3.121 \(\int \text{Erfc}(a+b x) \, dx\)

Optimal. Leaf size=37 \[ \frac{(a+b x) \text{Erfc}(a+b x)}{b}-\frac{e^{-(a+b x)^2}}{\sqrt{\pi } b} \]

[Out]

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

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Rubi [A]  time = 0.0067298, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6350} \[ \frac{(a+b x) \text{Erfc}(a+b x)}{b}-\frac{e^{-(a+b x)^2}}{\sqrt{\pi } b} \]

Antiderivative was successfully verified.

[In]

Int[Erfc[a + b*x],x]

[Out]

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

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

Mathematica [A]  time = 0.0407458, size = 42, normalized size = 1.14 \[ -\frac{a \text{Erf}(a+b x)}{b}+x \text{Erfc}(a+b x)-\frac{e^{-(a+b x)^2}}{\sqrt{\pi } b} \]

Antiderivative was successfully verified.

[In]

Integrate[Erfc[a + b*x],x]

[Out]

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

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Maple [A]  time = 0.042, size = 33, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ( \left ( bx+a \right ){\it erfc} \left ( bx+a \right ) -{\frac{{{\rm e}^{- \left ( bx+a \right ) ^{2}}}}{\sqrt{\pi }}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erfc(b*x+a),x)

[Out]

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

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Maxima [A]  time = 1.12613, size = 43, normalized size = 1.16 \begin{align*} \frac{{\left (b x + a\right )} \operatorname{erfc}\left (b x + a\right ) - \frac{e^{\left (-{\left (b x + a\right )}^{2}\right )}}{\sqrt{\pi }}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.07231, size = 119, normalized size = 3.22 \begin{align*} \frac{\pi b x -{\left (\pi b x + \pi a\right )} \operatorname{erf}\left (b x + a\right ) - \sqrt{\pi } e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{\pi b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.782964, size = 53, normalized size = 1.43 \begin{align*} \begin{cases} \frac{a \operatorname{erfc}{\left (a + b x \right )}}{b} + x \operatorname{erfc}{\left (a + b x \right )} - \frac{e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{\sqrt{\pi } b} & \text{for}\: b \neq 0 \\x \operatorname{erfc}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfc(b*x+a),x)

[Out]

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

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Giac [A]  time = 1.31475, size = 81, normalized size = 2.19 \begin{align*} -x \operatorname{erf}\left (b x + a\right ) + x + \frac{\frac{\sqrt{\pi } a \operatorname{erf}\left (-b{\left (x + \frac{a}{b}\right )}\right )}{b} - \frac{e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}}{\sqrt{\pi }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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