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

Optimal. Leaf size=36 \[ \frac{e^{b^2 x^2+c} \text{Erfc}(b x)}{2 b^2}+\frac{e^c x}{\sqrt{\pi } b} \]

[Out]

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

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Rubi [A]  time = 0.0347616, antiderivative size = 36, 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 = {6383, 8} \[ \frac{e^{b^2 x^2+c} \text{Erfc}(b x)}{2 b^2}+\frac{e^c x}{\sqrt{\pi } b} \]

Antiderivative was successfully verified.

[In]

Int[E^(c + b^2*x^2)*x*Erfc[b*x],x]

[Out]

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

Rule 6383

Int[E^((c_.) + (d_.)*(x_)^2)*Erfc[(a_.) + (b_.)*(x_)]*(x_), x_Symbol] :> Simp[(E^(c + d*x^2)*Erfc[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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int e^{c+b^2 x^2} x \text{erfc}(b x) \, dx &=\frac{e^{c+b^2 x^2} \text{erfc}(b x)}{2 b^2}+\frac{\int e^c \, dx}{b \sqrt{\pi }}\\ &=\frac{e^c x}{b \sqrt{\pi }}+\frac{e^{c+b^2 x^2} \text{erfc}(b x)}{2 b^2}\\ \end{align*}

Mathematica [A]  time = 0.0199603, size = 36, normalized size = 1. \[ \frac{e^{b^2 x^2+c} \text{Erfc}(b x)}{2 b^2}+\frac{e^c x}{\sqrt{\pi } b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.138, size = 51, normalized size = 1.4 \begin{align*}{\frac{2\,{{\rm e}^{{b}^{2}{x}^{2}+c}}{{\rm e}^{-{b}^{2}{x}^{2}}}xb+{{\rm e}^{{b}^{2}{x}^{2}+c}}{\it erfc} \left ( bx \right ) \sqrt{\pi }}{2\,{b}^{2}\sqrt{\pi }}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*exp(b^2*x^2+c)*exp(-b^2*x^2)*x*b+exp(b^2*x^2+c)*erfc(b*x)*Pi^(1/2))/Pi^(1/2)/b^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \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)*x*erfc(b*x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.10251, size = 97, normalized size = 2.69 \begin{align*} \frac{2 \, \sqrt{\pi } b x e^{c} +{\left (\pi - \pi \operatorname{erf}\left (b x\right )\right )} e^{\left (b^{2} x^{2} + c\right )}}{2 \, \pi b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 22.4207, size = 41, normalized size = 1.14 \begin{align*} \begin{cases} \frac{x e^{c}}{\sqrt{\pi } b} + \frac{e^{c} e^{b^{2} x^{2}} \operatorname{erfc}{\left (b x \right )}}{2 b^{2}} & \text{for}\: b \neq 0 \\\frac{x^{2} e^{c}}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x*exp(c)/(sqrt(pi)*b) + exp(c)*exp(b**2*x**2)*erfc(b*x)/(2*b**2), Ne(b, 0)), (x**2*exp(c)/2, True))

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

[Out]

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