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

Optimal. Leaf size=28 \[ \frac{\sqrt{\pi } e^c \text{Erf}(b x)^{n+1}}{2 b (n+1)} \]

[Out]

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

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Rubi [A]  time = 0.0367038, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {6373, 30} \[ \frac{\sqrt{\pi } e^c \text{Erf}(b x)^{n+1}}{2 b (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[E^(c - b^2*x^2)*Erf[b*x]^n,x]

[Out]

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

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)^n \, dx &=\frac{\left (e^c \sqrt{\pi }\right ) \operatorname{Subst}\left (\int x^n \, dx,x,\text{erf}(b x)\right )}{2 b}\\ &=\frac{e^c \sqrt{\pi } \text{erf}(b x)^{1+n}}{2 b (1+n)}\\ \end{align*}

Mathematica [A]  time = 0.0101597, size = 28, normalized size = 1. \[ \frac{\sqrt{\pi } e^c \text{Erf}(b x)^{n+1}}{2 b (n+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(c - b^2*x^2)*Erf[b*x]^n,x]

[Out]

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

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Maple [F]  time = 0.076, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{-{b}^{2}{x}^{2}+c}} \left ({\it Erf} \left ( bx \right ) \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{erf}\left (b x\right )^{n} 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)^n,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 3.31869, size = 65, normalized size = 2.32 \begin{align*} \frac{\sqrt{\pi } \operatorname{erf}\left (b x\right )^{n} \operatorname{erf}\left (b x\right ) e^{c}}{2 \,{\left (b n + b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 13.2319, size = 63, normalized size = 2.25 \begin{align*} \begin{cases} \tilde{\infty } x e^{c} & \text{for}\: b = 0 \wedge n = -1 \\0^{n} x e^{c} & \text{for}\: b = 0 \\\frac{\sqrt{\pi } e^{c} \log{\left (\operatorname{erf}{\left (b x \right )} \right )}}{2 b} & \text{for}\: n = -1 \\\frac{\sqrt{\pi } e^{c} \operatorname{erf}{\left (b x \right )} \operatorname{erf}^{n}{\left (b x \right )}}{2 b n + 2 b} & \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)**n,x)

[Out]

Piecewise((zoo*x*exp(c), Eq(b, 0) & Eq(n, -1)), (0**n*x*exp(c), Eq(b, 0)), (sqrt(pi)*exp(c)*log(erf(b*x))/(2*b
), Eq(n, -1)), (sqrt(pi)*exp(c)*erf(b*x)*erf(b*x)**n/(2*b*n + 2*b), True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{erf}\left (b x\right )^{n} 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)^n,x, algorithm="giac")

[Out]

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