3.1023 \(\int (1+x+x^2+x^3)^{-n} (1-x^4)^n \, dx\)

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

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Rubi [F]  time = 0.06, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \left (1+x+x^2+x^3\right )^{-n} \left (1-x^4\right )^n \, dx \]

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]

Rubi steps

\begin {align*} \int \left (1+x+x^2+x^3\right )^{-n} \left (1-x^4\right )^n \, dx &=\int \left (1+x+x^2+x^3\right )^{-n} \left (1-x^4\right )^n \, dx\\ \end {align*}

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Mathematica [A]  time = 0.04, 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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fricas [A]  time = 0.62, size = 31, normalized size = 0.91 \[ \frac {{\left (-x^{4} + 1\right )}^{n} {\left (x - 1\right )}}{{\left (x^{3} + x^{2} + x + 1\right )}^{n} {\left (n + 1\right )}} \]

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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giac [B]  time = 0.46, size = 81, normalized size = 2.38 \[ \frac {\frac {x e^{\left (n \log \left (x^{3} + x^{2} + x + 1\right ) + n \log \left (-x + 1\right )\right )}}{{\left (x^{3} + x^{2} + x + 1\right )}^{n}} - \frac {e^{\left (n \log \left (x^{3} + x^{2} + x + 1\right ) + n \log \left (-x + 1\right )\right )}}{{\left (x^{3} + x^{2} + x + 1\right )}^{n}}}{n + 1} \]

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="giac")

[Out]

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

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maple [A]  time = 0.00, size = 32, normalized size = 0.94 \[ \frac {\left (x -1\right ) \left (-x^{4}+1\right )^{n} \left (x^{3}+x^{2}+x +1\right )^{-n}}{n +1} \]

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]

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

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maxima [A]  time = 1.09, size = 16, normalized size = 0.47 \[ \frac {{\left (x - 1\right )} {\left (-x + 1\right )}^{n}}{n + 1} \]

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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mupad [B]  time = 3.45, size = 31, normalized size = 0.91 \[ \frac {{\left (1-x^4\right )}^n\,\left (x-1\right )}{\left (n+1\right )\,{\left (x^3+x^2+x+1\right )}^n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 72.39, size = 73, normalized size = 2.15 \[ \begin {cases} \frac {x \left (1 - x^{4}\right )^{n}}{n \left (x^{3} + x^{2} + x + 1\right )^{n} + \left (x^{3} + x^{2} + x + 1\right )^{n}} - \frac {\left (1 - x^{4}\right )^{n}}{n \left (x^{3} + x^{2} + x + 1\right )^{n} + \left (x^{3} + x^{2} + x + 1\right )^{n}} & \text {for}\: n \neq -1 \\- \log {\left (x - 1 \right )} & \text {otherwise} \end {cases} \]

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]

Piecewise((x*(1 - x**4)**n/(n*(x**3 + x**2 + x + 1)**n + (x**3 + x**2 + x + 1)**n) - (1 - x**4)**n/(n*(x**3 +
x**2 + x + 1)**n + (x**3 + x**2 + x + 1)**n), Ne(n, -1)), (-log(x - 1), True))

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