Optimal. Leaf size=80 \[ \frac {1}{27} a^3 \log (x)-\frac {1}{27} a^3 \log (1-a x)-\frac {a^2}{27 x}-\frac {\text {Li}_2(a x)}{9 x^3}-\frac {\text {Li}_3(a x)}{3 x^3}+\frac {\log (1-a x)}{27 x^3}-\frac {a}{54 x^2} \]
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Rubi [A] time = 0.05, antiderivative size = 80, 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} \[ -\frac {\text {PolyLog}(2,a x)}{9 x^3}-\frac {\text {PolyLog}(3,a x)}{3 x^3}-\frac {a^2}{27 x}+\frac {1}{27} a^3 \log (x)-\frac {1}{27} a^3 \log (1-a x)-\frac {a}{54 x^2}+\frac {\log (1-a x)}{27 x^3} \]
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^4} \, dx &=-\frac {\text {Li}_3(a x)}{3 x^3}+\frac {1}{3} \int \frac {\text {Li}_2(a x)}{x^4} \, dx\\ &=-\frac {\text {Li}_2(a x)}{9 x^3}-\frac {\text {Li}_3(a x)}{3 x^3}-\frac {1}{9} \int \frac {\log (1-a x)}{x^4} \, dx\\ &=\frac {\log (1-a x)}{27 x^3}-\frac {\text {Li}_2(a x)}{9 x^3}-\frac {\text {Li}_3(a x)}{3 x^3}+\frac {1}{27} a \int \frac {1}{x^3 (1-a x)} \, dx\\ &=\frac {\log (1-a x)}{27 x^3}-\frac {\text {Li}_2(a x)}{9 x^3}-\frac {\text {Li}_3(a x)}{3 x^3}+\frac {1}{27} a \int \left (\frac {1}{x^3}+\frac {a}{x^2}+\frac {a^2}{x}-\frac {a^3}{-1+a x}\right ) \, dx\\ &=-\frac {a}{54 x^2}-\frac {a^2}{27 x}+\frac {1}{27} a^3 \log (x)-\frac {1}{27} a^3 \log (1-a x)+\frac {\log (1-a x)}{27 x^3}-\frac {\text {Li}_2(a x)}{9 x^3}-\frac {\text {Li}_3(a x)}{3 x^3}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 25, normalized size = 0.31 \[ \frac {G_{5,5}^{2,4}\left (-a x\left |\begin {array}{c} 1,1,1,1,4 \\ 1,3,0,0,0 \\\end {array}\right .\right )}{x^3} \]
Warning: Unable to verify antiderivative.
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fricas [C] time = 0.71, size = 63, normalized size = 0.79 \[ -\frac {2 \, a^{3} x^{3} \log \left (a x - 1\right ) - 2 \, a^{3} x^{3} \log \relax (x) + 2 \, a^{2} x^{2} + a x + 6 \, {\rm Li}_2\left (a x\right ) - 2 \, \log \left (-a x + 1\right ) + 18 \, {\rm polylog}\left (3, a x\right )}{54 \, x^{3}} \]
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 \[ \int \frac {{\rm Li}_{3}(a x)}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 106, normalized size = 1.32 \[ a^{3} \left (\frac {64 a^{2} x^{2}+152 a x +832}{1728 a^{2} x^{2}}+\frac {\left (-64 a^{3} x^{3}+64\right ) \ln \left (-a x +1\right )}{1728 a^{3} x^{3}}-\frac {\polylog \left (2, a x \right )}{9 a^{3} x^{3}}-\frac {\polylog \left (3, a x \right )}{3 a^{3} x^{3}}-\frac {1}{27}+\frac {\ln \relax (x )}{27}+\frac {\ln \left (-a \right )}{27}-\frac {1}{2 x^{2} a^{2}}-\frac {1}{8 a x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 56, normalized size = 0.70 \[ \frac {1}{27} \, a^{3} \log \relax (x) - \frac {2 \, a^{2} x^{2} + a x + 2 \, {\left (a^{3} x^{3} - 1\right )} \log \left (-a x + 1\right ) + 6 \, {\rm Li}_2\left (a x\right ) + 18 \, {\rm Li}_{3}(a x)}{54 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.50, size = 62, normalized size = 0.78 \[ \frac {\ln \left (1-a\,x\right )}{27\,x^3}-\frac {\mathrm {polylog}\left (2,a\,x\right )}{9\,x^3}-\frac {\mathrm {polylog}\left (3,a\,x\right )}{3\,x^3}-\frac {x\,a^2+\frac {a}{2}}{27\,x^2}-\frac {a^3\,\mathrm {atan}\left (a\,x\,2{}\mathrm {i}-\mathrm {i}\right )\,2{}\mathrm {i}}{27} \]
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 \[ \int \frac {\operatorname {Li}_{3}\left (a x\right )}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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