Optimal. Leaf size=62 \[ \frac{x^4}{4}+\frac{1}{40} \left (15-7 \sqrt{5}\right ) \log \left (-2 x^4-\sqrt{5}+3\right )+\frac{1}{40} \left (15+7 \sqrt{5}\right ) \log \left (-2 x^4+\sqrt{5}+3\right ) \]
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Rubi [A] time = 0.045602, antiderivative size = 62, 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 = {1357, 703, 632, 31} \[ \frac{x^4}{4}+\frac{1}{40} \left (15-7 \sqrt{5}\right ) \log \left (-2 x^4-\sqrt{5}+3\right )+\frac{1}{40} \left (15+7 \sqrt{5}\right ) \log \left (-2 x^4+\sqrt{5}+3\right ) \]
Antiderivative was successfully verified.
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Rule 1357
Rule 703
Rule 632
Rule 31
Rubi steps
\begin{align*} \int \frac{x^{11}}{1-3 x^4+x^8} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x^2}{1-3 x+x^2} \, dx,x,x^4\right )\\ &=\frac{x^4}{4}+\frac{1}{4} \operatorname{Subst}\left (\int \frac{-1+3 x}{1-3 x+x^2} \, dx,x,x^4\right )\\ &=\frac{x^4}{4}+\frac{1}{40} \left (15-7 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{3}{2}+\frac{\sqrt{5}}{2}+x} \, dx,x,x^4\right )+\frac{1}{40} \left (15+7 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{3}{2}-\frac{\sqrt{5}}{2}+x} \, dx,x,x^4\right )\\ &=\frac{x^4}{4}+\frac{1}{40} \left (15-7 \sqrt{5}\right ) \log \left (3-\sqrt{5}-2 x^4\right )+\frac{1}{40} \left (15+7 \sqrt{5}\right ) \log \left (3+\sqrt{5}-2 x^4\right )\\ \end{align*}
Mathematica [A] time = 0.0328176, size = 56, normalized size = 0.9 \[ \frac{1}{40} \left (10 x^4+\left (15+7 \sqrt{5}\right ) \log \left (-2 x^4+\sqrt{5}+3\right )+\left (15-7 \sqrt{5}\right ) \log \left (2 x^4+\sqrt{5}-3\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 38, normalized size = 0.6 \begin{align*}{\frac{{x}^{4}}{4}}+{\frac{3\,\ln \left ({x}^{8}-3\,{x}^{4}+1 \right ) }{8}}-{\frac{7\,\sqrt{5}}{20}{\it Artanh} \left ({\frac{ \left ( 2\,{x}^{4}-3 \right ) \sqrt{5}}{5}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.46732, size = 68, normalized size = 1.1 \begin{align*} \frac{1}{4} \, x^{4} + \frac{7}{40} \, \sqrt{5} \log \left (\frac{2 \, x^{4} - \sqrt{5} - 3}{2 \, x^{4} + \sqrt{5} - 3}\right ) + \frac{3}{8} \, \log \left (x^{8} - 3 \, x^{4} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71304, size = 157, normalized size = 2.53 \begin{align*} \frac{1}{4} \, x^{4} + \frac{7}{40} \, \sqrt{5} \log \left (\frac{2 \, x^{8} - 6 \, x^{4} - \sqrt{5}{\left (2 \, x^{4} - 3\right )} + 7}{x^{8} - 3 \, x^{4} + 1}\right ) + \frac{3}{8} \, \log \left (x^{8} - 3 \, x^{4} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.136557, size = 58, normalized size = 0.94 \begin{align*} \frac{x^{4}}{4} + \left (\frac{3}{8} + \frac{7 \sqrt{5}}{40}\right ) \log{\left (x^{4} - \frac{3}{2} - \frac{\sqrt{5}}{2} \right )} + \left (\frac{3}{8} - \frac{7 \sqrt{5}}{40}\right ) \log{\left (x^{4} - \frac{3}{2} + \frac{\sqrt{5}}{2} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14956, size = 72, normalized size = 1.16 \begin{align*} \frac{1}{4} \, x^{4} + \frac{7}{40} \, \sqrt{5} \log \left (\frac{{\left | 2 \, x^{4} - \sqrt{5} - 3 \right |}}{{\left | 2 \, x^{4} + \sqrt{5} - 3 \right |}}\right ) + \frac{3}{8} \, \log \left ({\left | x^{8} - 3 \, x^{4} + 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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