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

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

[Out]

1/2/(1-x)^2

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Rubi [A]  time = 0.02, 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} \[ \frac {1}{2 (1-x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

1/(2*(1 - x)^2)

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

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Mathematica [A]  time = 0.00, 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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fricas [A]  time = 0.38, size = 12, normalized size = 1.09 \[ \frac {1}{2 \, {\left (x^{2} - 2 \, x + 1\right )}} \]

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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giac [A]  time = 0.16, 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]

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

[Out]

1/2/(x - 1)^2

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maple [A]  time = 0.04, size = 8, normalized size = 0.73 \[ \frac {1}{2 \left (x -1\right )^{2}} \]

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/(x-1)^2

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maxima [A]  time = 1.30, size = 12, normalized size = 1.09 \[ \frac {1}{2 \, {\left (x^{2} - 2 \, x + 1\right )}} \]

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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mupad [B]  time = 4.84, 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]

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

[Out]

1/(2*(x - 1)^2)

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sympy [A]  time = 0.21, 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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