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

Optimal. Leaf size=34 \[ -\frac{(1-x) \left (x^3+x^2+x+1\right )^{-n} \left (1-x^4\right )^n}{n+1} \]

[Out]

-(((1 - x)*(1 - x^4)^n)/((1 + n)*(1 + x + x^2 + x^3)^n))

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Rubi [F]  time = 0.105439, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0. \[ \text{Int}\left (\left (1+x+x^2+x^3\right )^{-n} \left (1-x^4\right )^n,x\right ) \]

Verification is Not applicable to the result.

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

[Out]

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

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Rubi in Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: GeneratorsError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: GeneratorsError

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Mathematica [A]  time = 0.0238496, size = 31, normalized size = 0.91 \[ \frac{(x-1) \left (x^3+x^2+x+1\right )^{-n} \left (1-x^4\right )^n}{n+1} \]

Antiderivative was successfully verified.

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

[Out]

((-1 + x)*(1 - x^4)^n)/((1 + n)*(1 + x + x^2 + x^3)^n)

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Maple [A]  time = 0.003, size = 32, normalized size = 0.9 \[{\frac{ \left ( -1+x \right ) \left ( -{x}^{4}+1 \right ) ^{n}}{ \left ( 1+n \right ) \left ({x}^{3}+{x}^{2}+x+1 \right ) ^{n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-1+x)/(1+n)*(-x^4+1)^n/((x^3+x^2+x+1)^n)

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Maxima [A]  time = 0.776766, size = 22, normalized size = 0.65 \[ \frac{{\left (x - 1\right )}{\left (-x + 1\right )}^{n}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x - 1)*(-x + 1)^n/(n + 1)

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Fricas [A]  time = 0.29238, size = 42, normalized size = 1.24 \[ \frac{{\left (-x^{4} + 1\right )}^{n}{\left (x - 1\right )}}{{\left (x^{3} + x^{2} + x + 1\right )}^{n}{\left (n + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-x^4 + 1)^n*(x - 1)/((x^3 + x^2 + x + 1)^n*(n + 1))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.254906, size = 39, normalized size = 1.15 \[ \frac{x e^{\left (n{\rm ln}\left (-x + 1\right )\right )} - e^{\left (n{\rm ln}\left (-x + 1\right )\right )}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x*e^(n*ln(-x + 1)) - e^(n*ln(-x + 1)))/(n + 1)