3.326 \(\int \frac{x^4}{16+x^{10}} \, dx\)

Optimal. Leaf size=12 \[ \frac{1}{20} \tan ^{-1}\left (\frac{x^5}{4}\right ) \]

[Out]

ArcTan[x^5/4]/20

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Rubi [A]  time = 0.0137458, antiderivative size = 12, 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 \[ \frac{1}{20} \tan ^{-1}\left (\frac{x^5}{4}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^4/(16 + x^10),x]

[Out]

ArcTan[x^5/4]/20

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Rubi in Sympy [A]  time = 1.23571, size = 7, normalized size = 0.58 \[ \frac{\operatorname{atan}{\left (\frac{x^{5}}{4} \right )}}{20} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(x**10+16),x)

[Out]

atan(x**5/4)/20

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Mathematica [A]  time = 0.00538723, size = 12, normalized size = 1. \[ \frac{1}{20} \tan ^{-1}\left (\frac{x^5}{4}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(16 + x^10),x]

[Out]

ArcTan[x^5/4]/20

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Maple [A]  time = 0.002, size = 9, normalized size = 0.8 \[{\frac{1}{20}\arctan \left ({\frac{{x}^{5}}{4}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(x^10+16),x)

[Out]

1/20*arctan(1/4*x^5)

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Maxima [A]  time = 1.52469, size = 11, normalized size = 0.92 \[ \frac{1}{20} \, \arctan \left (\frac{1}{4} \, x^{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(x^10 + 16),x, algorithm="maxima")

[Out]

1/20*arctan(1/4*x^5)

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Fricas [A]  time = 0.228837, size = 11, normalized size = 0.92 \[ \frac{1}{20} \, \arctan \left (\frac{1}{4} \, x^{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(x^10 + 16),x, algorithm="fricas")

[Out]

1/20*arctan(1/4*x^5)

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Sympy [A]  time = 0.141754, size = 7, normalized size = 0.58 \[ \frac{\operatorname{atan}{\left (\frac{x^{5}}{4} \right )}}{20} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(x**10+16),x)

[Out]

atan(x**5/4)/20

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GIAC/XCAS [A]  time = 0.214287, size = 11, normalized size = 0.92 \[ \frac{1}{20} \, \arctan \left (\frac{1}{4} \, x^{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(x^10 + 16),x, algorithm="giac")

[Out]

1/20*arctan(1/4*x^5)