3.2838 \(\int \frac{2+x}{\left (1+(2+x)^2\right )^2} \, dx\)

Optimal. Leaf size=13 \[ -\frac{1}{2 \left ((x+2)^2+1\right )} \]

[Out]

-1/(2*(1 + (2 + x)^2))

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Rubi [A]  time = 0.0085989, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{1}{2 \left ((x+2)^2+1\right )} \]

Antiderivative was successfully verified.

[In]  Int[(2 + x)/(1 + (2 + x)^2)^2,x]

[Out]

-1/(2*(1 + (2 + x)^2))

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Rubi in Sympy [A]  time = 2.15877, size = 10, normalized size = 0.77 \[ - \frac{1}{2 \left (\left (x + 2\right )^{2} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+x)/(1+(2+x)**2)**2,x)

[Out]

-1/(2*((x + 2)**2 + 1))

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Mathematica [A]  time = 0.00420682, size = 13, normalized size = 1. \[ -\frac{1}{2 \left ((x+2)^2+1\right )} \]

Antiderivative was successfully verified.

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

[Out]

-1/(2*(1 + (2 + x)^2))

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Maple [A]  time = 0.002, size = 13, normalized size = 1. \[ -{\frac{1}{2\,{x}^{2}+8\,x+10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+x)/(1+(2+x)^2)^2,x)

[Out]

-1/2/(x^2+4*x+5)

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Maxima [A]  time = 1.37199, size = 15, normalized size = 1.15 \[ -\frac{1}{2 \,{\left ({\left (x + 2\right )}^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 2)/((x + 2)^2 + 1)^2,x, algorithm="maxima")

[Out]

-1/2/((x + 2)^2 + 1)

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Fricas [A]  time = 0.201616, size = 16, normalized size = 1.23 \[ -\frac{1}{2 \,{\left (x^{2} + 4 \, x + 5\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 2)/((x + 2)^2 + 1)^2,x, algorithm="fricas")

[Out]

-1/2/(x^2 + 4*x + 5)

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Sympy [A]  time = 0.198018, size = 12, normalized size = 0.92 \[ - \frac{1}{2 x^{2} + 8 x + 10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2+x)/(1+(2+x)**2)**2,x)

[Out]

-1/(2*x**2 + 8*x + 10)

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GIAC/XCAS [A]  time = 0.214925, size = 16, normalized size = 1.23 \[ -\frac{1}{2 \,{\left (x^{2} + 4 \, x + 5\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 2)/((x + 2)^2 + 1)^2,x, algorithm="giac")

[Out]

-1/2/(x^2 + 4*x + 5)