3.78.83 \(\int \frac {2}{3+x} \, dx\)

Optimal. Leaf size=27 \[ \log \left (\frac {4 (3+x)^2}{3 x \left (4-\frac {-1+4 x}{x}\right )}\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 6, normalized size of antiderivative = 0.22, number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {12, 31} \begin {gather*} 2 \log (x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[2/(3 + x),x]

[Out]

2*Log[3 + x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 {gather*} \begin {aligned} \text {integral} &=2 \int \frac {1}{3+x} \, dx\\ &=2 \log (3+x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 6, normalized size = 0.22 \begin {gather*} 2 \log (3+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[2/(3 + x),x]

[Out]

2*Log[3 + x]

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fricas [A]  time = 0.73, size = 6, normalized size = 0.22 \begin {gather*} 2 \, \log \left (x + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/(3+x),x, algorithm="fricas")

[Out]

2*log(x + 3)

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giac [A]  time = 0.14, size = 7, normalized size = 0.26 \begin {gather*} 2 \, \log \left ({\left | x + 3 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/(3+x),x, algorithm="giac")

[Out]

2*log(abs(x + 3))

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maple [A]  time = 0.23, size = 7, normalized size = 0.26

method	result	size
default	\(2 \ln \left (3+x \right )\)	\(7\)
norman	\(2 \ln \left (3+x \right )\)	\(7\)
risch	\(2 \ln \left (3+x \right )\)	\(7\)
meijerg	\(2 \ln \left (1+\frac {x}{3}\right )\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2/(3+x),x,method=_RETURNVERBOSE)

[Out]

2*ln(3+x)

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maxima [A]  time = 0.36, size = 6, normalized size = 0.22 \begin {gather*} 2 \, \log \left (x + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/(3+x),x, algorithm="maxima")

[Out]

2*log(x + 3)

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mupad [B]  time = 0.04, size = 6, normalized size = 0.22 \begin {gather*} 2\,\ln \left (x+3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2/(x + 3),x)

[Out]

2*log(x + 3)

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sympy [A]  time = 0.05, size = 5, normalized size = 0.19 \begin {gather*} 2 \log {\left (x + 3 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/(3+x),x)

[Out]

2*log(x + 3)

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