Optimal. Leaf size=58 \[ \frac {1}{4} a^2 \log (x)-\frac {1}{4} a^2 \log (1-a x)-\frac {\text {Li}_2(a x)}{2 x^2}+\frac {\log (1-a x)}{4 x^2}-\frac {a}{4 x} \]
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Rubi [A] time = 0.03, antiderivative size = 58, normalized size of antiderivative = 1.00, 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 44
Rule 2395
Rule 6591
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*}
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Mathematica [A] time = 0.02, size = 50, normalized size = 0.86 \[ \frac {a^2 x^2 \log (x)-a^2 x^2 \log (1-a x)-2 \text {Li}_2(a x)-a x+\log (1-a x)}{4 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 47, normalized size = 0.81 \[ -\frac {a^{2} x^{2} \log \left (a x - 1\right ) - a^{2} x^{2} \log \relax (x) + a x + 2 \, {\rm Li}_2\left (a x\right ) - \log \left (-a x + 1\right )}{4 \, x^{2}} \]
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 \[ \int \frac {{\rm Li}_2\left (a x\right )}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 52, normalized size = 0.90 \[ -\frac {\polylog \left (2, a x \right )}{2 x^{2}}+\frac {a^{2} \ln \left (-a x \right )}{4}-\frac {a}{4 x}-\frac {a^{2} \ln \left (-a x +1\right )}{4}+\frac {\ln \left (-a x +1\right )}{4 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 40, normalized size = 0.69 \[ \frac {1}{4} \, a^{2} \log \relax (x) - \frac {a x + {\left (a^{2} x^{2} - 1\right )} \log \left (-a x + 1\right ) + 2 \, {\rm Li}_2\left (a x\right )}{4 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.30, size = 51, normalized size = 0.88 \[ \frac {a^2\,\ln \relax (x)}{2}+\frac {\ln \left (1-a\,x\right )}{4\,x^2}-\frac {a^2\,\ln \left (a\,x^2-x\right )}{4}-\frac {a}{4\,x}-\frac {\mathrm {polylog}\left (2,a\,x\right )}{2\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.97, size = 42, normalized size = 0.72 \[ \frac {a^{2} \log {\relax (x )}}{4} + \frac {a^{2} \operatorname {Li}_{1}\left (a x\right )}{4} - \frac {a}{4 x} - \frac {\operatorname {Li}_{1}\left (a x\right )}{4 x^{2}} - \frac {\operatorname {Li}_{2}\left (a x\right )}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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