3.373 \(\int \frac{x}{1+3 x^4+x^8} \, dx\)

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]

-(ArcTan[Sqrt[2/(3 + Sqrt[5])]*x^2]/Sqrt[10*(3 + Sqrt[5])]) + (Sqrt[(3 + Sqrt[5]
)/10]*ArcTan[Sqrt[(3 + Sqrt[5])/2]*x^2])/2

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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]

-(ArcTan[Sqrt[2/(3 + Sqrt[5])]*x^2]/Sqrt[10*(3 + Sqrt[5])]) + (Sqrt[(3 + Sqrt[5]
)/10]*ArcTan[Sqrt[(3 + Sqrt[5])/2]*x^2])/2

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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]

sqrt(10)*atan(sqrt(2)*x**2/sqrt(-sqrt(5) + 3))/(10*sqrt(-sqrt(5) + 3)) - sqrt(10
)*atan(sqrt(2)*x**2/sqrt(sqrt(5) + 3))/(10*sqrt(sqrt(5) + 3))

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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]

ArcTan[Sqrt[2/(3 - Sqrt[5])]*x^2]/Sqrt[10*(3 - Sqrt[5])] - ArcTan[Sqrt[2/(3 + Sq
rt[5])]*x^2]/Sqrt[10*(3 + Sqrt[5])]

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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]

-2/5*5^(1/2)/(2*5^(1/2)+2)*arctan(4*x^2/(2*5^(1/2)+2))+2/5*5^(1/2)/(-2+2*5^(1/2)
)*arctan(4*x^2/(-2+2*5^(1/2)))

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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]

integrate(x/(x^8 + 3*x^4 + 1), x)

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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]

1/5*sqrt(1/2)*sqrt(sqrt(5)*(3*sqrt(5) - 5))*arctan(1/2*sqrt(1/2)*sqrt(sqrt(5)*(3
*sqrt(5) - 5))*(sqrt(5) + 3)/(sqrt(5)*x^2 + sqrt(5)*sqrt(1/10)*sqrt(sqrt(5)*(sqr
t(5)*(2*x^4 + 3) + 5)))) + 1/5*sqrt(1/2)*sqrt(sqrt(5)*(3*sqrt(5) + 5))*arctan(1/
2*sqrt(1/2)*sqrt(sqrt(5)*(3*sqrt(5) + 5))*(sqrt(5) - 3)/(sqrt(5)*x^2 + sqrt(5)*s
qrt(1/10)*sqrt(sqrt(5)*(sqrt(5)*(2*x^4 + 3) - 5))))

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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]

2*(sqrt(5)/40 + 1/8)*atan(2*x**2/(-1 + sqrt(5))) - 2*(-sqrt(5)/40 + 1/8)*atan(2*
x**2/(1 + sqrt(5)))

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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]

1/20*(sqrt(5) - 5)*arctan(2*x^2/(sqrt(5) + 1)) + 1/20*(sqrt(5) + 5)*arctan(2*x^2
/(sqrt(5) - 1))