3.294 \(\int \frac{x^3}{1-2 x^4+x^8} \, dx\)

Optimal. Leaf size=13 \[ \frac{1}{4 \left (1-x^4\right )} \]

[Out]

1/(4*(1 - x^4))

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Rubi [A]  time = 0.00732921, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{1}{4 \left (1-x^4\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(1 - 2*x^4 + x^8),x]

[Out]

1/(4*(1 - x^4))

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Rubi in Sympy [A]  time = 2.85154, size = 7, normalized size = 0.54 \[ \frac{1}{4 \left (- x^{4} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(x**8-2*x**4+1),x)

[Out]

1/(4*(-x**4 + 1))

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Mathematica [A]  time = 0.00414282, size = 11, normalized size = 0.85 \[ -\frac{1}{4 \left (x^4-1\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(1 - 2*x^4 + x^8),x]

[Out]

-1/(4*(-1 + x^4))

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Maple [A]  time = 0.005, size = 10, normalized size = 0.8 \[ -{\frac{1}{4\,{x}^{4}-4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(x^8-2*x^4+1),x)

[Out]

-1/4/(x^4-1)

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Maxima [A]  time = 0.770972, size = 12, normalized size = 0.92 \[ -\frac{1}{4 \,{\left (x^{4} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(x^8 - 2*x^4 + 1),x, algorithm="maxima")

[Out]

-1/4/(x^4 - 1)

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Fricas [A]  time = 0.241493, size = 12, normalized size = 0.92 \[ -\frac{1}{4 \,{\left (x^{4} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(x^8 - 2*x^4 + 1),x, algorithm="fricas")

[Out]

-1/4/(x^4 - 1)

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Sympy [A]  time = 0.219153, size = 8, normalized size = 0.62 \[ - \frac{1}{4 x^{4} - 4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(x**8-2*x**4+1),x)

[Out]

-1/(4*x**4 - 4)

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GIAC/XCAS [A]  time = 0.274711, size = 12, normalized size = 0.92 \[ -\frac{1}{4 \,{\left (x^{4} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(x^8 - 2*x^4 + 1),x, algorithm="giac")

[Out]

-1/4/(x^4 - 1)