\(\int e^{-b^2 x^2} x^3 \text {erf}(b x) \, dx\) [76]

Optimal result

Integrand size = 18, antiderivative size = 90 \[ \int e^{-b^2 x^2} x^3 \text {erf}(b x) \, dx=-\frac {e^{-2 b^2 x^2} x}{4 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{2 b^4}-\frac {e^{-b^2 x^2} x^2 \text {erf}(b x)}{2 b^2}+\frac {5 \text {erf}\left (\sqrt {2} b x\right )}{8 \sqrt {2} b^4} \]

[Out]

-1/2*erf(b*x)/b^4/exp(b^2*x^2)-1/2*x^2*erf(b*x)/b^2/exp(b^2*x^2)+5/16*erf(b*x*2^(1/2))/b^4*2^(1/2)-1/4*x/b^3/e
xp(2*b^2*x^2)/Pi^(1/2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6520, 6517, 2236, 2243} \[ \int e^{-b^2 x^2} x^3 \text {erf}(b x) \, dx=\frac {5 \text {erf}\left (\sqrt {2} b x\right )}{8 \sqrt {2} b^4}-\frac {x^2 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{2 b^4}-\frac {x e^{-2 b^2 x^2}}{4 \sqrt {\pi } b^3} \]

[In]

Int[(x^3*Erf[b*x])/E^(b^2*x^2),x]

[Out]

-1/4*x/(b^3*E^(2*b^2*x^2)*Sqrt[Pi]) - Erf[b*x]/(2*b^4*E^(b^2*x^2)) - (x^2*Erf[b*x])/(2*b^2*E^(b^2*x^2)) + (5*E
rf[Sqrt[2]*b*x])/(8*Sqrt[2]*b^4)

Rule 2236

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F],
2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 2243

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m
- n + 1)*(F^(a + b*(c + d*x)^n)/(b*d*n*Log[F])), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^(m - n)*F^(
a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2*((m + 1)/n)] && LtQ[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

Rule 6517

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

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^2 \text {erf}(b x)}{2 b^2}+\frac {\int e^{-b^2 x^2} x \text {erf}(b x) \, dx}{b^2}+\frac {\int e^{-2 b^2 x^2} x^2 \, dx}{b \sqrt {\pi }} \\ & = -\frac {e^{-2 b^2 x^2} x}{4 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{2 b^4}-\frac {e^{-b^2 x^2} x^2 \text {erf}(b x)}{2 b^2}+\frac {\int e^{-2 b^2 x^2} \, dx}{4 b^3 \sqrt {\pi }}+\frac {\int e^{-2 b^2 x^2} \, dx}{b^3 \sqrt {\pi }} \\ & = -\frac {e^{-2 b^2 x^2} x}{4 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{2 b^4}-\frac {e^{-b^2 x^2} x^2 \text {erf}(b x)}{2 b^2}+\frac {5 \text {erf}\left (\sqrt {2} b x\right )}{8 \sqrt {2} b^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.76 \[ \int e^{-b^2 x^2} x^3 \text {erf}(b x) \, dx=\frac {-\frac {4 b e^{-2 b^2 x^2} x}{\sqrt {\pi }}-8 e^{-b^2 x^2} \left (1+b^2 x^2\right ) \text {erf}(b x)+5 \sqrt {2} \text {erf}\left (\sqrt {2} b x\right )}{16 b^4} \]

[In]

Integrate[(x^3*Erf[b*x])/E^(b^2*x^2),x]

[Out]

((-4*b*x)/(E^(2*b^2*x^2)*Sqrt[Pi]) - (8*(1 + b^2*x^2)*Erf[b*x])/E^(b^2*x^2) + 5*Sqrt[2]*Erf[Sqrt[2]*b*x])/(16*
b^4)

Maple [A] (verified)

Time = 0.91 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.92

method	result	size
default	\(\frac {\frac {\operatorname {erf}\left (b x \right ) \left (-\frac {x^{2} {\mathrm e}^{-b^{2} x^{2}} b^{2}}{2}-\frac {{\mathrm e}^{-b^{2} x^{2}}}{2}\right )}{b^{3}}-\frac {-\frac {5 \sqrt {2}\, \sqrt {\pi }\, \operatorname {erf}\left (b x \sqrt {2}\right )}{16}+\frac {{\mathrm e}^{-2 b^{2} x^{2}} b x}{4}}{\sqrt {\pi }\, b^{3}}}{b}\)	\(83\)

[In]

int(x^3*erf(b*x)/exp(b^2*x^2),x,method=_RETURNVERBOSE)

[Out]

(erf(b*x)/b^3*(-1/2*b^2*x^2/exp(b^2*x^2)-1/2/exp(b^2*x^2))-1/Pi^(1/2)/b^3*(-5/16*2^(1/2)*Pi^(1/2)*erf(b*x*2^(1
/2))+1/4/exp(b^2*x^2)^2*b*x))/b

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.84 \[ \int e^{-b^2 x^2} x^3 \text {erf}(b x) \, dx=-\frac {4 \, \sqrt {\pi } b^{2} x e^{\left (-2 \, b^{2} x^{2}\right )} - 5 \, \sqrt {2} \pi \sqrt {b^{2}} \operatorname {erf}\left (\sqrt {2} \sqrt {b^{2}} x\right ) + 8 \, {\left (\pi b^{3} x^{2} + \pi b\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{16 \, \pi b^{5}} \]

[In]

integrate(x^3*erf(b*x)/exp(b^2*x^2),x, algorithm="fricas")

[Out]

-1/16*(4*sqrt(pi)*b^2*x*e^(-2*b^2*x^2) - 5*sqrt(2)*pi*sqrt(b^2)*erf(sqrt(2)*sqrt(b^2)*x) + 8*(pi*b^3*x^2 + pi*
b)*erf(b*x)*e^(-b^2*x^2))/(pi*b^5)

Sympy [F]

\[ \int e^{-b^2 x^2} x^3 \text {erf}(b x) \, dx=\int x^{3} e^{- b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}\, dx \]

[In]

integrate(x**3*erf(b*x)/exp(b**2*x**2),x)

[Out]

Integral(x**3*exp(-b**2*x**2)*erf(b*x), x)

Maxima [F]

\[ \int e^{-b^2 x^2} x^3 \text {erf}(b x) \, dx=\int { x^{3} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} \,d x } \]

[In]

integrate(x^3*erf(b*x)/exp(b^2*x^2),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.04 \[ \int e^{-b^2 x^2} x^3 \text {erf}(b x) \, dx=-\frac {{\left (b^{2} x^{2} + 1\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{2 \, b^{4}} - \frac {b^{2} {\left (\frac {4 \, x e^{\left (-2 \, b^{2} x^{2}\right )}}{b^{2}} + \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {2} b x\right )}{b^{3}}\right )} + \frac {4 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {2} b x\right )}{b}}{16 \, \sqrt {\pi } b^{3}} \]

[In]

integrate(x^3*erf(b*x)/exp(b^2*x^2),x, algorithm="giac")

[Out]

-1/2*(b^2*x^2 + 1)*erf(b*x)*e^(-b^2*x^2)/b^4 - 1/16*(b^2*(4*x*e^(-2*b^2*x^2)/b^2 + sqrt(2)*sqrt(pi)*erf(-sqrt(
2)*b*x)/b^3) + 4*sqrt(2)*sqrt(pi)*erf(-sqrt(2)*b*x)/b)/(sqrt(pi)*b^3)

Mupad [B] (verification not implemented)

Time = 5.29 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.18 \[ \int e^{-b^2 x^2} x^3 \text {erf}(b x) \, dx=\frac {\sqrt {2}\,\mathrm {erf}\left (\sqrt {2}\,x\,\sqrt {b^2}\right )}{4\,b\,{\left (b^2\right )}^{3/2}}-\frac {\mathrm {erfi}\left (\sqrt {2}\,x\,\sqrt {-b^2}\right )}{4\,b\,{\left (-2\,b^2\right )}^{3/2}}-\mathrm {erf}\left (b\,x\right )\,\left (\frac {{\mathrm {e}}^{-b^2\,x^2}}{2\,b^4}+\frac {x^2\,{\mathrm {e}}^{-b^2\,x^2}}{2\,b^2}\right )-\frac {x\,{\mathrm {e}}^{-2\,b^2\,x^2}}{4\,b^3\,\sqrt {\pi }} \]

[In]

int(x^3*exp(-b^2*x^2)*erf(b*x),x)

[Out]

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