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

Optimal. Leaf size=43 \[ -\frac{\sqrt{\pi } x \left (a x^n\right )^{-1/n} \text{Erf}\left (\frac{\sqrt{-\log \left (a x^n\right )}}{\sqrt{n}}\right )}{\sqrt{n}} \]

[Out]

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

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Rubi [A]  time = 0.0308945, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2300, 2180, 2205} \[ -\frac{\sqrt{\pi } x \left (a x^n\right )^{-1/n} \text{Erf}\left (\frac{\sqrt{-\log \left (a x^n\right )}}{\sqrt{n}}\right )}{\sqrt{n}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2300

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]

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

Mathematica [A]  time = 0.0097883, size = 62, normalized size = 1.44 \[ \frac{\sqrt{\pi } x \left (a x^n\right )^{-1/n} \sqrt{\log \left (a x^n\right )} \text{Erfi}\left (\frac{\sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{\sqrt{n} \sqrt{-\log \left (a x^n\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.045, size = 0, normalized size = 0. \begin{align*} \int{\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/(-ln(a*x^n))^(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(-log(a*x^n)), x)

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

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(-log(a*x**n)), x)

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Giac [A]  time = 1.27486, size = 43, normalized size = 1. \begin{align*} \frac{\sqrt{\pi } \operatorname{erf}\left (-\frac{\sqrt{-n \log \left (x\right ) - \log \left (a\right )}}{\sqrt{n}}\right )}{a^{\left (\frac{1}{n}\right )} \sqrt{n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(pi)*erf(-sqrt(-n*log(x) - log(a))/sqrt(n))/(a^(1/n)*sqrt(n))