3.184 \(\int \frac{\left (1+x+x^2+x^3\right )^3}{\left (1-x^4\right )^3} \, dx\)

Optimal. Leaf size=11 \[ \frac{1}{2 (1-x)^2} \]

[Out]

1/(2*(1 - x)^2)

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Rubi [A]  time = 0.0163131, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{1}{2 (1-x)^2} \]

Antiderivative was successfully verified.

[In]  Int[(1 + x + x^2 + x^3)^3/(1 - x^4)^3,x]

[Out]

1/(2*(1 - x)^2)

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Rubi in Sympy [A]  time = 4.29654, size = 7, normalized size = 0.64 \[ \frac{1}{2 \left (- x + 1\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(2*(-x + 1)**2)

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Mathematica [A]  time = 0.00271378, size = 9, normalized size = 0.82 \[ \frac{1}{2 (x-1)^2} \]

Antiderivative was successfully verified.

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

[Out]

1/(2*(-1 + x)^2)

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Maple [A]  time = 0.003, size = 8, normalized size = 0.7 \[{\frac{1}{2\, \left ( -1+x \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^3+x^2+x+1)^3/(-x^4+1)^3,x)

[Out]

1/2/(-1+x)^2

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Maxima [A]  time = 5.949, size = 16, normalized size = 1.45 \[ \frac{1}{2 \,{\left (x^{2} - 2 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2/(x^2 - 2*x + 1)

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Fricas [A]  time = 0.220694, size = 16, normalized size = 1.45 \[ \frac{1}{2 \,{\left (x^{2} - 2 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2/(x^2 - 2*x + 1)

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Sympy [A]  time = 0.11103, size = 10, normalized size = 0.91 \[ \frac{1}{2 x^{2} - 4 x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(2*x**2 - 4*x + 2)

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GIAC/XCAS [A]  time = 0.21115, size = 9, normalized size = 0.82 \[ \frac{1}{2 \,{\left (x - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2/(x - 1)^2