Optimal. Leaf size=66 \[ \frac {4}{25} a^{5/2} \tanh ^{-1}\left (\sqrt {a} x\right )-\frac {4 a^2}{25 x}-\frac {\text {Li}_2\left (a x^2\right )}{5 x^5}-\frac {4 a}{75 x^3}+\frac {2 \log \left (1-a x^2\right )}{25 x^5} \]
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Rubi [A] time = 0.04, antiderivative size = 66, normalized size of antiderivative = 1.00, 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} \[ -\frac {\text {PolyLog}\left (2,a x^2\right )}{5 x^5}-\frac {4 a^2}{25 x}+\frac {4}{25} a^{5/2} \tanh ^{-1}\left (\sqrt {a} x\right )-\frac {4 a}{75 x^3}+\frac {2 \log \left (1-a x^2\right )}{25 x^5} \]
Antiderivative was successfully verified.
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Rule 206
Rule 325
Rule 2455
Rule 6591
Rubi steps
\begin {align*} \int \frac {\text {Li}_2\left (a x^2\right )}{x^6} \, dx &=-\frac {\text {Li}_2\left (a x^2\right )}{5 x^5}-\frac {2}{5} \int \frac {\log \left (1-a x^2\right )}{x^6} \, dx\\ &=\frac {2 \log \left (1-a x^2\right )}{25 x^5}-\frac {\text {Li}_2\left (a x^2\right )}{5 x^5}+\frac {1}{25} (4 a) \int \frac {1}{x^4 \left (1-a x^2\right )} \, dx\\ &=-\frac {4 a}{75 x^3}+\frac {2 \log \left (1-a x^2\right )}{25 x^5}-\frac {\text {Li}_2\left (a x^2\right )}{5 x^5}+\frac {1}{25} \left (4 a^2\right ) \int \frac {1}{x^2 \left (1-a x^2\right )} \, dx\\ &=-\frac {4 a}{75 x^3}-\frac {4 a^2}{25 x}+\frac {2 \log \left (1-a x^2\right )}{25 x^5}-\frac {\text {Li}_2\left (a x^2\right )}{5 x^5}+\frac {1}{25} \left (4 a^3\right ) \int \frac {1}{1-a x^2} \, dx\\ &=-\frac {4 a}{75 x^3}-\frac {4 a^2}{25 x}+\frac {4}{25} a^{5/2} \tanh ^{-1}\left (\sqrt {a} x\right )+\frac {2 \log \left (1-a x^2\right )}{25 x^5}-\frac {\text {Li}_2\left (a x^2\right )}{5 x^5}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 47, normalized size = 0.71 \[ -\frac {4 a x^2 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};a x^2\right )+15 \text {Li}_2\left (a x^2\right )-6 \log \left (1-a x^2\right )}{75 x^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 132, normalized size = 2.00 \[ \left [\frac {6 \, a^{\frac {5}{2}} x^{5} \log \left (\frac {a x^{2} + 2 \, \sqrt {a} x + 1}{a x^{2} - 1}\right ) - 12 \, a^{2} x^{4} - 4 \, a x^{2} - 15 \, {\rm Li}_2\left (a x^{2}\right ) + 6 \, \log \left (-a x^{2} + 1\right )}{75 \, x^{5}}, -\frac {12 \, \sqrt {-a} a^{2} x^{5} \arctan \left (\sqrt {-a} x\right ) + 12 \, a^{2} x^{4} + 4 \, a x^{2} + 15 \, {\rm Li}_2\left (a x^{2}\right ) - 6 \, \log \left (-a x^{2} + 1\right )}{75 \, x^{5}}\right ] \]
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 \[ \int \frac {{\rm Li}_2\left (a x^{2}\right )}{x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 53, normalized size = 0.80 \[ -\frac {4 a}{75 x^{3}}-\frac {4 a^{2}}{25 x}+\frac {4 a^{\frac {5}{2}} \arctanh \left (x \sqrt {a}\right )}{25}+\frac {2 \ln \left (-a \,x^{2}+1\right )}{25 x^{5}}-\frac {\polylog \left (2, a \,x^{2}\right )}{5 x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 65, normalized size = 0.98 \[ -\frac {2}{25} \, a^{\frac {5}{2}} \log \left (\frac {a x - \sqrt {a}}{a x + \sqrt {a}}\right ) - \frac {12 \, a^{2} x^{4} + 4 \, a x^{2} + 15 \, {\rm Li}_2\left (a x^{2}\right ) - 6 \, \log \left (-a x^{2} + 1\right )}{75 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.34, size = 58, normalized size = 0.88 \[ \frac {2\,\ln \left (1-a\,x^2\right )}{25\,x^5}-\frac {4\,a^2\,x^2+\frac {4\,a}{3}}{25\,x^3}-\frac {\mathrm {polylog}\left (2,a\,x^2\right )}{5\,x^5}-\frac {a^{5/2}\,\mathrm {atan}\left (\sqrt {a}\,x\,1{}\mathrm {i}\right )\,4{}\mathrm {i}}{25} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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