3.3.71 \(\int \frac {1}{2+2 x} \, dx\) [271]

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

[Out]

1/2*ln(1+x)

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Rubi [A]
time = 0.00, antiderivative size = 8, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {1}{2} \log (x+1) \end {gather*}

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*}

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Mathematica [A]
time = 0.00, size = 10, normalized size = 1.25 \begin {gather*} \frac {1}{2} \log (2+2 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 2*x)^(-1),x]

[Out]

Log[2 + 2*x]/2

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Mathics [A]
time = 1.55, size = 8, normalized size = 1.00 \begin {gather*} \frac {\text {Log}\left [2+2 x\right ]}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[1/(2 + 2*x),x]')

[Out]

Log[2 + 2 x] / 2

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Maple [A]
time = 0.08, size = 9, normalized size = 1.12

method	result	size
meijerg	\(\frac {\ln \left (1+x \right )}{2}\)	\(7\)
risch	\(\frac {\ln \left (1+x \right )}{2}\)	\(7\)
default	\(\frac {\ln \left (2+2 x \right )}{2}\)	\(9\)
norman	\(\frac {\ln \left (2+2 x \right )}{2}\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2+2*x),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.24, size = 6, normalized size = 0.75 \begin {gather*} \frac {1}{2} \, \log \left (x + 1\right ) \end {gather*}

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 = 0.30, size = 6, normalized size = 0.75 \begin {gather*} \frac {1}{2} \, \log \left (x + 1\right ) \end {gather*}

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 = 0.03, size = 7, normalized size = 0.88 \begin {gather*} \frac {\log {\left (2 x + 2 \right )}}{2} \end {gather*}

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 = 0.00, size = 8, normalized size = 1.00 \begin {gather*} \frac {\ln \left |x+1\right |}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(abs(x + 1))

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Mupad [B]
time = 0.15, size = 6, normalized size = 0.75 \begin {gather*} \frac {\ln \left (x+1\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2*x + 2),x)

[Out]

log(x + 1)/2

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