3.18 \(\int \text{Erf}(a+b x) \, dx\)

Optimal. Leaf size=36 \[ \frac{(a+b x) \text{Erf}(a+b x)}{b}+\frac{e^{-(a+b x)^2}}{\sqrt{\pi } b} \]

[Out]

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

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Rubi [A]  time = 0.0069472, antiderivative size = 36, 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 = {6349} \[ \frac{(a+b x) \text{Erf}(a+b x)}{b}+\frac{e^{-(a+b x)^2}}{\sqrt{\pi } b} \]

Antiderivative was successfully verified.

[In]

Int[Erf[a + b*x],x]

[Out]

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

Rule 6349

Int[Erf[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[((a + b*x)*Erf[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{erf}(a+b x) \, dx &=\frac{e^{-(a+b x)^2}}{b \sqrt{\pi }}+\frac{(a+b x) \text{erf}(a+b x)}{b}\\ \end{align*}

Mathematica [A]  time = 0.0402724, size = 35, normalized size = 0.97 \[ \left (\frac{a}{b}+x\right ) \text{Erf}(a+b x)+\frac{e^{-(a+b x)^2}}{\sqrt{\pi } b} \]

Antiderivative was successfully verified.

[In]

Integrate[Erf[a + b*x],x]

[Out]

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

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Maple [A]  time = 0.047, size = 32, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ({\it Erf} \left ( bx+a \right ) \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(erf(b*x+a),x)

[Out]

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

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Maxima [A]  time = 1.07452, size = 42, normalized size = 1.17 \begin{align*} \frac{{\left (b x + a\right )} \operatorname{erf}\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(erf(b*x+a),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.5475, size = 107, normalized size = 2.97 \begin{align*} \frac{{\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(erf(b*x+a),x, algorithm="fricas")

[Out]

((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 = 0.819628, size = 53, normalized size = 1.47 \begin{align*} \begin{cases} \frac{a \operatorname{erf}{\left (a + b x \right )}}{b} + x \operatorname{erf}{\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{erf}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.30072, size = 80, normalized size = 2.22 \begin{align*} x \operatorname{erf}\left (b x + a\right ) - \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(erf(b*x+a),x, algorithm="giac")

[Out]

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