3.20 \(\int \frac{1}{x \log (x)} \, dx\)

Optimal. Leaf size=3 \[ \log (\log (x)) \]

[Out]

Log[Log[x]]

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Rubi [A]  time = 0.0123612, antiderivative size = 3, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2302, 29} \[ \log (\log (x)) \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Log[x]),x]

[Out]

Log[Log[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]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

\begin{align*} \int \frac{1}{x \log (x)} \, dx &=\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\log (x)\right )\\ &=\log (\log (x))\\ \end{align*}

Mathematica [A]  time = 0.0041385, size = 3, normalized size = 1. \[ \log (\log (x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[Log[x]]

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Maple [A]  time = 0., size = 4, normalized size = 1.3 \begin{align*} \ln \left ( \ln \left ( x \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/ln(x),x)

[Out]

ln(ln(x))

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Maxima [A]  time = 0.925341, size = 4, normalized size = 1.33 \begin{align*} \log \left (\log \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(log(x))

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Fricas [A]  time = 2.15694, size = 18, normalized size = 6. \begin{align*} \log \left (\log \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(log(x))

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Sympy [A]  time = 0.08814, size = 3, normalized size = 1. \begin{align*} \log{\left (\log{\left (x \right )} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(log(x))

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Giac [A]  time = 1.04621, size = 5, normalized size = 1.67 \begin{align*} \log \left ({\left | \log \left (x\right ) \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/log(x),x, algorithm="giac")

[Out]

log(abs(log(x)))