Optimal. Leaf size=70 \[ -\frac{2 \text{PolyLog}\left (2,a x^2\right )}{9 x^3}-\frac{\text{PolyLog}\left (3,a x^2\right )}{3 x^3}+\frac{8}{27} a^{3/2} \tanh ^{-1}\left (\sqrt{a} x\right )+\frac{4 \log \left (1-a x^2\right )}{27 x^3}-\frac{8 a}{27 x} \]
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Rubi [A] time = 0.0409967, antiderivative size = 70, normalized size of antiderivative = 1., 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 = {6591, 2455, 325, 206} \[ -\frac{2 \text{PolyLog}\left (2,a x^2\right )}{9 x^3}-\frac{\text{PolyLog}\left (3,a x^2\right )}{3 x^3}+\frac{8}{27} a^{3/2} \tanh ^{-1}\left (\sqrt{a} x\right )+\frac{4 \log \left (1-a x^2\right )}{27 x^3}-\frac{8 a}{27 x} \]
Antiderivative was successfully verified.
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Rule 6591
Rule 2455
Rule 325
Rule 206
Rubi steps
\begin{align*} \int \frac{\text{Li}_3\left (a x^2\right )}{x^4} \, dx &=-\frac{\text{Li}_3\left (a x^2\right )}{3 x^3}+\frac{2}{3} \int \frac{\text{Li}_2\left (a x^2\right )}{x^4} \, dx\\ &=-\frac{2 \text{Li}_2\left (a x^2\right )}{9 x^3}-\frac{\text{Li}_3\left (a x^2\right )}{3 x^3}-\frac{4}{9} \int \frac{\log \left (1-a x^2\right )}{x^4} \, dx\\ &=\frac{4 \log \left (1-a x^2\right )}{27 x^3}-\frac{2 \text{Li}_2\left (a x^2\right )}{9 x^3}-\frac{\text{Li}_3\left (a x^2\right )}{3 x^3}+\frac{1}{27} (8 a) \int \frac{1}{x^2 \left (1-a x^2\right )} \, dx\\ &=-\frac{8 a}{27 x}+\frac{4 \log \left (1-a x^2\right )}{27 x^3}-\frac{2 \text{Li}_2\left (a x^2\right )}{9 x^3}-\frac{\text{Li}_3\left (a x^2\right )}{3 x^3}+\frac{1}{27} \left (8 a^2\right ) \int \frac{1}{1-a x^2} \, dx\\ &=-\frac{8 a}{27 x}+\frac{8}{27} a^{3/2} \tanh ^{-1}\left (\sqrt{a} x\right )+\frac{4 \log \left (1-a x^2\right )}{27 x^3}-\frac{2 \text{Li}_2\left (a x^2\right )}{9 x^3}-\frac{\text{Li}_3\left (a x^2\right )}{3 x^3}\\ \end{align*}
Mathematica [A] time = 0.088813, size = 61, normalized size = 0.87 \[ -\frac{6 \text{PolyLog}\left (2,a x^2\right )+9 \text{PolyLog}\left (3,a x^2\right )-8 a^{3/2} x^3 \tanh ^{-1}\left (\sqrt{a} x\right )+8 a x^2-4 \log \left (1-a x^2\right )}{27 x^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.177, size = 125, normalized size = 1.8 \begin{align*} -{\frac{{a}^{2}}{2} \left ( -{\frac{16}{27\,x}{\frac{1}{\sqrt{-a}}}}-{\frac{8\,ax}{27} \left ( \ln \left ( 1-\sqrt{a{x}^{2}} \right ) -\ln \left ( 1+\sqrt{a{x}^{2}} \right ) \right ){\frac{1}{\sqrt{-a}}}{\frac{1}{\sqrt{a{x}^{2}}}}}+{\frac{8\,\ln \left ( -a{x}^{2}+1 \right ) }{27\,{x}^{3}a}{\frac{1}{\sqrt{-a}}}}-{\frac{4\,{\it polylog} \left ( 2,a{x}^{2} \right ) }{9\,{x}^{3}a}{\frac{1}{\sqrt{-a}}}}-{\frac{2\,{\it polylog} \left ( 3,a{x}^{2} \right ) }{3\,{x}^{3}a}{\frac{1}{\sqrt{-a}}}} \right ){\frac{1}{\sqrt{-a}}}} \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 [C] time = 3.22669, size = 462, normalized size = 6.6 \begin{align*} \left [\frac{4 \, a^{\frac{3}{2}} x^{3} \log \left (\frac{a x^{2} + 2 \, \sqrt{a} x + 1}{a x^{2} - 1}\right ) - 8 \, a x^{2} - 6 \,{\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 ) + 4 \, \log \left (-a x^{2} + 1\right ) - 9 \,{\rm polylog}\left (3, a x^{2}\right )}{27 \, x^{3}}, -\frac{8 \, \sqrt{-a} a x^{3} \arctan \left (\sqrt{-a} x\right ) + 8 \, a x^{2} + 6 \,{\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 ) - 4 \, \log \left (-a x^{2} + 1\right ) + 9 \,{\rm polylog}\left (3, a x^{2}\right )}{27 \, x^{3}}\right ] \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^{4}}\, 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^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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