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

Optimal result

Integrand size = 2, antiderivative size = 18 \[ \int \text {erf}(x) \, dx=\frac {e^{-x^2}}{\sqrt {\pi }}+x \text {erf}(x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6484} \[ \int \text {erf}(x) \, dx=x \text {erf}(x)+\frac {e^{-x^2}}{\sqrt {\pi }} \]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \text {erf}(x) \, dx=\frac {e^{-x^2}}{\sqrt {\pi }}+x \text {erf}(x) \]

[In]

Integrate[Erf[x],x]

[Out]

1/(E^x^2*Sqrt[Pi]) + x*Erf[x]

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89

method	result	size
default	\(x \,\operatorname {erf}\left (x \right )+\frac {{\mathrm e}^{-x^{2}}}{\sqrt {\pi }}\)	\(16\)
parts	\(x \,\operatorname {erf}\left (x \right )+\frac {{\mathrm e}^{-x^{2}}}{\sqrt {\pi }}\)	\(16\)
parallelrisch	\(\frac {x \sqrt {\pi }\, \operatorname {erf}\left (x \right )+{\mathrm e}^{-x^{2}}}{\sqrt {\pi }}\)	\(19\)
meijerg	\(\frac {-2+2 \,{\mathrm e}^{-x^{2}}+2 x \sqrt {\pi }\, \operatorname {erf}\left (x \right )}{2 \sqrt {\pi }}\)	\(24\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \text {erf}(x) \, dx=\frac {\pi x \operatorname {erf}\left (x\right ) + \sqrt {\pi } e^{\left (-x^{2}\right )}}{\pi } \]

[In]

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

[Out]

(pi*x*erf(x) + sqrt(pi)*e^(-x^2))/pi

Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \text {erf}(x) \, dx=x \operatorname {erf}{\left (x \right )} + \frac {e^{- x^{2}}}{\sqrt {\pi }} \]

[In]

integrate(erf(x),x)

[Out]

x*erf(x) + exp(-x**2)/sqrt(pi)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \text {erf}(x) \, dx=x \operatorname {erf}\left (x\right ) + \frac {e^{\left (-x^{2}\right )}}{\sqrt {\pi }} \]

[In]

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

[Out]

x*erf(x) + e^(-x^2)/sqrt(pi)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \text {erf}(x) \, dx=x \operatorname {erf}\left (x\right ) + \frac {e^{\left (-x^{2}\right )}}{\sqrt {\pi }} \]

[In]

integrate(erf(x),x, algorithm="giac")

[Out]

x*erf(x) + e^(-x^2)/sqrt(pi)

Mupad [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \text {erf}(x) \, dx=\frac {{\mathrm {e}}^{-x^2}}{\sqrt {\pi }}+x\,\mathrm {erf}\left (x\right ) \]

[In]

int(erf(x),x)

[Out]

exp(-x^2)/pi^(1/2) + x*erf(x)