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

Optimal. Leaf size=9 \[ -\log (1-\log (x)) \]

[Out]

-Log[1 - Log[x]]

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Rubi [A]  time = 0.0202668, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2302, 29} \[ -\log (1-\log (x)) \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(1 - Log[x])),x]

[Out]

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

Mathematica [A]  time = 0.008645, size = 7, normalized size = 0.78 \[ -\log (\log (x)-1) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(1 - Log[x])),x]

[Out]

-Log[-1 + Log[x]]

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Maple [A]  time = 0.001, size = 10, normalized size = 1.1 \begin{align*} -\ln \left ( 1-\ln \left ( x \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-ln(1-ln(x))

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Maxima [A]  time = 0.926967, size = 9, normalized size = 1. \begin{align*} -\log \left (\log \left (x\right ) - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(log(x) - 1)

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Fricas [A]  time = 2.29388, size = 24, normalized size = 2.67 \begin{align*} -\log \left (\log \left (x\right ) - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(log(x) - 1)

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Sympy [A]  time = 0.0888, size = 7, normalized size = 0.78 \begin{align*} - \log{\left (\log{\left (x \right )} - 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(log(x) - 1)

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Giac [B]  time = 1.06416, size = 30, normalized size = 3.33 \begin{align*} -\frac{1}{2} \, \log \left (\frac{1}{4} \, \pi ^{2}{\left (\mathrm{sgn}\left (x\right ) - 1\right )}^{2} +{\left (\log \left ({\left | x \right |}\right ) - 1\right )}^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(1/4*pi^2*(sgn(x) - 1)^2 + (log(abs(x)) - 1)^2)