3.628 \(\int \frac {\log (\log (x))}{x} \, dx\)

Optimal. Leaf size=11 \[ \log (x) \log (\log (x))-\log (x) \]

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Rubi [A]  time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2521} \[ \log (x) \log (\log (x))-\log (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Log[x] + Log[x]*Log[Log[x]]

Rule 2521

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*} \int \frac {\log (\log (x))}{x} \, dx &=-\log (x)+\log (x) \log (\log (x))\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 11, normalized size = 1.00 \[ \log (x) \log (\log (x))-\log (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Log[x] + Log[x]*Log[Log[x]]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log (\log (x))}{x} \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[Log[Log[x]]/x,x]

[Out]

Could not integrate

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fricas [A]  time = 1.04, size = 11, normalized size = 1.00 \[ \log \relax (x) \log \left (\log \relax (x)\right ) - \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x)*log(log(x)) - log(x)

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giac [A]  time = 1.65, size = 11, normalized size = 1.00 \[ \log \relax (x) \log \left (\log \relax (x)\right ) - \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x)*log(log(x)) - log(x)

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maple [A]  time = 0.02, size = 12, normalized size = 1.09

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.42, size = 11, normalized size = 1.00 \[ \log \relax (x) \log \left (\log \relax (x)\right ) - \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x)*log(log(x)) - log(x)

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mupad [B]  time = 0.32, size = 8, normalized size = 0.73 \[ \ln \relax (x)\,\left (\ln \left (\ln \relax (x)\right )-1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x)*(log(log(x)) - 1)

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sympy [A]  time = 0.26, size = 10, normalized size = 0.91 \[ \log {\relax (x )} \log {\left (\log {\relax (x )} \right )} - \log {\relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x)*log(log(x)) - log(x)

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