Optimal. Leaf size=70 \[ -\frac {8 a}{27 x}+\frac {8}{27} a^{3/2} \tanh ^{-1}\left (\sqrt {a} x\right )+\frac {4 \log \left (1-a x^2\right )}{27 x^3}-\frac {2 \text {PolyLog}\left (2,a x^2\right )}{9 x^3}-\frac {\text {PolyLog}\left (3,a x^2\right )}{3 x^3} \]
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Rubi [A]
time = 0.03, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6726, 2505,
331, 212} \begin {gather*} \frac {8}{27} a^{3/2} \tanh ^{-1}\left (\sqrt {a} x\right )-\frac {2 \text {Li}_2\left (a x^2\right )}{9 x^3}-\frac {\text {Li}_3\left (a x^2\right )}{3 x^3}+\frac {4 \log \left (1-a x^2\right )}{27 x^3}-\frac {8 a}{27 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 331
Rule 2505
Rule 6726
Rubi steps
\begin {align*} \int \frac {\text {Li}_3\left (a x^2\right )}{x^4} \, dx &=-\frac {\text {Li}_3\left (a x^2\right )}{3 x^3}+\frac {2}{3} \int \frac {\text {Li}_2\left (a x^2\right )}{x^4} \, dx\\ &=-\frac {2 \text {Li}_2\left (a x^2\right )}{9 x^3}-\frac {\text {Li}_3\left (a x^2\right )}{3 x^3}-\frac {4}{9} \int \frac {\log \left (1-a x^2\right )}{x^4} \, dx\\ &=\frac {4 \log \left (1-a x^2\right )}{27 x^3}-\frac {2 \text {Li}_2\left (a x^2\right )}{9 x^3}-\frac {\text {Li}_3\left (a x^2\right )}{3 x^3}+\frac {1}{27} (8 a) \int \frac {1}{x^2 \left (1-a x^2\right )} \, dx\\ &=-\frac {8 a}{27 x}+\frac {4 \log \left (1-a x^2\right )}{27 x^3}-\frac {2 \text {Li}_2\left (a x^2\right )}{9 x^3}-\frac {\text {Li}_3\left (a x^2\right )}{3 x^3}+\frac {1}{27} \left (8 a^2\right ) \int \frac {1}{1-a x^2} \, dx\\ &=-\frac {8 a}{27 x}+\frac {8}{27} a^{3/2} \tanh ^{-1}\left (\sqrt {a} x\right )+\frac {4 \log \left (1-a x^2\right )}{27 x^3}-\frac {2 \text {Li}_2\left (a x^2\right )}{9 x^3}-\frac {\text {Li}_3\left (a x^2\right )}{3 x^3}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 61, normalized size = 0.87 \begin {gather*} -\frac {8 a x^2-8 a^{3/2} x^3 \tanh ^{-1}\left (\sqrt {a} x\right )-4 \log \left (1-a x^2\right )+6 \text {PolyLog}\left (2,a x^2\right )+9 \text {PolyLog}\left (3,a x^2\right )}{27 x^3} \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. \(124\) vs.
\(2(56)=112\).
time = 0.12, size = 125, normalized size = 1.79
method | result | size |
meijerg | \(-\frac {a^{2} \left (-\frac {16}{27 x \sqrt {-a}}-\frac {8 x a \left (\ln \left (1-\sqrt {a \,x^{2}}\right )-\ln \left (1+\sqrt {a \,x^{2}}\right )\right )}{27 \sqrt {-a}\, \sqrt {a \,x^{2}}}+\frac {8 \ln \left (-a \,x^{2}+1\right )}{27 x^{3} \sqrt {-a}\, a}-\frac {4 \polylog \left (2, a \,x^{2}\right )}{9 x^{3} \sqrt {-a}\, a}-\frac {2 \polylog \left (3, a \,x^{2}\right )}{3 x^{3} \sqrt {-a}\, a}\right )}{2 \sqrt {-a}}\) | \(125\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 66, normalized size = 0.94 \begin {gather*} -\frac {4}{27} \, a^{\frac {3}{2}} \log \left (\frac {a x - \sqrt {a}}{a x + \sqrt {a}}\right ) - \frac {8 \, a x^{2} + 6 \, {\rm Li}_2\left (a x^{2}\right ) - 4 \, \log \left (-a x^{2} + 1\right ) + 9 \, {\rm Li}_{3}(a x^{2})}{27 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 132, normalized size = 1.89 \begin {gather*} \left [\frac {4 \, a^{\frac {3}{2}} x^{3} \log \left (\frac {a x^{2} + 2 \, \sqrt {a} x + 1}{a x^{2} - 1}\right ) - 8 \, a x^{2} - 6 \, {\rm Li}_2\left (a x^{2}\right ) + 4 \, \log \left (-a x^{2} + 1\right ) - 9 \, {\rm polylog}\left (3, a x^{2}\right )}{27 \, x^{3}}, -\frac {8 \, \sqrt {-a} a x^{3} \arctan \left (\sqrt {-a} x\right ) + 8 \, a x^{2} + 6 \, {\rm Li}_2\left (a x^{2}\right ) - 4 \, \log \left (-a x^{2} + 1\right ) + 9 \, {\rm polylog}\left (3, a x^{2}\right )}{27 \, x^{3}}\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^{4}}\, 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 = 0.77, size = 59, normalized size = 0.84 \begin {gather*} \frac {4\,\ln \left (1-a\,x^2\right )}{27\,x^3}-\frac {\mathrm {polylog}\left (3,a\,x^2\right )}{3\,x^3}-\frac {8\,a}{27\,x}-\frac {2\,\mathrm {polylog}\left (2,a\,x^2\right )}{9\,x^3}-\frac {a^{3/2}\,\mathrm {atan}\left (\sqrt {a}\,x\,1{}\mathrm {i}\right )\,8{}\mathrm {i}}{27} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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