Optimal. Leaf size=80 \[ \frac {8}{125} a^{5/2} \tanh ^{-1}\left (\sqrt {a} x\right )-\frac {8 a^2}{125 x}-\frac {2 \text {Li}_2\left (a x^2\right )}{25 x^5}-\frac {\text {Li}_3\left (a x^2\right )}{5 x^5}-\frac {8 a}{375 x^3}+\frac {4 \log \left (1-a x^2\right )}{125 x^5} \]
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Rubi [A] time = 0.05, antiderivative size = 80, normalized size of antiderivative = 1.00, 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 206
Rule 325
Rule 2455
Rule 6591
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*}
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Mathematica [A] time = 0.10, size = 69, normalized size = 0.86 \[ -\frac {-24 a^{5/2} x^5 \tanh ^{-1}\left (\sqrt {a} x\right )+24 a^2 x^4+30 \text {Li}_2\left (a x^2\right )+75 \text {Li}_3\left (a x^2\right )+8 a x^2-12 \log \left (1-a x^2\right )}{375 x^5} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.81, size = 150, normalized size = 1.88 \[ \left [\frac {12 \, a^{\frac {5}{2}} x^{5} \log \left (\frac {a x^{2} + 2 \, \sqrt {a} x + 1}{a x^{2} - 1}\right ) - 24 \, a^{2} x^{4} - 8 \, a x^{2} - 30 \, {\rm Li}_2\left (a x^{2}\right ) + 12 \, \log \left (-a x^{2} + 1\right ) - 75 \, {\rm polylog}\left (3, a x^{2}\right )}{375 \, x^{5}}, -\frac {24 \, \sqrt {-a} a^{2} x^{5} \arctan \left (\sqrt {-a} x\right ) + 24 \, a^{2} x^{4} + 8 \, a x^{2} + 30 \, {\rm Li}_2\left (a x^{2}\right ) - 12 \, \log \left (-a x^{2} + 1\right ) + 75 \, {\rm polylog}\left (3, a x^{2}\right )}{375 \, x^{5}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm Li}_{3}(a x^{2})}{x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.17, size = 138, normalized size = 1.72 \[ \frac {a^{3} \left (-\frac {16}{375 x^{3} \left (-a \right )^{\frac {3}{2}}}-\frac {16 a}{125 x \left (-a \right )^{\frac {3}{2}}}-\frac {8 x \,a^{2} \left (\ln \left (1-\sqrt {a \,x^{2}}\right )-\ln \left (1+\sqrt {a \,x^{2}}\right )\right )}{125 \left (-a \right )^{\frac {3}{2}} \sqrt {a \,x^{2}}}+\frac {8 \ln \left (-a \,x^{2}+1\right )}{125 x^{5} \left (-a \right )^{\frac {3}{2}} a}-\frac {4 \polylog \left (2, a \,x^{2}\right )}{25 x^{5} \left (-a \right )^{\frac {3}{2}} a}-\frac {2 \polylog \left (3, a \,x^{2}\right )}{5 x^{5} \left (-a \right )^{\frac {3}{2}} a}\right )}{2 \sqrt {-a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.13, size = 74, normalized size = 0.92 \[ -\frac {4}{125} \, a^{\frac {5}{2}} \log \left (\frac {a x - \sqrt {a}}{a x + \sqrt {a}}\right ) - \frac {24 \, a^{2} x^{4} + 8 \, a x^{2} + 30 \, {\rm Li}_2\left (a x^{2}\right ) - 12 \, \log \left (-a x^{2} + 1\right ) + 75 \, {\rm Li}_{3}(a x^{2})}{375 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.03, size = 70, normalized size = 0.88 \[ \frac {4\,\ln \left (1-a\,x^2\right )}{125\,x^5}-\frac {\mathrm {polylog}\left (3,a\,x^2\right )}{5\,x^5}-\frac {8\,a^2\,x^2+\frac {8\,a}{3}}{125\,x^3}-\frac {2\,\mathrm {polylog}\left (2,a\,x^2\right )}{25\,x^5}-\frac {a^{5/2}\,\mathrm {atan}\left (\sqrt {a}\,x\,1{}\mathrm {i}\right )\,8{}\mathrm {i}}{125} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {Li}_{3}\left (a x^{2}\right )}{x^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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