Optimal. Leaf size=54 \[ 8 \sqrt {a} \tanh ^{-1}\left (\sqrt {a} x\right )+\frac {4 \log \left (1-a x^2\right )}{x}-\frac {2 \text {PolyLog}\left (2,a x^2\right )}{x}-\frac {\text {PolyLog}\left (3,a x^2\right )}{x} \]
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Rubi [A]
time = 0.03, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6726, 2505,
212} \begin {gather*} -\frac {2 \text {Li}_2\left (a x^2\right )}{x}-\frac {\text {Li}_3\left (a x^2\right )}{x}+\frac {4 \log \left (1-a x^2\right )}{x}+8 \sqrt {a} \tanh ^{-1}\left (\sqrt {a} x\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2505
Rule 6726
Rubi steps
\begin {align*} \int \frac {\text {Li}_3\left (a x^2\right )}{x^2} \, dx &=-\frac {\text {Li}_3\left (a x^2\right )}{x}+2 \int \frac {\text {Li}_2\left (a x^2\right )}{x^2} \, dx\\ &=-\frac {2 \text {Li}_2\left (a x^2\right )}{x}-\frac {\text {Li}_3\left (a x^2\right )}{x}-4 \int \frac {\log \left (1-a x^2\right )}{x^2} \, dx\\ &=\frac {4 \log \left (1-a x^2\right )}{x}-\frac {2 \text {Li}_2\left (a x^2\right )}{x}-\frac {\text {Li}_3\left (a x^2\right )}{x}+(8 a) \int \frac {1}{1-a x^2} \, dx\\ &=8 \sqrt {a} \tanh ^{-1}\left (\sqrt {a} x\right )+\frac {4 \log \left (1-a x^2\right )}{x}-\frac {2 \text {Li}_2\left (a x^2\right )}{x}-\frac {\text {Li}_3\left (a x^2\right )}{x}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 50, normalized size = 0.93 \begin {gather*} \frac {8 \sqrt {a} x \tanh ^{-1}\left (\sqrt {a} x\right )+4 \log \left (1-a x^2\right )-2 \text {PolyLog}\left (2,a x^2\right )-\text {PolyLog}\left (3,a x^2\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(111\) vs.
\(2(50)=100\).
time = 0.12, size = 112, normalized size = 2.07
method | result | size |
meijerg | \(\frac {a \left (-\frac {8 x \sqrt {-a}\, \left (\ln \left (1-\sqrt {a \,x^{2}}\right )-\ln \left (1+\sqrt {a \,x^{2}}\right )\right )}{\sqrt {a \,x^{2}}}+\frac {8 \sqrt {-a}\, \ln \left (-a \,x^{2}+1\right )}{x a}-\frac {4 \sqrt {-a}\, \polylog \left (2, a \,x^{2}\right )}{x a}-\frac {2 \sqrt {-a}\, \polylog \left (3, a \,x^{2}\right )}{x a}\right )}{2 \sqrt {-a}}\) | \(112\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 58, normalized size = 1.07 \begin {gather*} -4 \, \sqrt {a} \log \left (\frac {a x - \sqrt {a}}{a x + \sqrt {a}}\right ) - \frac {2 \, {\rm Li}_2\left (a x^{2}\right ) - 4 \, \log \left (-a x^{2} + 1\right ) + {\rm Li}_{3}(a x^{2})}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.52, size = 112, normalized size = 2.07 \begin {gather*} \left [\frac {4 \, \sqrt {a} x \log \left (\frac {a x^{2} + 2 \, \sqrt {a} x + 1}{a x^{2} - 1}\right ) - 2 \, {\rm Li}_2\left (a x^{2}\right ) + 4 \, \log \left (-a x^{2} + 1\right ) - {\rm polylog}\left (3, a x^{2}\right )}{x}, -\frac {8 \, \sqrt {-a} x \arctan \left (\sqrt {-a} x\right ) + 2 \, {\rm Li}_2\left (a x^{2}\right ) - 4 \, \log \left (-a x^{2} + 1\right ) + {\rm polylog}\left (3, a x^{2}\right )}{x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {Li}_{3}\left (a x^{2}\right )}{x^{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.55, size = 53, normalized size = 0.98 \begin {gather*} \frac {4\,\ln \left (1-a\,x^2\right )}{x}-\frac {\mathrm {polylog}\left (3,a\,x^2\right )}{x}-\frac {2\,\mathrm {polylog}\left (2,a\,x^2\right )}{x}-\sqrt {a}\,\mathrm {atan}\left (\sqrt {a}\,x\,1{}\mathrm {i}\right )\,8{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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