\(\int \frac {1}{x \log ^{\frac {3}{2}}(a x^n)} \, dx\) [139]

Optimal result

Integrand size = 14, antiderivative size = 15 \[ \int \frac {1}{x \log ^{\frac {3}{2}}\left (a x^n\right )} \, dx=-\frac {2}{n \sqrt {\log \left (a x^n\right )}} \]

[Out]

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

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} \[ \int \frac {1}{x \log ^{\frac {3}{2}}\left (a x^n\right )} \, dx=-\frac {2}{n \sqrt {\log \left (a x^n\right )}} \]

[In]

Int[1/(x*Log[a*x^n]^(3/2)),x]

[Out]

-2/(n*Sqrt[Log[a*x^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}{x^{3/2}} \, dx,x,\log \left (a x^n\right )\right )}{n} \\ & = -\frac {2}{n \sqrt {\log \left (a x^n\right )}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \log ^{\frac {3}{2}}\left (a x^n\right )} \, dx=-\frac {2}{n \sqrt {\log \left (a x^n\right )}} \]

[In]

Integrate[1/(x*Log[a*x^n]^(3/2)),x]

[Out]

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

Maple [A] (verified)

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

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

[In]

int(1/x/ln(a*x^n)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.60 \[ \int \frac {1}{x \log ^{\frac {3}{2}}\left (a x^n\right )} \, dx=-\frac {2 \, \sqrt {n \log \left (x\right ) + \log \left (a\right )}}{n^{2} \log \left (x\right ) + n \log \left (a\right )} \]

[In]

integrate(1/x/log(a*x^n)^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 1.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.60 \[ \int \frac {1}{x \log ^{\frac {3}{2}}\left (a x^n\right )} \, dx=\begin {cases} - \frac {2}{n \sqrt {\log {\left (a x^{n} \right )}}} & \text {for}\: n \neq 0 \\\frac {\log {\left (x \right )}}{\log {\left (a \right )}^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]

[In]

integrate(1/x/ln(a*x**n)**(3/2),x)

[Out]

Piecewise((-2/(n*sqrt(log(a*x**n))), Ne(n, 0)), (log(x)/log(a)**(3/2), True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x \log ^{\frac {3}{2}}\left (a x^n\right )} \, dx=-\frac {2}{n \sqrt {\log \left (a x^{n}\right )}} \]

[In]

integrate(1/x/log(a*x^n)^(3/2),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x \log ^{\frac {3}{2}}\left (a x^n\right )} \, dx=-\frac {2}{\sqrt {n \log \left (x\right ) + \log \left (a\right )} n} \]

[In]

integrate(1/x/log(a*x^n)^(3/2),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 \[ \int \frac {1}{x \log ^{\frac {3}{2}}\left (a x^n\right )} \, dx=-\frac {2}{n\,\sqrt {\ln \left (a\,x^n\right )}} \]

[In]

int(1/(x*log(a*x^n)^(3/2)),x)

[Out]

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