3.369 \(\int \frac{x^9}{1+3 x^4+x^8} \, dx\)

Optimal. Leaf size=90 \[ \frac{x^2}{2}-\frac{1}{2} \sqrt{\frac{1}{5} \left (9+4 \sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right )+\frac{1}{2} \sqrt{\frac{1}{5} \left (9-4 \sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x^2\right ) \]

[Out]

x^2/2 - (Sqrt[(9 + 4*Sqrt[5])/5]*ArcTan[Sqrt[2/(3 + Sqrt[5])]*x^2])/2 + (Sqrt[(9
- 4*Sqrt[5])/5]*ArcTan[Sqrt[(3 + Sqrt[5])/2]*x^2])/2

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Rubi [A]  time = 0.265382, antiderivative size = 90, 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 \[ \frac{x^2}{2}-\frac{1}{2} \sqrt{\frac{1}{5} \left (9+4 \sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right )+\frac{1}{2} \sqrt{\frac{1}{5} \left (9-4 \sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x^2\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^9/(1 + 3*x^4 + x^8),x]

[Out]

x^2/2 - (Sqrt[(9 + 4*Sqrt[5])/5]*ArcTan[Sqrt[2/(3 + Sqrt[5])]*x^2])/2 + (Sqrt[(9
- 4*Sqrt[5])/5]*ArcTan[Sqrt[(3 + Sqrt[5])/2]*x^2])/2

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Rubi in Sympy [A]  time = 20.1815, size = 102, normalized size = 1.13 \[ \frac{x^{2}}{2} - \frac{\sqrt{2} \left (- \frac{7 \sqrt{5}}{10} + \frac{3}{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} x^{2}}{\sqrt{- \sqrt{5} + 3}} \right )}}{2 \sqrt{- \sqrt{5} + 3}} - \frac{\sqrt{2} \left (\frac{3}{2} + \frac{7 \sqrt{5}}{10}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} x^{2}}{\sqrt{\sqrt{5} + 3}} \right )}}{2 \sqrt{\sqrt{5} + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9/(x**8+3*x**4+1),x)

[Out]

x**2/2 - sqrt(2)*(-7*sqrt(5)/10 + 3/2)*atan(sqrt(2)*x**2/sqrt(-sqrt(5) + 3))/(2*
sqrt(-sqrt(5) + 3)) - sqrt(2)*(3/2 + 7*sqrt(5)/10)*atan(sqrt(2)*x**2/sqrt(sqrt(5
) + 3))/(2*sqrt(sqrt(5) + 3))

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Mathematica [A]  time = 0.264443, size = 97, normalized size = 1.08 \[ \frac{1}{40} \left (20 x^2-\sqrt{6-2 \sqrt{5}} \left (15+7 \sqrt{5}\right ) \tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right )+\sqrt{2 \left (3+\sqrt{5}\right )} \left (7 \sqrt{5}-15\right ) \tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x^2\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^9/(1 + 3*x^4 + x^8),x]

[Out]

(20*x^2 - Sqrt[6 - 2*Sqrt[5]]*(15 + 7*Sqrt[5])*ArcTan[Sqrt[2/(3 + Sqrt[5])]*x^2]
+ Sqrt[2*(3 + Sqrt[5])]*(-15 + 7*Sqrt[5])*ArcTan[Sqrt[(3 + Sqrt[5])/2]*x^2])/40

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Maple [B]  time = 0.044, size = 117, normalized size = 1.3 \[{\frac{{x}^{2}}{2}}-{\frac{7\,\sqrt{5}}{10+10\,\sqrt{5}}\arctan \left ( 4\,{\frac{{x}^{2}}{2\,\sqrt{5}+2}} \right ) }-3\,{\frac{1}{2\,\sqrt{5}+2}\arctan \left ( 4\,{\frac{{x}^{2}}{2\,\sqrt{5}+2}} \right ) }+{\frac{7\,\sqrt{5}}{-10+10\,\sqrt{5}}\arctan \left ( 4\,{\frac{{x}^{2}}{-2+2\,\sqrt{5}}} \right ) }-3\,{\frac{1}{-2+2\,\sqrt{5}}\arctan \left ( 4\,{\frac{{x}^{2}}{-2+2\,\sqrt{5}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9/(x^8+3*x^4+1),x)

[Out]

1/2*x^2-7/5*5^(1/2)/(2*5^(1/2)+2)*arctan(4*x^2/(2*5^(1/2)+2))-3/(2*5^(1/2)+2)*ar
ctan(4*x^2/(2*5^(1/2)+2))+7/5*5^(1/2)/(-2+2*5^(1/2))*arctan(4*x^2/(-2+2*5^(1/2))
)-3/(-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. \[ \frac{1}{2} \, x^{2} - \int \frac{{\left (3 \, x^{4} + 1\right )} x}{x^{8} + 3 \, x^{4} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(x^8 + 3*x^4 + 1),x, algorithm="maxima")

[Out]

1/2*x^2 - integrate((3*x^4 + 1)*x/(x^8 + 3*x^4 + 1), x)

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Fricas [A]  time = 0.265717, size = 203, normalized size = 2.26 \[ \frac{1}{2} \, x^{2} - \frac{1}{5} \, \sqrt{\sqrt{5}{\left (9 \, \sqrt{5} - 20\right )}} \arctan \left (\frac{\sqrt{\sqrt{5}{\left (9 \, \sqrt{5} - 20\right )}}{\left (\sqrt{5} + 3\right )}}{2 \,{\left (\sqrt{5} x^{2} + \sqrt{5} \sqrt{\frac{1}{10}} \sqrt{\sqrt{5}{\left (\sqrt{5}{\left (2 \, x^{4} + 3\right )} - 5\right )}}\right )}}\right ) - \frac{1}{5} \, \sqrt{\sqrt{5}{\left (9 \, \sqrt{5} + 20\right )}} \arctan \left (\frac{\sqrt{\sqrt{5}{\left (9 \, \sqrt{5} + 20\right )}}{\left (\sqrt{5} - 3\right )}}{2 \,{\left (\sqrt{5} x^{2} + \sqrt{5} \sqrt{\frac{1}{10}} \sqrt{\sqrt{5}{\left (\sqrt{5}{\left (2 \, x^{4} + 3\right )} + 5\right )}}\right )}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(x^8 + 3*x^4 + 1),x, algorithm="fricas")

[Out]

1/2*x^2 - 1/5*sqrt(sqrt(5)*(9*sqrt(5) - 20))*arctan(1/2*sqrt(sqrt(5)*(9*sqrt(5)
- 20))*(sqrt(5) + 3)/(sqrt(5)*x^2 + sqrt(5)*sqrt(1/10)*sqrt(sqrt(5)*(sqrt(5)*(2*
x^4 + 3) - 5)))) - 1/5*sqrt(sqrt(5)*(9*sqrt(5) + 20))*arctan(1/2*sqrt(sqrt(5)*(9
*sqrt(5) + 20))*(sqrt(5) - 3)/(sqrt(5)*x^2 + sqrt(5)*sqrt(1/10)*sqrt(sqrt(5)*(sq
rt(5)*(2*x^4 + 3) + 5))))

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Sympy [A]  time = 0.617601, size = 54, normalized size = 0.6 \[ \frac{x^{2}}{2} + 2 \left (- \frac{\sqrt{5}}{10} + \frac{1}{4}\right ) \operatorname{atan}{\left (\frac{2 x^{2}}{-1 + \sqrt{5}} \right )} - 2 \left (\frac{\sqrt{5}}{10} + \frac{1}{4}\right ) \operatorname{atan}{\left (\frac{2 x^{2}}{1 + \sqrt{5}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**9/(x**8+3*x**4+1),x)

[Out]

x**2/2 + 2*(-sqrt(5)/10 + 1/4)*atan(2*x**2/(-1 + sqrt(5))) - 2*(sqrt(5)/10 + 1/4
)*atan(2*x**2/(1 + sqrt(5)))

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GIAC/XCAS [A]  time = 0.281519, size = 89, normalized size = 0.99 \[ \frac{1}{2} \, x^{2} - \frac{1}{20} \,{\left (3 \, x^{4}{\left (\sqrt{5} - 5\right )} + \sqrt{5} - 5\right )} \arctan \left (\frac{2 \, x^{2}}{\sqrt{5} + 1}\right ) - \frac{1}{20} \,{\left (3 \, x^{4}{\left (\sqrt{5} + 5\right )} + \sqrt{5} + 5\right )} \arctan \left (\frac{2 \, x^{2}}{\sqrt{5} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(x^8 + 3*x^4 + 1),x, algorithm="giac")

[Out]

1/2*x^2 - 1/20*(3*x^4*(sqrt(5) - 5) + sqrt(5) - 5)*arctan(2*x^2/(sqrt(5) + 1)) -
1/20*(3*x^4*(sqrt(5) + 5) + sqrt(5) + 5)*arctan(2*x^2/(sqrt(5) - 1))