\(\int \text {erf}(b x) \, dx\) [11]

Optimal result

Integrand size = 4, antiderivative size = 26 \[ \int \text {erf}(b x) \, dx=\frac {e^{-b^2 x^2}}{b \sqrt {\pi }}+x \text {erf}(b x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6484} \[ \int \text {erf}(b x) \, dx=\frac {e^{-b^2 x^2}}{\sqrt {\pi } b}+x \text {erf}(b x) \]

[In]

Int[Erf[b*x],x]

[Out]

1/(b*E^(b^2*x^2)*Sqrt[Pi]) + x*Erf[b*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^{-b^2 x^2}}{b \sqrt {\pi }}+x \text {erf}(b x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \text {erf}(b x) \, dx=\frac {e^{-b^2 x^2}}{b \sqrt {\pi }}+x \text {erf}(b x) \]

[In]

Integrate[Erf[b*x],x]

[Out]

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

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92

method	result	size
parts	\(x \,\operatorname {erf}\left (b x \right )+\frac {{\mathrm e}^{-b^{2} x^{2}}}{\sqrt {\pi }\, b}\)	\(24\)
derivativedivides	\(\frac {\operatorname {erf}\left (b x \right ) b x +\frac {{\mathrm e}^{-b^{2} x^{2}}}{\sqrt {\pi }}}{b}\)	\(26\)
default	\(\frac {\operatorname {erf}\left (b x \right ) b x +\frac {{\mathrm e}^{-b^{2} x^{2}}}{\sqrt {\pi }}}{b}\)	\(26\)
parallelrisch	\(\frac {b x \,\operatorname {erf}\left (b x \right ) \sqrt {\pi }+{\mathrm e}^{-b^{2} x^{2}}}{\sqrt {\pi }\, b}\)	\(28\)
meijerg	\(\frac {-2+2 \,{\mathrm e}^{-b^{2} x^{2}}+2 b x \,\operatorname {erf}\left (b x \right ) \sqrt {\pi }}{2 \sqrt {\pi }\, b}\)	\(33\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \text {erf}(b x) \, dx=\frac {\pi b x \operatorname {erf}\left (b x\right ) + \sqrt {\pi } e^{\left (-b^{2} x^{2}\right )}}{\pi b} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \text {erf}(b x) \, dx=\begin {cases} x \operatorname {erf}{\left (b x \right )} + \frac {e^{- b^{2} x^{2}}}{\sqrt {\pi } b} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

integrate(erf(b*x),x)

[Out]

Piecewise((x*erf(b*x) + exp(-b**2*x**2)/(sqrt(pi)*b), Ne(b, 0)), (0, True))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \text {erf}(b x) \, dx=\frac {b x \operatorname {erf}\left (b x\right ) + \frac {e^{\left (-b^{2} x^{2}\right )}}{\sqrt {\pi }}}{b} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \text {erf}(b x) \, dx=x \operatorname {erf}\left (b x\right ) + \frac {e^{\left (-b^{2} x^{2}\right )}}{\sqrt {\pi } b} \]

[In]

integrate(erf(b*x),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.12 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \text {erf}(b x) \, dx=x\,\mathrm {erf}\left (b\,x\right )+\frac {{\mathrm {e}}^{-b^2\,x^2}}{b\,\sqrt {\pi }} \]

[In]

int(erf(b*x),x)

[Out]

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