Optimal. Leaf size=46 \[ -\frac{\text{PolyLog}(2,a x)}{x}-\frac{\text{PolyLog}(3,a x)}{x}+a \log (x)-a \log (1-a x)+\frac{\log (1-a x)}{x} \]
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Rubi [A] time = 0.0304426, antiderivative size = 46, normalized size of antiderivative = 1., 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 = {6591, 2395, 36, 29, 31} \[ -\frac{\text{PolyLog}(2,a x)}{x}-\frac{\text{PolyLog}(3,a x)}{x}+a \log (x)-a \log (1-a x)+\frac{\log (1-a x)}{x} \]
Antiderivative was successfully verified.
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Rule 6591
Rule 2395
Rule 36
Rule 29
Rule 31
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*}
Mathematica [A] time = 0.0324637, size = 44, normalized size = 0.96 \[ -\frac{\text{PolyLog}(2,a x)+\text{PolyLog}(3,a x)-a x \log (-a x)+a x \log (1-a x)-\log (1-a x)}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.08, size = 57, normalized size = 1.2 \begin{align*} a \left ( \ln \left ( x \right ) +\ln \left ( -a \right ) +{\frac{ \left ( -8\,ax+8 \right ) \ln \left ( -ax+1 \right ) }{8\,ax}}-{\frac{{\it polylog} \left ( 2,ax \right ) }{ax}}-{\frac{{\it polylog} \left ( 3,ax \right ) }{ax}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.995112, size = 45, normalized size = 0.98 \begin{align*} 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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.66231, size = 162, normalized size = 3.52 \begin{align*} -\frac{a x \log \left (a x - 1\right ) - a x \log \left (x\right ) +{\rm \%iint}\left (a, x, -\frac{\log \left (-a x + 1\right )}{a}, -\frac{\log \left (-a x + 1\right )}{x}\right ) - \log \left (-a x + 1\right ) +{\rm polylog}\left (3, a x\right )}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{Li}_{3}\left (a x\right )}{x^{2}}\, dx \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 \frac{{\rm Li}_{3}(a x)}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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