3.20.26 \(\int \frac {x}{(2+2 x+x^2)^2} \, dx\)

Optimal. Leaf size=26 \[ -\frac {x+2}{2 \left (x^2+2 x+2\right )}-\frac {1}{2} \tan ^{-1}(x+1) \]

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Rubi [A]  time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {638, 617, 204} \begin {gather*} -\frac {x+2}{2 \left (x^2+2 x+2\right )}-\frac {1}{2} \tan ^{-1}(x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(2 + 2*x + x^2)^2,x]

[Out]

-(2 + x)/(2*(2 + 2*x + x^2)) - ArcTan[1 + x]/2

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 638

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[((2*p + 3)*(2*c*d - b*e))/((p + 1)*(b^2
- 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 28, normalized size = 1.08 \begin {gather*} \frac {-x-2}{2 \left (x^2+2 x+2\right )}-\frac {1}{2} \tan ^{-1}(x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(2 + 2*x + x^2)^2,x]

[Out]

(-2 - x)/(2*(2 + 2*x + x^2)) - ArcTan[1 + x]/2

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\left (2+2 x+x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x/(2 + 2*x + x^2)^2,x]

[Out]

IntegrateAlgebraic[x/(2 + 2*x + x^2)^2, x]

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fricas [A]  time = 0.38, size = 28, normalized size = 1.08 \begin {gather*} -\frac {{\left (x^{2} + 2 \, x + 2\right )} \arctan \left (x + 1\right ) + x + 2}{2 \, {\left (x^{2} + 2 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((x^2 + 2*x + 2)*arctan(x + 1) + x + 2)/(x^2 + 2*x + 2)

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giac [A]  time = 0.15, size = 22, normalized size = 0.85 \begin {gather*} -\frac {x + 2}{2 \, {\left (x^{2} + 2 \, x + 2\right )}} - \frac {1}{2} \, \arctan \left (x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(x + 2)/(x^2 + 2*x + 2) - 1/2*arctan(x + 1)

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maple [A]  time = 0.04, size = 25, normalized size = 0.96 \begin {gather*} -\frac {\arctan \left (x +1\right )}{2}+\frac {-2 x -4}{4 x^{2}+8 x +8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^2+2*x+2)^2,x)

[Out]

1/4*(-2*x-4)/(x^2+2*x+2)-1/2*arctan(x+1)

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maxima [A]  time = 1.89, size = 22, normalized size = 0.85 \begin {gather*} -\frac {x + 2}{2 \, {\left (x^{2} + 2 \, x + 2\right )}} - \frac {1}{2} \, \arctan \left (x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(x + 2)/(x^2 + 2*x + 2) - 1/2*arctan(x + 1)

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mupad [B]  time = 0.04, size = 24, normalized size = 0.92 \begin {gather*} -\frac {\mathrm {atan}\left (x+1\right )}{2}-\frac {\frac {x}{2}+1}{x^2+2\,x+2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(2*x + x^2 + 2)^2,x)

[Out]

- atan(x + 1)/2 - (x/2 + 1)/(2*x + x^2 + 2)

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sympy [A]  time = 0.11, size = 20, normalized size = 0.77 \begin {gather*} \frac {- x - 2}{2 x^{2} + 4 x + 4} - \frac {\operatorname {atan}{\left (x + 1 \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-x - 2)/(2*x**2 + 4*x + 4) - atan(x + 1)/2

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