\(\int x^2 \text {erf}(b x)^2 \, dx\) [30]

Optimal result

Integrand size = 10, antiderivative size = 113 \[ \int x^2 \text {erf}(b x)^2 \, dx=\frac {e^{-2 b^2 x^2} x}{3 b^2 \pi }+\frac {2 e^{-b^2 x^2} \text {erf}(b x)}{3 b^3 \sqrt {\pi }}+\frac {2 e^{-b^2 x^2} x^2 \text {erf}(b x)}{3 b \sqrt {\pi }}+\frac {1}{3} x^3 \text {erf}(b x)^2-\frac {5 \text {erf}\left (\sqrt {2} b x\right )}{6 b^3 \sqrt {2 \pi }} \]

[Out]

1/3*x/b^2/exp(2*b^2*x^2)/Pi+1/3*x^3*erf(b*x)^2+2/3*erf(b*x)/b^3/exp(b^2*x^2)/Pi^(1/2)+2/3*x^2*erf(b*x)/b/exp(b
^2*x^2)/Pi^(1/2)-5/12*erf(b*x*2^(1/2))/b^3*2^(1/2)/Pi^(1/2)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6499, 6520, 6517, 2236, 2243} \[ \int x^2 \text {erf}(b x)^2 \, dx=-\frac {5 \text {erf}\left (\sqrt {2} b x\right )}{6 \sqrt {2 \pi } b^3}+\frac {2 x^2 e^{-b^2 x^2} \text {erf}(b x)}{3 \sqrt {\pi } b}+\frac {x e^{-2 b^2 x^2}}{3 \pi b^2}+\frac {2 e^{-b^2 x^2} \text {erf}(b x)}{3 \sqrt {\pi } b^3}+\frac {1}{3} x^3 \text {erf}(b x)^2 \]

[In]

Int[x^2*Erf[b*x]^2,x]

[Out]

x/(3*b^2*E^(2*b^2*x^2)*Pi) + (2*Erf[b*x])/(3*b^3*E^(b^2*x^2)*Sqrt[Pi]) + (2*x^2*Erf[b*x])/(3*b*E^(b^2*x^2)*Sqr
t[Pi]) + (x^3*Erf[b*x]^2)/3 - (5*Erf[Sqrt[2]*b*x])/(6*b^3*Sqrt[2*Pi])

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 6499

Int[Erf[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(Erf[b*x]^2/(m + 1)), x] - Dist[4*(b/(Sqrt[Pi]*(
m + 1))), Int[(x^(m + 1)*Erf[b*x])/E^(b^2*x^2), x], x] /; FreeQ[b, x] && (IGtQ[m, 0] || ILtQ[(m + 1)/2, 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 {1}{3} x^3 \text {erf}(b x)^2-\frac {(4 b) \int e^{-b^2 x^2} x^3 \text {erf}(b x) \, dx}{3 \sqrt {\pi }} \\ & = \frac {2 e^{-b^2 x^2} x^2 \text {erf}(b x)}{3 b \sqrt {\pi }}+\frac {1}{3} x^3 \text {erf}(b x)^2-\frac {4 \int e^{-2 b^2 x^2} x^2 \, dx}{3 \pi }-\frac {4 \int e^{-b^2 x^2} x \text {erf}(b x) \, dx}{3 b \sqrt {\pi }} \\ & = \frac {e^{-2 b^2 x^2} x}{3 b^2 \pi }+\frac {2 e^{-b^2 x^2} \text {erf}(b x)}{3 b^3 \sqrt {\pi }}+\frac {2 e^{-b^2 x^2} x^2 \text {erf}(b x)}{3 b \sqrt {\pi }}+\frac {1}{3} x^3 \text {erf}(b x)^2-\frac {\int e^{-2 b^2 x^2} \, dx}{3 b^2 \pi }-\frac {4 \int e^{-2 b^2 x^2} \, dx}{3 b^2 \pi } \\ & = \frac {e^{-2 b^2 x^2} x}{3 b^2 \pi }+\frac {2 e^{-b^2 x^2} \text {erf}(b x)}{3 b^3 \sqrt {\pi }}+\frac {2 e^{-b^2 x^2} x^2 \text {erf}(b x)}{3 b \sqrt {\pi }}+\frac {1}{3} x^3 \text {erf}(b x)^2-\frac {\sqrt {\frac {2}{\pi }} \text {erf}\left (\sqrt {2} b x\right )}{3 b^3}-\frac {\text {erf}\left (\sqrt {2} b x\right )}{6 b^3 \sqrt {2 \pi }} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.78 \[ \int x^2 \text {erf}(b x)^2 \, dx=\frac {4 b e^{-2 b^2 x^2} x+8 e^{-b^2 x^2} \sqrt {\pi } \left (1+b^2 x^2\right ) \text {erf}(b x)+4 b^3 \pi x^3 \text {erf}(b x)^2-5 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} b x\right )}{12 b^3 \pi } \]

[In]

Integrate[x^2*Erf[b*x]^2,x]

[Out]

((4*b*x)/E^(2*b^2*x^2) + (8*Sqrt[Pi]*(1 + b^2*x^2)*Erf[b*x])/E^(b^2*x^2) + 4*b^3*Pi*x^3*Erf[b*x]^2 - 5*Sqrt[2*
Pi]*Erf[Sqrt[2]*b*x])/(12*b^3*Pi)

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.84

method	result	size
derivativedivides	\(\frac {\frac {\operatorname {erf}\left (b x \right )^{2} b^{3} x^{3}}{3}-\frac {4 \,\operatorname {erf}\left (b x \right ) \left (-\frac {x^{2} {\mathrm e}^{-b^{2} x^{2}} b^{2}}{2}-\frac {{\mathrm e}^{-b^{2} x^{2}}}{2}\right )}{3 \sqrt {\pi }}+\frac {-\frac {5 \sqrt {2}\, \sqrt {\pi }\, \operatorname {erf}\left (b x \sqrt {2}\right )}{12}+\frac {{\mathrm e}^{-2 b^{2} x^{2}} b x}{3}}{\pi }}{b^{3}}\)	\(95\)
default	\(\frac {\frac {\operatorname {erf}\left (b x \right )^{2} b^{3} x^{3}}{3}-\frac {4 \,\operatorname {erf}\left (b x \right ) \left (-\frac {x^{2} {\mathrm e}^{-b^{2} x^{2}} b^{2}}{2}-\frac {{\mathrm e}^{-b^{2} x^{2}}}{2}\right )}{3 \sqrt {\pi }}+\frac {-\frac {5 \sqrt {2}\, \sqrt {\pi }\, \operatorname {erf}\left (b x \sqrt {2}\right )}{12}+\frac {{\mathrm e}^{-2 b^{2} x^{2}} b x}{3}}{\pi }}{b^{3}}\)	\(95\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.80 \[ \int x^2 \text {erf}(b x)^2 \, dx=\frac {4 \, \pi b^{4} x^{3} \operatorname {erf}\left (b x\right )^{2} + 4 \, b^{2} x e^{\left (-2 \, b^{2} x^{2}\right )} + 8 \, \sqrt {\pi } {\left (b^{3} x^{2} + b\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} - 5 \, \sqrt {2} \sqrt {\pi } \sqrt {b^{2}} \operatorname {erf}\left (\sqrt {2} \sqrt {b^{2}} x\right )}{12 \, \pi b^{4}} \]

[In]

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

[Out]

1/12*(4*pi*b^4*x^3*erf(b*x)^2 + 4*b^2*x*e^(-2*b^2*x^2) + 8*sqrt(pi)*(b^3*x^2 + b)*erf(b*x)*e^(-b^2*x^2) - 5*sq
rt(2)*sqrt(pi)*sqrt(b^2)*erf(sqrt(2)*sqrt(b^2)*x))/(pi*b^4)

Sympy [F]

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

[In]

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

[Out]

Integral(x**2*erf(b*x)**2, x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.98 \[ \int x^2 \text {erf}(b x)^2 \, dx=\frac {1}{3} \, x^{3} \operatorname {erf}\left (b x\right )^{2} + \frac {b {\left (\frac {8 \, {\left (b^{2} x^{2} + 1\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{b^{4}} + \frac {b^{2} {\left (\frac {4 \, x e^{\left (-2 \, b^{2} x^{2}\right )}}{b^{2}} + \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {2} b x\right )}{b^{3}}\right )} + \frac {4 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {2} b x\right )}{b}}{\sqrt {\pi } b^{3}}\right )}}{12 \, \sqrt {\pi }} \]

[In]

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

[Out]

1/3*x^3*erf(b*x)^2 + 1/12*b*(8*(b^2*x^2 + 1)*erf(b*x)*e^(-b^2*x^2)/b^4 + (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))/sqrt(pi)

Mupad [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.80 \[ \int x^2 \text {erf}(b x)^2 \, dx=\frac {x^3\,{\mathrm {erf}\left (b\,x\right )}^2}{3}+\frac {\frac {2\,\sqrt {\pi }\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erf}\left (b\,x\right )}{3}-\frac {5\,\sqrt {2}\,\sqrt {\pi }\,\mathrm {erf}\left (\sqrt {2}\,b\,x\right )}{12}+\frac {b\,x\,{\mathrm {e}}^{-2\,b^2\,x^2}}{3}+\frac {2\,b^2\,x^2\,\sqrt {\pi }\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erf}\left (b\,x\right )}{3}}{b^3\,\pi } \]

[In]

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

[Out]

(x^3*erf(b*x)^2)/3 + ((2*pi^(1/2)*exp(-b^2*x^2)*erf(b*x))/3 - (5*2^(1/2)*pi^(1/2)*erf(2^(1/2)*b*x))/12 + (b*x*
exp(-2*b^2*x^2))/3 + (2*b^2*x^2*pi^(1/2)*exp(-b^2*x^2)*erf(b*x))/3)/(b^3*pi)