Optimal. Leaf size=88 \[ -\frac {1}{54} a^3 \log \left (1-a x^2\right )+\frac {1}{27} a^3 \log (x)-\frac {a^2}{54 x^2}-\frac {\text {Li}_2\left (a x^2\right )}{18 x^6}-\frac {\text {Li}_3\left (a x^2\right )}{6 x^6}-\frac {a}{108 x^4}+\frac {\log \left (1-a x^2\right )}{54 x^6} \]
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Rubi [A] time = 0.07, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6591, 2454, 2395, 44} \[ -\frac {\text {PolyLog}\left (2,a x^2\right )}{18 x^6}-\frac {\text {PolyLog}\left (3,a x^2\right )}{6 x^6}-\frac {a^2}{54 x^2}-\frac {1}{54} a^3 \log \left (1-a x^2\right )+\frac {1}{27} a^3 \log (x)-\frac {a}{108 x^4}+\frac {\log \left (1-a x^2\right )}{54 x^6} \]
Antiderivative was successfully verified.
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Rule 44
Rule 2395
Rule 2454
Rule 6591
Rubi steps
\begin {align*} \int \frac {\text {Li}_3\left (a x^2\right )}{x^7} \, dx &=-\frac {\text {Li}_3\left (a x^2\right )}{6 x^6}+\frac {1}{3} \int \frac {\text {Li}_2\left (a x^2\right )}{x^7} \, dx\\ &=-\frac {\text {Li}_2\left (a x^2\right )}{18 x^6}-\frac {\text {Li}_3\left (a x^2\right )}{6 x^6}-\frac {1}{9} \int \frac {\log \left (1-a x^2\right )}{x^7} \, dx\\ &=-\frac {\text {Li}_2\left (a x^2\right )}{18 x^6}-\frac {\text {Li}_3\left (a x^2\right )}{6 x^6}-\frac {1}{18} \operatorname {Subst}\left (\int \frac {\log (1-a x)}{x^4} \, dx,x,x^2\right )\\ &=\frac {\log \left (1-a x^2\right )}{54 x^6}-\frac {\text {Li}_2\left (a x^2\right )}{18 x^6}-\frac {\text {Li}_3\left (a x^2\right )}{6 x^6}+\frac {1}{54} a \operatorname {Subst}\left (\int \frac {1}{x^3 (1-a x)} \, dx,x,x^2\right )\\ &=\frac {\log \left (1-a x^2\right )}{54 x^6}-\frac {\text {Li}_2\left (a x^2\right )}{18 x^6}-\frac {\text {Li}_3\left (a x^2\right )}{6 x^6}+\frac {1}{54} a \operatorname {Subst}\left (\int \left (\frac {1}{x^3}+\frac {a}{x^2}+\frac {a^2}{x}-\frac {a^3}{-1+a x}\right ) \, dx,x,x^2\right )\\ &=-\frac {a}{108 x^4}-\frac {a^2}{54 x^2}+\frac {1}{27} a^3 \log (x)-\frac {1}{54} a^3 \log \left (1-a x^2\right )+\frac {\log \left (1-a x^2\right )}{54 x^6}-\frac {\text {Li}_2\left (a x^2\right )}{18 x^6}-\frac {\text {Li}_3\left (a x^2\right )}{6 x^6}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 30, normalized size = 0.34 \[ \frac {G_{5,5}^{2,4}\left (-a x^2|\begin {array}{c} 1,1,1,1,4 \\ 1,3,0,0,0 \\\end {array}\right )}{2 x^6} \]
Warning: Unable to verify antiderivative.
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fricas [C] time = 0.58, size = 73, normalized size = 0.83 \[ -\frac {2 \, a^{3} x^{6} \log \left (a x^{2} - 1\right ) - 4 \, a^{3} x^{6} \log \relax (x) + 2 \, a^{2} x^{4} + a x^{2} + 6 \, {\rm Li}_2\left (a x^{2}\right ) - 2 \, \log \left (-a x^{2} + 1\right ) + 18 \, {\rm polylog}\left (3, a x^{2}\right )}{108 \, x^{6}} \]
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 \[ \int \frac {{\rm Li}_{3}(a x^{2})}{x^{7}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 115, normalized size = 1.31 \[ \frac {a^{3} \left (\frac {64 a^{2} x^{4}+152 a \,x^{2}+832}{1728 a^{2} x^{4}}+\frac {\left (-64 a^{3} x^{6}+64\right ) \ln \left (-a \,x^{2}+1\right )}{1728 a^{3} x^{6}}-\frac {\polylog \left (2, a \,x^{2}\right )}{9 a^{3} x^{6}}-\frac {\polylog \left (3, a \,x^{2}\right )}{3 a^{3} x^{6}}-\frac {1}{27}+\frac {2 \ln \relax (x )}{27}+\frac {\ln \left (-a \right )}{27}-\frac {1}{2 x^{4} a^{2}}-\frac {1}{8 a \,x^{2}}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 64, normalized size = 0.73 \[ \frac {1}{27} \, a^{3} \log \relax (x) - \frac {2 \, a^{2} x^{4} + a x^{2} + 2 \, {\left (a^{3} x^{6} - 1\right )} \log \left (-a x^{2} + 1\right ) + 6 \, {\rm Li}_2\left (a x^{2}\right ) + 18 \, {\rm Li}_{3}(a x^{2})}{108 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.02, size = 73, normalized size = 0.83 \[ \frac {a^3\,\ln \relax (x)}{27}-\frac {\mathrm {polylog}\left (2,a\,x^2\right )}{18\,x^6}-\frac {\mathrm {polylog}\left (3,a\,x^2\right )}{6\,x^6}-\frac {a^3\,\ln \left (a\,x^2-1\right )}{54}-\frac {a}{108\,x^4}+\frac {\ln \left (1-a\,x^2\right )}{54\,x^6}-\frac {a^2}{54\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {Li}_{3}\left (a x^{2}\right )}{x^{7}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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