\(\int \frac {1}{x^3 \sqrt {\log (a x^n)}} \, dx\) [134]

Optimal result

Integrand size = 14, antiderivative size = 51 \[ \int \frac {1}{x^3 \sqrt {\log \left (a x^n\right )}} \, dx=\frac {\sqrt {\frac {\pi }{2}} \left (a x^n\right )^{2/n} \text {erf}\left (\frac {\sqrt {2} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{\sqrt {n} x^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2347, 2211, 2236} \[ \int \frac {1}{x^3 \sqrt {\log \left (a x^n\right )}} \, dx=\frac {\sqrt {\frac {\pi }{2}} \left (a x^n\right )^{2/n} \text {erf}\left (\frac {\sqrt {2} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{\sqrt {n} x^2} \]

[In]

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

[Out]

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

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 2236

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

Rule 2347

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

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.18 \[ \int \frac {1}{x^3 \sqrt {\log \left (a x^n\right )}} \, dx=-\frac {\left (a x^n\right )^{2/n} \Gamma \left (\frac {1}{2},\frac {2 \log \left (a x^n\right )}{n}\right ) \sqrt {\frac {\log \left (a x^n\right )}{n}}}{\sqrt {2} x^2 \sqrt {\log \left (a x^n\right )}} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(1/(x**3*sqrt(log(a*x**n))), x)

Maxima [F]

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

[In]

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

[Out]

integrate(1/(x^3*sqrt(log(a*x^n))), x)

Giac [F]

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

[In]

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

[Out]

integrate(1/(x^3*sqrt(log(a*x^n))), x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(1/(x^3*log(a*x^n)^(1/2)), x)