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

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6485} \begin {gather*} x \text {Erfc}(b x)-\frac {e^{-b^2 x^2}}{\sqrt {\pi } b} \end {gather*}

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

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Mathematica [A]
time = 0.00, size = 27, normalized size = 1.00 \begin {gather*} -\frac {e^{-b^2 x^2}}{b \sqrt {\pi }}+x \text {Erfc}(b x) \end {gather*}

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 = 0.18, size = 27, normalized size = 1.00

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.25, size = 26, normalized size = 0.96 \begin {gather*} \frac {b x \operatorname {erfc}\left (b x\right ) - \frac {e^{\left (-b^{2} x^{2}\right )}}{\sqrt {\pi }}}{b} \end {gather*}

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 = 0.35, size = 35, normalized size = 1.30 \begin {gather*} -\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 {gather*}

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 = 0.15, size = 24, normalized size = 0.89 \begin {gather*} \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 {gather*}

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 = 0.40, size = 26, normalized size = 0.96 \begin {gather*} -x \operatorname {erf}\left (b x\right ) + x - \frac {e^{\left (-b^{2} x^{2}\right )}}{\sqrt {\pi } b} \end {gather*}

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)

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Mupad [B]
time = 0.10, size = 24, normalized size = 0.89 \begin {gather*} x\,\mathrm {erfc}\left (b\,x\right )-\frac {{\mathrm {e}}^{-b^2\,x^2}}{b\,\sqrt {\pi }} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erfc(b*x),x)

[Out]

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

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