Optimal. Leaf size=29 \[ x \text {Li}_2(a x)-\frac {(1-a x) \log (1-a x)}{a}-x \]
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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 = {6586, 2389, 2295} \[ x \text {PolyLog}(2,a x)-\frac {(1-a x) \log (1-a x)}{a}-x \]
Antiderivative was successfully verified.
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Rule 2295
Rule 2389
Rule 6586
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 {\operatorname {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 \[ x \text {Li}_2(a x)+\left (x-\frac {1}{a}\right ) \log (1-a x)-x \]
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 29, normalized size = 1.00 \[ \frac {a x {\rm Li}_2\left (a x\right ) - a x + {\left (a x - 1\right )} \log \left (-a x + 1\right )}{a} \]
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 \[ \int {\rm Li}_2\left (a x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 36, normalized size = 1.24 \[ \ln \left (-a x +1\right ) x +x \polylog \left (2, a x \right )-x -\frac {\ln \left (-a x +1\right )}{a}+\frac {1}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 29, normalized size = 1.00 \[ \frac {a x {\rm Li}_2\left (a x\right ) - a x + {\left (a x - 1\right )} \log \left (-a x + 1\right )}{a} \]
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 \[ x\,\mathrm {polylog}\left (2,a\,x\right )-\frac {\ln \left (1-a\,x\right )}{a}-x+x\,\ln \left (1-a\,x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.94, size = 22, normalized size = 0.76 \[ \begin {cases} - x \operatorname {Li}_{1}\left (a x\right ) + x \operatorname {Li}_{2}\left (a x\right ) - x + \frac {\operatorname {Li}_{1}\left (a x\right )}{a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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