\(\int e^{c+b^2 x^2} x \text {erfc}(b x) \, dx\) [169]

Optimal result

Integrand size = 17, antiderivative size = 36 \[ \int e^{c+b^2 x^2} x \text {erfc}(b x) \, dx=\frac {e^c x}{b \sqrt {\pi }}+\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{2 b^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {6518, 8} \[ \int e^{c+b^2 x^2} x \text {erfc}(b x) \, dx=\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{2 b^2}+\frac {e^c x}{\sqrt {\pi } b} \]

[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 8

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

Rule 6518

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]

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00 \[ \int e^{c+b^2 x^2} x \text {erfc}(b x) \, dx=\frac {e^c x}{b \sqrt {\pi }}+\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{2 b^2} \]

[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)

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.42

method	result	size
default	\(\frac {2 \,{\mathrm e}^{b^{2} x^{2}+c} x \,{\mathrm e}^{-b^{2} x^{2}} b +{\mathrm e}^{b^{2} x^{2}+c} \operatorname {erfc}\left (b x \right ) \sqrt {\pi }}{2 \sqrt {\pi }\, b^{2}}\)	\(51\)
parallelrisch	\(\frac {2 \,{\mathrm e}^{b^{2} x^{2}+c} x \,{\mathrm e}^{-b^{2} x^{2}} b +{\mathrm e}^{b^{2} x^{2}+c} \operatorname {erfc}\left (b x \right ) \sqrt {\pi }}{2 \sqrt {\pi }\, b^{2}}\)	\(51\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.06 \[ \int e^{c+b^2 x^2} x \text {erfc}(b x) \, dx=\frac {2 \, \sqrt {\pi } b x e^{c} + {\left (\pi - \pi \operatorname {erf}\left (b x\right )\right )} e^{\left (b^{2} x^{2} + c\right )}}{2 \, \pi b^{2}} \]

[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)

Sympy [A] (verification not implemented)

Time = 1.18 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.14 \[ \int e^{c+b^2 x^2} x \text {erfc}(b x) \, dx=\begin {cases} \frac {x e^{c}}{\sqrt {\pi } b} + \frac {e^{c} e^{b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}}{2 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{2} e^{c}}{2} & \text {otherwise} \end {cases} \]

[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))

Maxima [F]

\[ \int e^{c+b^2 x^2} x \text {erfc}(b x) \, dx=\int { x \operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} \,d x } \]

[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)

Giac [F]

\[ \int e^{c+b^2 x^2} x \text {erfc}(b x) \, dx=\int { x \operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} \,d x } \]

[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)

Mupad [B] (verification not implemented)

Time = 4.84 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.83 \[ \int e^{c+b^2 x^2} x \text {erfc}(b x) \, dx=\frac {x\,{\mathrm {e}}^c}{b\,\sqrt {\pi }}+\frac {{\mathrm {e}}^{b^2\,x^2}\,{\mathrm {e}}^c\,\mathrm {erfc}\left (b\,x\right )}{2\,b^2} \]

[In]

int(x*exp(c + b^2*x^2)*erfc(b*x),x)

[Out]

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