### 3.132 $$\int \frac{1}{x \sqrt{\log (x)}} \, dx$$

Optimal. Leaf size=8 $2 \sqrt{\log (x)}$

[Out]

2*Sqrt[Log[x]]

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Rubi [A]  time = 0.0139242, antiderivative size = 8, 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, 30} $2 \sqrt{\log (x)}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/(x*Sqrt[Log[x]]),x]

[Out]

2*Sqrt[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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{1}{x \sqrt{\log (x)}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\sqrt{x}} \, dx,x,\log (x)\right )\\ &=2 \sqrt{\log (x)}\\ \end{align*}

Mathematica [A]  time = 0.0015343, size = 8, normalized size = 1. $2 \sqrt{\log (x)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

2*Sqrt[Log[x]]

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Maple [A]  time = 0.004, size = 7, normalized size = 0.9 \begin{align*} 2\,\sqrt{\ln \left ( x \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/ln(x)^(1/2),x)

[Out]

2*ln(x)^(1/2)

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Maxima [A]  time = 1.07431, size = 8, normalized size = 1. \begin{align*} 2 \, \sqrt{\log \left (x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/log(x)^(1/2),x, algorithm="maxima")

[Out]

2*sqrt(log(x))

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Fricas [A]  time = 2.02241, size = 22, normalized size = 2.75 \begin{align*} 2 \, \sqrt{\log \left (x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/log(x)^(1/2),x, algorithm="fricas")

[Out]

2*sqrt(log(x))

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Sympy [A]  time = 0.431645, size = 7, normalized size = 0.88 \begin{align*} 2 \sqrt{\log{\left (x \right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/ln(x)**(1/2),x)

[Out]

2*sqrt(log(x))

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Giac [A]  time = 1.21724, size = 8, normalized size = 1. \begin{align*} 2 \, \sqrt{\log \left (x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/log(x)^(1/2),x, algorithm="giac")

[Out]

2*sqrt(log(x))