Optimal. Leaf size=46 \[ a \log (x)-a \log (1-a x)+\frac {\log (1-a x)}{x}-\frac {\text {PolyLog}(2,a x)}{x}-\frac {\text {PolyLog}(3,a x)}{x} \]
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Rubi [A]
time = 0.02, 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 = {6726, 2442, 36,
29, 31} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 2442
Rule 6726
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 = 0.03, size = 44, normalized size = 0.96 \begin {gather*} -\frac {-a x \log (-a x)-\log (1-a x)+a x \log (1-a x)+\text {PolyLog}(2,a x)+\text {PolyLog}(3,a x)}{x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 57, normalized size = 1.24
method | result | size |
meijerg | \(a \left (\frac {\left (-8 a x +8\right ) \ln \left (-a x +1\right )}{8 a x}-\frac {\polylog \left (2, a x \right )}{a x}-\frac {\polylog \left (3, a x \right )}{a x}+\ln \left (x \right )+\ln \left (-a \right )\right )\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 33, normalized size = 0.72 \begin {gather*} a \log \left (x\right ) - \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 39, normalized size = 0.85 \begin {gather*} -\frac {a x \log \left (a x - 1\right ) - a x \log \left (x\right ) + {\rm Li}_2\left (a x\right ) - \log \left (-a x + 1\right ) + {\rm polylog}\left (3, a x\right )}{x} \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\right )}{x^{2}}\, 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 = 0.89, size = 36, normalized size = 0.78 \begin {gather*} 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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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