3.328 \(\int \frac{x^{11}}{1+x^4+x^8} \, dx\)

Optimal. Leaf size=44 \[ \frac{x^4}{4}-\frac{\tan ^{-1}\left (\frac{2 x^4+1}{\sqrt{3}}\right )}{4 \sqrt{3}}-\frac{1}{8} \log \left (x^8+x^4+1\right ) \]

[Out]

x^4/4 - ArcTan[(1 + 2*x^4)/Sqrt[3]]/(4*Sqrt[3]) - Log[1 + x^4 + x^8]/8

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Rubi [A]  time = 0.0778663, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429 \[ \frac{x^4}{4}-\frac{\tan ^{-1}\left (\frac{2 x^4+1}{\sqrt{3}}\right )}{4 \sqrt{3}}-\frac{1}{8} \log \left (x^8+x^4+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^11/(1 + x^4 + x^8),x]

[Out]

x^4/4 - ArcTan[(1 + 2*x^4)/Sqrt[3]]/(4*Sqrt[3]) - Log[1 + x^4 + x^8]/8

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Rubi in Sympy [A]  time = 11.4976, size = 39, normalized size = 0.89 \[ \frac{x^{4}}{4} - \frac{\log{\left (x^{8} + x^{4} + 1 \right )}}{8} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x^{4}}{3} + \frac{1}{3}\right ) \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11/(x**8+x**4+1),x)

[Out]

x**4/4 - log(x**8 + x**4 + 1)/8 - sqrt(3)*atan(sqrt(3)*(2*x**4/3 + 1/3))/12

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Mathematica [A]  time = 0.0182896, size = 44, normalized size = 1. \[ \frac{x^4}{4}-\frac{\tan ^{-1}\left (\frac{2 x^4+1}{\sqrt{3}}\right )}{4 \sqrt{3}}-\frac{1}{8} \log \left (x^8+x^4+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^11/(1 + x^4 + x^8),x]

[Out]

x^4/4 - ArcTan[(1 + 2*x^4)/Sqrt[3]]/(4*Sqrt[3]) - Log[1 + x^4 + x^8]/8

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Maple [A]  time = 0.004, size = 36, normalized size = 0.8 \[{\frac{{x}^{4}}{4}}-{\frac{\ln \left ({x}^{8}+{x}^{4}+1 \right ) }{8}}-{\frac{\sqrt{3}}{12}\arctan \left ({\frac{ \left ( 2\,{x}^{4}+1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11/(x^8+x^4+1),x)

[Out]

1/4*x^4-1/8*ln(x^8+x^4+1)-1/12*arctan(1/3*(2*x^4+1)*3^(1/2))*3^(1/2)

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Maxima [A]  time = 0.822668, size = 47, normalized size = 1.07 \[ \frac{1}{4} \, x^{4} - \frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{4} + 1\right )}\right ) - \frac{1}{8} \, \log \left (x^{8} + x^{4} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(x^8 + x^4 + 1),x, algorithm="maxima")

[Out]

1/4*x^4 - 1/12*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^4 + 1)) - 1/8*log(x^8 + x^4 + 1)

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Fricas [A]  time = 0.25262, size = 58, normalized size = 1.32 \[ \frac{1}{24} \, \sqrt{3}{\left (2 \, \sqrt{3} x^{4} - \sqrt{3} \log \left (x^{8} + x^{4} + 1\right ) - 2 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{4} + 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(x^8 + x^4 + 1),x, algorithm="fricas")

[Out]

1/24*sqrt(3)*(2*sqrt(3)*x^4 - sqrt(3)*log(x^8 + x^4 + 1) - 2*arctan(1/3*sqrt(3)*
(2*x^4 + 1)))

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Sympy [A]  time = 0.332669, size = 42, normalized size = 0.95 \[ \frac{x^{4}}{4} - \frac{\log{\left (x^{8} + x^{4} + 1 \right )}}{8} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x^{4}}{3} + \frac{\sqrt{3}}{3} \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**11/(x**8+x**4+1),x)

[Out]

x**4/4 - log(x**8 + x**4 + 1)/8 - sqrt(3)*atan(2*sqrt(3)*x**4/3 + sqrt(3)/3)/12

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GIAC/XCAS [A]  time = 0.265038, size = 47, normalized size = 1.07 \[ \frac{1}{4} \, x^{4} - \frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{4} + 1\right )}\right ) - \frac{1}{8} \,{\rm ln}\left (x^{8} + x^{4} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(x^8 + x^4 + 1),x, algorithm="giac")

[Out]

1/4*x^4 - 1/12*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^4 + 1)) - 1/8*ln(x^8 + x^4 + 1)