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

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

[Out]

ln(ln(x))

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Rubi [A]  time = 0.01, antiderivative size = 3, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2302, 29} \[ \log (\log (x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[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 {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*}

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Mathematica [A]  time = 0.00, size = 3, normalized size = 1.00 \[ \log (\log (x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[Log[x]]

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fricas [A]  time = 0.40, size = 3, normalized size = 1.00 \[ \log \left (\log \relax (x)\right ) \]

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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giac [A]  time = 1.23, size = 4, normalized size = 1.33 \[ \log \left ({\left | \log \relax (x) \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(log(x)))

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maple [A]  time = 0.00, size = 4, normalized size = 1.33 \[ \ln \left (\ln \relax (x )\right ) \]

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 = 0.41, size = 3, normalized size = 1.00 \[ \log \left (\log \relax (x)\right ) \]

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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mupad [B]  time = 0.12, size = 3, normalized size = 1.00 \[ \ln \left (\ln \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*log(x)),x)

[Out]

log(log(x))

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sympy [A]  time = 0.10, size = 3, normalized size = 1.00 \[ \log {\left (\log {\relax (x )} \right )} \]

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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