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

Optimal. Leaf size=11 \[ \frac {1}{3 (1-x)^3} \]

[Out]

1/3/(1-x)^3

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Rubi [A]  time = 0.02, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1586, 32} \[ \frac {1}{3 (1-x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

1/(3*(1 - x)^3)

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 9, normalized size = 0.82 \[ -\frac {1}{3 (x-1)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*1/(-1 + x)^3

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fricas [B]  time = 0.39, size = 17, normalized size = 1.55 \[ -\frac {1}{3 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3/(x^3 - 3*x^2 + 3*x - 1)

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giac [A]  time = 0.16, size = 7, normalized size = 0.64 \[ -\frac {1}{3 \, {\left (x - 1\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3/(x - 1)^3

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maple [A]  time = 0.04, size = 8, normalized size = 0.73 \[ -\frac {1}{3 \left (x -1\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3/(x-1)^3

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maxima [B]  time = 1.32, size = 17, normalized size = 1.55 \[ -\frac {1}{3 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3/(x^3 - 3*x^2 + 3*x - 1)

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mupad [B]  time = 4.81, size = 7, normalized size = 0.64 \[ -\frac {1}{3\,{\left (x-1\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(3*(x - 1)^3)

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sympy [B]  time = 0.15, size = 17, normalized size = 1.55 \[ - \frac {1}{3 x^{3} - 9 x^{2} + 9 x - 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(3*x**3 - 9*x**2 + 9*x - 3)

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