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

Optimal. Leaf size=25 \[ -\log \left (\frac {x^2}{x-\frac {1}{4} (1-x) x}\right )+\log (\log (3)) \]

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Rubi [A]  time = 0.00, antiderivative size = 9, normalized size of antiderivative = 0.36, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {12, 615} \begin {gather*} \log (x+3)-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-3/(3*x + x^2),x]

[Out]

-Log[x] + 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 615

Int[((b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[Log[x]/b, x] - Simp[Log[RemoveContent[b + c*x, x]]/b,
x] /; FreeQ[{b, c}, x]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 17, normalized size = 0.68 \begin {gather*} -3 \left (\frac {\log (x)}{3}-\frac {1}{3} \log (3+x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-3/(3*x + x^2),x]

[Out]

-3*(Log[x]/3 - Log[3 + x]/3)

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fricas [A]  time = 0.47, size = 9, normalized size = 0.36 \begin {gather*} \log \left (x + 3\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x + 3) - log(x)

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giac [A]  time = 0.16, size = 11, normalized size = 0.44 \begin {gather*} \log \left ({\left | x + 3 \right |}\right ) - \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(x + 3)) - log(abs(x))

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maple [A]  time = 0.44, size = 10, normalized size = 0.40

method	result	size
default	\(-\ln \relax (x )+\ln \left (3+x \right )\)	\(10\)
norman	\(-\ln \relax (x )+\ln \left (3+x \right )\)	\(10\)
risch	\(-\ln \relax (x )+\ln \left (3+x \right )\)	\(10\)
meijerg	\(-\ln \relax (x )+\ln \relax (3)+\ln \left (1+\frac {x}{3}\right )\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-ln(x)+ln(3+x)

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maxima [A]  time = 0.49, size = 9, normalized size = 0.36 \begin {gather*} \log \left (x + 3\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x + 3) - log(x)

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mupad [B]  time = 1.79, size = 8, normalized size = 0.32 \begin {gather*} 2\,\mathrm {atanh}\left (\frac {2\,x}{3}+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-3/(3*x + x^2),x)

[Out]

2*atanh((2*x)/3 + 1)

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sympy [A]  time = 0.09, size = 7, normalized size = 0.28 \begin {gather*} - \log {\relax (x )} + \log {\left (x + 3 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x) + log(x + 3)

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