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

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Rubi [F]  time = 0.0647157, 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., 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*}

Mathematica [A]  time = 0.0339632, 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., size = 32, normalized size = 0.9 \begin{align*}{\frac{ \left ( x-1 \right ) \left ( -{x}^{4}+1 \right ) ^{n}}{ \left ( 1+n \right ) \left ({x}^{3}+{x}^{2}+x+1 \right ) ^{n}}} \end{align*}

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

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Maxima [A]  time = 1.51408, size = 22, normalized size = 0.65 \begin{align*} \frac{{\left (x - 1\right )}{\left (-x + 1\right )}^{n}}{n + 1} \end{align*}

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 = 1.75946, size = 73, normalized size = 2.15 \begin{align*} \frac{{\left (-x^{4} + 1\right )}^{n}{\left (x - 1\right )}}{{\left (x^{3} + x^{2} + x + 1\right )}^{n}{\left (n + 1\right )}} \end{align*}

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 [A]  time = 169.789, size = 73, normalized size = 2.15 \begin{align*} \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} \end{align*}

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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Giac [B]  time = 1.12934, size = 109, normalized size = 3.21 \begin{align*} \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} \end{align*}

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)