3.187 \(\int \frac{x}{4+4 x+x^2} \, dx\)

Optimal. Leaf size=12 \[ \frac{2}{x+2}+\log (x+2) \]

[Out]

2/(2 + x) + Log[2 + x]

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Rubi [A]  time = 0.0138489, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{2}{x+2}+\log (x+2) \]

Antiderivative was successfully verified.

[In]  Int[x/(4 + 4*x + x^2),x]

[Out]

2/(2 + x) + Log[2 + x]

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Rubi in Sympy [A]  time = 2.28869, size = 8, normalized size = 0.67 \[ \log{\left (x + 2 \right )} + \frac{2}{x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(x**2+4*x+4),x)

[Out]

log(x + 2) + 2/(x + 2)

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Mathematica [A]  time = 0.0049511, size = 12, normalized size = 1. \[ \frac{2}{x+2}+\log (x+2) \]

Antiderivative was successfully verified.

[In]  Integrate[x/(4 + 4*x + x^2),x]

[Out]

2/(2 + x) + Log[2 + x]

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Maple [A]  time = 0.009, size = 13, normalized size = 1.1 \[ 2\, \left ( 2+x \right ) ^{-1}+\ln \left ( 2+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(x^2+4*x+4),x)

[Out]

2/(2+x)+ln(2+x)

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Maxima [A]  time = 1.32862, size = 16, normalized size = 1.33 \[ \frac{2}{x + 2} + \log \left (x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x^2 + 4*x + 4),x, algorithm="maxima")

[Out]

2/(x + 2) + log(x + 2)

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Fricas [A]  time = 0.202383, size = 22, normalized size = 1.83 \[ \frac{{\left (x + 2\right )} \log \left (x + 2\right ) + 2}{x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x^2 + 4*x + 4),x, algorithm="fricas")

[Out]

((x + 2)*log(x + 2) + 2)/(x + 2)

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Sympy [A]  time = 0.069793, size = 8, normalized size = 0.67 \[ \log{\left (x + 2 \right )} + \frac{2}{x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x**2+4*x+4),x)

[Out]

log(x + 2) + 2/(x + 2)

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GIAC/XCAS [A]  time = 0.209304, size = 18, normalized size = 1.5 \[ \frac{2}{x + 2} +{\rm ln}\left ({\left | x + 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x^2 + 4*x + 4),x, algorithm="giac")

[Out]

2/(x + 2) + ln(abs(x + 2))