Optimal. Leaf size=56 \[ \frac {4}{9} a^{3/2} \tanh ^{-1}\left (\sqrt {a} x\right )-\frac {\text {Li}_2\left (a x^2\right )}{3 x^3}+\frac {2 \log \left (1-a x^2\right )}{9 x^3}-\frac {4 a}{9 x} \]
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Rubi [A] time = 0.03, antiderivative size = 56, normalized size of antiderivative = 1.00, 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 206
Rule 325
Rule 2455
Rule 6591
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*}
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Mathematica [C] time = 0.01, size = 47, normalized size = 0.84 \[ -\frac {4 a x^2 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};a x^2\right )+3 \text {Li}_2\left (a x^2\right )-2 \log \left (1-a x^2\right )}{9 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 114, normalized size = 2.04 \[ \left [\frac {2 \, a^{\frac {3}{2}} x^{3} \log \left (\frac {a x^{2} + 2 \, \sqrt {a} x + 1}{a x^{2} - 1}\right ) - 4 \, a x^{2} - 3 \, {\rm Li}_2\left (a x^{2}\right ) + 2 \, \log \left (-a x^{2} + 1\right )}{9 \, x^{3}}, -\frac {4 \, \sqrt {-a} a x^{3} \arctan \left (\sqrt {-a} x\right ) + 4 \, a x^{2} + 3 \, {\rm Li}_2\left (a x^{2}\right ) - 2 \, \log \left (-a x^{2} + 1\right )}{9 \, x^{3}}\right ] \]
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 \[ \int \frac {{\rm Li}_2\left (a x^{2}\right )}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 45, normalized size = 0.80 \[ -\frac {4 a}{9 x}+\frac {4 a^{\frac {3}{2}} \arctanh \left (x \sqrt {a}\right )}{9}+\frac {2 \ln \left (-a \,x^{2}+1\right )}{9 x^{3}}-\frac {\polylog \left (2, a \,x^{2}\right )}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 57, normalized size = 1.02 \[ -\frac {2}{9} \, a^{\frac {3}{2}} \log \left (\frac {a x - \sqrt {a}}{a x + \sqrt {a}}\right ) - \frac {4 \, a x^{2} + 3 \, {\rm Li}_2\left (a x^{2}\right ) - 2 \, \log \left (-a x^{2} + 1\right )}{9 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.33, size = 47, normalized size = 0.84 \[ \frac {2\,\ln \left (1-a\,x^2\right )}{9\,x^3}-\frac {4\,a}{9\,x}-\frac {\mathrm {polylog}\left (2,a\,x^2\right )}{3\,x^3}-\frac {a^{3/2}\,\mathrm {atan}\left (\sqrt {a}\,x\,1{}\mathrm {i}\right )\,4{}\mathrm {i}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 133.63, size = 275, normalized size = 4.91 \[ \begin {cases} - \frac {\pi ^{2}}{18 x^{3}} & \text {for}\: a = \frac {1}{x^{2}} \\0 & \text {for}\: a = 0 \\- \frac {4 a^{2} x^{5} \sqrt {\frac {1}{a}} \log {\left (x - \sqrt {\frac {1}{a}} \right )}}{9 x^{5} - \frac {9 x^{3}}{a}} - \frac {2 a^{2} x^{5} \sqrt {\frac {1}{a}} \operatorname {Li}_{1}\left (a x^{2}\right )}{9 x^{5} - \frac {9 x^{3}}{a}} - \frac {4 a x^{4}}{9 x^{5} - \frac {9 x^{3}}{a}} + \frac {4 a x^{3} \sqrt {\frac {1}{a}} \log {\left (x - \sqrt {\frac {1}{a}} \right )}}{9 x^{5} - \frac {9 x^{3}}{a}} + \frac {2 a x^{3} \sqrt {\frac {1}{a}} \operatorname {Li}_{1}\left (a x^{2}\right )}{9 x^{5} - \frac {9 x^{3}}{a}} - \frac {2 x^{2} \operatorname {Li}_{1}\left (a x^{2}\right )}{9 x^{5} - \frac {9 x^{3}}{a}} - \frac {3 x^{2} \operatorname {Li}_{2}\left (a x^{2}\right )}{9 x^{5} - \frac {9 x^{3}}{a}} + \frac {4 x^{2}}{9 x^{5} - \frac {9 x^{3}}{a}} + \frac {2 \operatorname {Li}_{1}\left (a x^{2}\right )}{9 a x^{5} - 9 x^{3}} + \frac {3 \operatorname {Li}_{2}\left (a x^{2}\right )}{9 a x^{5} - 9 x^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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