\(\int \frac {1}{x \sqrt {\log (x)}} \, dx\) [132]

Optimal result

Integrand size = 10, antiderivative size = 8 \[ \int \frac {1}{x \sqrt {\log (x)}} \, dx=2 \sqrt {\log (x)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2339, 30} \[ \int \frac {1}{x \sqrt {\log (x)}} \, dx=2 \sqrt {\log (x)} \]

[In]

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

[Out]

2*Sqrt[Log[x]]

Rule 30

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

Rule 2339

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {\log (x)}} \, dx=2 \sqrt {\log (x)} \]

[In]

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

[Out]

2*Sqrt[Log[x]]

Maple [A] (verified)

Time = 0.31 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.88

method	result	size
derivativedivides	\(2 \sqrt {\ln \left (x \right )}\)	\(7\)
default	\(2 \sqrt {\ln \left (x \right )}\)	\(7\)

[In]

int(1/x/ln(x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x \sqrt {\log (x)}} \, dx=2 \, \sqrt {\log \left (x\right )} \]

[In]

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

[Out]

2*sqrt(log(x))

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x \sqrt {\log (x)}} \, dx=2 \sqrt {\log {\left (x \right )}} \]

[In]

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

[Out]

2*sqrt(log(x))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x \sqrt {\log (x)}} \, dx=2 \, \sqrt {\log \left (x\right )} \]

[In]

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

[Out]

2*sqrt(log(x))

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x \sqrt {\log (x)}} \, dx=2 \, \sqrt {\log \left (x\right )} \]

[In]

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

[Out]

2*sqrt(log(x))

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x \sqrt {\log (x)}} \, dx=2\,\sqrt {\ln \left (x\right )} \]

[In]

int(1/(x*log(x)^(1/2)),x)

[Out]

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