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

Optimal. Leaf size=63 \[ -\frac{x e^{-b^2 x^2} \text{Erf}(b x)}{2 b^2}+\frac{\sqrt{\pi } \text{Erf}(b x)^2}{8 b^3}-\frac{e^{-2 b^2 x^2}}{4 \sqrt{\pi } b^3} \]

[Out]

-1/(4*b^3*E^(2*b^2*x^2)*Sqrt[Pi]) - (x*Erf[b*x])/(2*b^2*E^(b^2*x^2)) + (Sqrt[Pi]*Erf[b*x]^2)/(8*b^3)

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Rubi [A]  time = 0.0684107, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6385, 6373, 30, 2209} \[ -\frac{x e^{-b^2 x^2} \text{Erf}(b x)}{2 b^2}+\frac{\sqrt{\pi } \text{Erf}(b x)^2}{8 b^3}-\frac{e^{-2 b^2 x^2}}{4 \sqrt{\pi } b^3} \]

Antiderivative was successfully verified.

[In]

Int[(x^2*Erf[b*x])/E^(b^2*x^2),x]

[Out]

-1/(4*b^3*E^(2*b^2*x^2)*Sqrt[Pi]) - (x*Erf[b*x])/(2*b^2*E^(b^2*x^2)) + (Sqrt[Pi]*Erf[b*x]^2)/(8*b^3)

Rule 6385

Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[(x^(m - 1)*E^(c + d*x^2)*Erf
[a + b*x])/(2*d), x] + (-Dist[(m - 1)/(2*d), Int[x^(m - 2)*E^(c + d*x^2)*Erf[a + b*x], x], x] - Dist[b/(d*Sqrt
[Pi]), Int[x^(m - 1)*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2), x], x]) /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 1]

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]

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int e^{-b^2 x^2} x^2 \text{erf}(b x) \, dx &=-\frac{e^{-b^2 x^2} x \text{erf}(b x)}{2 b^2}+\frac{\int e^{-b^2 x^2} \text{erf}(b x) \, dx}{2 b^2}+\frac{\int e^{-2 b^2 x^2} x \, dx}{b \sqrt{\pi }}\\ &=-\frac{e^{-2 b^2 x^2}}{4 b^3 \sqrt{\pi }}-\frac{e^{-b^2 x^2} x \text{erf}(b x)}{2 b^2}+\frac{\sqrt{\pi } \operatorname{Subst}(\int x \, dx,x,\text{erf}(b x))}{4 b^3}\\ &=-\frac{e^{-2 b^2 x^2}}{4 b^3 \sqrt{\pi }}-\frac{e^{-b^2 x^2} x \text{erf}(b x)}{2 b^2}+\frac{\sqrt{\pi } \text{erf}(b x)^2}{8 b^3}\\ \end{align*}

Mathematica [A]  time = 0.0387112, size = 56, normalized size = 0.89 \[ -\frac{4 b x e^{-b^2 x^2} \text{Erf}(b x)+\frac{2 e^{-2 b^2 x^2}}{\sqrt{\pi }}-\sqrt{\pi } \text{Erf}(b x)^2}{8 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^2*Erf[b*x])/E^(b^2*x^2),x]

[Out]

-(2/(E^(2*b^2*x^2)*Sqrt[Pi]) + (4*b*x*Erf[b*x])/E^(b^2*x^2) - Sqrt[Pi]*Erf[b*x]^2)/(8*b^3)

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Maple [F]  time = 0.236, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2}{\it Erf} \left ( bx \right ) }{{{\rm e}^{{b}^{2}{x}^{2}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*erf(b*x)/exp(b^2*x^2),x)

[Out]

int(x^2*erf(b*x)/exp(b^2*x^2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{-\frac{e^{\left (-2 \, b^{2} x^{2}\right )}}{4 \, b^{2}}}{\sqrt{\pi } b} - \frac{4 \, b x \operatorname{erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} - \sqrt{\pi } \operatorname{erf}\left (b x\right )^{2}}{8 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*erf(b*x)/exp(b^2*x^2),x, algorithm="maxima")

[Out]

integrate(x*e^(-2*b^2*x^2), x)/(sqrt(pi)*b) - 1/8*(4*b*x*erf(b*x)*e^(-b^2*x^2) - sqrt(pi)*erf(b*x)^2)/b^3

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Fricas [A]  time = 3.09933, size = 127, normalized size = 2.02 \begin{align*} -\frac{4 \, \pi b x \operatorname{erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} - \sqrt{\pi }{\left (\pi \operatorname{erf}\left (b x\right )^{2} - 2 \, e^{\left (-2 \, b^{2} x^{2}\right )}\right )}}{8 \, \pi b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*erf(b*x)/exp(b^2*x^2),x, algorithm="fricas")

[Out]

-1/8*(4*pi*b*x*erf(b*x)*e^(-b^2*x^2) - sqrt(pi)*(pi*erf(b*x)^2 - 2*e^(-2*b^2*x^2)))/(pi*b^3)

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Sympy [A]  time = 12.2537, size = 60, normalized size = 0.95 \begin{align*} \begin{cases} - \frac{x e^{- b^{2} x^{2}} \operatorname{erf}{\left (b x \right )}}{2 b^{2}} + \frac{\sqrt{\pi } \operatorname{erf}^{2}{\left (b x \right )}}{8 b^{3}} - \frac{e^{- 2 b^{2} x^{2}}}{4 \sqrt{\pi } b^{3}} & \text{for}\: b \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*erf(b*x)/exp(b**2*x**2),x)

[Out]

Piecewise((-x*exp(-b**2*x**2)*erf(b*x)/(2*b**2) + sqrt(pi)*erf(b*x)**2/(8*b**3) - exp(-2*b**2*x**2)/(4*sqrt(pi
)*b**3), Ne(b, 0)), (0, True))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*erf(b*x)/exp(b^2*x^2),x, algorithm="giac")

[Out]

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