Optimal. Leaf size=36 \[ a \log (x)-a \log (1-a x)+\frac {\log (1-a x)}{x}-\frac {\text {PolyLog}(2,a x)}{x} \]
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Rubi [A]
time = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps
used = 5, 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}+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}_2(a x)}{x^2} \, dx &=-\frac {\text {Li}_2(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}+a \int \frac {1}{x (1-a x)} \, dx\\ &=\frac {\log (1-a x)}{x}-\frac {\text {Li}_2(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}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 36, normalized size = 1.00 \begin {gather*} a \log (x)-a \log (1-a x)+\frac {\log (1-a x)}{x}-\frac {\text {PolyLog}(2,a x)}{x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.39, size = 42, normalized size = 1.17
method | result | size |
derivativedivides | \(a \left (-\frac {\polylog \left (2, a x \right )}{a x}+\ln \left (-a x \right )+\frac {\ln \left (-a x +1\right ) \left (-a x +1\right )}{a x}\right )\) | \(42\) |
default | \(a \left (-\frac {\polylog \left (2, a x \right )}{a x}+\ln \left (-a x \right )+\frac {\ln \left (-a x +1\right ) \left (-a x +1\right )}{a x}\right )\) | \(42\) |
meijerg | \(a \left (\frac {\left (-4 a x +4\right ) \ln \left (-a x +1\right )}{4 a x}-\frac {\polylog \left (2, a x \right )}{a x}+\ln \left (x \right )+\ln \left (-a \right )\right )\) | \(44\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 28, normalized size = 0.78 \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 )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 34, normalized size = 0.94 \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 )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.37, size = 24, normalized size = 0.67 \begin {gather*} a \log {\left (x \right )} + a \operatorname {Li}_{1}\left (a x\right ) - \frac {\operatorname {Li}_{1}\left (a x\right )}{x} - \frac {\operatorname {Li}_{2}\left (a x\right )}{x} \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 = 0.20, size = 34, normalized size = 0.94 \begin {gather*} \frac {\ln \left (1-a\,x\right )-\mathrm {polylog}\left (2,a\,x\right )}{x}+a\,\ln \left (x\right )-a\,\ln \left (1-a\,x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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