3.167 \(\int \frac {1-x^2}{(1+x+x^2)^2} \, dx\)

Optimal. Leaf size=10 \[ \frac {x}{x^2+x+1} \]

[Out]

x/(x^2+x+1)

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Rubi [A]  time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {1588} \[ \frac {x}{x^2+x+1} \]

Antiderivative was successfully verified.

[In]

Int[(1 - x^2)/(1 + x + x^2)^2,x]

[Out]

x/(1 + x + x^2)

Rule 1588

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*x^(p - q
+ 1)*Qq^(m + 1))/((p + m*q + 1)*Coeff[Qq, x, q]), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 10, normalized size = 1.00 \[ \frac {x}{x^2+x+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/(1 + x + x^2)

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fricas [A]  time = 0.82, size = 10, normalized size = 1.00 \[ \frac {x}{x^{2} + x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/(x^2 + x + 1)

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giac [A]  time = 0.15, size = 8, normalized size = 0.80 \[ \frac {1}{x + \frac {1}{x} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(x + 1/x + 1)

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maple [A]  time = 0.00, size = 11, normalized size = 1.10 \[ \frac {x}{x^{2}+x +1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/(x^2+x+1)

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maxima [A]  time = 0.43, size = 10, normalized size = 1.00 \[ \frac {x}{x^{2} + x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/(x^2 + x + 1)

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mupad [B]  time = 0.05, size = 10, normalized size = 1.00 \[ \frac {x}{x^2+x+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^2 - 1)/(x + x^2 + 1)^2,x)

[Out]

x/(x + x^2 + 1)

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sympy [A]  time = 0.10, size = 7, normalized size = 0.70 \[ \frac {x}{x^{2} + x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/(x**2 + x + 1)

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