\(\int \text {erfc}(a+b x) \, dx\) [121]

Optimal result

Integrand size = 6, antiderivative size = 37 \[ \int \text {erfc}(a+b x) \, dx=-\frac {e^{-(a+b x)^2}}{b \sqrt {\pi }}+\frac {(a+b x) \text {erfc}(a+b x)}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6485} \[ \int \text {erfc}(a+b x) \, dx=\frac {(a+b x) \text {erfc}(a+b x)}{b}-\frac {e^{-(a+b x)^2}}{\sqrt {\pi } b} \]

[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 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^{-(a+b x)^2}}{b \sqrt {\pi }}+\frac {(a+b x) \text {erfc}(a+b x)}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.14 \[ \int \text {erfc}(a+b x) \, dx=-\frac {e^{-(a+b x)^2}}{b \sqrt {\pi }}-\frac {a \text {erf}(a+b x)}{b}+x \text {erfc}(a+b x) \]

[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]

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.89

method	result	size
derivativedivides	\(\frac {\left (b x +a \right ) \operatorname {erfc}\left (b x +a \right )-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}}}{\sqrt {\pi }}}{b}\)	\(33\)
default	\(\frac {\left (b x +a \right ) \operatorname {erfc}\left (b x +a \right )-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}}}{\sqrt {\pi }}}{b}\)	\(33\)
parallelrisch	\(\frac {x \,\operatorname {erfc}\left (b x +a \right ) \sqrt {\pi }\, b +a \,\operatorname {erfc}\left (b x +a \right ) \sqrt {\pi }-{\mathrm e}^{-\left (b x +a \right )^{2}}}{\sqrt {\pi }\, b}\)	\(44\)
parts	\(x \,\operatorname {erfc}\left (b x +a \right )+\frac {2 b \left (-\frac {{\mathrm e}^{-b^{2} x^{2}-2 a b x -a^{2}}}{2 b^{2}}-\frac {a \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{2 b^{2}}\right )}{\sqrt {\pi }}\)	\(57\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.43 \[ \int \text {erfc}(a+b x) \, dx=\frac {\pi b x - {\left (\pi b x + \pi a\right )} \operatorname {erf}\left (b x + a\right ) - \sqrt {\pi } e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{\pi b} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.43 \[ \int \text {erfc}(a+b x) \, dx=\begin {cases} \frac {a \operatorname {erfc}{\left (a + b x \right )}}{b} + x \operatorname {erfc}{\left (a + b x \right )} - \frac {e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{\sqrt {\pi } b} & \text {for}\: b \neq 0 \\x \operatorname {erfc}{\left (a \right )} & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.86 \[ \int \text {erfc}(a+b x) \, dx=\frac {{\left (b x + a\right )} \operatorname {erfc}\left (b x + a\right ) - \frac {e^{\left (-{\left (b x + a\right )}^{2}\right )}}{\sqrt {\pi }}}{b} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.62 \[ \int \text {erfc}(a+b x) \, dx=-x \operatorname {erf}\left (b x + a\right ) + x + \frac {\frac {\sqrt {\pi } a \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{b} - \frac {e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}}{\sqrt {\pi }} \]

[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)

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.32 \[ \int \text {erfc}(a+b x) \, dx=x\,\mathrm {erfc}\left (a+b\,x\right )+\frac {a\,\mathrm {erfc}\left (a+b\,x\right )}{b}-\frac {{\mathrm {e}}^{-b^2\,x^2}\,{\mathrm {e}}^{-a^2}\,{\mathrm {e}}^{-2\,a\,b\,x}}{b\,\sqrt {\pi }} \]

[In]

int(erfc(a + b*x),x)

[Out]

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