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

Optimal. Leaf size=11 \[ \log \left (\frac {\log (2)}{9 \log (x)}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 5, normalized size of antiderivative = 0.45, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, 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 = 5, normalized size = 0.45 \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 = 0.66, size = 5, normalized size = 0.45 \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 = 0.22, size = 6, normalized size = 0.55 \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 = 6, normalized size = 0.55

method	result	size
derivativedivides	\(-\ln \left (\ln \relax (x )\right )\)	\(6\)
default	\(-\ln \left (\ln \relax (x )\right )\)	\(6\)
norman	\(-\ln \left (\ln \relax (x )\right )\)	\(6\)
risch	\(-\ln \left (\ln \relax (x )\right )\)	\(6\)

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 = 0.49, size = 5, normalized size = 0.45 \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 = 1.06, size = 5, normalized size = 0.45 \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 = 5, normalized size = 0.45 \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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