### 3.264 $$\int \frac{1}{\sqrt{-\log (\frac{a}{x^2})}} \, dx$$

Optimal. Leaf size=39 $\sqrt{\frac{\pi }{2}} x \sqrt{\frac{a}{x^2}} \text{Erfi}\left (\frac{\sqrt{-\log \left (\frac{a}{x^2}\right )}}{\sqrt{2}}\right )$

[Out]

Sqrt[Pi/2]*Sqrt[a/x^2]*x*Erfi[Sqrt[-Log[a/x^2]]/Sqrt[2]]

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Rubi [A]  time = 0.0246292, antiderivative size = 39, 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, 2204} $\sqrt{\frac{\pi }{2}} x \sqrt{\frac{a}{x^2}} \text{Erfi}\left (\frac{\sqrt{-\log \left (\frac{a}{x^2}\right )}}{\sqrt{2}}\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[1/Sqrt[-Log[a/x^2]],x]

[Out]

Sqrt[Pi/2]*Sqrt[a/x^2]*x*Erfi[Sqrt[-Log[a/x^2]]/Sqrt[2]]

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 2204

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]

Rubi steps

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

Mathematica [A]  time = 0.0124686, size = 60, normalized size = 1.54 $-\frac{\sqrt{\frac{\pi }{2}} x \sqrt{\frac{a}{x^2}} \sqrt{\log \left (\frac{a}{x^2}\right )} \text{Erf}\left (\frac{\sqrt{\log \left (\frac{a}{x^2}\right )}}{\sqrt{2}}\right )}{\sqrt{-\log \left (\frac{a}{x^2}\right )}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/Sqrt[-Log[a/x^2]],x]

[Out]

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

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Maple [F]  time = 0.016, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt{-\ln \left ({\frac{a}{{x}^{2}}} \right ) }}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(-log(a/x^2)), 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^2))^(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 (\frac{a}{x^{2}} \right )}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(-log(a/x**2)), x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(-log(a/x^2)), x)