3.1491 \(\int \frac{x^3}{1+x^8} \, dx\)

Optimal. Leaf size=8 \[ \frac{1}{4} \tan ^{-1}\left (x^4\right ) \]

[Out]

ArcTan[x^4]/4

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Rubi [A]  time = 0.0144181, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{1}{4} \tan ^{-1}\left (x^4\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^3/(1 + x^8),x]

[Out]

ArcTan[x^4]/4

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Rubi in Sympy [A]  time = 2.56164, size = 5, normalized size = 0.62 \[ \frac{\operatorname{atan}{\left (x^{4} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(x**8+1),x)

[Out]

atan(x**4)/4

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Mathematica [A]  time = 0.00631838, size = 8, normalized size = 1. \[ \frac{1}{4} \tan ^{-1}\left (x^4\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(1 + x^8),x]

[Out]

ArcTan[x^4]/4

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Maple [A]  time = 0.001, size = 7, normalized size = 0.9 \[{\frac{\arctan \left ({x}^{4} \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(x^8+1),x)

[Out]

1/4*arctan(x^4)

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Maxima [A]  time = 1.58507, size = 8, normalized size = 1. \[ \frac{1}{4} \, \arctan \left (x^{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*arctan(x^4)

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Fricas [A]  time = 0.214333, size = 8, normalized size = 1. \[ \frac{1}{4} \, \arctan \left (x^{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*arctan(x^4)

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Sympy [A]  time = 0.238005, size = 5, normalized size = 0.62 \[ \frac{\operatorname{atan}{\left (x^{4} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(x**8+1),x)

[Out]

atan(x**4)/4

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GIAC/XCAS [A]  time = 0.242136, size = 8, normalized size = 1. \[ \frac{1}{4} \, \arctan \left (x^{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*arctan(x^4)