3.48 \(\int e^{c-b^2 x^2} \text{Erf}(b x) \, dx\)

Optimal. Leaf size=21 \[ \frac{\sqrt{\pi } e^c \text{Erf}(b x)^2}{4 b} \]

[Out]

(E^c*Sqrt[Pi]*Erf[b*x]^2)/(4*b)

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Rubi [A]  time = 0.01828, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {6373, 30} \[ \frac{\sqrt{\pi } e^c \text{Erf}(b x)^2}{4 b} \]

Antiderivative was successfully verified.

[In]

Int[E^(c - b^2*x^2)*Erf[b*x],x]

[Out]

(E^c*Sqrt[Pi]*Erf[b*x]^2)/(4*b)

Rule 6373

Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(b_.)*(x_)]^(n_.), x_Symbol] :> Dist[(E^c*Sqrt[Pi])/(2*b), Subst[Int[x^n, x],
x, Erf[b*x]], x] /; FreeQ[{b, c, d, n}, x] && EqQ[d, -b^2]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int e^{c-b^2 x^2} \text{erf}(b x) \, dx &=\frac{\left (e^c \sqrt{\pi }\right ) \operatorname{Subst}(\int x \, dx,x,\text{erf}(b x))}{2 b}\\ &=\frac{e^c \sqrt{\pi } \text{erf}(b x)^2}{4 b}\\ \end{align*}

Mathematica [A]  time = 0.0050945, size = 21, normalized size = 1. \[ \frac{\sqrt{\pi } e^c \text{Erf}(b x)^2}{4 b} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(c - b^2*x^2)*Erf[b*x],x]

[Out]

(E^c*Sqrt[Pi]*Erf[b*x]^2)/(4*b)

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Maple [A]  time = 0.111, size = 17, normalized size = 0.8 \begin{align*}{\frac{{{\rm e}^{c}} \left ({\it Erf} \left ( bx \right ) \right ) ^{2}\sqrt{\pi }}{4\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-b^2*x^2+c)*erf(b*x),x)

[Out]

1/4*exp(c)*erf(b*x)^2*Pi^(1/2)/b

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Maxima [A]  time = 1.00857, size = 22, normalized size = 1.05 \begin{align*} \frac{\sqrt{\pi } \operatorname{erf}\left (b x\right )^{2} e^{c}}{4 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-b^2*x^2+c)*erf(b*x),x, algorithm="maxima")

[Out]

1/4*sqrt(pi)*erf(b*x)^2*e^c/b

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Fricas [A]  time = 2.71171, size = 42, normalized size = 2. \begin{align*} \frac{\sqrt{\pi } \operatorname{erf}\left (b x\right )^{2} e^{c}}{4 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-b^2*x^2+c)*erf(b*x),x, algorithm="fricas")

[Out]

1/4*sqrt(pi)*erf(b*x)^2*e^c/b

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Sympy [A]  time = 1.18538, size = 19, normalized size = 0.9 \begin{align*} \begin{cases} \frac{\sqrt{\pi } e^{c} \operatorname{erf}^{2}{\left (b x \right )}}{4 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(exp(-b**2*x**2+c)*erf(b*x),x)

[Out]

Piecewise((sqrt(pi)*exp(c)*erf(b*x)**2/(4*b), Ne(b, 0)), (0, True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{erf}\left (b x\right ) e^{\left (-b^{2} x^{2} + c\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-b^2*x^2+c)*erf(b*x),x, algorithm="giac")

[Out]

integrate(erf(b*x)*e^(-b^2*x^2 + c), x)