Optimal. Leaf size=70 \[ \frac {1}{8} a^2 \log (x)-\frac {1}{8} a^2 \log (1-a x)-\frac {\text {Li}_2(a x)}{4 x^2}-\frac {\text {Li}_3(a x)}{2 x^2}+\frac {\log (1-a x)}{8 x^2}-\frac {a}{8 x} \]
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Rubi [A] time = 0.04, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6591, 2395, 44} \[ -\frac {\text {PolyLog}(2,a x)}{4 x^2}-\frac {\text {PolyLog}(3,a x)}{2 x^2}+\frac {1}{8} a^2 \log (x)-\frac {1}{8} a^2 \log (1-a x)+\frac {\log (1-a x)}{8 x^2}-\frac {a}{8 x} \]
Antiderivative was successfully verified.
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Rule 44
Rule 2395
Rule 6591
Rubi steps
\begin {align*} \int \frac {\text {Li}_3(a x)}{x^3} \, dx &=-\frac {\text {Li}_3(a x)}{2 x^2}+\frac {1}{2} \int \frac {\text {Li}_2(a x)}{x^3} \, dx\\ &=-\frac {\text {Li}_2(a x)}{4 x^2}-\frac {\text {Li}_3(a x)}{2 x^2}-\frac {1}{4} \int \frac {\log (1-a x)}{x^3} \, dx\\ &=\frac {\log (1-a x)}{8 x^2}-\frac {\text {Li}_2(a x)}{4 x^2}-\frac {\text {Li}_3(a x)}{2 x^2}+\frac {1}{8} a \int \frac {1}{x^2 (1-a x)} \, dx\\ &=\frac {\log (1-a x)}{8 x^2}-\frac {\text {Li}_2(a x)}{4 x^2}-\frac {\text {Li}_3(a x)}{2 x^2}+\frac {1}{8} a \int \left (\frac {1}{x^2}+\frac {a}{x}-\frac {a^2}{-1+a x}\right ) \, dx\\ &=-\frac {a}{8 x}+\frac {1}{8} a^2 \log (x)-\frac {1}{8} a^2 \log (1-a x)+\frac {\log (1-a x)}{8 x^2}-\frac {\text {Li}_2(a x)}{4 x^2}-\frac {\text {Li}_3(a x)}{2 x^2}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 25, normalized size = 0.36 \[ \frac {G_{5,5}^{2,4}\left (-a x\left |\begin {array}{c} 1,1,1,1,3 \\ 1,2,0,0,0 \\\end {array}\right .\right )}{x^2} \]
Warning: Unable to verify antiderivative.
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fricas [C] time = 0.62, size = 54, normalized size = 0.77 \[ -\frac {a^{2} x^{2} \log \left (a x - 1\right ) - a^{2} x^{2} \log \relax (x) + a x + 2 \, {\rm Li}_2\left (a x\right ) - \log \left (-a x + 1\right ) + 4 \, {\rm polylog}\left (3, a x\right )}{8 \, x^{2}} \]
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 \[ \int \frac {{\rm Li}_{3}(a x)}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 90, normalized size = 1.29 \[ -a^{2} \left (-\frac {81 a x +378}{432 a x}-\frac {\left (-27 a^{2} x^{2}+27\right ) \ln \left (-a x +1\right )}{216 a^{2} x^{2}}+\frac {\polylog \left (2, a x \right )}{4 a^{2} x^{2}}+\frac {\polylog \left (3, a x \right )}{2 a^{2} x^{2}}+\frac {3}{16}-\frac {\ln \relax (x )}{8}-\frac {\ln \left (-a \right )}{8}+\frac {1}{a x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 47, normalized size = 0.67 \[ \frac {1}{8} \, a^{2} \log \relax (x) - \frac {a x + {\left (a^{2} x^{2} - 1\right )} \log \left (-a x + 1\right ) + 2 \, {\rm Li}_2\left (a x\right ) + 4 \, {\rm Li}_{3}(a x)}{8 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.27, size = 46, normalized size = 0.66 \[ \frac {a^2\,\mathrm {atanh}\left (2\,a\,x-1\right )}{4}-\frac {\frac {a\,x}{8}-\frac {\ln \left (1-a\,x\right )}{8}+\frac {\mathrm {polylog}\left (2,a\,x\right )}{4}+\frac {\mathrm {polylog}\left (3,a\,x\right )}{2}}{x^2} \]
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 \[ \int \frac {\operatorname {Li}_{3}\left (a x\right )}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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