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

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

[Out]

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

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Rubi [A]  time = 0.0183673, 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 = {6374, 30} \[ -\frac{\sqrt{\pi } \text{Erfc}(b x)^2}{4 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6374

Int[E^((c_.) + (d_.)*(x_)^2)*Erfc[(b_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(E^c*Sqrt[Pi])/(2*b), Subst[Int[x^n, x
], x, Erfc[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{erfc}(b x) \, dx &=-\frac{\sqrt{\pi } \operatorname{Subst}(\int x \, dx,x,\text{erfc}(b x))}{2 b}\\ &=-\frac{\sqrt{\pi } \text{erfc}(b x)^2}{4 b}\\ \end{align*}

Mathematica [A]  time = 0.0045196, size = 18, normalized size = 1. \[ -\frac{\sqrt{\pi } \text{Erfc}(b x)^2}{4 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.071, size = 22, normalized size = 1.2 \begin{align*}{\frac{\sqrt{\pi }}{2\,b} \left ( -{\frac{ \left ({\it Erf} \left ( bx \right ) \right ) ^{2}}{2}}+{\it Erf} \left ( bx \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.995533, size = 19, normalized size = 1.06 \begin{align*} -\frac{\sqrt{\pi } \operatorname{erfc}\left (b x\right )^{2}}{4 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.02123, size = 58, normalized size = 3.22 \begin{align*} -\frac{\sqrt{\pi }{\left (\operatorname{erf}\left (b x\right )^{2} - 2 \, \operatorname{erf}\left (b x\right )\right )}}{4 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.49723, size = 17, normalized size = 0.94 \begin{align*} \begin{cases} - \frac{\sqrt{\pi } \operatorname{erfc}^{2}{\left (b x \right )}}{4 b} & \text{for}\: b \neq 0 \\x & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

integrate(erfc(b*x)*e^(-b^2*x^2), x)