3.276 \(\int \text{Erf}(x) \, dx\)

Optimal. Leaf size=18 \[ x \text{Erf}(x)+\frac{e^{-x^2}}{\sqrt{\pi }} \]

[Out]

1/(E^x^2*Sqrt[Pi]) + x*Erf[x]

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Rubi [A]  time = 0.0053284, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 2, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6349} \[ x \text{Erf}(x)+\frac{e^{-x^2}}{\sqrt{\pi }} \]

Antiderivative was successfully verified.

[In]

Int[Erf[x],x]

[Out]

1/(E^x^2*Sqrt[Pi]) + x*Erf[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}(x) \, dx &=\frac{e^{-x^2}}{\sqrt{\pi }}+x \text{erf}(x)\\ \end{align*}

Mathematica [A]  time = 0.0066766, size = 18, normalized size = 1. \[ x \text{Erf}(x)+\frac{e^{-x^2}}{\sqrt{\pi }} \]

Antiderivative was successfully verified.

[In]

Integrate[Erf[x],x]

[Out]

1/(E^x^2*Sqrt[Pi]) + x*Erf[x]

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Maple [A]  time = 0.006, size = 16, normalized size = 0.9 \begin{align*} x{\it Erf} \left ( x \right ) +{\frac{{{\rm e}^{-{x}^{2}}}}{\sqrt{\pi }}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erf(x),x)

[Out]

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

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Maxima [A]  time = 0.928914, size = 20, normalized size = 1.11 \begin{align*} x \operatorname{erf}\left (x\right ) + \frac{e^{\left (-x^{2}\right )}}{\sqrt{\pi }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*erf(x) + e^(-x^2)/sqrt(pi)

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Fricas [A]  time = 1.80279, size = 51, normalized size = 2.83 \begin{align*} \frac{\pi x \operatorname{erf}\left (x\right ) + \sqrt{\pi } e^{\left (-x^{2}\right )}}{\pi } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(pi*x*erf(x) + sqrt(pi)*e^(-x^2))/pi

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Sympy [A]  time = 0.307698, size = 15, normalized size = 0.83 \begin{align*} x \operatorname{erf}{\left (x \right )} + \frac{e^{- x^{2}}}{\sqrt{\pi }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erf(x),x)

[Out]

x*erf(x) + exp(-x**2)/sqrt(pi)

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Giac [A]  time = 1.07719, size = 20, normalized size = 1.11 \begin{align*} x \operatorname{erf}\left (x\right ) + \frac{e^{\left (-x^{2}\right )}}{\sqrt{\pi }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erf(x),x, algorithm="giac")

[Out]

x*erf(x) + e^(-x^2)/sqrt(pi)