3.3.76 \(\int \text {erf}(x) \, dx\) [276]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6484} \begin {gather*} x \text {Erf}(x)+\frac {e^{-x^2}}{\sqrt {\pi }} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Erf[x],x]

[Out]

1/(E^x^2*Sqrt[Pi]) + x*Erf[x]

Rule 6484

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*}

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Mathematica [A]
time = 0.01, size = 18, normalized size = 1.00 \begin {gather*} \frac {e^{-x^2}}{\sqrt {\pi }}+x \text {erf}(x) \end {gather*}

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 = 0.01, size = 16, normalized size = 0.89

method	result	size
default	\(x \erf \left (x \right )+\frac {{\mathrm e}^{-x^{2}}}{\sqrt {\pi }}\)	\(16\)
meijerg	\(\frac {-2+2 \,{\mathrm e}^{-x^{2}}+2 x \sqrt {\pi }\, \erf \left (x \right )}{2 \sqrt {\pi }}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erf(x),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 3.16, size = 15, normalized size = 0.83 \begin {gather*} x \operatorname {erf}\left (x\right ) + \frac {e^{\left (-x^{2}\right )}}{\sqrt {\pi }} \end {gather*}

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 = 0.95, size = 20, normalized size = 1.11 \begin {gather*} \frac {\pi x \operatorname {erf}\left (x\right ) + \sqrt {\pi } e^{\left (-x^{2}\right )}}{\pi } \end {gather*}

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 = 0.13, size = 15, normalized size = 0.83 \begin {gather*} x \operatorname {erf}{\left (x \right )} + \frac {e^{- x^{2}}}{\sqrt {\pi }} \end {gather*}

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 = 0.44, size = 15, normalized size = 0.83 \begin {gather*} x \operatorname {erf}\left (x\right ) + \frac {e^{\left (-x^{2}\right )}}{\sqrt {\pi }} \end {gather*}

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)

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Mupad [B]
time = 0.15, size = 15, normalized size = 0.83 \begin {gather*} \frac {{\mathrm {e}}^{-x^2}}{\sqrt {\pi }}+x\,\mathrm {erf}\left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erf(x),x)

[Out]

exp(-x^2)/pi^(1/2) + x*erf(x)

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