3.84 \(\int \frac{e^{-b^2 x^2} \text{Erf}(b x)}{x^2} \, dx\)

Optimal. Leaf size=52 \[ -\frac{e^{-b^2 x^2} \text{Erf}(b x)}{x}+\frac{b \text{ExpIntegralEi}\left (-2 b^2 x^2\right )}{\sqrt{\pi }}-\frac{1}{2} \sqrt{\pi } b \text{Erf}(b x)^2 \]

[Out]

-(Erf[b*x]/(E^(b^2*x^2)*x)) - (b*Sqrt[Pi]*Erf[b*x]^2)/2 + (b*ExpIntegralEi[-2*b^2*x^2])/Sqrt[Pi]

________________________________________________________________________________________

Rubi [A]  time = 0.0705583, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6391, 6373, 30, 2210} \[ -\frac{e^{-b^2 x^2} \text{Erf}(b x)}{x}+\frac{b \text{Ei}\left (-2 b^2 x^2\right )}{\sqrt{\pi }}-\frac{1}{2} \sqrt{\pi } b \text{Erf}(b x)^2 \]

Antiderivative was successfully verified.

[In]

Int[Erf[b*x]/(E^(b^2*x^2)*x^2),x]

[Out]

-(Erf[b*x]/(E^(b^2*x^2)*x)) - (b*Sqrt[Pi]*Erf[b*x]^2)/2 + (b*ExpIntegralEi[-2*b^2*x^2])/Sqrt[Pi]

Rule 6391

Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[(x^(m + 1)*E^(c + d*x^2)*Erf
[a + b*x])/(m + 1), x] + (-Dist[(2*d)/(m + 1), Int[x^(m + 2)*E^(c + d*x^2)*Erf[a + b*x], x], x] - Dist[(2*b)/(
(m + 1)*Sqrt[Pi]), Int[x^(m + 1)*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2), x], x]) /; FreeQ[{a, b, c, d}, x] &&
ILtQ[m, -1]

Rule 6373

Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(b_.)*(x_)]^(n_.), x_Symbol] :> Dist[(E^c*Sqrt[Pi])/(2*b), Subst[Int[x^n, x],
x, Erf[b*x]], x] /; FreeQ[{b, c, d, n}, x] && EqQ[d, -b^2]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2210

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Simp[(F^a*ExpIntegralEi[
b*(c + d*x)^n*Log[F]])/(f*n), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps

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

Mathematica [A]  time = 0.0152362, size = 52, normalized size = 1. \[ -\frac{e^{-b^2 x^2} \text{Erf}(b x)}{x}+\frac{b \text{ExpIntegralEi}\left (-2 b^2 x^2\right )}{\sqrt{\pi }}-\frac{1}{2} \sqrt{\pi } b \text{Erf}(b x)^2 \]

Antiderivative was successfully verified.

[In]

Integrate[Erf[b*x]/(E^(b^2*x^2)*x^2),x]

[Out]

-(Erf[b*x]/(E^(b^2*x^2)*x)) - (b*Sqrt[Pi]*Erf[b*x]^2)/2 + (b*ExpIntegralEi[-2*b^2*x^2])/Sqrt[Pi]

________________________________________________________________________________________

Maple [F]  time = 0.296, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it Erf} \left ( bx \right ) }{{{\rm e}^{{b}^{2}{x}^{2}}}{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erf(b*x)/exp(b^2*x^2)/x^2,x)

[Out]

int(erf(b*x)/exp(b^2*x^2)/x^2,x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erf(b*x)/exp(b^2*x^2)/x^2,x, algorithm="maxima")

[Out]

integrate(erf(b*x)*e^(-b^2*x^2)/x^2, x)

________________________________________________________________________________________

Fricas [A]  time = 2.59884, size = 130, normalized size = 2.5 \begin{align*} -\frac{2 \, \pi \operatorname{erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} + \sqrt{\pi }{\left (\pi b x \operatorname{erf}\left (b x\right )^{2} - 2 \, b x{\rm Ei}\left (-2 \, b^{2} x^{2}\right )\right )}}{2 \, \pi x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erf(b*x)/exp(b^2*x^2)/x^2,x, algorithm="fricas")

[Out]

-1/2*(2*pi*erf(b*x)*e^(-b^2*x^2) + sqrt(pi)*(pi*b*x*erf(b*x)^2 - 2*b*x*Ei(-2*b^2*x^2)))/(pi*x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{- b^{2} x^{2}} \operatorname{erf}{\left (b x \right )}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erf(b*x)/exp(b**2*x**2)/x**2,x)

[Out]

Integral(exp(-b**2*x**2)*erf(b*x)/x**2, x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erf(b*x)/exp(b^2*x^2)/x^2,x, algorithm="giac")

[Out]

integrate(erf(b*x)*e^(-b^2*x^2)/x^2, x)