Optimal. Leaf size=46 \[ -\frac {\text {Li}_2(a x)}{x}-\frac {\text {Li}_3(a x)}{x}+a \log (x)-a \log (1-a x)+\frac {\log (1-a x)}{x} \]
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Rubi [A] time = 0.03, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {6591, 2395, 36, 29, 31} \[ -\frac {\text {PolyLog}(2,a x)}{x}-\frac {\text {PolyLog}(3,a x)}{x}+a \log (x)-a \log (1-a x)+\frac {\log (1-a x)}{x} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 2395
Rule 6591
Rubi steps
\begin {align*} \int \frac {\text {Li}_3(a x)}{x^2} \, dx &=-\frac {\text {Li}_3(a x)}{x}+\int \frac {\text {Li}_2(a x)}{x^2} \, dx\\ &=-\frac {\text {Li}_2(a x)}{x}-\frac {\text {Li}_3(a x)}{x}-\int \frac {\log (1-a x)}{x^2} \, dx\\ &=\frac {\log (1-a x)}{x}-\frac {\text {Li}_2(a x)}{x}-\frac {\text {Li}_3(a x)}{x}+a \int \frac {1}{x (1-a x)} \, dx\\ &=\frac {\log (1-a x)}{x}-\frac {\text {Li}_2(a x)}{x}-\frac {\text {Li}_3(a x)}{x}+a \int \frac {1}{x} \, dx+a^2 \int \frac {1}{1-a x} \, dx\\ &=a \log (x)-a \log (1-a x)+\frac {\log (1-a x)}{x}-\frac {\text {Li}_2(a x)}{x}-\frac {\text {Li}_3(a x)}{x}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 44, normalized size = 0.96 \[ -\frac {\text {Li}_2(a x)+\text {Li}_3(a x)-a x \log (-a x)+a x \log (1-a x)-\log (1-a x)}{x} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.49, size = 39, normalized size = 0.85 \[ -\frac {a x \log \left (a x - 1\right ) - a x \log \relax (x) + {\rm Li}_2\left (a x\right ) - \log \left (-a x + 1\right ) + {\rm polylog}\left (3, a x\right )}{x} \]
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 \[ \int \frac {{\rm Li}_{3}(a x)}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 57, normalized size = 1.24 \[ a \left (\frac {\left (-8 a x +8\right ) \ln \left (-a x +1\right )}{8 a x}-\frac {\polylog \left (2, a x \right )}{a x}-\frac {\polylog \left (3, a x \right )}{a x}+\ln \relax (x )+\ln \left (-a \right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 33, normalized size = 0.72 \[ a \log \relax (x) - \frac {{\left (a x - 1\right )} \log \left (-a x + 1\right ) + {\rm Li}_2\left (a x\right ) + {\rm Li}_{3}(a x)}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.89, size = 36, normalized size = 0.78 \[ 2\,a\,\mathrm {atanh}\left (2\,a\,x-1\right )-\frac {\mathrm {polylog}\left (2,a\,x\right )-\ln \left (1-a\,x\right )+\mathrm {polylog}\left (3,a\,x\right )}{x} \]
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 \[ \int \frac {\operatorname {Li}_{3}\left (a x\right )}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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