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

Optimal. Leaf size=21 \[ -\frac{\sqrt{\pi } e^c \text{Erfc}(b x)^2}{4 b} \]

[Out]

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

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Rubi [A]  time = 0.0177382, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {6374, 30} \[ -\frac{\sqrt{\pi } e^c \text{Erfc}(b x)^2}{4 b} \]

Antiderivative was successfully verified.

[In]

Int[E^(c - b^2*x^2)*Erfc[b*x],x]

[Out]

-(E^c*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^{c-b^2 x^2} \text{erfc}(b x) \, dx &=-\frac{\left (e^c \sqrt{\pi }\right ) \operatorname{Subst}(\int x \, dx,x,\text{erfc}(b x))}{2 b}\\ &=-\frac{e^c \sqrt{\pi } \text{erfc}(b x)^2}{4 b}\\ \end{align*}

Mathematica [A]  time = 0.0053808, size = 21, normalized size = 1. \[ -\frac{\sqrt{\pi } e^c \text{Erfc}(b x)^2}{4 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.107, size = 30, normalized size = 1.4 \begin{align*}{\frac{1}{b} \left ({\frac{{{\rm e}^{c}}\sqrt{\pi }{\it Erf} \left ( bx \right ) }{2}}-{\frac{{{\rm e}^{c}}\sqrt{\pi } \left ({\it Erf} \left ( bx \right ) \right ) ^{2}}{4}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-b^2*x^2+c)*erfc(b*x),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-b^2*x^2+c)*erfc(b*x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.16173, size = 63, normalized size = 3. \begin{align*} -\frac{\sqrt{\pi }{\left (\operatorname{erf}\left (b x\right )^{2} - 2 \, \operatorname{erf}\left (b x\right )\right )} e^{c}}{4 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-b^2*x^2+c)*erfc(b*x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.691919, size = 24, normalized size = 1.14 \begin{align*} \begin{cases} - \frac{\sqrt{\pi } e^{c} \operatorname{erfc}^{2}{\left (b x \right )}}{4 b} & \text{for}\: b \neq 0 \\x e^{c} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-b**2*x**2+c)*erfc(b*x),x)

[Out]

Piecewise((-sqrt(pi)*exp(c)*erfc(b*x)**2/(4*b), Ne(b, 0)), (x*exp(c), 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} + c\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-b^2*x^2+c)*erfc(b*x),x, algorithm="giac")

[Out]

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