Optimal. Leaf size=42 \[ -\frac {\text {Li}_2\left (a x^2\right )}{x}+\frac {2 \log \left (1-a x^2\right )}{x}+4 \sqrt {a} \tanh ^{-1}\left (\sqrt {a} x\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6591, 2455, 206} \[ -\frac {\text {PolyLog}\left (2,a x^2\right )}{x}+\frac {2 \log \left (1-a x^2\right )}{x}+4 \sqrt {a} \tanh ^{-1}\left (\sqrt {a} x\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 2455
Rule 6591
Rubi steps
\begin {align*} \int \frac {\text {Li}_2\left (a x^2\right )}{x^2} \, dx &=-\frac {\text {Li}_2\left (a x^2\right )}{x}-2 \int \frac {\log \left (1-a x^2\right )}{x^2} \, dx\\ &=\frac {2 \log \left (1-a x^2\right )}{x}-\frac {\text {Li}_2\left (a x^2\right )}{x}+(4 a) \int \frac {1}{1-a x^2} \, dx\\ &=4 \sqrt {a} \tanh ^{-1}\left (\sqrt {a} x\right )+\frac {2 \log \left (1-a x^2\right )}{x}-\frac {\text {Li}_2\left (a x^2\right )}{x}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 41, normalized size = 0.98 \[ \frac {-\text {Li}_2\left (a x^2\right )+2 \log \left (1-a x^2\right )+4 \sqrt {a} x \tanh ^{-1}\left (\sqrt {a} x\right )}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.00, size = 94, normalized size = 2.24 \[ \left [\frac {2 \, \sqrt {a} x \log \left (\frac {a x^{2} + 2 \, \sqrt {a} x + 1}{a x^{2} - 1}\right ) - {\rm Li}_2\left (a x^{2}\right ) + 2 \, \log \left (-a x^{2} + 1\right )}{x}, -\frac {4 \, \sqrt {-a} x \arctan \left (\sqrt {-a} x\right ) + {\rm Li}_2\left (a x^{2}\right ) - 2 \, \log \left (-a x^{2} + 1\right )}{x}\right ] \]
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 \[ \int \frac {{\rm Li}_2\left (a x^{2}\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 39, normalized size = 0.93 \[ \frac {2 \ln \left (-a \,x^{2}+1\right )}{x}-\frac {\polylog \left (2, a \,x^{2}\right )}{x}+4 \arctanh \left (x \sqrt {a}\right ) \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 49, normalized size = 1.17 \[ -2 \, \sqrt {a} \log \left (\frac {a x - \sqrt {a}}{a x + \sqrt {a}}\right ) - \frac {{\rm Li}_2\left (a x^{2}\right ) - 2 \, \log \left (-a x^{2} + 1\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.26, size = 38, normalized size = 0.90 \[ 4\,\sqrt {a}\,\mathrm {atanh}\left (\sqrt {a}\,x\right )-\frac {\mathrm {polylog}\left (2,a\,x^2\right )}{x}+\frac {2\,\ln \left (1-a\,x^2\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 34.46, size = 184, normalized size = 4.38 \[ \begin {cases} - \frac {\pi ^{2}}{6 x} & \text {for}\: a = \frac {1}{x^{2}} \\0 & \text {for}\: a = 0 \\- \frac {4 a x^{3} \sqrt {\frac {1}{a}} \log {\left (x - \sqrt {\frac {1}{a}} \right )}}{x^{3} - \frac {x}{a}} - \frac {2 a x^{3} \sqrt {\frac {1}{a}} \operatorname {Li}_{1}\left (a x^{2}\right )}{x^{3} - \frac {x}{a}} - \frac {2 x^{2} \operatorname {Li}_{1}\left (a x^{2}\right )}{x^{3} - \frac {x}{a}} - \frac {x^{2} \operatorname {Li}_{2}\left (a x^{2}\right )}{x^{3} - \frac {x}{a}} + \frac {4 x \sqrt {\frac {1}{a}} \log {\left (x - \sqrt {\frac {1}{a}} \right )}}{x^{3} - \frac {x}{a}} + \frac {2 x \sqrt {\frac {1}{a}} \operatorname {Li}_{1}\left (a x^{2}\right )}{x^{3} - \frac {x}{a}} + \frac {2 \operatorname {Li}_{1}\left (a x^{2}\right )}{a x^{3} - x} + \frac {\operatorname {Li}_{2}\left (a x^{2}\right )}{a x^{3} - x} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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