Optimal. Leaf size=75 \[ \frac{1}{2} \sqrt{\frac{1}{10} \left (3+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x^2\right )-\frac{\tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right )}{\sqrt{10 \left (3+\sqrt{5}\right )}} \]
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Rubi [A] time = 0.058168, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {1359, 1093, 203} \[ \frac{1}{2} \sqrt{\frac{1}{10} \left (3+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x^2\right )-\frac{\tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right )}{\sqrt{10 \left (3+\sqrt{5}\right )}} \]
Antiderivative was successfully verified.
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Rule 1359
Rule 1093
Rule 203
Rubi steps
\begin{align*} \int \frac{x}{1+3 x^4+x^8} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1+3 x^2+x^4} \, dx,x,x^2\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{3}{2}-\frac{\sqrt{5}}{2}+x^2} \, dx,x,x^2\right )}{2 \sqrt{5}}-\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{3}{2}+\frac{\sqrt{5}}{2}+x^2} \, dx,x,x^2\right )}{2 \sqrt{5}}\\ &=-\frac{\tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right )}{\sqrt{10 \left (3+\sqrt{5}\right )}}+\frac{1}{2} \sqrt{\frac{1}{10} \left (3+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0383284, size = 74, normalized size = 0.99 \[ \frac{\tan ^{-1}\left (\sqrt{\frac{2}{3-\sqrt{5}}} x^2\right )}{\sqrt{10 \left (3-\sqrt{5}\right )}}-\frac{\tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right )}{\sqrt{10 \left (3+\sqrt{5}\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 60, normalized size = 0.8 \begin{align*}{\frac{2\,\sqrt{5}}{-10+10\,\sqrt{5}}\arctan \left ( 4\,{\frac{{x}^{2}}{-2+2\,\sqrt{5}}} \right ) }-{\frac{2\,\sqrt{5}}{10+10\,\sqrt{5}}\arctan \left ( 4\,{\frac{{x}^{2}}{2+2\,\sqrt{5}}} \right ) } \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*} \int \frac{x}{x^{8} + 3 \, x^{4} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.54003, size = 417, normalized size = 5.56 \begin{align*} \frac{1}{10} \, \sqrt{10} \sqrt{-\sqrt{5} + 3} \arctan \left (-\frac{1}{10} \, \sqrt{10} \sqrt{5} x^{2} \sqrt{-\sqrt{5} + 3} + \frac{1}{20} \, \sqrt{10} \sqrt{5} \sqrt{2} \sqrt{2 \, x^{4} + \sqrt{5} + 3} \sqrt{-\sqrt{5} + 3}\right ) - \frac{1}{10} \, \sqrt{10} \sqrt{\sqrt{5} + 3} \arctan \left (-\frac{1}{20} \,{\left (2 \, \sqrt{10} \sqrt{5} x^{2} - \sqrt{10} \sqrt{5} \sqrt{2} \sqrt{2 \, x^{4} - \sqrt{5} + 3}\right )} \sqrt{\sqrt{5} + 3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.194831, size = 49, normalized size = 0.65 \begin{align*} 2 \left (\frac{\sqrt{5}}{40} + \frac{1}{8}\right ) \operatorname{atan}{\left (\frac{2 x^{2}}{-1 + \sqrt{5}} \right )} - 2 \left (\frac{1}{8} - \frac{\sqrt{5}}{40}\right ) \operatorname{atan}{\left (\frac{2 x^{2}}{1 + \sqrt{5}} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30564, size = 55, normalized size = 0.73 \begin{align*} \frac{1}{20} \,{\left (\sqrt{5} - 5\right )} \arctan \left (\frac{2 \, x^{2}}{\sqrt{5} + 1}\right ) + \frac{1}{20} \,{\left (\sqrt{5} + 5\right )} \arctan \left (\frac{2 \, x^{2}}{\sqrt{5} - 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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