3.23 \(\int \frac{1}{x (1+\log (\frac{x}{a}))} \, dx\)

Optimal. Leaf size=9 \[ \log \left (\log \left (\frac{x}{a}\right )+1\right ) \]

[Out]

Log[1 + Log[x/a]]

________________________________________________________________________________________

Rubi [A]  time = 0.0180361, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2302, 29} \[ \log \left (\log \left (\frac{x}{a}\right )+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(1 + Log[x/a])),x]

[Out]

Log[1 + Log[x/a]]

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

Mathematica [A]  time = 0.0161651, size = 9, normalized size = 1. \[ \log \left (\log \left (\frac{x}{a}\right )+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(1 + Log[x/a])),x]

[Out]

Log[1 + Log[x/a]]

________________________________________________________________________________________

Maple [A]  time = 0., size = 10, normalized size = 1.1 \begin{align*} \ln \left ( 1+\ln \left ({\frac{x}{a}} \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(1+ln(x/a))

________________________________________________________________________________________

Maxima [A]  time = 0.931764, size = 12, normalized size = 1.33 \begin{align*} \log \left (\log \left (\frac{x}{a}\right ) + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(log(x/a) + 1)

________________________________________________________________________________________

Fricas [A]  time = 2.27744, size = 26, normalized size = 2.89 \begin{align*} \log \left (\log \left (\frac{x}{a}\right ) + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(log(x/a) + 1)

________________________________________________________________________________________

Sympy [A]  time = 0.095479, size = 7, normalized size = 0.78 \begin{align*} \log{\left (\log{\left (\frac{x}{a} \right )} + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(log(x/a) + 1)

________________________________________________________________________________________

Giac [B]  time = 1.05264, size = 77, normalized size = 8.56 \begin{align*} \frac{1}{2} \, \log \left (\frac{1}{4} \,{\left (\pi{\left (\mathrm{sgn}\left (a\right ) - 1\right )} + \pi{\left (\mathrm{sgn}\left (x\right ) - 1\right )} + 4 \, \pi \left \lfloor -\frac{\pi{\left (\mathrm{sgn}\left (a\right ) - 1\right )} + \pi{\left (\mathrm{sgn}\left (x\right ) - 1\right )}}{4 \, \pi } + \frac{1}{2} \right \rfloor \right )}^{2} +{\left (\log \left (\frac{{\left | x \right |}}{{\left | a \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/a)),x, algorithm="giac")

[Out]

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