\(\int \frac {\log (\log (x))}{x} \, dx\) [17]

Optimal result

Integrand size = 7, antiderivative size = 11 \[ \int \frac {\log (\log (x))}{x} \, dx=-\log (x)+\log (x) \log (\log (x)) \]

[Out]

-ln(x)+ln(x)*ln(ln(x))

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2601} \[ \int \frac {\log (\log (x))}{x} \, dx=\log (x) \log (\log (x))-\log (x) \]

[In]

Int[Log[Log[x]]/x,x]

[Out]

-Log[x] + Log[x]*Log[Log[x]]

Rule 2601

Int[((a_.) + Log[Log[(d_.)*(x_)^(n_.)]^(p_.)*(c_.)]*(b_.))/(x_), x_Symbol] :> Simp[Log[d*x^n]*((a + b*Log[c*Lo
g[d*x^n]^p])/n), x] - Simp[b*p*Log[x], x] /; FreeQ[{a, b, c, d, n, p}, x]

Rubi steps \begin{align*} \text {integral}& = -\log (x)+\log (x) \log (\log (x)) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {\log (\log (x))}{x} \, dx=-\log (x)+\log (x) \log (\log (x)) \]

[In]

Integrate[Log[Log[x]]/x,x]

[Out]

-Log[x] + Log[x]*Log[Log[x]]

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09

method	result	size
derivativedivides	\(-\ln \left (x \right )+\ln \left (x \right ) \ln \left (\ln \left (x \right )\right )\)	\(12\)
default	\(-\ln \left (x \right )+\ln \left (x \right ) \ln \left (\ln \left (x \right )\right )\)	\(12\)
norman	\(-\ln \left (x \right )+\ln \left (x \right ) \ln \left (\ln \left (x \right )\right )\)	\(12\)
risch	\(-\ln \left (x \right )+\ln \left (x \right ) \ln \left (\ln \left (x \right )\right )\)	\(12\)

[In]

int(ln(ln(x))/x,x,method=_RETURNVERBOSE)

[Out]

-ln(x)+ln(x)*ln(ln(x))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {\log (\log (x))}{x} \, dx=\log \left (x\right ) \log \left (\log \left (x\right )\right ) - \log \left (x\right ) \]

[In]

integrate(log(log(x))/x,x, algorithm="fricas")

[Out]

log(x)*log(log(x)) - log(x)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.91 \[ \int \frac {\log (\log (x))}{x} \, dx=\log {\left (x \right )} \log {\left (\log {\left (x \right )} \right )} - \log {\left (x \right )} \]

[In]

integrate(ln(ln(x))/x,x)

[Out]

log(x)*log(log(x)) - log(x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {\log (\log (x))}{x} \, dx=\log \left (x\right ) \log \left (\log \left (x\right )\right ) - \log \left (x\right ) \]

[In]

integrate(log(log(x))/x,x, algorithm="maxima")

[Out]

log(x)*log(log(x)) - log(x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {\log (\log (x))}{x} \, dx=\log \left (x\right ) \log \left (\log \left (x\right )\right ) - \log \left (x\right ) \]

[In]

integrate(log(log(x))/x,x, algorithm="giac")

[Out]

log(x)*log(log(x)) - log(x)

Mupad [B] (verification not implemented)

Time = 15.86 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.73 \[ \int \frac {\log (\log (x))}{x} \, dx=\ln \left (x\right )\,\left (\ln \left (\ln \left (x\right )\right )-1\right ) \]

[In]

int(log(log(x))/x,x)

[Out]

log(x)*(log(log(x)) - 1)