Optimal. Leaf size=49 \[ -\frac{1}{6} \log \left (1-x^2\right )+\frac{1}{12} \log \left (x^4+x^2+1\right )-\frac{\tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{2 \sqrt{3}} \]
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Rubi [A] time = 0.0367543, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {275, 292, 31, 634, 618, 204, 628} \[ -\frac{1}{6} \log \left (1-x^2\right )+\frac{1}{12} \log \left (x^4+x^2+1\right )-\frac{\tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{2 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 275
Rule 292
Rule 31
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x^3}{1-x^6} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{1-x^3} \, dx,x,x^2\right )\\ &=\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{1-x} \, dx,x,x^2\right )-\frac{1}{6} \operatorname{Subst}\left (\int \frac{1-x}{1+x+x^2} \, dx,x,x^2\right )\\ &=-\frac{1}{6} \log \left (1-x^2\right )+\frac{1}{12} \operatorname{Subst}\left (\int \frac{1+2 x}{1+x+x^2} \, dx,x,x^2\right )-\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{1+x+x^2} \, dx,x,x^2\right )\\ &=-\frac{1}{6} \log \left (1-x^2\right )+\frac{1}{12} \log \left (1+x^2+x^4\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 x^2\right )\\ &=-\frac{\tan ^{-1}\left (\frac{1+2 x^2}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{1}{6} \log \left (1-x^2\right )+\frac{1}{12} \log \left (1+x^2+x^4\right )\\ \end{align*}
Mathematica [A] time = 0.0082118, size = 73, normalized size = 1.49 \[ \frac{1}{12} \left (\log \left (x^2-x+1\right )+\log \left (x^2+x+1\right )-2 \log (1-x)-2 \log (x+1)-2 \sqrt{3} \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 66, normalized size = 1.4 \begin{align*} -{\frac{\ln \left ( -1+x \right ) }{6}}+{\frac{\ln \left ({x}^{2}+x+1 \right ) }{12}}+{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }+{\frac{\ln \left ({x}^{2}-x+1 \right ) }{12}}-{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }-{\frac{\ln \left ( 1+x \right ) }{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.51912, size = 51, normalized size = 1.04 \begin{align*} -\frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} + 1\right )}\right ) + \frac{1}{12} \, \log \left (x^{4} + x^{2} + 1\right ) - \frac{1}{6} \, \log \left (x^{2} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46309, size = 123, normalized size = 2.51 \begin{align*} -\frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} + 1\right )}\right ) + \frac{1}{12} \, \log \left (x^{4} + x^{2} + 1\right ) - \frac{1}{6} \, \log \left (x^{2} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.133131, size = 46, normalized size = 0.94 \begin{align*} - \frac{\log{\left (x^{2} - 1 \right )}}{6} + \frac{\log{\left (x^{4} + x^{2} + 1 \right )}}{12} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x^{2}}{3} + \frac{\sqrt{3}}{3} \right )}}{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20975, size = 53, normalized size = 1.08 \begin{align*} -\frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} + 1\right )}\right ) + \frac{1}{12} \, \log \left (x^{4} + x^{2} + 1\right ) - \frac{1}{6} \, \log \left ({\left | x^{2} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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