Optimal. Leaf size=29 \[ x \text{PolyLog}(2,a x)-\frac{(1-a x) \log (1-a x)}{a}-x \]
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Rubi [A] time = 0.0085078, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 5, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, 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 6586
Rule 2389
Rule 2295
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*}
Mathematica [A] time = 0.011601, size = 26, normalized size = 0.9 \[ x \text{PolyLog}(2,a x)+\left (x-\frac{1}{a}\right ) \log (1-a x)-x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 36, normalized size = 1.2 \begin{align*} x{\it polylog} \left ( 2,ax \right ) +\ln \left ( -ax+1 \right ) x-x-{\frac{\ln \left ( -ax+1 \right ) }{a}}+{a}^{-1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.980908, size = 39, normalized size = 1.34 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.63468, size = 70, normalized size = 2.41 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.20944, size = 22, normalized size = 0.76 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\rm Li}_2\left (a x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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