3.1359 \(\int \frac{x^8}{1+x^6} \, dx\)

Optimal. Leaf size=16 \[ \frac{x^3}{3}-\frac{1}{3} \tan ^{-1}\left (x^3\right ) \]

[Out]

x^3/3 - ArcTan[x^3]/3

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Rubi [A]  time = 0.0214251, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{x^3}{3}-\frac{1}{3} \tan ^{-1}\left (x^3\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^8/(1 + x^6),x]

[Out]

x^3/3 - ArcTan[x^3]/3

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Rubi in Sympy [A]  time = 4.73217, size = 10, normalized size = 0.62 \[ \frac{x^{3}}{3} - \frac{\operatorname{atan}{\left (x^{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8/(x**6+1),x)

[Out]

x**3/3 - atan(x**3)/3

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Mathematica [A]  time = 0.00846163, size = 16, normalized size = 1. \[ \frac{x^3}{3}-\frac{1}{3} \tan ^{-1}\left (x^3\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^8/(1 + x^6),x]

[Out]

x^3/3 - ArcTan[x^3]/3

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Maple [A]  time = 0.003, size = 13, normalized size = 0.8 \[{\frac{{x}^{3}}{3}}-{\frac{\arctan \left ({x}^{3} \right ) }{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(x^6+1),x)

[Out]

1/3*x^3-1/3*arctan(x^3)

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Maxima [A]  time = 1.59426, size = 16, normalized size = 1. \[ \frac{1}{3} \, x^{3} - \frac{1}{3} \, \arctan \left (x^{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(x^6 + 1),x, algorithm="maxima")

[Out]

1/3*x^3 - 1/3*arctan(x^3)

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Fricas [A]  time = 0.211003, size = 16, normalized size = 1. \[ \frac{1}{3} \, x^{3} - \frac{1}{3} \, \arctan \left (x^{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(x^6 + 1),x, algorithm="fricas")

[Out]

1/3*x^3 - 1/3*arctan(x^3)

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Sympy [A]  time = 0.221464, size = 10, normalized size = 0.62 \[ \frac{x^{3}}{3} - \frac{\operatorname{atan}{\left (x^{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8/(x**6+1),x)

[Out]

x**3/3 - atan(x**3)/3

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GIAC/XCAS [A]  time = 0.228468, size = 16, normalized size = 1. \[ \frac{1}{3} \, x^{3} - \frac{1}{3} \, \arctan \left (x^{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(x^6 + 1),x, algorithm="giac")

[Out]

1/3*x^3 - 1/3*arctan(x^3)