\(\int \frac {1}{\log ^{\frac {3}{2}}(a x^n)} \, dx\) [138]

Optimal result

Integrand size = 10, antiderivative size = 58 \[ \int \frac {1}{\log ^{\frac {3}{2}}\left (a x^n\right )} \, dx=\frac {2 \sqrt {\pi } x \left (a x^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{n^{3/2}}-\frac {2 x}{n \sqrt {\log \left (a x^n\right )}} \]

[Out]

2*x*erfi(ln(a*x^n)^(1/2)/n^(1/2))*Pi^(1/2)/n^(3/2)/((a*x^n)^(1/n))-2*x/n/ln(a*x^n)^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2334, 2337, 2211, 2235} \[ \int \frac {1}{\log ^{\frac {3}{2}}\left (a x^n\right )} \, dx=\frac {2 \sqrt {\pi } x \left (a x^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{n^{3/2}}-\frac {2 x}{n \sqrt {\log \left (a x^n\right )}} \]

[In]

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

[Out]

(2*Sqrt[Pi]*x*Erfi[Sqrt[Log[a*x^n]]/Sqrt[n]])/(n^(3/2)*(a*x^n)^n^(-1)) - (2*x)/(n*Sqrt[Log[a*x^n]])

Rule 2211

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - c*(
f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2235

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2
]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2334

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[x*((a + b*Log[c*x^n])^(p + 1)/(b*n*(p + 1)))
, x] - Dist[1/(b*n*(p + 1)), Int[(a + b*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && LtQ[p, -1] &&
IntegerQ[2*p]

Rule 2337

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[E^(x/n)*(a +
b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 x}{n \sqrt {\log \left (a x^n\right )}}+\frac {2 \int \frac {1}{\sqrt {\log \left (a x^n\right )}} \, dx}{n} \\ & = -\frac {2 x}{n \sqrt {\log \left (a x^n\right )}}+\frac {\left (2 x \left (a x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{n}}}{\sqrt {x}} \, dx,x,\log \left (a x^n\right )\right )}{n^2} \\ & = -\frac {2 x}{n \sqrt {\log \left (a x^n\right )}}+\frac {\left (4 x \left (a x^n\right )^{-1/n}\right ) \text {Subst}\left (\int e^{\frac {x^2}{n}} \, dx,x,\sqrt {\log \left (a x^n\right )}\right )}{n^2} \\ & = \frac {2 \sqrt {\pi } x \left (a x^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{n^{3/2}}-\frac {2 x}{n \sqrt {\log \left (a x^n\right )}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.19 \[ \int \frac {1}{\log ^{\frac {3}{2}}\left (a x^n\right )} \, dx=-\frac {2 x \left (a x^n\right )^{-1/n} \left (\left (a x^n\right )^{\frac {1}{n}}-\Gamma \left (\frac {1}{2},-\frac {\log \left (a x^n\right )}{n}\right ) \sqrt {-\frac {\log \left (a x^n\right )}{n}}\right )}{n \sqrt {\log \left (a x^n\right )}} \]

[In]

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

[Out]

(-2*x*((a*x^n)^n^(-1) - Gamma[1/2, -(Log[a*x^n]/n)]*Sqrt[-(Log[a*x^n]/n)]))/(n*(a*x^n)^n^(-1)*Sqrt[Log[a*x^n]]
)

Maple [F]

\[\int \frac {1}{\ln \left (a \,x^{n}\right )^{\frac {3}{2}}}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{\log ^{\frac {3}{2}}\left (a x^n\right )} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F]

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

[In]

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

[Out]

Integral(log(a*x**n)**(-3/2), x)

Maxima [F]

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

[In]

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

[Out]

integrate(log(a*x^n)^(-3/2), x)

Giac [F]

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

[In]

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

[Out]

integrate(log(a*x^n)^(-3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\log ^{\frac {3}{2}}\left (a x^n\right )} \, dx=\int \frac {1}{{\ln \left (a\,x^n\right )}^{3/2}} \,d x \]

[In]

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

[Out]

int(1/log(a*x^n)^(3/2), x)