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

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

[Out]

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

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Rubi [A]  time = 0.0324727, 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 = {6382, 2205} \[ \frac{\text{Erf}\left (\sqrt{2} b x\right )}{2 \sqrt{2} b^2}-\frac{e^{-b^2 x^2} \text{Erf}(b x)}{2 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6382

Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(a_.) + (b_.)*(x_)]*(x_), x_Symbol] :> Simp[(E^(c + d*x^2)*Erf[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{erf}(b x) \, dx &=-\frac{e^{-b^2 x^2} \text{erf}(b x)}{2 b^2}+\frac{\int e^{-2 b^2 x^2} \, dx}{b \sqrt{\pi }}\\ &=-\frac{e^{-b^2 x^2} \text{erf}(b x)}{2 b^2}+\frac{\text{erf}\left (\sqrt{2} b x\right )}{2 \sqrt{2} b^2}\\ \end{align*}

Mathematica [A]  time = 0.0171781, size = 39, normalized size = 0.91 \[ \frac{\sqrt{2} \text{Erf}\left (\sqrt{2} b x\right )-2 e^{-b^2 x^2} \text{Erf}(b x)}{4 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.109, size = 39, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ( -{\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*erf(b*x)/exp(b^2*x^2),x)

[Out]

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

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Maxima [A]  time = 1.54507, size = 46, normalized size = 1.07 \begin{align*} -\frac{\operatorname{erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{2 \, b^{2}} + \frac{\sqrt{2} \operatorname{erf}\left (\sqrt{2} b x\right )}{4 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.63868, size = 112, normalized size = 2.6 \begin{align*} -\frac{2 \, b \operatorname{erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} - \sqrt{2} \sqrt{b^{2}} \operatorname{erf}\left (\sqrt{2} \sqrt{b^{2}} x\right )}{4 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.30691, size = 47, normalized size = 1.09 \begin{align*} -\frac{\operatorname{erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{2 \, b^{2}} - \frac{\sqrt{2} \operatorname{erf}\left (-\sqrt{2} b x\right )}{4 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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