3.180 \(\int e^{-b^2 x^2} x \text{Erfc}(b x) \, dx\)

Optimal. Leaf size=43 \[ -\frac{\text{Erf}\left (\sqrt{2} b x\right )}{2 \sqrt{2} b^2}-\frac{e^{-b^2 x^2} \text{Erfc}(b x)}{2 b^2} \]

[Out]

-Erf[Sqrt[2]*b*x]/(2*Sqrt[2]*b^2) - Erfc[b*x]/(2*b^2*E^(b^2*x^2))

________________________________________________________________________________________

Rubi [A]  time = 0.0310541, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {6383, 2205} \[ -\frac{\text{Erf}\left (\sqrt{2} b x\right )}{2 \sqrt{2} b^2}-\frac{e^{-b^2 x^2} \text{Erfc}(b x)}{2 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Erf[Sqrt[2]*b*x]/(2*Sqrt[2]*b^2) - Erfc[b*x]/(2*b^2*E^(b^2*x^2))

Rule 6383

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

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 e^{-b^2 x^2} x \text{erfc}(b x) \, dx &=-\frac{e^{-b^2 x^2} \text{erfc}(b x)}{2 b^2}-\frac{\int e^{-2 b^2 x^2} \, dx}{b \sqrt{\pi }}\\ &=-\frac{\text{erf}\left (\sqrt{2} b x\right )}{2 \sqrt{2} b^2}-\frac{e^{-b^2 x^2} \text{erfc}(b x)}{2 b^2}\\ \end{align*}

Mathematica [A]  time = 0.0225946, size = 39, normalized size = 0.91 \[ -\frac{2 e^{-b^2 x^2} \text{Erfc}(b x)+\sqrt{2} \text{Erf}\left (\sqrt{2} b x\right )}{4 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[2]*Erf[Sqrt[2]*b*x] + (2*Erfc[b*x])/E^(b^2*x^2))/(4*b^2)

________________________________________________________________________________________

Maple [A]  time = 0.113, size = 53, normalized size = 1.2 \begin{align*}{\frac{1}{b} \left ( -{\frac{{{\rm e}^{-{b}^{2}{x}^{2}}}}{2\,b}}+{\frac{{\it Erf} \left ( bx \right ){{\rm e}^{-{b}^{2}{x}^{2}}}}{2\,b}}-{\frac{\sqrt{2}{\it Erf} \left ( bx\sqrt{2} \right ) }{4\,b}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-1/2/b*exp(-b^2*x^2)+1/2*erf(b*x)/b*exp(-b^2*x^2)-1/4/b*2^(1/2)*erf(b*x*2^(1/2)))/b

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*erfc(b*x)*e^(-b^2*x^2), x)

________________________________________________________________________________________

Fricas [A]  time = 2.16247, size = 120, normalized size = 2.79 \begin{align*} -\frac{\sqrt{2} \sqrt{b^{2}} \operatorname{erf}\left (\sqrt{2} \sqrt{b^{2}} x\right ) - 2 \,{\left (b \operatorname{erf}\left (b x\right ) - b\right )} e^{\left (-b^{2} x^{2}\right )}}{4 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(sqrt(2)*sqrt(b^2)*erf(sqrt(2)*sqrt(b^2)*x) - 2*(b*erf(b*x) - b)*e^(-b^2*x^2))/b^3

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*exp(-b**2*x**2)*erfc(b*x), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*erfc(b*x)*e^(-b^2*x^2), x)