Optimal. Leaf size=78 \[ -\frac {1}{16} a^2 \log \left (1-a x^2\right )+\frac {1}{8} a^2 \log (x)-\frac {\text {Li}_2\left (a x^2\right )}{8 x^4}-\frac {\text {Li}_3\left (a x^2\right )}{4 x^4}-\frac {a}{16 x^2}+\frac {\log \left (1-a x^2\right )}{16 x^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 78, 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} \[ -\frac {\text {PolyLog}\left (2,a x^2\right )}{8 x^4}-\frac {\text {PolyLog}\left (3,a x^2\right )}{4 x^4}-\frac {1}{16} a^2 \log \left (1-a x^2\right )+\frac {1}{8} a^2 \log (x)-\frac {a}{16 x^2}+\frac {\log \left (1-a x^2\right )}{16 x^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 44
Rule 2395
Rule 2454
Rule 6591
Rubi steps
\begin {align*} \int \frac {\text {Li}_3\left (a x^2\right )}{x^5} \, dx &=-\frac {\text {Li}_3\left (a x^2\right )}{4 x^4}+\frac {1}{2} \int \frac {\text {Li}_2\left (a x^2\right )}{x^5} \, dx\\ &=-\frac {\text {Li}_2\left (a x^2\right )}{8 x^4}-\frac {\text {Li}_3\left (a x^2\right )}{4 x^4}-\frac {1}{4} \int \frac {\log \left (1-a x^2\right )}{x^5} \, dx\\ &=-\frac {\text {Li}_2\left (a x^2\right )}{8 x^4}-\frac {\text {Li}_3\left (a x^2\right )}{4 x^4}-\frac {1}{8} \operatorname {Subst}\left (\int \frac {\log (1-a x)}{x^3} \, dx,x,x^2\right )\\ &=\frac {\log \left (1-a x^2\right )}{16 x^4}-\frac {\text {Li}_2\left (a x^2\right )}{8 x^4}-\frac {\text {Li}_3\left (a x^2\right )}{4 x^4}+\frac {1}{16} a \operatorname {Subst}\left (\int \frac {1}{x^2 (1-a x)} \, dx,x,x^2\right )\\ &=\frac {\log \left (1-a x^2\right )}{16 x^4}-\frac {\text {Li}_2\left (a x^2\right )}{8 x^4}-\frac {\text {Li}_3\left (a x^2\right )}{4 x^4}+\frac {1}{16} a \operatorname {Subst}\left (\int \left (\frac {1}{x^2}+\frac {a}{x}-\frac {a^2}{-1+a x}\right ) \, dx,x,x^2\right )\\ &=-\frac {a}{16 x^2}+\frac {1}{8} a^2 \log (x)-\frac {1}{16} a^2 \log \left (1-a x^2\right )+\frac {\log \left (1-a x^2\right )}{16 x^4}-\frac {\text {Li}_2\left (a x^2\right )}{8 x^4}-\frac {\text {Li}_3\left (a x^2\right )}{4 x^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.01, size = 30, normalized size = 0.38 \[ \frac {G_{5,5}^{2,4}\left (-a x^2|\begin {array}{c} 1,1,1,1,3 \\ 1,2,0,0,0 \\\end {array}\right )}{2 x^4} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [C] time = 0.57, size = 64, normalized size = 0.82 \[ -\frac {a^{2} x^{4} \log \left (a x^{2} - 1\right ) - 2 \, a^{2} x^{4} \log \relax (x) + a x^{2} + 2 \, {\rm Li}_2\left (a x^{2}\right ) - \log \left (-a x^{2} + 1\right ) + 4 \, {\rm polylog}\left (3, a x^{2}\right )}{16 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm Li}_{3}(a x^{2})}{x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 98, normalized size = 1.26 \[ -\frac {a^{2} \left (-\frac {81 a \,x^{2}+378}{432 a \,x^{2}}-\frac {\left (-27 a^{2} x^{4}+27\right ) \ln \left (-a \,x^{2}+1\right )}{216 a^{2} x^{4}}+\frac {\polylog \left (2, a \,x^{2}\right )}{4 a^{2} x^{4}}+\frac {\polylog \left (3, a \,x^{2}\right )}{2 a^{2} x^{4}}+\frac {3}{16}-\frac {\ln \relax (x )}{4}-\frac {\ln \left (-a \right )}{8}+\frac {1}{a \,x^{2}}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.34, size = 55, normalized size = 0.71 \[ \frac {1}{8} \, a^{2} \log \relax (x) - \frac {a x^{2} + {\left (a^{2} x^{4} - 1\right )} \log \left (-a x^{2} + 1\right ) + 2 \, {\rm Li}_2\left (a x^{2}\right ) + 4 \, {\rm Li}_{3}(a x^{2})}{16 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.73, size = 65, normalized size = 0.83 \[ \frac {a^2\,\ln \relax (x)}{8}-\frac {\mathrm {polylog}\left (2,a\,x^2\right )}{8\,x^4}-\frac {\mathrm {polylog}\left (3,a\,x^2\right )}{4\,x^4}-\frac {a^2\,\ln \left (a\,x^2-1\right )}{16}-\frac {a}{16\,x^2}+\frac {\ln \left (1-a\,x^2\right )}{16\,x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {Li}_{3}\left (a x^{2}\right )}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________