3.39.39 \(\int \frac {1}{x \log (x) \log (-e^2 \log (x))} \, dx\)

Optimal. Leaf size=9 \[ \log \left (\log \left (-e^2 \log (x)\right )\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 9, normalized size of antiderivative = 1.00, 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 (-e^2 \log (x)\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*Log[x]*Log[-(E^2*Log[x])]),x]

[Out]

Log[Log[-(E^2*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 (-e^2 x\right )} \, dx,x,\log (x)\right )\\ &=\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (-e^2 \log (x)\right )\right )\\ &=\log \left (\log \left (-e^2 \log (x)\right )\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 8, normalized size = 0.89 \begin {gather*} \log (2+\log (-\log (x))) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Log[x]*Log[-(E^2*Log[x])]),x]

[Out]

Log[2 + Log[-Log[x]]]

________________________________________________________________________________________

fricas [A]  time = 0.70, size = 8, normalized size = 0.89 \begin {gather*} \log \left (\log \left (-e^{2} \log \relax (x)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/log(x)/log(-exp(2)*log(x)),x, algorithm="fricas")

[Out]

log(log(-e^2*log(x)))

________________________________________________________________________________________

giac [A]  time = 0.16, size = 9, normalized size = 1.00 \begin {gather*} \log \left ({\left | \log \left (-e^{2} \log \relax (x)\right ) \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/log(x)/log(-exp(2)*log(x)),x, algorithm="giac")

[Out]

log(abs(log(-e^2*log(x))))

________________________________________________________________________________________

maple [A]  time = 0.02, size = 9, normalized size = 1.00

method	result	size
derivativedivides	\(\ln \left (\ln \left (-{\mathrm e}^{2} \ln \relax (x )\right )\right )\)	\(9\)
default	\(\ln \left (\ln \left (-{\mathrm e}^{2} \ln \relax (x )\right )\right )\)	\(9\)
norman	\(\ln \left (\ln \left (-{\mathrm e}^{2} \ln \relax (x )\right )\right )\)	\(9\)
risch	\(\ln \left (\ln \left (-{\mathrm e}^{2} \ln \relax (x )\right )\right )\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/ln(x)/ln(-exp(2)*ln(x)),x,method=_RETURNVERBOSE)

[Out]

ln(ln(-exp(2)*ln(x)))

________________________________________________________________________________________

maxima [A]  time = 0.34, size = 8, normalized size = 0.89 \begin {gather*} \log \left (\log \left (-e^{2} \log \relax (x)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/log(x)/log(-exp(2)*log(x)),x, algorithm="maxima")

[Out]

log(log(-e^2*log(x)))

________________________________________________________________________________________

mupad [B]  time = 2.50, size = 8, normalized size = 0.89 \begin {gather*} \ln \left (\ln \left (-\ln \relax (x)\right )+2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*log(-exp(2)*log(x))*log(x)),x)

[Out]

log(log(-log(x)) + 2)

________________________________________________________________________________________

sympy [A]  time = 0.24, size = 10, normalized size = 1.11 \begin {gather*} \log {\left (\log {\left (- e^{2} \log {\relax (x )} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/ln(x)/ln(-exp(2)*ln(x)),x)

[Out]

log(log(-exp(2)*log(x)))

________________________________________________________________________________________