3.114 \(\int \text{Erfc}(b x) \, dx\)

Optimal. Leaf size=27 \[ x \text{Erfc}(b x)-\frac{e^{-b^2 x^2}}{\sqrt{\pi } b} \]

[Out]

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

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Rubi [A]  time = 0.0049246, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6350} \[ x \text{Erfc}(b x)-\frac{e^{-b^2 x^2}}{\sqrt{\pi } b} \]

Antiderivative was successfully verified.

[In]

Int[Erfc[b*x],x]

[Out]

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

Rule 6350

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

Mathematica [A]  time = 0.0021854, size = 27, normalized size = 1. \[ x \text{Erfc}(b x)-\frac{e^{-b^2 x^2}}{\sqrt{\pi } b} \]

Antiderivative was successfully verified.

[In]

Integrate[Erfc[b*x],x]

[Out]

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

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Maple [A]  time = 0.039, size = 27, normalized size = 1. \begin{align*}{\frac{1}{b} \left ( bx{\it erfc} \left ( bx \right ) -{\frac{{{\rm e}^{-{b}^{2}{x}^{2}}}}{\sqrt{\pi }}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erfc(b*x),x)

[Out]

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

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Maxima [A]  time = 1.03127, size = 35, normalized size = 1.3 \begin{align*} \frac{b x \operatorname{erfc}\left (b x\right ) - \frac{e^{\left (-b^{2} x^{2}\right )}}{\sqrt{\pi }}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.31983, size = 81, normalized size = 3. \begin{align*} -\frac{\pi b x \operatorname{erf}\left (b x\right ) - \pi b x + \sqrt{\pi } e^{\left (-b^{2} x^{2}\right )}}{\pi b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 0.346812, size = 24, normalized size = 0.89 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Giac [A]  time = 1.32128, size = 35, normalized size = 1.3 \begin{align*} -x \operatorname{erf}\left (b x\right ) + x - \frac{e^{\left (-b^{2} x^{2}\right )}}{\sqrt{\pi } b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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