Optimal. Leaf size=46 \[ \frac{x^4}{4}+\frac{1}{8} \log \left (x^8-x^4+1\right )+\frac{\tan ^{-1}\left (\frac{1-2 x^4}{\sqrt{3}}\right )}{4 \sqrt{3}} \]
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Rubi [A] time = 0.0401795, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {1357, 703, 634, 618, 204, 628} \[ \frac{x^4}{4}+\frac{1}{8} \log \left (x^8-x^4+1\right )+\frac{\tan ^{-1}\left (\frac{1-2 x^4}{\sqrt{3}}\right )}{4 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1357
Rule 703
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x^{11}}{1-x^4+x^8} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x^2}{1-x+x^2} \, dx,x,x^4\right )\\ &=\frac{x^4}{4}+\frac{1}{4} \operatorname{Subst}\left (\int \frac{-1+x}{1-x+x^2} \, dx,x,x^4\right )\\ &=\frac{x^4}{4}-\frac{1}{8} \operatorname{Subst}\left (\int \frac{1}{1-x+x^2} \, dx,x,x^4\right )+\frac{1}{8} \operatorname{Subst}\left (\int \frac{-1+2 x}{1-x+x^2} \, dx,x,x^4\right )\\ &=\frac{x^4}{4}+\frac{1}{8} \log \left (1-x^4+x^8\right )+\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+2 x^4\right )\\ &=\frac{x^4}{4}+\frac{\tan ^{-1}\left (\frac{1-2 x^4}{\sqrt{3}}\right )}{4 \sqrt{3}}+\frac{1}{8} \log \left (1-x^4+x^8\right )\\ \end{align*}
Mathematica [A] time = 0.0130415, size = 46, normalized size = 1. \[ \frac{x^4}{4}+\frac{1}{8} \log \left (x^8-x^4+1\right )-\frac{\tan ^{-1}\left (\frac{2 x^4-1}{\sqrt{3}}\right )}{4 \sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 38, normalized size = 0.8 \begin{align*}{\frac{{x}^{4}}{4}}+{\frac{\ln \left ({x}^{8}-{x}^{4}+1 \right ) }{8}}-{\frac{\sqrt{3}}{12}\arctan \left ({\frac{ \left ( 2\,{x}^{4}-1 \right ) \sqrt{3}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47832, size = 50, normalized size = 1.09 \begin{align*} \frac{1}{4} \, x^{4} - \frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{4} - 1\right )}\right ) + \frac{1}{8} \, \log \left (x^{8} - x^{4} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44576, size = 109, normalized size = 2.37 \begin{align*} \frac{1}{4} \, x^{4} - \frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{4} - 1\right )}\right ) + \frac{1}{8} \, \log \left (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 = 0.139121, size = 42, normalized size = 0.91 \begin{align*} \frac{x^{4}}{4} + \frac{\log{\left (x^{8} - x^{4} + 1 \right )}}{8} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x^{4}}{3} - \frac{\sqrt{3}}{3} \right )}}{12} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13786, size = 50, normalized size = 1.09 \begin{align*} \frac{1}{4} \, x^{4} - \frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{4} - 1\right )}\right ) + \frac{1}{8} \, \log \left (x^{8} - x^{4} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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