3.273 \(\int \frac{x^7}{1+2 x^4+x^8} \, dx\)

Optimal. Leaf size=22 \[ \frac{1}{4 \left (x^4+1\right )}+\frac{1}{4} \log \left (x^4+1\right ) \]

[Out]

1/(4*(1 + x^4)) + Log[1 + x^4]/4

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Rubi [A]  time = 0.026405, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{1}{4 \left (x^4+1\right )}+\frac{1}{4} \log \left (x^4+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^7/(1 + 2*x^4 + x^8),x]

[Out]

1/(4*(1 + x^4)) + Log[1 + x^4]/4

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Rubi in Sympy [A]  time = 4.80556, size = 15, normalized size = 0.68 \[ \frac{\log{\left (x^{4} + 1 \right )}}{4} + \frac{1}{4 \left (x^{4} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7/(x**8+2*x**4+1),x)

[Out]

log(x**4 + 1)/4 + 1/(4*(x**4 + 1))

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Mathematica [A]  time = 0.00798902, size = 18, normalized size = 0.82 \[ \frac{1}{4} \left (\frac{1}{x^4+1}+\log \left (x^4+1\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^7/(1 + 2*x^4 + x^8),x]

[Out]

((1 + x^4)^(-1) + Log[1 + x^4])/4

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Maple [A]  time = 0.009, size = 19, normalized size = 0.9 \[{\frac{1}{4\,{x}^{4}+4}}+{\frac{\ln \left ({x}^{4}+1 \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(x^8+2*x^4+1),x)

[Out]

1/4/(x^4+1)+1/4*ln(x^4+1)

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Maxima [A]  time = 0.792124, size = 24, normalized size = 1.09 \[ \frac{1}{4 \,{\left (x^{4} + 1\right )}} + \frac{1}{4} \, \log \left (x^{4} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(x^8 + 2*x^4 + 1),x, algorithm="maxima")

[Out]

1/4/(x^4 + 1) + 1/4*log(x^4 + 1)

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Fricas [A]  time = 0.245339, size = 31, normalized size = 1.41 \[ \frac{{\left (x^{4} + 1\right )} \log \left (x^{4} + 1\right ) + 1}{4 \,{\left (x^{4} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(x^8 + 2*x^4 + 1),x, algorithm="fricas")

[Out]

1/4*((x^4 + 1)*log(x^4 + 1) + 1)/(x^4 + 1)

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Sympy [A]  time = 0.241917, size = 15, normalized size = 0.68 \[ \frac{\log{\left (x^{4} + 1 \right )}}{4} + \frac{1}{4 x^{4} + 4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7/(x**8+2*x**4+1),x)

[Out]

log(x**4 + 1)/4 + 1/(4*x**4 + 4)

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GIAC/XCAS [A]  time = 0.275785, size = 24, normalized size = 1.09 \[ \frac{1}{4 \,{\left (x^{4} + 1\right )}} + \frac{1}{4} \,{\rm ln}\left (x^{4} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(x^8 + 2*x^4 + 1),x, algorithm="giac")

[Out]

1/4/(x^4 + 1) + 1/4*ln(x^4 + 1)