Optimal. Leaf size=40 \[ x \text {Li}_2\left (a x^2\right )+2 x \log \left (1-a x^2\right )+\frac {4 \tanh ^{-1}\left (\sqrt {a} x\right )}{\sqrt {a}}-4 x \]
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Rubi [A] time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {6586, 2448, 321, 206} \[ x \text {PolyLog}\left (2,a x^2\right )+2 x \log \left (1-a x^2\right )+\frac {4 \tanh ^{-1}\left (\sqrt {a} x\right )}{\sqrt {a}}-4 x \]
Antiderivative was successfully verified.
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Rule 206
Rule 321
Rule 2448
Rule 6586
Rubi steps
\begin {align*} \int \text {Li}_2\left (a x^2\right ) \, dx &=x \text {Li}_2\left (a x^2\right )+2 \int \log \left (1-a x^2\right ) \, dx\\ &=2 x \log \left (1-a x^2\right )+x \text {Li}_2\left (a x^2\right )+(4 a) \int \frac {x^2}{1-a x^2} \, dx\\ &=-4 x+2 x \log \left (1-a x^2\right )+x \text {Li}_2\left (a x^2\right )+4 \int \frac {1}{1-a x^2} \, dx\\ &=-4 x+\frac {4 \tanh ^{-1}\left (\sqrt {a} x\right )}{\sqrt {a}}+2 x \log \left (1-a x^2\right )+x \text {Li}_2\left (a x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 39, normalized size = 0.98 \[ x \text {Li}_2\left (a x^2\right )+2 x \left (\log \left (1-a x^2\right )-2\right )+\frac {4 \tanh ^{-1}\left (\sqrt {a} x\right )}{\sqrt {a}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 107, normalized size = 2.68 \[ \left [\frac {a x {\rm Li}_2\left (a x^{2}\right ) + 2 \, a x \log \left (-a x^{2} + 1\right ) - 4 \, a x + 2 \, \sqrt {a} \log \left (\frac {a x^{2} + 2 \, \sqrt {a} x + 1}{a x^{2} - 1}\right )}{a}, \frac {a x {\rm Li}_2\left (a x^{2}\right ) + 2 \, a x \log \left (-a x^{2} + 1\right ) - 4 \, a x - 4 \, \sqrt {-a} \arctan \left (\sqrt {-a} x\right )}{a}\right ] \]
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 \[ \int {\rm Li}_2\left (a x^{2}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 37, normalized size = 0.92 \[ -4 x +2 x \ln \left (-a \,x^{2}+1\right )+x \polylog \left (2, a \,x^{2}\right )+\frac {4 \arctanh \left (x \sqrt {a}\right )}{\sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 49, normalized size = 1.22 \[ x {\rm Li}_2\left (a x^{2}\right ) + 2 \, x \log \left (-a x^{2} + 1\right ) - 4 \, x - \frac {2 \, \log \left (\frac {a x - \sqrt {a}}{a x + \sqrt {a}}\right )}{\sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.24, size = 39, normalized size = 0.98 \[ 2\,x\,\ln \left (1-a\,x^2\right )-4\,x+x\,\mathrm {polylog}\left (2,a\,x^2\right )-\frac {\mathrm {atan}\left (\sqrt {a}\,x\,1{}\mathrm {i}\right )\,4{}\mathrm {i}}{\sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 9.92, size = 60, normalized size = 1.50 \[ \begin {cases} - 2 x \operatorname {Li}_{1}\left (a x^{2}\right ) + x \operatorname {Li}_{2}\left (a x^{2}\right ) - 4 x - \frac {4 \log {\left (x - \sqrt {\frac {1}{a}} \right )}}{a \sqrt {\frac {1}{a}}} - \frac {2 \operatorname {Li}_{1}\left (a x^{2}\right )}{a \sqrt {\frac {1}{a}}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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