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

Optimal. Leaf size=24 \[ (a+x) \text{Erf}(a+x)+\frac{e^{-(a+x)^2}}{\sqrt{\pi }} \]

[Out]

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

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Rubi [A]  time = 0.0076567, antiderivative size = 24, 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 = {6349} \[ (a+x) \text{Erf}(a+x)+\frac{e^{-(a+x)^2}}{\sqrt{\pi }} \]

Antiderivative was successfully verified.

[In]

Int[Erf[a + x],x]

[Out]

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

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

Mathematica [A]  time = 0.0276084, size = 24, normalized size = 1. \[ (a+x) \text{Erf}(a+x)+\frac{e^{-(a+x)^2}}{\sqrt{\pi }} \]

Antiderivative was successfully verified.

[In]

Integrate[Erf[a + x],x]

[Out]

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

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Maple [A]  time = 0.001, size = 22, normalized size = 0.9 \begin{align*} \left ( a+x \right ){\it Erf} \left ( a+x \right ) +{\frac{{{\rm e}^{- \left ( a+x \right ) ^{2}}}}{\sqrt{\pi }}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erf(a+x),x)

[Out]

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

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Maxima [A]  time = 0.931253, size = 28, normalized size = 1.17 \begin{align*}{\left (a + x\right )} \operatorname{erf}\left (a + x\right ) + \frac{e^{\left (-{\left (a + x\right )}^{2}\right )}}{\sqrt{\pi }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erf(a+x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.87621, size = 88, normalized size = 3.67 \begin{align*} \frac{{\left (\pi a + \pi x\right )} \operatorname{erf}\left (a + x\right ) + \sqrt{\pi } e^{\left (-a^{2} - 2 \, a x - x^{2}\right )}}{\pi } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erf(a+x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.497251, size = 36, normalized size = 1.5 \begin{align*} a \operatorname{erf}{\left (a + x \right )} + x \operatorname{erf}{\left (a + x \right )} + \frac{e^{- a^{2}} e^{- x^{2}} e^{- 2 a x}}{\sqrt{\pi }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erf(a+x),x)

[Out]

a*erf(a + x) + x*erf(a + x) + exp(-a**2)*exp(-x**2)*exp(-2*a*x)/sqrt(pi)

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Giac [A]  time = 1.07223, size = 50, normalized size = 2.08 \begin{align*} x \operatorname{erf}\left (a + x\right ) + \frac{\sqrt{\pi } a \operatorname{erf}\left (a + x\right ) + e^{\left (-a^{2} - 2 \, a x - x^{2}\right )}}{\sqrt{\pi }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erf(a+x),x, algorithm="giac")

[Out]

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