Optimal. Leaf size=75 \[ \frac{1}{2} \sqrt{\frac{1}{10} \left (3+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x^2\right )-\frac{\tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right )}{\sqrt{10 \left (3+\sqrt{5}\right )}} \]
[Out]
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Rubi [A] time = 0.103794, 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 \[ \frac{1}{2} \sqrt{\frac{1}{10} \left (3+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x^2\right )-\frac{\tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right )}{\sqrt{10 \left (3+\sqrt{5}\right )}} \]
Antiderivative was successfully verified.
[In] Int[x/(1 + 3*x^4 + x^8),x]
[Out]
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Rubi in Sympy [A] time = 7.37182, size = 73, normalized size = 0.97 \[ \frac{\sqrt{10} \operatorname{atan}{\left (\frac{\sqrt{2} x^{2}}{\sqrt{- \sqrt{5} + 3}} \right )}}{10 \sqrt{- \sqrt{5} + 3}} - \frac{\sqrt{10} \operatorname{atan}{\left (\frac{\sqrt{2} x^{2}}{\sqrt{\sqrt{5} + 3}} \right )}}{10 \sqrt{\sqrt{5} + 3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(x**8+3*x**4+1),x)
[Out]
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Mathematica [A] time = 0.0621362, 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.
[In] Integrate[x/(1 + 3*x^4 + x^8),x]
[Out]
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Maple [A] time = 0.018, size = 60, normalized size = 0.8 \[ -{\frac{2\,\sqrt{5}}{10+10\,\sqrt{5}}\arctan \left ( 4\,{\frac{{x}^{2}}{2\,\sqrt{5}+2}} \right ) }+{\frac{2\,\sqrt{5}}{-10+10\,\sqrt{5}}\arctan \left ( 4\,{\frac{{x}^{2}}{-2+2\,\sqrt{5}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(x^8+3*x^4+1),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{x^{8} + 3 \, x^{4} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(x^8 + 3*x^4 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.280995, size = 212, normalized size = 2.83 \[ \frac{1}{5} \, \sqrt{\frac{1}{2}} \sqrt{\sqrt{5}{\left (3 \, \sqrt{5} - 5\right )}} \arctan \left (\frac{\sqrt{\frac{1}{2}} \sqrt{\sqrt{5}{\left (3 \, \sqrt{5} - 5\right )}}{\left (\sqrt{5} + 3\right )}}{2 \,{\left (\sqrt{5} x^{2} + \sqrt{5} \sqrt{\frac{1}{10}} \sqrt{\sqrt{5}{\left (\sqrt{5}{\left (2 \, x^{4} + 3\right )} + 5\right )}}\right )}}\right ) + \frac{1}{5} \, \sqrt{\frac{1}{2}} \sqrt{\sqrt{5}{\left (3 \, \sqrt{5} + 5\right )}} \arctan \left (\frac{\sqrt{\frac{1}{2}} \sqrt{\sqrt{5}{\left (3 \, \sqrt{5} + 5\right )}}{\left (\sqrt{5} - 3\right )}}{2 \,{\left (\sqrt{5} x^{2} + \sqrt{5} \sqrt{\frac{1}{10}} \sqrt{\sqrt{5}{\left (\sqrt{5}{\left (2 \, x^{4} + 3\right )} - 5\right )}}\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(x^8 + 3*x^4 + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.593474, size = 49, normalized size = 0.65 \[ 2 \left (\frac{\sqrt{5}}{40} + \frac{1}{8}\right ) \operatorname{atan}{\left (\frac{2 x^{2}}{-1 + \sqrt{5}} \right )} - 2 \left (- \frac{\sqrt{5}}{40} + \frac{1}{8}\right ) \operatorname{atan}{\left (\frac{2 x^{2}}{1 + \sqrt{5}} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(x**8+3*x**4+1),x)
[Out]
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GIAC/XCAS [A] time = 0.284052, size = 55, normalized size = 0.73 \[ \frac{1}{20} \,{\left (\sqrt{5} - 5\right )} \arctan \left (\frac{2 \, x^{2}}{\sqrt{5} + 1}\right ) + \frac{1}{20} \,{\left (\sqrt{5} + 5\right )} \arctan \left (\frac{2 \, x^{2}}{\sqrt{5} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(x^8 + 3*x^4 + 1),x, algorithm="giac")
[Out]