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 = 0.0181356, antiderivative size = 11, normalized size of antiderivative = 1., 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 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]

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]

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*}

Mathematica [A]  time = 0.0013108, 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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Maple [A]  time = 0.002, size = 8, normalized size = 0.7 \begin{align*}{\frac{1}{2\, \left ( -1+x \right ) ^{2}}} \end{align*}

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/(-1+x)^2

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Maxima [A]  time = 0.936703, size = 16, normalized size = 1.45 \begin{align*} \frac{1}{2 \,{\left (x^{2} - 2 \, x + 1\right )}} \end{align*}

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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Fricas [A]  time = 1.66084, size = 28, normalized size = 2.55 \begin{align*} \frac{1}{2 \,{\left (x^{2} - 2 \, x + 1\right )}} \end{align*}

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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Sympy [A]  time = 0.096645, size = 10, normalized size = 0.91 \begin{align*} \frac{1}{2 x^{2} - 4 x + 2} \end{align*}

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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Giac [A]  time = 1.06814, size = 9, normalized size = 0.82 \begin{align*} \frac{1}{2 \,{\left (x - 1\right )}^{2}} \end{align*}

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