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 = 0.0008358, 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, Rules used = {31} \[ \frac{1}{2} \log (x+1) \]

Antiderivative was successfully verified.

[In]

Int[(2 + 2*x)^(-1),x]

[Out]

Log[1 + x]/2

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{1}{2+2 x} \, dx &=\frac{1}{2} \log (1+x)\\ \end{align*}

Mathematica [A]  time = 0.0011513, 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., size = 9, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( 2+2\,x \right ) }{2}} \end{align*}

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 = 1.0238, size = 8, normalized size = 1. \begin{align*} \frac{1}{2} \, \log \left (x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*x+2),x, algorithm="maxima")

[Out]

1/2*log(x + 1)

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Fricas [A]  time = 1.53517, size = 22, normalized size = 2.75 \begin{align*} \frac{1}{2} \, \log \left (x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*x+2),x, algorithm="fricas")

[Out]

1/2*log(x + 1)

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Sympy [A]  time = 0.066004, size = 7, normalized size = 0.88 \begin{align*} \frac{\log{\left (2 x + 2 \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*x+2),x)

[Out]

log(2*x + 2)/2

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Giac [A]  time = 1.16433, size = 9, normalized size = 1.12 \begin{align*} \frac{1}{2} \, \log \left ({\left | x + 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*x+2),x, algorithm="giac")

[Out]

1/2*log(abs(x + 1))