Optimal. Leaf size=63 \[ -\frac {\text {Li}_2\left (a x^2\right )}{2 x^2}-\frac {\text {Li}_3\left (a x^2\right )}{2 x^2}-\frac {1}{2} a \log \left (1-a x^2\right )+\frac {\log \left (1-a x^2\right )}{2 x^2}+a \log (x) \]
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Rubi [A] time = 0.05, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {6591, 2454, 2395, 36, 29, 31} \[ -\frac {\text {PolyLog}\left (2,a x^2\right )}{2 x^2}-\frac {\text {PolyLog}\left (3,a x^2\right )}{2 x^2}-\frac {1}{2} a \log \left (1-a x^2\right )+\frac {\log \left (1-a x^2\right )}{2 x^2}+a \log (x) \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 2395
Rule 2454
Rule 6591
Rubi steps
\begin {align*} \int \frac {\text {Li}_3\left (a x^2\right )}{x^3} \, dx &=-\frac {\text {Li}_3\left (a x^2\right )}{2 x^2}+\int \frac {\text {Li}_2\left (a x^2\right )}{x^3} \, dx\\ &=-\frac {\text {Li}_2\left (a x^2\right )}{2 x^2}-\frac {\text {Li}_3\left (a x^2\right )}{2 x^2}-\int \frac {\log \left (1-a x^2\right )}{x^3} \, dx\\ &=-\frac {\text {Li}_2\left (a x^2\right )}{2 x^2}-\frac {\text {Li}_3\left (a x^2\right )}{2 x^2}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {\log (1-a x)}{x^2} \, dx,x,x^2\right )\\ &=\frac {\log \left (1-a x^2\right )}{2 x^2}-\frac {\text {Li}_2\left (a x^2\right )}{2 x^2}-\frac {\text {Li}_3\left (a x^2\right )}{2 x^2}+\frac {1}{2} a \operatorname {Subst}\left (\int \frac {1}{x (1-a x)} \, dx,x,x^2\right )\\ &=\frac {\log \left (1-a x^2\right )}{2 x^2}-\frac {\text {Li}_2\left (a x^2\right )}{2 x^2}-\frac {\text {Li}_3\left (a x^2\right )}{2 x^2}+\frac {1}{2} a \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{2} a^2 \operatorname {Subst}\left (\int \frac {1}{1-a x} \, dx,x,x^2\right )\\ &=a \log (x)-\frac {1}{2} a \log \left (1-a x^2\right )+\frac {\log \left (1-a x^2\right )}{2 x^2}-\frac {\text {Li}_2\left (a x^2\right )}{2 x^2}-\frac {\text {Li}_3\left (a x^2\right )}{2 x^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 60, normalized size = 0.95 \[ -\frac {\text {Li}_2\left (a x^2\right )+\text {Li}_3\left (a x^2\right )-a x^2 \log \left (-a x^2\right )+a x^2 \log \left (1-a x^2\right )-\log \left (1-a x^2\right )}{2 x^2} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.57, size = 51, normalized size = 0.81 \[ -\frac {a x^{2} \log \left (a x^{2} - 1\right ) - 2 \, a x^{2} \log \relax (x) + {\rm Li}_2\left (a x^{2}\right ) - \log \left (-a x^{2} + 1\right ) + {\rm polylog}\left (3, a x^{2}\right )}{2 \, x^{2}} \]
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 \[ \int \frac {{\rm Li}_{3}(a x^{2})}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 68, normalized size = 1.08 \[ \frac {a \left (\frac {\left (-8 a \,x^{2}+8\right ) \ln \left (-a \,x^{2}+1\right )}{8 a \,x^{2}}-\frac {\polylog \left (2, a \,x^{2}\right )}{a \,x^{2}}-\frac {\polylog \left (3, a \,x^{2}\right )}{a \,x^{2}}+2 \ln \relax (x )+\ln \left (-a \right )\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 41, normalized size = 0.65 \[ a \log \relax (x) - \frac {{\left (a x^{2} - 1\right )} \log \left (-a x^{2} + 1\right ) + {\rm Li}_2\left (a x^{2}\right ) + {\rm Li}_{3}(a x^{2})}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.29, size = 54, normalized size = 0.86 \[ -\frac {\mathrm {polylog}\left (2,a\,x^2\right )-\ln \left (1-a\,x^2\right )+\mathrm {polylog}\left (3,a\,x^2\right )-3\,a\,x^2\,\ln \relax (x)+a\,x^2\,\ln \left (x\,\left (a\,x^2-1\right )\right )}{2\,x^2} \]
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 \[ \int \frac {\operatorname {Li}_{3}\left (a x^{2}\right )}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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