Optimal. Leaf size=78 \[ \frac {1}{16} a^4 \log (x)-\frac {1}{16} a^4 \log (1-a x)-\frac {a^3}{16 x}-\frac {a^2}{32 x^2}-\frac {\text {Li}_2(a x)}{4 x^4}+\frac {\log (1-a x)}{16 x^4}-\frac {a}{48 x^3} \]
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Rubi [A] time = 0.04, antiderivative size = 78, 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} \[ -\frac {\text {PolyLog}(2,a x)}{4 x^4}-\frac {a^2}{32 x^2}-\frac {a^3}{16 x}+\frac {1}{16} a^4 \log (x)-\frac {1}{16} a^4 \log (1-a x)-\frac {a}{48 x^3}+\frac {\log (1-a x)}{16 x^4} \]
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^5} \, dx &=-\frac {\text {Li}_2(a x)}{4 x^4}-\frac {1}{4} \int \frac {\log (1-a x)}{x^5} \, dx\\ &=\frac {\log (1-a x)}{16 x^4}-\frac {\text {Li}_2(a x)}{4 x^4}+\frac {1}{16} a \int \frac {1}{x^4 (1-a x)} \, dx\\ &=\frac {\log (1-a x)}{16 x^4}-\frac {\text {Li}_2(a x)}{4 x^4}+\frac {1}{16} a \int \left (\frac {1}{x^4}+\frac {a}{x^3}+\frac {a^2}{x^2}+\frac {a^3}{x}-\frac {a^4}{-1+a x}\right ) \, dx\\ &=-\frac {a}{48 x^3}-\frac {a^2}{32 x^2}-\frac {a^3}{16 x}+\frac {1}{16} a^4 \log (x)-\frac {1}{16} a^4 \log (1-a x)+\frac {\log (1-a x)}{16 x^4}-\frac {\text {Li}_2(a x)}{4 x^4}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 60, normalized size = 0.77 \[ -\frac {-6 a^4 x^4 \log (x)+6 \left (a^4 x^4-1\right ) \log (1-a x)+a x \left (6 a^2 x^2+3 a x+2\right )+24 \text {Li}_2(a x)}{96 x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 65, normalized size = 0.83 \[ -\frac {6 \, a^{4} x^{4} \log \left (a x - 1\right ) - 6 \, a^{4} x^{4} \log \relax (x) + 6 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + 2 \, a x + 24 \, {\rm Li}_2\left (a x\right ) - 6 \, \log \left (-a x + 1\right )}{96 \, x^{4}} \]
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 \[ \int \frac {{\rm Li}_2\left (a x\right )}{x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 68, normalized size = 0.87 \[ -\frac {\polylog \left (2, a x \right )}{4 x^{4}}-\frac {a^{2}}{32 x^{2}}+\frac {a^{4} \ln \left (-a x \right )}{16}-\frac {a}{48 x^{3}}-\frac {a^{3}}{16 x}-\frac {a^{4} \ln \left (-a x +1\right )}{16}+\frac {\ln \left (-a x +1\right )}{16 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 58, normalized size = 0.74 \[ \frac {1}{16} \, a^{4} \log \relax (x) - \frac {6 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + 2 \, a x + 6 \, {\left (a^{4} x^{4} - 1\right )} \log \left (-a x + 1\right ) + 24 \, {\rm Li}_2\left (a x\right )}{96 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.44, size = 60, normalized size = 0.77 \[ \frac {\ln \left (1-a\,x\right )}{16\,x^4}-\frac {\mathrm {polylog}\left (2,a\,x\right )}{4\,x^4}-\frac {a^3\,x^2+\frac {a^2\,x}{2}+\frac {a}{3}}{16\,x^3}-\frac {a^4\,\mathrm {atan}\left (a\,x\,2{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.53, size = 60, normalized size = 0.77 \[ \frac {a^{4} \log {\relax (x )}}{16} + \frac {a^{4} \operatorname {Li}_{1}\left (a x\right )}{16} - \frac {a^{3}}{16 x} - \frac {a^{2}}{32 x^{2}} - \frac {a}{48 x^{3}} - \frac {\operatorname {Li}_{1}\left (a x\right )}{16 x^{4}} - \frac {\operatorname {Li}_{2}\left (a x\right )}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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