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]

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

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Rubi [A]  time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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*}

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Mathematica [A]  time = 0.01, size = 26, normalized size = 1.00 \[ \frac {e^{-b^2 x^2}}{\sqrt {\pi } b}+x \text {erf}(b x) \]

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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fricas [A]  time = 0.38, size = 29, normalized size = 1.12 \[ \frac {\pi b x \operatorname {erf}\left (b x\right ) + \sqrt {\pi } e^{\left (-b^{2} x^{2}\right )}}{\pi b} \]

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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giac [A]  time = 0.44, size = 23, normalized size = 0.88 \[ x \operatorname {erf}\left (b x\right ) + \frac {e^{\left (-b^{2} x^{2}\right )}}{\sqrt {\pi } b} \]

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)

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maple [A]  time = 0.01, size = 26, normalized size = 1.00 \[ \frac {b x \erf \left (b x \right )+\frac {{\mathrm e}^{-b^{2} x^{2}}}{\sqrt {\pi }}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erf(b*x),x)

[Out]

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

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maxima [A]  time = 0.38, size = 25, normalized size = 0.96 \[ \frac {b x \operatorname {erf}\left (b x\right ) + \frac {e^{\left (-b^{2} x^{2}\right )}}{\sqrt {\pi }}}{b} \]

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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mupad [B]  time = 0.08, size = 23, normalized size = 0.88 \[ x\,\mathrm {erf}\left (b\,x\right )+\frac {{\mathrm {e}}^{-b^2\,x^2}}{b\,\sqrt {\pi }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erf(b*x),x)

[Out]

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

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sympy [A]  time = 0.39, size = 24, normalized size = 0.92 \[ \begin {cases} x \operatorname {erf}{\left (b x \right )} + \frac {e^{- b^{2} x^{2}}}{\sqrt {\pi } b} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]

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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