3.90.21 \(\int -\frac {1}{x \log (x) \log (\frac {3}{\log (x)})} \, dx\)

Optimal. Leaf size=13 \[ \log \left (\frac {3 \log \left (\frac {3}{\log (x)}\right )}{e^3}\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 8, normalized size of antiderivative = 0.62, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2302, 29} \begin {gather*} \log \left (\log \left (\frac {3}{\log (x)}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[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 2302

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

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 8, normalized size = 0.62 \begin {gather*} \log \left (\log \left (\frac {3}{\log (x)}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.44, size = 8, normalized size = 0.62 \begin {gather*} \log \left (\log \left (\frac {3}{\log \relax (x)}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [B]  time = 0.12, size = 27, normalized size = 2.08 \begin {gather*} \frac {1}{2} \, \log \left (\frac {1}{4} \, \pi ^{2} {\left (\mathrm {sgn}\left (\log \relax (x)\right ) - 1\right )}^{2} + {\left (\log \relax (3) - \log \left ({\left | \log \relax (x) \right |}\right )\right )}^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 9, normalized size = 0.69

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.37, size = 8, normalized size = 0.62 \begin {gather*} \log \left (\log \left (\frac {3}{\log \relax (x)}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 7.04, size = 8, normalized size = 0.62 \begin {gather*} \ln \left (\ln \left (\frac {3}{\ln \relax (x)}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.24, size = 7, normalized size = 0.54 \begin {gather*} \log {\left (\log {\left (\frac {3}{\log {\relax (x )}} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________