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

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

[Out]

-(1 - x)^3/3

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Rubi [A]  time = 0.0122637, 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^4)^2/(1 + x + x^2 + x^3)^2,x]

[Out]

-(1 - x)^3/3

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

Mathematica [A]  time = 0.0005153, size = 14, normalized size = 1.27 \[ \frac{x^3}{3}-x^2+x \]

Antiderivative was successfully verified.

[In]

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

[Out]

x - x^2 + x^3/3

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Maple [A]  time = 0.001, size = 8, normalized size = 0.7 \begin{align*}{\frac{ \left ( -1+x \right ) ^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.939669, size = 16, normalized size = 1.45 \begin{align*} \frac{1}{3} \, x^{3} - x^{2} + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3 - x^2 + x

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Fricas [A]  time = 1.68165, size = 26, normalized size = 2.36 \begin{align*} \frac{1}{3} \, x^{3} - x^{2} + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3 - x^2 + x

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Sympy [A]  time = 0.066356, size = 8, normalized size = 0.73 \begin{align*} \frac{x^{3}}{3} - x^{2} + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**3/3 - x**2 + x

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Giac [A]  time = 1.06477, size = 16, normalized size = 1.45 \begin{align*} \frac{1}{3} \, x^{3} - x^{2} + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3 - x^2 + x