\(\int \frac {1}{x (3+\log (x))} \, dx\) [129]

Optimal result

Integrand size = 10, antiderivative size = 5 \[ \int \frac {1}{x (3+\log (x))} \, dx=\log (3+\log (x)) \]

[Out]

ln(3+ln(x))

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2339, 29} \[ \int \frac {1}{x (3+\log (x))} \, dx=\log (\log (x)+3) \]

[In]

Int[1/(x*(3 + Log[x])),x]

[Out]

Log[3 + Log[x]]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 2339

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{x} \, dx,x,3+\log (x)\right ) \\ & = \log (3+\log (x)) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x (3+\log (x))} \, dx=\log (3+\log (x)) \]

[In]

Integrate[1/(x*(3 + Log[x])),x]

[Out]

Log[3 + Log[x]]

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.20

method	result	size
derivativedivides	\(\ln \left (3+\ln \left (x \right )\right )\)	\(6\)
default	\(\ln \left (3+\ln \left (x \right )\right )\)	\(6\)
norman	\(\ln \left (3+\ln \left (x \right )\right )\)	\(6\)
risch	\(\ln \left (3+\ln \left (x \right )\right )\)	\(6\)
parallelrisch	\(\ln \left (3+\ln \left (x \right )\right )\)	\(6\)

[In]

int(1/x/(3+ln(x)),x,method=_RETURNVERBOSE)

[Out]

ln(3+ln(x))

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x (3+\log (x))} \, dx=\log \left (\log \left (x\right ) + 3\right ) \]

[In]

integrate(1/x/(3+log(x)),x, algorithm="fricas")

[Out]

log(log(x) + 3)

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x (3+\log (x))} \, dx=\log {\left (\log {\left (x \right )} + 3 \right )} \]

[In]

integrate(1/x/(3+ln(x)),x)

[Out]

log(log(x) + 3)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x (3+\log (x))} \, dx=\log \left (\log \left (x\right ) + 3\right ) \]

[In]

integrate(1/x/(3+log(x)),x, algorithm="maxima")

[Out]

log(log(x) + 3)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 22 vs. \(2 (5) = 10\).

Time = 0.32 (sec) , antiderivative size = 22, normalized size of antiderivative = 4.40 \[ \int \frac {1}{x (3+\log (x))} \, dx=\frac {1}{2} \, \log \left (\frac {1}{4} \, \pi ^{2} {\left (\mathrm {sgn}\left (x\right ) - 1\right )}^{2} + {\left (\log \left ({\left | x \right |}\right ) + 3\right )}^{2}\right ) \]

[In]

integrate(1/x/(3+log(x)),x, algorithm="giac")

[Out]

1/2*log(1/4*pi^2*(sgn(x) - 1)^2 + (log(abs(x)) + 3)^2)

Mupad [B] (verification not implemented)

Time = 1.79 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x (3+\log (x))} \, dx=\ln \left (\ln \left (x\right )+3\right ) \]

[In]

int(1/(x*(log(x) + 3)),x)

[Out]

log(log(x) + 3)