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

Optimal. Leaf size=9 \[ x-\frac{x^2}{2} \]

[Out]

x - x^2/2

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Rubi [A]  time = 0.0086773, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {1586} \[ x-\frac{x^2}{2} \]

Antiderivative was successfully verified.

[In]

Int[(1 - x^4)/(1 + x + x^2 + x^3),x]

[Out]

x - x^2/2

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]

Rubi steps

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

Mathematica [A]  time = 0.0004395, size = 9, normalized size = 1. \[ x-\frac{x^2}{2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x - x^2/2

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Maple [A]  time = 0.001, size = 8, normalized size = 0.9 \begin{align*} x-{\frac{{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x-1/2*x^2

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Maxima [A]  time = 0.93638, size = 9, normalized size = 1. \begin{align*} -\frac{1}{2} \, x^{2} + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*x^2 + x

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Fricas [A]  time = 1.67103, size = 19, normalized size = 2.11 \begin{align*} -\frac{1}{2} \, x^{2} + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*x^2 + x

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Sympy [A]  time = 0.057147, size = 5, normalized size = 0.56 \begin{align*} - \frac{x^{2}}{2} + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x**2/2 + x

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Giac [A]  time = 1.0496, size = 9, normalized size = 1. \begin{align*} -\frac{1}{2} \, x^{2} + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*x^2 + x