\(\int \frac {2 x}{3+x^2} \, dx\) [6561]

Optimal result

Integrand size = 10, antiderivative size = 20 \[ \int \frac {2 x}{3+x^2} \, dx=3+\log \left (3 e^{-e^4} x \left (\frac {3}{x}+x\right )\right ) \]

[Out]

3+ln(3*x/exp(exp(4))*(x+3/x))

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.30, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {12, 266} \[ \int \frac {2 x}{3+x^2} \, dx=\log \left (x^2+3\right ) \]

[In]

Int[(2*x)/(3 + x^2),x]

[Out]

Log[3 + x^2]

Rule 12

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

Rule 266

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

Rubi steps \begin{align*} \text {integral}& = 2 \int \frac {x}{3+x^2} \, dx \\ & = \log \left (3+x^2\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.30 \[ \int \frac {2 x}{3+x^2} \, dx=\log \left (3+x^2\right ) \]

[In]

Integrate[(2*x)/(3 + x^2),x]

[Out]

Log[3 + x^2]

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.35

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

[In]

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

[Out]

ln(x^2+3)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.30 \[ \int \frac {2 x}{3+x^2} \, dx=\log \left (x^{2} + 3\right ) \]

[In]

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

[Out]

log(x^2 + 3)

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.25 \[ \int \frac {2 x}{3+x^2} \, dx=\log {\left (x^{2} + 3 \right )} \]

[In]

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

[Out]

log(x**2 + 3)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.30 \[ \int \frac {2 x}{3+x^2} \, dx=\log \left (x^{2} + 3\right ) \]

[In]

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

[Out]

log(x^2 + 3)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.30 \[ \int \frac {2 x}{3+x^2} \, dx=\log \left (x^{2} + 3\right ) \]

[In]

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

[Out]

log(x^2 + 3)

Mupad [B] (verification not implemented)

Time = 11.83 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.30 \[ \int \frac {2 x}{3+x^2} \, dx=\ln \left (x^2+3\right ) \]

[In]

int((2*x)/(x^2 + 3),x)

[Out]

log(x^2 + 3)