Optimal. Leaf size=68 \[ \frac {1}{9} a^3 \log (x)-\frac {1}{9} a^3 \log (1-a x)-\frac {a^2}{9 x}-\frac {\text {Li}_2(a x)}{3 x^3}+\frac {\log (1-a x)}{9 x^3}-\frac {a}{18 x^2} \]
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Rubi [A] time = 0.04, antiderivative size = 68, 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} \[ -\frac {\text {PolyLog}(2,a x)}{3 x^3}-\frac {a^2}{9 x}+\frac {1}{9} a^3 \log (x)-\frac {1}{9} a^3 \log (1-a x)-\frac {a}{18 x^2}+\frac {\log (1-a x)}{9 x^3} \]
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^4} \, dx &=-\frac {\text {Li}_2(a x)}{3 x^3}-\frac {1}{3} \int \frac {\log (1-a x)}{x^4} \, dx\\ &=\frac {\log (1-a x)}{9 x^3}-\frac {\text {Li}_2(a x)}{3 x^3}+\frac {1}{9} a \int \frac {1}{x^3 (1-a x)} \, dx\\ &=\frac {\log (1-a x)}{9 x^3}-\frac {\text {Li}_2(a x)}{3 x^3}+\frac {1}{9} a \int \left (\frac {1}{x^3}+\frac {a}{x^2}+\frac {a^2}{x}-\frac {a^3}{-1+a x}\right ) \, dx\\ &=-\frac {a}{18 x^2}-\frac {a^2}{9 x}+\frac {1}{9} a^3 \log (x)-\frac {1}{9} a^3 \log (1-a x)+\frac {\log (1-a x)}{9 x^3}-\frac {\text {Li}_2(a x)}{3 x^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 52, normalized size = 0.76 \[ -\frac {-2 a^3 x^3 \log (x)+2 \left (a^3 x^3-1\right ) \log (1-a x)+6 \text {Li}_2(a x)+a x (2 a x+1)}{18 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 56, normalized size = 0.82 \[ -\frac {2 \, a^{3} x^{3} \log \left (a x - 1\right ) - 2 \, a^{3} x^{3} \log \relax (x) + 2 \, a^{2} x^{2} + a x + 6 \, {\rm Li}_2\left (a x\right ) - 2 \, \log \left (-a x + 1\right )}{18 \, x^{3}} \]
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 \[ \int \frac {{\rm Li}_2\left (a x\right )}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 60, normalized size = 0.88 \[ -\frac {\polylog \left (2, a x \right )}{3 x^{3}}-\frac {a}{18 x^{2}}+\frac {a^{3} \ln \left (-a x \right )}{9}-\frac {a^{2}}{9 x}-\frac {a^{3} \ln \left (-a x +1\right )}{9}+\frac {\ln \left (-a x +1\right )}{9 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 49, normalized size = 0.72 \[ \frac {1}{9} \, a^{3} \log \relax (x) - \frac {2 \, a^{2} x^{2} + a x + 2 \, {\left (a^{3} x^{3} - 1\right )} \log \left (-a x + 1\right ) + 6 \, {\rm Li}_2\left (a x\right )}{18 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.32, size = 57, normalized size = 0.84 \[ \frac {2\,a^3\,\ln \relax (x)}{9}-\frac {\frac {a\,x}{18}-\frac {\ln \left (1-a\,x\right )}{9}+\frac {\mathrm {polylog}\left (2,a\,x\right )}{3}+\frac {a^2\,x^2}{9}}{x^3}-\frac {a^3\,\ln \left (a\,x^2-x\right )}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.30, size = 51, normalized size = 0.75 \[ \frac {a^{3} \log {\relax (x )}}{9} + \frac {a^{3} \operatorname {Li}_{1}\left (a x\right )}{9} - \frac {a^{2}}{9 x} - \frac {a}{18 x^{2}} - \frac {\operatorname {Li}_{1}\left (a x\right )}{9 x^{3}} - \frac {\operatorname {Li}_{2}\left (a x\right )}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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