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

Optimal. Leaf size=20 \[ 3+\log \left (3 e^{-e^4} x \left (\frac {3}{x}+x\right )\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 6, normalized size of antiderivative = 0.30, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {12, 260} \begin {gather*} \log \left (x^2+3\right ) \end {gather*}

Antiderivative was successfully verified.

[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 260

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

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Mathematica [A]  time = 0.00, size = 6, normalized size = 0.30 \begin {gather*} \log \left (3+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*x)/(3 + x^2),x]

[Out]

Log[3 + x^2]

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fricas [A]  time = 0.66, size = 6, normalized size = 0.30 \begin {gather*} \log \left (x^{2} + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x^2 + 3)

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giac [A]  time = 0.25, size = 6, normalized size = 0.30 \begin {gather*} \log \left (x^{2} + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x^2 + 3)

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maple [A]  time = 0.40, size = 7, normalized size = 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	\(\ln \left (x^{2}+3\right )\)	\(7\)
meijerg	\(\ln \left (1+\frac {x^{2}}{3}\right )\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(x^2+3)

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maxima [A]  time = 0.37, size = 6, normalized size = 0.30 \begin {gather*} \log \left (x^{2} + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x^2 + 3)

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mupad [B]  time = 4.05, size = 6, normalized size = 0.30 \begin {gather*} \ln \left (x^2+3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x)/(x^2 + 3),x)

[Out]

log(x^2 + 3)

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sympy [A]  time = 0.07, size = 5, normalized size = 0.25 \begin {gather*} \log {\left (x^{2} + 3 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x**2 + 3)

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