### 3.129 $$\int \frac{1}{x (3+\log (x))} \, dx$$

Optimal. Leaf size=5 $\log (\log (x)+3)$

[Out]

Log[3 + Log[x]]

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Rubi [A]  time = 0.0178284, antiderivative size = 5, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.2, Rules used = {2302, 29} $\log (\log (x)+3)$

Antiderivative was successfully veriﬁed.

[In]

Int[1/(x*(3 + Log[x])),x]

[Out]

Log[3 + 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 (3+\log (x))} \, dx &=\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,3+\log (x)\right )\\ &=\log (3+\log (x))\\ \end{align*}

Mathematica [A]  time = 0.0091176, size = 5, normalized size = 1. $\log (\log (x)+3)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/(x*(3 + Log[x])),x]

[Out]

Log[3 + Log[x]]

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Maple [A]  time = 0.006, size = 6, normalized size = 1.2 \begin{align*} \ln \left ( 3+\ln \left ( x \right ) \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(3+ln(x)),x)

[Out]

ln(3+ln(x))

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Maxima [A]  time = 1.08247, size = 7, normalized size = 1.4 \begin{align*} \log \left (\log \left (x\right ) + 3\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(3+log(x)),x, algorithm="maxima")

[Out]

log(log(x) + 3)

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Fricas [A]  time = 1.98059, size = 23, normalized size = 4.6 \begin{align*} \log \left (\log \left (x\right ) + 3\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(3+log(x)),x, algorithm="fricas")

[Out]

log(log(x) + 3)

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Sympy [A]  time = 0.095218, size = 5, normalized size = 1. \begin{align*} \log{\left (\log{\left (x \right )} + 3 \right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(3+ln(x)),x)

[Out]

log(log(x) + 3)

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Giac [B]  time = 1.27814, size = 30, normalized size = 6. \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 ) + 3\right )}^{2}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(3+log(x)),x, algorithm="giac")

[Out]

1/2*log(1/4*pi^2*(sgn(x) - 1)^2 + (log(abs(x)) + 3)^2)