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

Optimal result

Integrand size = 14, antiderivative size = 15 \[ \int \frac {1}{x \sqrt {\log \left (a x^n\right )}} \, dx=\frac {2 \sqrt {\log \left (a x^n\right )}}{n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2339, 30} \[ \int \frac {1}{x \sqrt {\log \left (a x^n\right )}} \, dx=\frac {2 \sqrt {\log \left (a x^n\right )}}{n} \]

[In]

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

[Out]

(2*Sqrt[Log[a*x^n]])/n

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {\log \left (a x^n\right )}} \, dx=\frac {2 \sqrt {\log \left (a x^n\right )}}{n} \]

[In]

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

[Out]

(2*Sqrt[Log[a*x^n]])/n

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93

method	result	size
derivativedivides	\(\frac {2 \sqrt {\ln \left (a \,x^{n}\right )}}{n}\)	\(14\)
default	\(\frac {2 \sqrt {\ln \left (a \,x^{n}\right )}}{n}\)	\(14\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x \sqrt {\log \left (a x^n\right )}} \, dx=\frac {2 \, \sqrt {n \log \left (x\right ) + \log \left (a\right )}}{n} \]

[In]

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

[Out]

2*sqrt(n*log(x) + log(a))/n

Sympy [A] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.47 \[ \int \frac {1}{x \sqrt {\log \left (a x^n\right )}} \, dx=\begin {cases} \frac {2 \sqrt {\log {\left (a x^{n} \right )}}}{n} & \text {for}\: n \neq 0 \\\frac {\log {\left (x \right )}}{\sqrt {\log {\left (a \right )}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((2*sqrt(log(a*x**n))/n, Ne(n, 0)), (log(x)/sqrt(log(a)), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x \sqrt {\log \left (a x^n\right )}} \, dx=\frac {2 \, \sqrt {\log \left (a x^{n}\right )}}{n} \]

[In]

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

[Out]

2*sqrt(log(a*x^n))/n

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x \sqrt {\log \left (a x^n\right )}} \, dx=\frac {2 \, \sqrt {n \log \left (x\right ) + \log \left (a\right )}}{n} \]

[In]

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

[Out]

2*sqrt(n*log(x) + log(a))/n

Mupad [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x \sqrt {\log \left (a x^n\right )}} \, dx=\frac {2\,\sqrt {\ln \left (a\,x^n\right )}}{n} \]

[In]

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

[Out]

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