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

Optimal. Leaf size=18 \[ \frac{\sqrt{\pi } \text{Erf}(b x)^2}{4 b} \]

[Out]

(Sqrt[Pi]*Erf[b*x]^2)/(4*b)

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Rubi [A]  time = 0.0174227, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {6373, 30} \[ \frac{\sqrt{\pi } \text{Erf}(b x)^2}{4 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[Pi]*Erf[b*x]^2)/(4*b)

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]

Rubi steps

\begin{align*} \int e^{-b^2 x^2} \text{erf}(b x) \, dx &=\frac{\sqrt{\pi } \operatorname{Subst}(\int x \, dx,x,\text{erf}(b x))}{2 b}\\ &=\frac{\sqrt{\pi } \text{erf}(b x)^2}{4 b}\\ \end{align*}

Mathematica [A]  time = 0.0040364, size = 18, normalized size = 1. \[ \frac{\sqrt{\pi } \text{Erf}(b x)^2}{4 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[Pi]*Erf[b*x]^2)/(4*b)

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Maple [A]  time = 0.075, size = 15, normalized size = 0.8 \begin{align*}{\frac{ \left ({\it Erf} \left ( bx \right ) \right ) ^{2}\sqrt{\pi }}{4\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*erf(b*x)^2*Pi^(1/2)/b

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Maxima [A]  time = 1.01892, size = 19, normalized size = 1.06 \begin{align*} \frac{\sqrt{\pi } \operatorname{erf}\left (b x\right )^{2}}{4 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.80272, size = 36, normalized size = 2. \begin{align*} \frac{\sqrt{\pi } \operatorname{erf}\left (b x\right )^{2}}{4 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.61369, size = 53, normalized size = 2.94 \begin{align*} \begin{cases} - \frac{\sqrt{\pi } \operatorname{erf}^{2}{\left (x \sqrt{b^{2}} \right )}}{4 b} + \frac{\sqrt{\pi } \sqrt{b^{2}} \operatorname{erf}{\left (b x \right )} \operatorname{erf}{\left (x \sqrt{b^{2}} \right )}}{2 b^{2}} & \text{for}\: b \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-sqrt(pi)*erf(x*sqrt(b**2))**2/(4*b) + sqrt(pi)*sqrt(b**2)*erf(b*x)*erf(x*sqrt(b**2))/(2*b**2), Ne(
b, 0)), (0, True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{erf}\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(erf(b*x)/exp(b^2*x^2),x, algorithm="giac")

[Out]

integrate(erf(b*x)*e^(-b^2*x^2), x)