\(\int \frac {2}{3+x} \, dx\) [7783]

Optimal result

Integrand size = 7, antiderivative size = 27 \[ \int \frac {2}{3+x} \, dx=\log \left (\frac {4 (3+x)^2}{3 x \left (4-\frac {-1+4 x}{x}\right )}\right ) \]

[Out]

ln(4/(12-3*(-1+4*x)/x)/x*(3+x)^2)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.22, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {12, 31} \[ \int \frac {2}{3+x} \, dx=2 \log (x+3) \]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.22 \[ \int \frac {2}{3+x} \, dx=2 \log (3+x) \]

[In]

Integrate[2/(3 + x),x]

[Out]

2*Log[3 + x]

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.26

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

[In]

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

[Out]

2*ln(3+x)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.22 \[ \int \frac {2}{3+x} \, dx=2 \, \log \left (x + 3\right ) \]

[In]

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

[Out]

2*log(x + 3)

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.19 \[ \int \frac {2}{3+x} \, dx=2 \log {\left (x + 3 \right )} \]

[In]

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

[Out]

2*log(x + 3)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.22 \[ \int \frac {2}{3+x} \, dx=2 \, \log \left (x + 3\right ) \]

[In]

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

[Out]

2*log(x + 3)

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.26 \[ \int \frac {2}{3+x} \, dx=2 \, \log \left ({\left | x + 3 \right |}\right ) \]

[In]

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

[Out]

2*log(abs(x + 3))

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.22 \[ \int \frac {2}{3+x} \, dx=2\,\ln \left (x+3\right ) \]

[In]

int(2/(x + 3),x)

[Out]

2*log(x + 3)