Optimal. Leaf size=80 \[ -\frac{2 \text{PolyLog}\left (2,a x^2\right )}{25 x^5}-\frac{\text{PolyLog}\left (3,a x^2\right )}{5 x^5}-\frac{8 a^2}{125 x}+\frac{8}{125} a^{5/2} \tanh ^{-1}\left (\sqrt{a} x\right )-\frac{8 a}{375 x^3}+\frac{4 \log \left (1-a x^2\right )}{125 x^5} \]
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Rubi [A] time = 0.0469064, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {6591, 2455, 325, 206} \[ -\frac{2 \text{PolyLog}\left (2,a x^2\right )}{25 x^5}-\frac{\text{PolyLog}\left (3,a x^2\right )}{5 x^5}-\frac{8 a^2}{125 x}+\frac{8}{125} a^{5/2} \tanh ^{-1}\left (\sqrt{a} x\right )-\frac{8 a}{375 x^3}+\frac{4 \log \left (1-a x^2\right )}{125 x^5} \]
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^6} \, dx &=-\frac{\text{Li}_3\left (a x^2\right )}{5 x^5}+\frac{2}{5} \int \frac{\text{Li}_2\left (a x^2\right )}{x^6} \, dx\\ &=-\frac{2 \text{Li}_2\left (a x^2\right )}{25 x^5}-\frac{\text{Li}_3\left (a x^2\right )}{5 x^5}-\frac{4}{25} \int \frac{\log \left (1-a x^2\right )}{x^6} \, dx\\ &=\frac{4 \log \left (1-a x^2\right )}{125 x^5}-\frac{2 \text{Li}_2\left (a x^2\right )}{25 x^5}-\frac{\text{Li}_3\left (a x^2\right )}{5 x^5}+\frac{1}{125} (8 a) \int \frac{1}{x^4 \left (1-a x^2\right )} \, dx\\ &=-\frac{8 a}{375 x^3}+\frac{4 \log \left (1-a x^2\right )}{125 x^5}-\frac{2 \text{Li}_2\left (a x^2\right )}{25 x^5}-\frac{\text{Li}_3\left (a x^2\right )}{5 x^5}+\frac{1}{125} \left (8 a^2\right ) \int \frac{1}{x^2 \left (1-a x^2\right )} \, dx\\ &=-\frac{8 a}{375 x^3}-\frac{8 a^2}{125 x}+\frac{4 \log \left (1-a x^2\right )}{125 x^5}-\frac{2 \text{Li}_2\left (a x^2\right )}{25 x^5}-\frac{\text{Li}_3\left (a x^2\right )}{5 x^5}+\frac{1}{125} \left (8 a^3\right ) \int \frac{1}{1-a x^2} \, dx\\ &=-\frac{8 a}{375 x^3}-\frac{8 a^2}{125 x}+\frac{8}{125} a^{5/2} \tanh ^{-1}\left (\sqrt{a} x\right )+\frac{4 \log \left (1-a x^2\right )}{125 x^5}-\frac{2 \text{Li}_2\left (a x^2\right )}{25 x^5}-\frac{\text{Li}_3\left (a x^2\right )}{5 x^5}\\ \end{align*}
Mathematica [A] time = 0.100693, size = 69, normalized size = 0.86 \[ -\frac{30 \text{PolyLog}\left (2,a x^2\right )+75 \text{PolyLog}\left (3,a x^2\right )+24 a^2 x^4-24 a^{5/2} x^5 \tanh ^{-1}\left (\sqrt{a} x\right )+8 a x^2-12 \log \left (1-a x^2\right )}{375 x^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.18, size = 138, normalized size = 1.7 \begin{align*}{\frac{{a}^{3}}{2} \left ( -{\frac{16}{375\,{x}^{3}} \left ( -a \right ) ^{-{\frac{3}{2}}}}-{\frac{16\,a}{125\,x} \left ( -a \right ) ^{-{\frac{3}{2}}}}-{\frac{8\,{a}^{2}x}{125} \left ( \ln \left ( 1-\sqrt{a{x}^{2}} \right ) -\ln \left ( 1+\sqrt{a{x}^{2}} \right ) \right ) \left ( -a \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{a{x}^{2}}}}}+{\frac{8\,\ln \left ( -a{x}^{2}+1 \right ) }{125\,a{x}^{5}} \left ( -a \right ) ^{-{\frac{3}{2}}}}-{\frac{4\,{\it polylog} \left ( 2,a{x}^{2} \right ) }{25\,a{x}^{5}} \left ( -a \right ) ^{-{\frac{3}{2}}}}-{\frac{2\,{\it polylog} \left ( 3,a{x}^{2} \right ) }{5\,a{x}^{5}} \left ( -a \right ) ^{-{\frac{3}{2}}}} \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 = 3.24127, size = 513, normalized size = 6.41 \begin{align*} \left [\frac{12 \, a^{\frac{5}{2}} x^{5} \log \left (\frac{a x^{2} + 2 \, \sqrt{a} x + 1}{a x^{2} - 1}\right ) - 24 \, a^{2} x^{4} - 8 \, a x^{2} - 30 \,{\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 ) + 12 \, \log \left (-a x^{2} + 1\right ) - 75 \,{\rm polylog}\left (3, a x^{2}\right )}{375 \, x^{5}}, -\frac{24 \, \sqrt{-a} a^{2} x^{5} \arctan \left (\sqrt{-a} x\right ) + 24 \, a^{2} x^{4} + 8 \, a x^{2} + 30 \,{\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 ) - 12 \, \log \left (-a x^{2} + 1\right ) + 75 \,{\rm polylog}\left (3, a x^{2}\right )}{375 \, x^{5}}\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^{6}}\, 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^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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