\(\int e^{-b^2 x^2} x \text {erfc}(b x) \, dx\) [180]

Optimal result

Integrand size = 16, antiderivative size = 43 \[ \int e^{-b^2 x^2} x \text {erfc}(b x) \, dx=-\frac {\text {erf}\left (\sqrt {2} b x\right )}{2 \sqrt {2} b^2}-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{2 b^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6518, 2236} \[ \int e^{-b^2 x^2} x \text {erfc}(b x) \, dx=-\frac {\text {erf}\left (\sqrt {2} b x\right )}{2 \sqrt {2} b^2}-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{2 b^2} \]

[In]

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

[Out]

-1/2*Erf[Sqrt[2]*b*x]/(Sqrt[2]*b^2) - Erfc[b*x]/(2*b^2*E^(b^2*x^2))

Rule 2236

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F],
2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.91 \[ \int e^{-b^2 x^2} x \text {erfc}(b x) \, dx=-\frac {\sqrt {2} \text {erf}\left (\sqrt {2} b x\right )+2 e^{-b^2 x^2} \text {erfc}(b x)}{4 b^2} \]

[In]

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

[Out]

-1/4*(Sqrt[2]*Erf[Sqrt[2]*b*x] + (2*Erfc[b*x])/E^(b^2*x^2))/b^2

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.23

method	result	size
default	\(\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}}}{2 b}+\frac {\operatorname {erf}\left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}}{2 b}-\frac {\sqrt {2}\, \operatorname {erf}\left (b x \sqrt {2}\right )}{4 b}}{b}\)	\(53\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.09 \[ \int e^{-b^2 x^2} x \text {erfc}(b x) \, dx=-\frac {\sqrt {2} \sqrt {b^{2}} \operatorname {erf}\left (\sqrt {2} \sqrt {b^{2}} x\right ) - 2 \, {\left (b \operatorname {erf}\left (b x\right ) - b\right )} e^{\left (-b^{2} x^{2}\right )}}{4 \, b^{3}} \]

[In]

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

[Out]

-1/4*(sqrt(2)*sqrt(b^2)*erf(sqrt(2)*sqrt(b^2)*x) - 2*(b*erf(b*x) - b)*e^(-b^2*x^2))/b^3

Sympy [F]

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

[In]

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

[Out]

Integral(x*exp(-b**2*x**2)*erfc(b*x), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int e^{-b^2 x^2} x \text {erfc}(b x) \, dx=\int x\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right ) \,d x \]

[In]

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

[Out]

int(x*exp(-b^2*x^2)*erfc(b*x), x)