3.2251 \(\int \frac{x^2}{\left (2+2 x+x^2\right )^2} \, dx\)

Optimal. Leaf size=15 \[ \frac{1}{x^2+2 x+2}+\tan ^{-1}(x+1) \]

[Out]

(2 + 2*x + x^2)^(-1) + ArcTan[1 + x]

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Rubi [A]  time = 0.0255814, antiderivative size = 23, normalized size of antiderivative = 1.53, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \tan ^{-1}(x+1)-\frac{x (x+2)}{2 \left (x^2+2 x+2\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(2 + 2*x + x^2)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 4.6084, size = 20, normalized size = 1.33 \[ - \frac{x \left (2 x + 4\right )}{4 \left (x^{2} + 2 x + 2\right )} + \operatorname{atan}{\left (x + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0157934, size = 15, normalized size = 1. \[ \frac{1}{x^2+2 x+2}+\tan ^{-1}(x+1) \]

Antiderivative was successfully verified.

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

[Out]

(2 + 2*x + x^2)^(-1) + ArcTan[1 + x]

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Maple [A]  time = 0.007, size = 16, normalized size = 1.1 \[ \left ({x}^{2}+2\,x+2 \right ) ^{-1}+\arctan \left ( 1+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(x^2+2*x+2)^2,x)

[Out]

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

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Maxima [A]  time = 0.757027, size = 20, normalized size = 1.33 \[ \frac{1}{x^{2} + 2 \, x + 2} + \arctan \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(x^2 + 2*x + 2) + arctan(x + 1)

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Fricas [A]  time = 0.208205, size = 35, normalized size = 2.33 \[ \frac{{\left (x^{2} + 2 \, x + 2\right )} \arctan \left (x + 1\right ) + 1}{x^{2} + 2 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.237658, size = 14, normalized size = 0.93 \[ \operatorname{atan}{\left (x + 1 \right )} + \frac{1}{x^{2} + 2 x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan(x + 1) + 1/(x**2 + 2*x + 2)

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GIAC/XCAS [A]  time = 0.203897, size = 20, normalized size = 1.33 \[ \frac{1}{x^{2} + 2 \, x + 2} + \arctan \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(x^2 + 2*x + 2) + arctan(x + 1)