\(\int e^{-b^2 x^2} \text {erfc}(b x) \, dx\) [186]

Optimal result

Integrand size = 15, antiderivative size = 18 \[ \int e^{-b^2 x^2} \text {erfc}(b x) \, dx=-\frac {\sqrt {\pi } \text {erfc}(b x)^2}{4 b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {6509, 30} \[ \int e^{-b^2 x^2} \text {erfc}(b x) \, dx=-\frac {\sqrt {\pi } \text {erfc}(b x)^2}{4 b} \]

[In]

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

[Out]

-1/4*(Sqrt[Pi]*Erfc[b*x]^2)/b

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6509

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]

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

Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int e^{-b^2 x^2} \text {erfc}(b x) \, dx=-\frac {\sqrt {\pi } \text {erfc}(b x)^2}{4 b} \]

[In]

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

[Out]

-1/4*(Sqrt[Pi]*Erfc[b*x]^2)/b

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22

method	result	size
default	\(\frac {\sqrt {\pi }\, \left (-\frac {\operatorname {erf}\left (b x \right )^{2}}{2}+\operatorname {erf}\left (b x \right )\right )}{2 b}\)	\(22\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.17 \[ \int e^{-b^2 x^2} \text {erfc}(b x) \, dx=-\frac {\sqrt {\pi } {\left (\operatorname {erf}\left (b x\right )^{2} - 2 \, \operatorname {erf}\left (b x\right )\right )}}{4 \, b} \]

[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

Sympy [A] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int e^{-b^2 x^2} \text {erfc}(b x) \, dx=\begin {cases} - \frac {\sqrt {\pi } \operatorname {erfc}^{2}{\left (b x \right )}}{4 b} & \text {for}\: b \neq 0 \\x & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int e^{-b^2 x^2} \text {erfc}(b x) \, dx=-\frac {\sqrt {\pi } \operatorname {erfc}\left (b x\right )^{2}}{4 \, b} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

integrate(erfc(b*x)*e^(-b^2*x^2), x)

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int e^{-b^2 x^2} \text {erfc}(b x) \, dx=-\frac {\sqrt {\pi }\,{\mathrm {erfc}\left (b\,x\right )}^2}{4\,b} \]

[In]

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

[Out]

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