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

Optimal. Leaf size=7 \[ \frac{1}{1-x} \]

[Out]

(1 - x)^(-1)

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Rubi [A]  time = 0.015811, antiderivative size = 7, 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} \[ \frac{1}{1-x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1 - x)^(-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]

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

Mathematica [A]  time = 0.0008222, size = 7, normalized size = 1. \[ -\frac{1}{x-1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-1 + x)^(-1)

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Maple [A]  time = 0.001, size = 8, normalized size = 1.1 \begin{align*} - \left ( -1+x \right ) ^{-1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(-1+x)

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Maxima [A]  time = 0.960074, size = 9, normalized size = 1.29 \begin{align*} -\frac{1}{x - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(x - 1)

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Fricas [A]  time = 1.66394, size = 16, normalized size = 2.29 \begin{align*} -\frac{1}{x - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(x - 1)

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Sympy [A]  time = 0.080331, size = 5, normalized size = 0.71 \begin{align*} - \frac{1}{x - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(x - 1)

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Giac [A]  time = 1.05921, size = 9, normalized size = 1.29 \begin{align*} -\frac{1}{x - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(x - 1)