^2\right )}{9 x^3}-\frac{\text{Li}_2\left (a x^2\right )}{3 x^3}+\frac{1}{9} (4 a) \int \frac{1}{x^2 \left (1-a x^2\right )} \, dx\\ &=-\frac{4 a}{9 x}+\frac{2 \log \left (1-a x^2\right )}{9 x^3}-\frac{\text{Li}_2\left (a x^2\right )}{3 x^3}+\frac{1}{9} \left (4 a^2\right ) \int \frac{1}{1-a x^2} \, dx\\ &=-\frac{4 a}{9 x}+\frac{4}{9} a^{3/2} \tanh ^{-1}\left (\sqrt{a} x\right )+\frac{2 \log \left (1-a x^2\right )}{9 x^3}-\frac{\text{Li}_2\left (a x^2\right )}{3 x^3}\\ \end{align*}
Mathematica [C] time = 0.0136452, size = 47, normalized size = 0.84 \[ -\frac{4 a x^2 \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},a x^2\right )+3 \text{PolyLog}\left (2,a x^2\right )-2 \log \left (1-a x^2\right )}{9 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 45, normalized size = 0.8 \begin{align*} -{\frac{4\,a}{9\,x}}+{\frac{4}{9}{a}^{{\frac{3}{2}}}{\it Artanh} \left ( x\sqrt{a} \right ) }+{\frac{2\,\ln \left ( -a{x}^{2}+1 \right ) }{9\,{x}^{3}}}-{\frac{{\it polylog} \left ( 2,a{x}^{2} \right ) }{3\,{x}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.74604, size = 289, normalized size = 5.16 \begin{align*} \left [\frac{2 \, a^{\frac{3}{2}} x^{3} \log \left (\frac{a x^{2} + 2 \, \sqrt{a} x + 1}{a x^{2} - 1}\right ) - 4 \, a x^{2} - 3 \,{\rm Li}_2\left (a x^{2}\right ) + 2 \, \log \left (-a x^{2} + 1\right )}{9 \, x^{3}}, -\frac{4 \, \sqrt{-a} a x^{3} \arctan \left (\sqrt{-a} x\right ) + 4 \, a x^{2} + 3 \,{\rm Li}_2\left (a x^{2}\right ) - 2 \, \log \left (-a x^{2} + 1\right )}{9 \, x^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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}_2\left (a x^{2}\right )}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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