3.11 \(\int \text{Erf}(b x) \, dx\)

Optimal. Leaf size=26 \[ \frac{e^{-b^2 x^2}}{\sqrt{\pi } b}+x \text{Erf}(b x) \]

[Out]

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

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Rubi [A]  time = 0.005257, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6349} \[ \frac{e^{-b^2 x^2}}{\sqrt{\pi } b}+x \text{Erf}(b x) \]

Antiderivative was successfully verified.

[In]

Int[Erf[b*x],x]

[Out]

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

Mathematica [A]  time = 0.009682, size = 26, normalized size = 1. \[ \frac{e^{-b^2 x^2}}{\sqrt{\pi } b}+x \text{Erf}(b x) \]

Antiderivative was successfully verified.

[In]

Integrate[Erf[b*x],x]

[Out]

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

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Maple [A]  time = 0.044, size = 26, normalized size = 1. \begin{align*}{\frac{1}{b} \left ({\it Erf} \left ( bx \right ) bx+{\frac{{{\rm e}^{-{b}^{2}{x}^{2}}}}{\sqrt{\pi }}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erf(b*x),x)

[Out]

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

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Maxima [A]  time = 1.08567, size = 34, normalized size = 1.31 \begin{align*} \frac{b x \operatorname{erf}\left (b x\right ) + \frac{e^{\left (-b^{2} x^{2}\right )}}{\sqrt{\pi }}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.52973, size = 68, normalized size = 2.62 \begin{align*} \frac{\pi b x \operatorname{erf}\left (b x\right ) + \sqrt{\pi } e^{\left (-b^{2} x^{2}\right )}}{\pi b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.36341, size = 24, normalized size = 0.92 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Giac [A]  time = 1.23808, size = 31, normalized size = 1.19 \begin{align*} x \operatorname{erf}\left (b x\right ) + \frac{e^{\left (-b^{2} x^{2}\right )}}{\sqrt{\pi } b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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