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

Optimal. Leaf size=22 \[ 3+\log (x)+\frac {1}{5} (-3-5 \log (\log (4)))+\log \left (\frac {\log (x)}{x}\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 3, normalized size of antiderivative = 0.14, 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} \begin {gather*} \log (\log (x)) \end {gather*}

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

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Mathematica [A]  time = 0.00, size = 3, normalized size = 0.14 \begin {gather*} \log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[Log[x]]

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fricas [A]  time = 0.62, size = 3, normalized size = 0.14 \begin {gather*} \log \left (\log \relax (x)\right ) \end {gather*}

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 = 0.23, size = 4, normalized size = 0.18 \begin {gather*} \log \left ({\left | \log \relax (x) \right |}\right ) \end {gather*}

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 = 0.02, size = 4, normalized size = 0.18

method	result	size
derivativedivides	\(\ln \left (\ln \relax (x )\right )\)	\(4\)
default	\(\ln \left (\ln \relax (x )\right )\)	\(4\)
norman	\(\ln \left (\ln \relax (x )\right )\)	\(4\)
risch	\(\ln \left (\ln \relax (x )\right )\)	\(4\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/ln(x),x,method=_RETURNVERBOSE)

[Out]

ln(ln(x))

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maxima [A]  time = 0.46, size = 3, normalized size = 0.14 \begin {gather*} \log \left (\log \relax (x)\right ) \end {gather*}

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 = 1.12, size = 3, normalized size = 0.14 \begin {gather*} \ln \left (\ln \relax (x)\right ) \end {gather*}

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 = 0.08, size = 3, normalized size = 0.14 \begin {gather*} \log {\left (\log {\relax (x )} \right )} \end {gather*}

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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