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

Optimal result

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

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

Mathematica [A] (verified)

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

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

Output:

((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)
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.32, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6918, 6939, 2641, 2634, 6936, 2634}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x^2 \text {erf}(b x)^2 \, dx\)

\(\Big \downarrow \) 6918

\(\displaystyle \frac {1}{3} x^3 \text {erf}(b x)^2-\frac {4 b \int e^{-b^2 x^2} x^3 \text {erf}(b x)dx}{3 \sqrt {\pi }}\)

\(\Big \downarrow \) 6939

\(\displaystyle \frac {1}{3} x^3 \text {erf}(b x)^2-\frac {4 b \left (\frac {\int e^{-b^2 x^2} x \text {erf}(b x)dx}{b^2}+\frac {\int e^{-2 b^2 x^2} x^2dx}{\sqrt {\pi } b}-\frac {x^2 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}\right )}{3 \sqrt {\pi }}\)

\(\Big \downarrow \) 2641

\(\displaystyle \frac {1}{3} x^3 \text {erf}(b x)^2-\frac {4 b \left (\frac {\int e^{-b^2 x^2} x \text {erf}(b x)dx}{b^2}+\frac {\frac {\int e^{-2 b^2 x^2}dx}{4 b^2}-\frac {x e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}-\frac {x^2 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}\right )}{3 \sqrt {\pi }}\)

\(\Big \downarrow \) 2634

\(\displaystyle \frac {1}{3} x^3 \text {erf}(b x)^2-\frac {4 b \left (\frac {\int e^{-b^2 x^2} x \text {erf}(b x)dx}{b^2}-\frac {x^2 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}+\frac {\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} b x\right )}{8 b^3}-\frac {x e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}\right )}{3 \sqrt {\pi }}\)

\(\Big \downarrow \) 6936

\(\displaystyle \frac {1}{3} x^3 \text {erf}(b x)^2-\frac {4 b \left (\frac {\frac {\int e^{-2 b^2 x^2}dx}{\sqrt {\pi } b}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}}{b^2}-\frac {x^2 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}+\frac {\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} b x\right )}{8 b^3}-\frac {x e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}\right )}{3 \sqrt {\pi }}\)

\(\Big \downarrow \) 2634

\(\displaystyle \frac {1}{3} x^3 \text {erf}(b x)^2-\frac {4 b \left (-\frac {x^2 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}+\frac {\frac {\text {erf}\left (\sqrt {2} b x\right )}{2 \sqrt {2} b^2}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}}{b^2}+\frac {\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} b x\right )}{8 b^3}-\frac {x e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}\right )}{3 \sqrt {\pi }}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 2634

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] /; Fr 
eeQ[{F, a, b, c, d}, x] && NegQ[b]
 

rule 2641

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*L 
og[F])), x] - Simp[(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 6918

Int[Erf[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(Erf[b*x]^2/( 
m + 1)), x] - Simp[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 6936

Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(a_.) + (b_.)*(x_)]*(x_), x_Symbol] :> Sim 
p[E^(c + d*x^2)*(Erf[a + b*x]/(2*d)), x] - Simp[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 6939

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] + (-Simp[(m - 1)/(2 
*d)   Int[x^(m - 2)*E^(c + d*x^2)*Erf[a + b*x], x], x] - Simp[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]
 

Maple [A] (verified)

Time = 0.88 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.84

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

Input:

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

Output:

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(2^(1/2)*b*x)+1/4 
/exp(b^2*x^2)^2*b*x))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.80 \[ \int x^2 \text {erf}(b x)^2 \, dx=\frac {4 \, \pi b^{4} x^{3} \operatorname {erf}\left (b x\right )^{2} + 4 \, b^{2} x e^{\left (-2 \, b^{2} x^{2}\right )} + 8 \, \sqrt {\pi } {\left (b^{3} x^{2} + b\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} - 5 \, \sqrt {2} \sqrt {\pi } \sqrt {b^{2}} \operatorname {erf}\left (\sqrt {2} \sqrt {b^{2}} x\right )}{12 \, \pi b^{4}} \] Input:

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

Output:

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*sqrt(2)*sqrt(pi)*sqrt(b^2)*erf(sqrt(2)*s 
qrt(b^2)*x))/(pi*b^4)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-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 [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 3.97 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.80 \[ \int x^2 \text {erf}(b x)^2 \, dx=\frac {x^3\,{\mathrm {erf}\left (b\,x\right )}^2}{3}+\frac {\frac {2\,\sqrt {\pi }\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erf}\left (b\,x\right )}{3}-\frac {5\,\sqrt {2}\,\sqrt {\pi }\,\mathrm {erf}\left (\sqrt {2}\,b\,x\right )}{12}+\frac {b\,x\,{\mathrm {e}}^{-2\,b^2\,x^2}}{3}+\frac {2\,b^2\,x^2\,\sqrt {\pi }\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erf}\left (b\,x\right )}{3}}{b^3\,\pi } \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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