\(\int \frac {e^{-\frac {2019}{4 x^2}}}{x^2} \, dx\) [201]

Optimal result

Integrand size = 13, antiderivative size = 24 \[ \int \frac {e^{-\frac {2019}{4 x^2}}}{x^2} \, dx=-\sqrt {\frac {\pi }{2019}} \text {erf}\left (\frac {\sqrt {2019}}{2 x}\right ) \]

[Out]

-1/2019*2019^(1/2)*Pi^(1/2)*erf(1/2*2019^(1/2)/x)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2242, 2236} \[ \int \frac {e^{-\frac {2019}{4 x^2}}}{x^2} \, dx=-\sqrt {\frac {\pi }{2019}} \text {erf}\left (\frac {\sqrt {2019}}{2 x}\right ) \]

[In]

Int[1/(E^(2019/(4*x^2))*x^2),x]

[Out]

-(Sqrt[Pi/2019]*Erf[Sqrt[2019]/(2*x)])

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 2242

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Dist[1/(d*(m + 1))
, Subst[Int[F^(a + b*x^2), x], x, (c + d*x)^(m + 1)], x] /; FreeQ[{F, a, b, c, d, m, n}, x] && EqQ[n, 2*(m + 1
)]

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int e^{-\frac {2019 x^2}{4}} \, dx,x,\frac {1}{x}\right ) \\ & = -\sqrt {\frac {\pi }{2019}} \text {erf}\left (\frac {\sqrt {2019}}{2 x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-\frac {2019}{4 x^2}}}{x^2} \, dx=-\sqrt {\frac {\pi }{2019}} \text {erf}\left (\frac {\sqrt {2019}}{2 x}\right ) \]

[In]

Integrate[1/(E^(2019/(4*x^2))*x^2),x]

[Out]

-(Sqrt[Pi/2019]*Erf[Sqrt[2019]/(2*x)])

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.75

method	result	size
derivativedivides	\(-\frac {\sqrt {2019}\, \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {2019}}{2 x}\right )}{2019}\)	\(18\)
default	\(-\frac {\sqrt {2019}\, \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {2019}}{2 x}\right )}{2019}\)	\(18\)
meijerg	\(-\frac {\sqrt {2019}\, \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {2019}}{2 x}\right )}{2019}\)	\(18\)
risch	\(-\frac {\sqrt {2019}\, \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {2019}}{2 x}\right )}{2019}\)	\(18\)

[In]

int(exp(-2019/4/x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

-1/2019*2019^(1/2)*Pi^(1/2)*erf(1/2*2019^(1/2)/x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.71 \[ \int \frac {e^{-\frac {2019}{4 x^2}}}{x^2} \, dx=-\frac {1}{2019} \, \sqrt {2019} \sqrt {\pi } \operatorname {erf}\left (\frac {\sqrt {2019}}{2 \, x}\right ) \]

[In]

integrate(exp(-2019/4/x^2)/x^2,x, algorithm="fricas")

[Out]

-1/2019*sqrt(2019)*sqrt(pi)*erf(1/2*sqrt(2019)/x)

Sympy [A] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {e^{-\frac {2019}{4 x^2}}}{x^2} \, dx=- \frac {\sqrt {2019} \sqrt {\pi } \operatorname {erf}{\left (\frac {\sqrt {2019}}{2 x} \right )}}{2019} \]

[In]

integrate(exp(-2019/4/x**2)/x**2,x)

[Out]

-sqrt(2019)*sqrt(pi)*erf(sqrt(2019)/(2*x))/2019

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21 \[ \int \frac {e^{-\frac {2019}{4 x^2}}}{x^2} \, dx=-\frac {\sqrt {2019} \sqrt {\pi } \sqrt {x^{2}} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {2019} \sqrt {\frac {1}{x^{2}}}\right ) - 1\right )}}{2019 \, x} \]

[In]

integrate(exp(-2019/4/x^2)/x^2,x, algorithm="maxima")

[Out]

-1/2019*sqrt(2019)*sqrt(pi)*sqrt(x^2)*(erf(1/2*sqrt(2019)*sqrt(x^(-2))) - 1)/x

Giac [F]

\[ \int \frac {e^{-\frac {2019}{4 x^2}}}{x^2} \, dx=\int { \frac {e^{\left (-\frac {2019}{4 \, x^{2}}\right )}}{x^{2}} \,d x } \]

[In]

integrate(exp(-2019/4/x^2)/x^2,x, algorithm="giac")

[Out]

integrate(e^(-2019/4/x^2)/x^2, x)

Mupad [B] (verification not implemented)

Time = 15.66 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.71 \[ \int \frac {e^{-\frac {2019}{4 x^2}}}{x^2} \, dx=-\frac {\sqrt {2019}\,\sqrt {\pi }\,\mathrm {erf}\left (\frac {\sqrt {2019}}{2\,x}\right )}{2019} \]

[In]

int(exp(-2019/(4*x^2))/x^2,x)

[Out]

-(2019^(1/2)*pi^(1/2)*erf(2019^(1/2)/(2*x)))/2019