\(\int e^{-b^2 x^2} x^2 \text {erf}(b x) \, dx\) [82]

Optimal result

Integrand size = 18, antiderivative size = 63 \[ \int e^{-b^2 x^2} x^2 \text {erf}(b x) \, dx=-\frac {e^{-2 b^2 x^2}}{4 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} x \text {erf}(b x)}{2 b^2}+\frac {\sqrt {\pi } \text {erf}(b x)^2}{8 b^3} \]

[Out]

-1/2*x*erf(b*x)/b^2/exp(b^2*x^2)-1/4/b^3/exp(2*b^2*x^2)/Pi^(1/2)+1/8*erf(b*x)^2*Pi^(1/2)/b^3

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6520, 6508, 30, 2240} \[ \int e^{-b^2 x^2} x^2 \text {erf}(b x) \, dx=\frac {\sqrt {\pi } \text {erf}(b x)^2}{8 b^3}-\frac {x e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}-\frac {e^{-2 b^2 x^2}}{4 \sqrt {\pi } b^3} \]

[In]

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

[Out]

-1/4*1/(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 30

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

Rule 2240

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]

Rule 6508

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 6520

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]

Rubi steps \begin{align*} \text {integral}& = -\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 } \text {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] (verified)

Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.89 \[ \int e^{-b^2 x^2} x^2 \text {erf}(b x) \, dx=-\frac {\frac {2 e^{-2 b^2 x^2}}{\sqrt {\pi }}+4 b e^{-b^2 x^2} x \text {erf}(b x)-\sqrt {\pi } \text {erf}(b x)^2}{8 b^3} \]

[In]

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

[Out]

-1/8*(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)/b^3

Maple [F]

\[\int x^{2} \operatorname {erf}\left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}d x\]

[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)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.83 \[ \int e^{-b^2 x^2} x^2 \text {erf}(b x) \, dx=-\frac {4 \, \pi b x \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} - \sqrt {\pi } {\left (\pi \operatorname {erf}\left (b x\right )^{2} - 2 \, e^{\left (-2 \, b^{2} x^{2}\right )}\right )}}{8 \, \pi b^{3}} \]

[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)

Sympy [A] (verification not implemented)

Time = 1.23 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.95 \[ \int e^{-b^2 x^2} x^2 \text {erf}(b x) \, dx=\begin {cases} - \frac {x e^{- b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{2 b^{2}} + \frac {\sqrt {\pi } \operatorname {erf}^{2}{\left (b x \right )}}{8 b^{3}} - \frac {e^{- 2 b^{2} x^{2}}}{4 \sqrt {\pi } b^{3}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]

[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))

Maxima [F]

\[ \int e^{-b^2 x^2} x^2 \text {erf}(b x) \, dx=\int { x^{2} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} \,d x } \]

[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

Giac [F]

\[ \int e^{-b^2 x^2} x^2 \text {erf}(b x) \, dx=\int { x^{2} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} \,d x } \]

[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)

Mupad [B] (verification not implemented)

Time = 5.26 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.27 \[ \int e^{-b^2 x^2} x^2 \text {erf}(b x) \, dx=-\mathrm {erf}\left (b\,x\right )\,\left (\frac {\sqrt {\pi }\,\mathrm {erfi}\left (x\,\sqrt {-b^2}\right )}{4\,{\left (-b^2\right )}^{3/2}}+\frac {x\,{\mathrm {e}}^{-b^2\,x^2}}{2\,b^2}\right )-\frac {2\,{\mathrm {e}}^{-2\,b^2\,x^2}-\pi \,{\mathrm {erfi}\left (x\,\sqrt {-b^2}\right )}^2}{8\,b^3\,\sqrt {\pi }} \]

[In]

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

[Out]

- erf(b*x)*((pi^(1/2)*erfi(x*(-b^2)^(1/2)))/(4*(-b^2)^(3/2)) + (x*exp(-b^2*x^2))/(2*b^2)) - (2*exp(-2*b^2*x^2)
- pi*erfi(x*(-b^2)^(1/2))^2)/(8*b^3*pi^(1/2))