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

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

[Out]

-Log[x] + Log[x]*Log[Log[x]]

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Rubi [A]  time = 0.0072318, antiderivative size = 11, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0041991, size = 11, normalized size = 1. \[ \log (x) \log (\log (x))-\log (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Log[x] + Log[x]*Log[Log[x]]

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Maple [A]  time = 0.002, size = 12, normalized size = 1.1 \begin{align*} -\ln \left ( x \right ) +\ln \left ( x \right ) \ln \left ( \ln \left ( x \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(ln(x))/x,x)

[Out]

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

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Maxima [A]  time = 0.950311, size = 15, normalized size = 1.36 \begin{align*} \log \left (x\right ) \log \left (\log \left (x\right )\right ) - \log \left (x\right ) \end{align*}

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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Fricas [A]  time = 2.45788, size = 39, normalized size = 3.55 \begin{align*} \log \left (x\right ) \log \left (\log \left (x\right )\right ) - \log \left (x\right ) \end{align*}

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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Sympy [A]  time = 0.274747, size = 10, normalized size = 0.91 \begin{align*} \log{\left (x \right )} \log{\left (\log{\left (x \right )} \right )} - \log{\left (x \right )} \end{align*}

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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Giac [A]  time = 1.06199, size = 15, normalized size = 1.36 \begin{align*} \log \left (x\right ) \log \left (\log \left (x\right )\right ) - \log \left (x\right ) \end{align*}

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)