Optimal. Leaf size=64 \[ -\frac{\text{PolyLog}\left (2,a x^2\right )}{4 x^4}-\frac{1}{8} a^2 \log \left (1-a x^2\right )+\frac{1}{4} a^2 \log (x)-\frac{a}{8 x^2}+\frac{\log \left (1-a x^2\right )}{8 x^4} \]
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Rubi [A] time = 0.0500673, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {6591, 2454, 2395, 44} \[ -\frac{\text{PolyLog}\left (2,a x^2\right )}{4 x^4}-\frac{1}{8} a^2 \log \left (1-a x^2\right )+\frac{1}{4} a^2 \log (x)-\frac{a}{8 x^2}+\frac{\log \left (1-a x^2\right )}{8 x^4} \]
Antiderivative was successfully verified.
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Rule 6591
Rule 2454
Rule 2395
Rule 44
Rubi steps
\begin{align*} \int \frac{\text{Li}_2\left (a x^2\right )}{x^5} \, dx &=-\frac{\text{Li}_2\left (a x^2\right )}{4 x^4}-\frac{1}{2} \int \frac{\log \left (1-a x^2\right )}{x^5} \, dx\\ &=-\frac{\text{Li}_2\left (a x^2\right )}{4 x^4}-\frac{1}{4} \operatorname{Subst}\left (\int \frac{\log (1-a x)}{x^3} \, dx,x,x^2\right )\\ &=\frac{\log \left (1-a x^2\right )}{8 x^4}-\frac{\text{Li}_2\left (a x^2\right )}{4 x^4}+\frac{1}{8} 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 )}{8 x^4}-\frac{\text{Li}_2\left (a x^2\right )}{4 x^4}+\frac{1}{8} 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}{8 x^2}+\frac{1}{4} a^2 \log (x)-\frac{1}{8} a^2 \log \left (1-a x^2\right )+\frac{\log \left (1-a x^2\right )}{8 x^4}-\frac{\text{Li}_2\left (a x^2\right )}{4 x^4}\\ \end{align*}
Mathematica [A] time = 0.0283499, size = 51, normalized size = 0.8 \[ -\frac{2 \text{PolyLog}\left (2,a x^2\right )-2 a^2 x^4 \log (x)+\left (a^2 x^4-1\right ) \log \left (1-a x^2\right )+a x^2}{8 x^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 54, normalized size = 0.8 \begin{align*} -{\frac{{\it polylog} \left ( 2,a{x}^{2} \right ) }{4\,{x}^{4}}}+{\frac{\ln \left ( -a{x}^{2}+1 \right ) }{8\,{x}^{4}}}-{\frac{a}{8\,{x}^{2}}}+{\frac{{a}^{2}\ln \left ( x \right ) }{4}}-{\frac{{a}^{2}\ln \left ( a{x}^{2}-1 \right ) }{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.987978, size = 62, normalized size = 0.97 \begin{align*} \frac{1}{4} \, a^{2} \log \left (x\right ) - \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 )}{8 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.64939, size = 131, normalized size = 2.05 \begin{align*} -\frac{a^{2} x^{4} \log \left (a x^{2} - 1\right ) - 2 \, a^{2} x^{4} \log \left (x\right ) + a x^{2} + 2 \,{\rm Li}_2\left (a x^{2}\right ) - \log \left (-a x^{2} + 1\right )}{8 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 14.9968, size = 49, normalized size = 0.77 \begin{align*} \frac{a^{2} \log{\left (x \right )}}{4} + \frac{a^{2} \operatorname{Li}_{1}\left (a x^{2}\right )}{8} - \frac{a}{8 x^{2}} - \frac{\operatorname{Li}_{1}\left (a x^{2}\right )}{8 x^{4}} - \frac{\operatorname{Li}_{2}\left (a x^{2}\right )}{4 x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Li}_2\left (a x^{2}\right )}{x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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