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-1/2*x^2

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Rubi [A]  time = 0.01, antiderivative size = 9, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 9, normalized size = 1.00 \[ x-\frac {x^2}{2} \]

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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fricas [A]  time = 0.50, size = 7, normalized size = 0.78 \[ -\frac {1}{2} \, x^{2} + x \]

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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giac [A]  time = 0.15, size = 7, normalized size = 0.78 \[ -\frac {1}{2} \, x^{2} + x \]

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

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maple [A]  time = 0.04, size = 8, normalized size = 0.89 \[ -\frac {1}{2} x^{2}+x \]

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 = 1.29, size = 7, normalized size = 0.78 \[ -\frac {1}{2} \, x^{2} + x \]

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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mupad [B]  time = 0.02, size = 6, normalized size = 0.67 \[ -\frac {x\,\left (x-2\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^4 - 1)/(x + x^2 + x^3 + 1),x)

[Out]

-(x*(x - 2))/2

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sympy [A]  time = 0.07, size = 5, normalized size = 0.56 \[ - \frac {x^{2}}{2} + x \]

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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