Optimal. Leaf size=57 \[ \frac{x^2}{2}+\frac{\log \left (x^4-\sqrt{3} x^2+1\right )}{4 \sqrt{3}}-\frac{\log \left (x^4+\sqrt{3} x^2+1\right )}{4 \sqrt{3}} \]
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Rubi [A] time = 0.0431245, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1359, 1122, 1164, 628} \[ \frac{x^2}{2}+\frac{\log \left (x^4-\sqrt{3} x^2+1\right )}{4 \sqrt{3}}-\frac{\log \left (x^4+\sqrt{3} x^2+1\right )}{4 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1359
Rule 1122
Rule 1164
Rule 628
Rubi steps
\begin{align*} \int \frac{x^9}{1-x^4+x^8} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^4}{1-x^2+x^4} \, dx,x,x^2\right )\\ &=\frac{x^2}{2}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1-x^2}{1-x^2+x^4} \, dx,x,x^2\right )\\ &=\frac{x^2}{2}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{3}+2 x}{-1-\sqrt{3} x-x^2} \, dx,x,x^2\right )}{4 \sqrt{3}}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{3}-2 x}{-1+\sqrt{3} x-x^2} \, dx,x,x^2\right )}{4 \sqrt{3}}\\ &=\frac{x^2}{2}+\frac{\log \left (1-\sqrt{3} x^2+x^4\right )}{4 \sqrt{3}}-\frac{\log \left (1+\sqrt{3} x^2+x^4\right )}{4 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0158855, size = 55, normalized size = 0.96 \[ \frac{1}{12} \left (6 x^2+\sqrt{3} \log \left (-x^4+\sqrt{3} x^2-1\right )-\sqrt{3} \log \left (x^4+\sqrt{3} x^2+1\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 44, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}}{2}}+{\frac{\ln \left ( 1+{x}^{4}-{x}^{2}\sqrt{3} \right ) \sqrt{3}}{12}}-{\frac{\ln \left ( 1+{x}^{4}+{x}^{2}\sqrt{3} \right ) \sqrt{3}}{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, x^{2} + \int \frac{{\left (x^{4} - 1\right )} x}{x^{8} - x^{4} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45492, size = 117, normalized size = 2.05 \begin{align*} \frac{1}{2} \, x^{2} + \frac{1}{12} \, \sqrt{3} \log \left (\frac{x^{8} + 5 \, x^{4} - 2 \, \sqrt{3}{\left (x^{6} + x^{2}\right )} + 1}{x^{8} - x^{4} + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.125956, size = 48, normalized size = 0.84 \begin{align*} \frac{x^{2}}{2} + \frac{\sqrt{3} \log{\left (x^{4} - \sqrt{3} x^{2} + 1 \right )}}{12} - \frac{\sqrt{3} \log{\left (x^{4} + \sqrt{3} x^{2} + 1 \right )}}{12} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.27482, size = 348, normalized size = 6.11 \begin{align*} \frac{1}{2} \, x^{2} - \frac{1}{48} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x + \sqrt{6} - \sqrt{2}}{\sqrt{6} + \sqrt{2}}\right ) - \frac{1}{48} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x - \sqrt{6} + \sqrt{2}}{\sqrt{6} + \sqrt{2}}\right ) - \frac{1}{48} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x + \sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) - \frac{1}{48} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x - \sqrt{6} - \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) - \frac{1}{96} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )} \log \left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{6} + \sqrt{2}\right )} + 1\right ) + \frac{1}{96} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )} \log \left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{6} + \sqrt{2}\right )} + 1\right ) - \frac{1}{96} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )} \log \left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 1\right ) + \frac{1}{96} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )} \log \left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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