Optimal. Leaf size=58 \[ -\frac{\text{PolyLog}(2,a x)}{2 x^2}+\frac{1}{4} a^2 \log (x)-\frac{1}{4} a^2 \log (1-a x)+\frac{\log (1-a x)}{4 x^2}-\frac{a}{4 x} \]
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Rubi [A] time = 0.0332573, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6591, 2395, 44} \[ -\frac{\text{PolyLog}(2,a x)}{2 x^2}+\frac{1}{4} a^2 \log (x)-\frac{1}{4} a^2 \log (1-a x)+\frac{\log (1-a x)}{4 x^2}-\frac{a}{4 x} \]
Antiderivative was successfully verified.
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Rule 6591
Rule 2395
Rule 44
Rubi steps
\begin{align*} \int \frac{\text{Li}_2(a x)}{x^3} \, dx &=-\frac{\text{Li}_2(a x)}{2 x^2}-\frac{1}{2} \int \frac{\log (1-a x)}{x^3} \, dx\\ &=\frac{\log (1-a x)}{4 x^2}-\frac{\text{Li}_2(a x)}{2 x^2}+\frac{1}{4} a \int \frac{1}{x^2 (1-a x)} \, dx\\ &=\frac{\log (1-a x)}{4 x^2}-\frac{\text{Li}_2(a x)}{2 x^2}+\frac{1}{4} a \int \left (\frac{1}{x^2}+\frac{a}{x}-\frac{a^2}{-1+a x}\right ) \, dx\\ &=-\frac{a}{4 x}+\frac{1}{4} a^2 \log (x)-\frac{1}{4} a^2 \log (1-a x)+\frac{\log (1-a x)}{4 x^2}-\frac{\text{Li}_2(a x)}{2 x^2}\\ \end{align*}
Mathematica [A] time = 0.0221772, size = 50, normalized size = 0.86 \[ \frac{-2 \text{PolyLog}(2,a x)+a^2 x^2 \log (x)-a^2 x^2 \log (1-a x)-a x+\log (1-a x)}{4 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.118, size = 52, normalized size = 0.9 \begin{align*} -{\frac{{\it polylog} \left ( 2,ax \right ) }{2\,{x}^{2}}}-{\frac{a}{4\,x}}+{\frac{{a}^{2}\ln \left ( -ax \right ) }{4}}-{\frac{{a}^{2}\ln \left ( -ax+1 \right ) }{4}}+{\frac{\ln \left ( -ax+1 \right ) }{4\,{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.981041, size = 54, normalized size = 0.93 \begin{align*} \frac{1}{4} \, a^{2} \log \left (x\right ) - \frac{a x +{\left (a^{2} x^{2} - 1\right )} \log \left (-a x + 1\right ) + 2 \,{\rm Li}_2\left (a x\right )}{4 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.62905, size = 117, normalized size = 2.02 \begin{align*} -\frac{a^{2} x^{2} \log \left (a x - 1\right ) - a^{2} x^{2} \log \left (x\right ) + a x + 2 \,{\rm Li}_2\left (a x\right ) - \log \left (-a x + 1\right )}{4 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.96484, size = 42, normalized size = 0.72 \begin{align*} \frac{a^{2} \log{\left (x \right )}}{4} + \frac{a^{2} \operatorname{Li}_{1}\left (a x\right )}{4} - \frac{a}{4 x} - \frac{\operatorname{Li}_{1}\left (a x\right )}{4 x^{2}} - \frac{\operatorname{Li}_{2}\left (a x\right )}{2 x^{2}} \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}_2\left (a x\right )}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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