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 = 0.0193226, 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} \[ \frac{1}{3 (1-x)^3} \]

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

Mathematica [A]  time = 0.0010176, 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 + x)^3)

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Maple [A]  time = 0., size = 8, normalized size = 0.7 \begin{align*} -{\frac{1}{3\, \left ( -1+x \right ) ^{3}}} \end{align*}

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/(-1+x)^3

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Maxima [B]  time = 0.941818, size = 23, normalized size = 2.09 \begin{align*} -\frac{1}{3 \,{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} \end{align*}

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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Fricas [B]  time = 1.69488, size = 41, normalized size = 3.73 \begin{align*} -\frac{1}{3 \,{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} \end{align*}

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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Sympy [B]  time = 0.102664, size = 17, normalized size = 1.55 \begin{align*} - \frac{1}{3 x^{3} - 9 x^{2} + 9 x - 3} \end{align*}

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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Giac [A]  time = 1.05314, size = 9, normalized size = 0.82 \begin{align*} -\frac{1}{3 \,{\left (x - 1\right )}^{3}} \end{align*}

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