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

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

[Out]

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

________________________________________________________________________________________

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

Antiderivative was successfully verified.

[In]

Integrate[Erf[x],x]

[Out]

1/(E^x^2*Sqrt[Pi]) + x*Erf[x]

________________________________________________________________________________________

fricas [A]  time = 0.41, size = 20, normalized size = 1.11 \[ \frac {\pi x \operatorname {erf}\relax (x) + \sqrt {\pi } e^{\left (-x^{2}\right )}}{\pi } \]

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

________________________________________________________________________________________

giac [A]  time = 1.07, size = 15, normalized size = 0.83 \[ x \operatorname {erf}\relax (x) + \frac {e^{\left (-x^{2}\right )}}{\sqrt {\pi }} \]

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)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 16, normalized size = 0.89 \[ x \erf \relax (x )+\frac {{\mathrm e}^{-x^{2}}}{\sqrt {\pi }} \]

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)

________________________________________________________________________________________

maxima [A]  time = 0.42, size = 15, normalized size = 0.83 \[ x \operatorname {erf}\relax (x) + \frac {e^{\left (-x^{2}\right )}}{\sqrt {\pi }} \]

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)

________________________________________________________________________________________

mupad [B]  time = 0.15, size = 15, normalized size = 0.83 \[ \frac {{\mathrm {e}}^{-x^2}}{\sqrt {\pi }}+x\,\mathrm {erf}\relax (x) \]

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)

________________________________________________________________________________________

sympy [A]  time = 0.31, size = 15, normalized size = 0.83 \[ x \operatorname {erf}{\relax (x )} + \frac {e^{- x^{2}}}{\sqrt {\pi }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erf(x),x)

[Out]

x*erf(x) + exp(-x**2)/sqrt(pi)

________________________________________________________________________________________