3.274 \(\int \frac{x^5}{1+2 x^4+x^8} \, dx\)

Optimal. Leaf size=23 \[ \frac{1}{4} \tan ^{-1}\left (x^2\right )-\frac{x^2}{4 \left (x^4+1\right )} \]

[Out]

-x^2/(4*(1 + x^4)) + ArcTan[x^2]/4

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Rubi [A]  time = 0.0267787, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{1}{4} \tan ^{-1}\left (x^2\right )-\frac{x^2}{4 \left (x^4+1\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(1 + 2*x^4 + x^8),x]

[Out]

-x^2/(4*(1 + x^4)) + ArcTan[x^2]/4

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Rubi in Sympy [A]  time = 5.84579, size = 15, normalized size = 0.65 \[ - \frac{x^{2}}{4 \left (x^{4} + 1\right )} + \frac{\operatorname{atan}{\left (x^{2} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(x**8+2*x**4+1),x)

[Out]

-x**2/(4*(x**4 + 1)) + atan(x**2)/4

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Mathematica [A]  time = 0.0138274, size = 23, normalized size = 1. \[ \frac{1}{4} \tan ^{-1}\left (x^2\right )-\frac{x^2}{4 \left (x^4+1\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(1 + 2*x^4 + x^8),x]

[Out]

-x^2/(4*(1 + x^4)) + ArcTan[x^2]/4

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Maple [A]  time = 0.009, size = 20, normalized size = 0.9 \[ -{\frac{{x}^{2}}{4\,{x}^{4}+4}}+{\frac{\arctan \left ({x}^{2} \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(x^8+2*x^4+1),x)

[Out]

-1/4*x^2/(x^4+1)+1/4*arctan(x^2)

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Maxima [A]  time = 0.844045, size = 26, normalized size = 1.13 \[ -\frac{x^{2}}{4 \,{\left (x^{4} + 1\right )}} + \frac{1}{4} \, \arctan \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(x^8 + 2*x^4 + 1),x, algorithm="maxima")

[Out]

-1/4*x^2/(x^4 + 1) + 1/4*arctan(x^2)

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Fricas [A]  time = 0.245134, size = 32, normalized size = 1.39 \[ -\frac{x^{2} -{\left (x^{4} + 1\right )} \arctan \left (x^{2}\right )}{4 \,{\left (x^{4} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(x^8 + 2*x^4 + 1),x, algorithm="fricas")

[Out]

-1/4*(x^2 - (x^4 + 1)*arctan(x^2))/(x^4 + 1)

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Sympy [A]  time = 0.322669, size = 15, normalized size = 0.65 \[ - \frac{x^{2}}{4 x^{4} + 4} + \frac{\operatorname{atan}{\left (x^{2} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(x**8+2*x**4+1),x)

[Out]

-x**2/(4*x**4 + 4) + atan(x**2)/4

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GIAC/XCAS [A]  time = 0.288737, size = 26, normalized size = 1.13 \[ -\frac{x^{2}}{4 \,{\left (x^{4} + 1\right )}} + \frac{1}{4} \, \arctan \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(x^8 + 2*x^4 + 1),x, algorithm="giac")

[Out]

-1/4*x^2/(x^4 + 1) + 1/4*arctan(x^2)