\(\int -\frac {1}{x \log (x) \log (\frac {3}{\log (x)})} \, dx\) [8921]

Optimal result

Integrand size = 18, antiderivative size = 13 \[ \int -\frac {1}{x \log (x) \log \left (\frac {3}{\log (x)}\right )} \, dx=\log \left (\frac {3 \log \left (\frac {3}{\log (x)}\right )}{e^3}\right ) \]

[Out]

ln(3/exp(2)*exp(-1)*ln(3/ln(x)))

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.62, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2339, 29} \[ \int -\frac {1}{x \log (x) \log \left (\frac {3}{\log (x)}\right )} \, dx=\log \left (\log \left (\frac {3}{\log (x)}\right )\right ) \]

[In]

Int[-(1/(x*Log[x]*Log[3/Log[x]])),x]

[Out]

Log[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 \log \left (\frac {3}{x}\right )} \, dx,x,\log (x)\right ) \\ & = \text {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (\frac {3}{\log (x)}\right )\right ) \\ & = \log \left (\log \left (\frac {3}{\log (x)}\right )\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.62 \[ \int -\frac {1}{x \log (x) \log \left (\frac {3}{\log (x)}\right )} \, dx=\log \left (\log \left (\frac {3}{\log (x)}\right )\right ) \]

[In]

Integrate[-(1/(x*Log[x]*Log[3/Log[x]])),x]

[Out]

Log[Log[3/Log[x]]]

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69

method	result	size
derivativedivides	\(\ln \left (\ln \left (\frac {3}{\ln \left (x \right )}\right )\right )\)	\(9\)
default	\(\ln \left (\ln \left (\frac {3}{\ln \left (x \right )}\right )\right )\)	\(9\)
norman	\(\ln \left (\ln \left (\frac {3}{\ln \left (x \right )}\right )\right )\)	\(9\)
parallelrisch	\(\ln \left (\ln \left (\frac {3}{\ln \left (x \right )}\right )\right )\)	\(9\)
risch	\(\ln \left (\ln \left (\ln \left (x \right )\right )-\ln \left (3\right )\right )\)	\(10\)

[In]

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

[Out]

ln(ln(3/ln(x)))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.62 \[ \int -\frac {1}{x \log (x) \log \left (\frac {3}{\log (x)}\right )} \, dx=\log \left (\log \left (\frac {3}{\log \left (x\right )}\right )\right ) \]

[In]

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

[Out]

log(log(3/log(x)))

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.54 \[ \int -\frac {1}{x \log (x) \log \left (\frac {3}{\log (x)}\right )} \, dx=\log {\left (\log {\left (\frac {3}{\log {\left (x \right )}} \right )} \right )} \]

[In]

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

[Out]

log(log(3/log(x)))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.62 \[ \int -\frac {1}{x \log (x) \log \left (\frac {3}{\log (x)}\right )} \, dx=\log \left (\log \left (\frac {3}{\log \left (x\right )}\right )\right ) \]

[In]

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

[Out]

log(log(3/log(x)))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (12) = 24\).

Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 2.08 \[ \int -\frac {1}{x \log (x) \log \left (\frac {3}{\log (x)}\right )} \, dx=\frac {1}{2} \, \log \left (\frac {1}{4} \, \pi ^{2} {\left (\mathrm {sgn}\left (\log \left (x\right )\right ) - 1\right )}^{2} + {\left (\log \left (3\right ) - \log \left ({\left | \log \left (x\right ) \right |}\right )\right )}^{2}\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 12.95 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.62 \[ \int -\frac {1}{x \log (x) \log \left (\frac {3}{\log (x)}\right )} \, dx=\ln \left (\ln \left (\frac {3}{\ln \left (x\right )}\right )\right ) \]

[In]

int(-1/(x*log(3/log(x))*log(x)),x)

[Out]

log(log(3/log(x)))