\(\int -\frac {3}{3 x+x^2} \, dx\) [2750]

Optimal result

Integrand size = 11, antiderivative size = 25 \[ \int -\frac {3}{3 x+x^2} \, dx=-\log \left (\frac {x^2}{x-\frac {1}{4} (1-x) x}\right )+\log (\log (3)) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.36, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {12, 629} \[ \int -\frac {3}{3 x+x^2} \, dx=\log (x+3)-\log (x) \]

[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 629

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.68 \[ \int -\frac {3}{3 x+x^2} \, dx=-3 \left (\frac {\log (x)}{3}-\frac {1}{3} \log (3+x)\right ) \]

[In]

Integrate[-3/(3*x + x^2),x]

[Out]

-3*(Log[x]/3 - Log[3 + x]/3)

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.40

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.36 \[ \int -\frac {3}{3 x+x^2} \, dx=\log \left (x + 3\right ) - \log \left (x\right ) \]

[In]

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

[Out]

log(x + 3) - log(x)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.28 \[ \int -\frac {3}{3 x+x^2} \, dx=- \log {\left (x \right )} + \log {\left (x + 3 \right )} \]

[In]

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

[Out]

-log(x) + log(x + 3)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.36 \[ \int -\frac {3}{3 x+x^2} \, dx=\log \left (x + 3\right ) - \log \left (x\right ) \]

[In]

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

[Out]

log(x + 3) - log(x)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.44 \[ \int -\frac {3}{3 x+x^2} \, dx=\log \left ({\left | x + 3 \right |}\right ) - \log \left ({\left | x \right |}\right ) \]

[In]

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

[Out]

log(abs(x + 3)) - log(abs(x))

Mupad [B] (verification not implemented)

Time = 9.19 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.32 \[ \int -\frac {3}{3 x+x^2} \, dx=2\,\mathrm {atanh}\left (\frac {2\,x}{3}+1\right ) \]

[In]

int(-3/(3*x + x^2),x)

[Out]

2*atanh((2*x)/3 + 1)