3.271 \(\int \frac{1}{2+2 x} \, dx\)

Optimal. Leaf size=8 \[ \frac{1}{2} \log (x+1) \]

[Out]

Log[1 + x]/2

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Rubi [A]  time = 0.00474695, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{1}{2} \log (x+1) \]

Antiderivative was successfully verified.

[In]  Int[(2 + 2*x)^(-1),x]

[Out]

Log[1 + x]/2

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Rubi in Sympy [A]  time = 1.04575, size = 5, normalized size = 0.62 \[ \frac{\log{\left (x + 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(2+2*x),x)

[Out]

log(x + 1)/2

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Mathematica [A]  time = 0.00142296, size = 10, normalized size = 1.25 \[ \frac{1}{2} \log (2 x+2) \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + 2*x)^(-1),x]

[Out]

Log[2 + 2*x]/2

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Maple [A]  time = 0.001, size = 9, normalized size = 1.1 \[{\frac{\ln \left ( 2+2\,x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2+2*x),x)

[Out]

1/2*ln(2+2*x)

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Maxima [A]  time = 1.33907, size = 8, normalized size = 1. \[ \frac{1}{2} \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*log(x + 1)

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Fricas [A]  time = 0.208046, size = 8, normalized size = 1. \[ \frac{1}{2} \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*log(x + 1)

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Sympy [A]  time = 0.066099, size = 7, normalized size = 0.88 \[ \frac{\log{\left (2 x + 2 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(2+2*x),x)

[Out]

log(2*x + 2)/2

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GIAC/XCAS [A]  time = 0.205084, size = 9, normalized size = 1.12 \[ \frac{1}{2} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*ln(abs(x + 1))