3.133 \(\int \frac{1}{x^2 \sqrt{\log (a x^n)}} \, dx\)

Optimal. Leaf size=40 \[ \frac{\sqrt{\pi } \left (a x^n\right )^{\frac{1}{n}} \text{Erf}\left (\frac{\sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{\sqrt{n} x} \]

[Out]

(Sqrt[Pi]*(a*x^n)^n^(-1)*Erf[Sqrt[Log[a*x^n]]/Sqrt[n]])/(Sqrt[n]*x)

________________________________________________________________________________________

Rubi [A]  time = 0.0327074, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {2310, 2180, 2205} \[ \frac{\sqrt{\pi } \left (a x^n\right )^{\frac{1}{n}} \text{Erf}\left (\frac{\sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{\sqrt{n} x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[Pi]*(a*x^n)^n^(-1)*Erf[Sqrt[Log[a*x^n]]/Sqrt[n]])/(Sqrt[n]*x)

Rule 2310

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)*x)/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rule 2180

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] &&  !$UseGamma === True

Rule 2205

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]

Rubi steps

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

Mathematica [A]  time = 0.0342237, size = 52, normalized size = 1.3 \[ -\frac{\left (a x^n\right )^{\frac{1}{n}} \sqrt{\frac{\log \left (a x^n\right )}{n}} \text{Gamma}\left (\frac{1}{2},\frac{\log \left (a x^n\right )}{n}\right )}{x \sqrt{\log \left (a x^n\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.171, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}}{\frac{1}{\sqrt{\ln \left ( a{x}^{n} \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \sqrt{\log \left (a x^{n}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \sqrt{\log{\left (a x^{n} \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x**2*sqrt(log(a*x**n))), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \sqrt{\log \left (a x^{n}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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