\(\int \text {erfc}(b x) \, dx\) [114]

Optimal result

Integrand size = 4, antiderivative size = 27 \[ \int \text {erfc}(b x) \, dx=-\frac {e^{-b^2 x^2}}{b \sqrt {\pi }}+x \text {erfc}(b x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6485} \[ \int \text {erfc}(b x) \, dx=x \text {erfc}(b x)-\frac {e^{-b^2 x^2}}{\sqrt {\pi } b} \]

[In]

Int[Erfc[b*x],x]

[Out]

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

Rule 6485

Int[Erfc[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[(a + b*x)*(Erfc[a + b*x]/b), x] - Simp[1/(b*Sqrt[Pi]*E^(a + b*
x)^2), x] /; FreeQ[{a, b}, x]

Rubi steps \begin{align*} \text {integral}& = -\frac {e^{-b^2 x^2}}{b \sqrt {\pi }}+x \text {erfc}(b x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \text {erfc}(b x) \, dx=-\frac {e^{-b^2 x^2}}{b \sqrt {\pi }}+x \text {erfc}(b x) \]

[In]

Integrate[Erfc[b*x],x]

[Out]

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

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93

method	result	size
parts	\(x \,\operatorname {erfc}\left (b x \right )-\frac {{\mathrm e}^{-b^{2} x^{2}}}{\sqrt {\pi }\, b}\)	\(25\)
derivativedivides	\(\frac {b x \,\operatorname {erfc}\left (b x \right )-\frac {{\mathrm e}^{-b^{2} x^{2}}}{\sqrt {\pi }}}{b}\)	\(27\)
default	\(\frac {b x \,\operatorname {erfc}\left (b x \right )-\frac {{\mathrm e}^{-b^{2} x^{2}}}{\sqrt {\pi }}}{b}\)	\(27\)
parallelrisch	\(\frac {x \,\operatorname {erfc}\left (b x \right ) \sqrt {\pi }\, b -{\mathrm e}^{-b^{2} x^{2}}}{\sqrt {\pi }\, b}\)	\(30\)

[In]

int(erfc(b*x),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.30 \[ \int \text {erfc}(b x) \, dx=-\frac {\pi b x \operatorname {erf}\left (b x\right ) - \pi b x + \sqrt {\pi } e^{\left (-b^{2} x^{2}\right )}}{\pi b} \]

[In]

integrate(erfc(b*x),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \text {erfc}(b x) \, dx=\begin {cases} x \operatorname {erfc}{\left (b x \right )} - \frac {e^{- b^{2} x^{2}}}{\sqrt {\pi } b} & \text {for}\: b \neq 0 \\x & \text {otherwise} \end {cases} \]

[In]

integrate(erfc(b*x),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \text {erfc}(b x) \, dx=\frac {b x \operatorname {erfc}\left (b x\right ) - \frac {e^{\left (-b^{2} x^{2}\right )}}{\sqrt {\pi }}}{b} \]

[In]

integrate(erfc(b*x),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \text {erfc}(b x) \, dx=-x \operatorname {erf}\left (b x\right ) + x - \frac {e^{\left (-b^{2} x^{2}\right )}}{\sqrt {\pi } b} \]

[In]

integrate(erfc(b*x),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \text {erfc}(b x) \, dx=x\,\mathrm {erfc}\left (b\,x\right )-\frac {{\mathrm {e}}^{-b^2\,x^2}}{b\,\sqrt {\pi }} \]

[In]

int(erfc(b*x),x)

[Out]

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