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]

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

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Rubi [A]  time = 0.03, antiderivative size = 43, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.02, 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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fricas [A]  time = 0.48, size = 43, normalized size = 1.00 \[ -\frac {2 \, b \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} - \sqrt {2} \sqrt {b^{2}} \operatorname {erf}\left (\sqrt {2} \sqrt {b^{2}} x\right )}{4 \, b^{3}} \]

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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giac [A]  time = 0.26, size = 35, normalized size = 0.81 \[ -\frac {\operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{2 \, b^{2}} - \frac {\sqrt {2} \operatorname {erf}\left (-\sqrt {2} b x\right )}{4 \, b^{2}} \]

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

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maple [A]  time = 0.07, size = 39, normalized size = 0.91 \[ \frac {-\frac {\erf \left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}}{2 b}+\frac {\sqrt {2}\, \erf \left (b x \sqrt {2}\right )}{4 b}}{b} \]

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 = 0.45, size = 34, normalized size = 0.79 \[ -\frac {\operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{2 \, b^{2}} + \frac {\sqrt {2} \operatorname {erf}\left (\sqrt {2} b x\right )}{4 \, b^{2}} \]

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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mupad [B]  time = 0.18, size = 43, normalized size = 1.00 \[ \frac {\sqrt {2}\,\mathrm {erf}\left (\sqrt {2}\,x\,\sqrt {b^2}\right )}{4\,b\,\sqrt {b^2}}-\frac {{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erf}\left (b\,x\right )}{2\,b^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x e^{- b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}\, dx \]

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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