3.2246 \(\int \frac{x}{\left (2+2 x+x^2\right )^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0232807, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{x+2}{2 \left (x^2+2 x+2\right )}-\frac{1}{2} \tan ^{-1}(x+1) \]

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

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Rubi in Sympy [A]  time = 3.72278, size = 22, normalized size = 0.85 \[ - \frac{2 x + 4}{4 \left (x^{2} + 2 x + 2\right )} - \frac{\operatorname{atan}{\left (x + 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(x**2+2*x+2)**2,x)

[Out]

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

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Mathematica [A]  time = 0.0204332, size = 28, normalized size = 1.08 \[ \frac{-x-2}{2 \left (x^2+2 x+2\right )}-\frac{1}{2} \tan ^{-1}(x+1) \]

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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Maple [A]  time = 0.005, size = 25, normalized size = 1. \[{\frac{-2\,x-4}{4\,{x}^{2}+8\,x+8}}-{\frac{\arctan \left ( 1+x \right ) }{2}} \]

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(1+x)

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Maxima [A]  time = 0.752572, size = 30, normalized size = 1.15 \[ -\frac{x + 2}{2 \,{\left (x^{2} + 2 \, x + 2\right )}} - \frac{1}{2} \, \arctan \left (x + 1\right ) \]

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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Fricas [A]  time = 0.202441, size = 38, normalized size = 1.46 \[ -\frac{{\left (x^{2} + 2 \, x + 2\right )} \arctan \left (x + 1\right ) + x + 2}{2 \,{\left (x^{2} + 2 \, x + 2\right )}} \]

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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Sympy [A]  time = 0.249234, size = 20, normalized size = 0.77 \[ - \frac{x + 2}{2 x^{2} + 4 x + 4} - \frac{\operatorname{atan}{\left (x + 1 \right )}}{2} \]

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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GIAC/XCAS [A]  time = 0.2026, size = 30, normalized size = 1.15 \[ -\frac{x + 2}{2 \,{\left (x^{2} + 2 \, x + 2\right )}} - \frac{1}{2} \, \arctan \left (x + 1\right ) \]

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)