Optimal. Leaf size=42 \[ -\frac{\text{PolyLog}\left (2,a x^2\right )}{x}+\frac{2 \log \left (1-a x^2\right )}{x}+4 \sqrt{a} \tanh ^{-1}\left (\sqrt{a} x\right ) \]
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Rubi [A] time = 0.0261791, antiderivative size = 42, normalized size of antiderivative = 1., 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 6591
Rule 2455
Rule 206
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*}
Mathematica [A] time = 0.0177403, size = 41, normalized size = 0.98 \[ \frac{-\text{PolyLog}\left (2,a x^2\right )+2 \log \left (1-a x^2\right )+4 \sqrt{a} x \tanh ^{-1}\left (\sqrt{a} x\right )}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 39, normalized size = 0.9 \begin{align*} 2\,{\frac{\ln \left ( -a{x}^{2}+1 \right ) }{x}}-{\frac{{\it polylog} \left ( 2,a{x}^{2} \right ) }{x}}+4\,{\it Artanh} \left ( x\sqrt{a} \right ) \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 [A] time = 2.79204, size = 232, normalized size = 5.52 \begin{align*} \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 ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 84.144, size = 184, normalized size = 4.38 \begin{align*} \begin{cases} 0 & \text{for}\: a = 0 \\- \frac{\pi ^{2}}{6 x} & \text{for}\: a = \frac{1}{x^{2}} \\- \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} \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}_2\left (a x^{2}\right )}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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