Optimal. Leaf size=29 \[ -x-\frac {(1-a x) \log (1-a x)}{a}+x \text {PolyLog}(2,a x) \]
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Rubi [A]
time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6721, 2436,
2332} \begin {gather*} x \text {Li}_2(a x)-\frac {(1-a x) \log (1-a x)}{a}-x \end {gather*}
Antiderivative was successfully verified.
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Rule 2332
Rule 2436
Rule 6721
Rubi steps
\begin {align*} \int \text {Li}_2(a x) \, dx &=x \text {Li}_2(a x)+\int \log (1-a x) \, dx\\ &=x \text {Li}_2(a x)-\frac {\text {Subst}(\int \log (x) \, dx,x,1-a x)}{a}\\ &=-x-\frac {(1-a x) \log (1-a x)}{a}+x \text {Li}_2(a x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 26, normalized size = 0.90 \begin {gather*} -x+\left (-\frac {1}{a}+x\right ) \log (1-a x)+x \text {PolyLog}(2,a x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.29, size = 34, normalized size = 1.17
method | result | size |
meijerg | \(\frac {-a x -\frac {\left (-2 a x +2\right ) \ln \left (-a x +1\right )}{2}+a x \polylog \left (2, a x \right )}{a}\) | \(33\) |
derivativedivides | \(\frac {a x \polylog \left (2, a x \right )-\ln \left (-a x +1\right ) \left (-a x +1\right )+1-a x}{a}\) | \(34\) |
default | \(\frac {a x \polylog \left (2, a x \right )-\ln \left (-a x +1\right ) \left (-a x +1\right )+1-a x}{a}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 29, normalized size = 1.00 \begin {gather*} \frac {a x {\rm Li}_2\left (a x\right ) - a x + {\left (a x - 1\right )} \log \left (-a x + 1\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 29, normalized size = 1.00 \begin {gather*} \frac {a x {\rm Li}_2\left (a x\right ) - a x + {\left (a x - 1\right )} \log \left (-a x + 1\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.25, size = 22, normalized size = 0.76 \begin {gather*} \begin {cases} - x \operatorname {Li}_{1}\left (a x\right ) + x \operatorname {Li}_{2}\left (a x\right ) - x + \frac {\operatorname {Li}_{1}\left (a x\right )}{a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \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 = 0.26, size = 32, normalized size = 1.10 \begin {gather*} x\,\mathrm {polylog}\left (2,a\,x\right )-\frac {\ln \left (1-a\,x\right )}{a}-x+x\,\ln \left (1-a\,x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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