\(\int \frac {1}{\sqrt {-\log (\frac {a}{x^2})}} \, dx\) [264]

Optimal result

Integrand size = 12, antiderivative size = 39 \[ \int \frac {1}{\sqrt {-\log \left (\frac {a}{x^2}\right )}} \, dx=\sqrt {\frac {\pi }{2}} \sqrt {\frac {a}{x^2}} x \text {erfi}\left (\frac {\sqrt {-\log \left (\frac {a}{x^2}\right )}}{\sqrt {2}}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2337, 2211, 2235} \[ \int \frac {1}{\sqrt {-\log \left (\frac {a}{x^2}\right )}} \, dx=\sqrt {\frac {\pi }{2}} x \sqrt {\frac {a}{x^2}} \text {erfi}\left (\frac {\sqrt {-\log \left (\frac {a}{x^2}\right )}}{\sqrt {2}}\right ) \]

[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 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 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}& = -\left (\frac {1}{2} \left (\sqrt {\frac {a}{x^2}} x\right ) \text {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 ) \text {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] (verified)

Time = 0.01 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.54 \[ \int \frac {1}{\sqrt {-\log \left (\frac {a}{x^2}\right )}} \, dx=-\frac {\sqrt {\frac {\pi }{2}} \sqrt {\frac {a}{x^2}} x \text {erf}\left (\frac {\sqrt {\log \left (\frac {a}{x^2}\right )}}{\sqrt {2}}\right ) \sqrt {\log \left (\frac {a}{x^2}\right )}}{\sqrt {-\log \left (\frac {a}{x^2}\right )}} \]

[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]])

Maple [F]

\[\int \frac {1}{\sqrt {-\ln \left (\frac {a}{x^{2}}\right )}}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(1/sqrt(-log(a/x**2)), x)

Maxima [F]

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

[In]

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

[Out]

integrate(1/sqrt(-log(a/x^2)), x)

Giac [F]

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

[In]

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

[Out]

integrate(1/sqrt(-log(a/x^2)), x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(1/(-log(a/x^2))^(1/2), x)