3.1.22 \(\int x^5 \text {erf}(b x)^2 \, dx\) [22]

3.1.22.1 Optimal result

Integrand size = 10, antiderivative size = 178 \[ \int x^5 \text {erf}(b x)^2 \, dx=\frac {11 e^{-2 b^2 x^2}}{12 b^6 \pi }+\frac {7 e^{-2 b^2 x^2} x^2}{12 b^4 \pi }+\frac {e^{-2 b^2 x^2} x^4}{6 b^2 \pi }+\frac {5 e^{-b^2 x^2} x \text {erf}(b x)}{4 b^5 \sqrt {\pi }}+\frac {5 e^{-b^2 x^2} x^3 \text {erf}(b x)}{6 b^3 \sqrt {\pi }}+\frac {e^{-b^2 x^2} x^5 \text {erf}(b x)}{3 b \sqrt {\pi }}-\frac {5 \text {erf}(b x)^2}{16 b^6}+\frac {1}{6} x^6 \text {erf}(b x)^2 \]

output

11/12/b^6/exp(2*b^2*x^2)/Pi+7/12*x^2/b^4/exp(2*b^2*x^2)/Pi+1/6*x^4/b^2/exp 
(2*b^2*x^2)/Pi-5/16*erf(b*x)^2/b^6+1/6*x^6*erf(b*x)^2+5/4*x*erf(b*x)/b^5/e 
xp(b^2*x^2)/Pi^(1/2)+5/6*x^3*erf(b*x)/b^3/exp(b^2*x^2)/Pi^(1/2)+1/3*x^5*er 
f(b*x)/b/exp(b^2*x^2)/Pi^(1/2)
 

3.1.22.2 Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.60 \[ \int x^5 \text {erf}(b x)^2 \, dx=\frac {e^{-2 b^2 x^2} \left (44+28 b^2 x^2+8 b^4 x^4+4 b e^{b^2 x^2} \sqrt {\pi } x \left (15+10 b^2 x^2+4 b^4 x^4\right ) \text {erf}(b x)+e^{2 b^2 x^2} \pi \left (-15+8 b^6 x^6\right ) \text {erf}(b x)^2\right )}{48 b^6 \pi } \]

input

Integrate[x^5*Erf[b*x]^2,x]
 

output

(44 + 28*b^2*x^2 + 8*b^4*x^4 + 4*b*E^(b^2*x^2)*Sqrt[Pi]*x*(15 + 10*b^2*x^2 
 + 4*b^4*x^4)*Erf[b*x] + E^(2*b^2*x^2)*Pi*(-15 + 8*b^6*x^6)*Erf[b*x]^2)/(4 
8*b^6*E^(2*b^2*x^2)*Pi)
 

3.1.22.3 Rubi [A] (verified)

Time = 1.26 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.52, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.200, Rules used = {6918, 6939, 2641, 2641, 2638, 6939, 2641, 2638, 6939, 2638, 6927, 15}

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.

\(\displaystyle \int x^5 \text {erf}(b x)^2 \, dx\)

\(\Big \downarrow \) 6918

\(\displaystyle \frac {1}{6} x^6 \text {erf}(b x)^2-\frac {2 b \int e^{-b^2 x^2} x^6 \text {erf}(b x)dx}{3 \sqrt {\pi }}\)

\(\Big \downarrow \) 6939

\(\displaystyle \frac {1}{6} x^6 \text {erf}(b x)^2-\frac {2 b \left (\frac {5 \int e^{-b^2 x^2} x^4 \text {erf}(b x)dx}{2 b^2}+\frac {\int e^{-2 b^2 x^2} x^5dx}{\sqrt {\pi } b}-\frac {x^5 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}\right )}{3 \sqrt {\pi }}\)

\(\Big \downarrow \) 2641

\(\displaystyle \frac {1}{6} x^6 \text {erf}(b x)^2-\frac {2 b \left (\frac {5 \int e^{-b^2 x^2} x^4 \text {erf}(b x)dx}{2 b^2}+\frac {\frac {\int e^{-2 b^2 x^2} x^3dx}{b^2}-\frac {x^4 e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}-\frac {x^5 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}\right )}{3 \sqrt {\pi }}\)

\(\Big \downarrow \) 2641

\(\displaystyle \frac {1}{6} x^6 \text {erf}(b x)^2-\frac {2 b \left (\frac {5 \int e^{-b^2 x^2} x^4 \text {erf}(b x)dx}{2 b^2}+\frac {\frac {\frac {\int e^{-2 b^2 x^2} xdx}{2 b^2}-\frac {x^2 e^{-2 b^2 x^2}}{4 b^2}}{b^2}-\frac {x^4 e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}-\frac {x^5 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}\right )}{3 \sqrt {\pi }}\)

\(\Big \downarrow \) 2638

\(\displaystyle \frac {1}{6} x^6 \text {erf}(b x)^2-\frac {2 b \left (\frac {5 \int e^{-b^2 x^2} x^4 \text {erf}(b x)dx}{2 b^2}-\frac {x^5 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}+\frac {\frac {-\frac {x^2 e^{-2 b^2 x^2}}{4 b^2}-\frac {e^{-2 b^2 x^2}}{8 b^4}}{b^2}-\frac {x^4 e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}\right )}{3 \sqrt {\pi }}\)

\(\Big \downarrow \) 6939

\(\displaystyle \frac {1}{6} x^6 \text {erf}(b x)^2-\frac {2 b \left (\frac {5 \left (\frac {3 \int e^{-b^2 x^2} x^2 \text {erf}(b x)dx}{2 b^2}+\frac {\int e^{-2 b^2 x^2} x^3dx}{\sqrt {\pi } b}-\frac {x^3 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}\right )}{2 b^2}-\frac {x^5 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}+\frac {\frac {-\frac {x^2 e^{-2 b^2 x^2}}{4 b^2}-\frac {e^{-2 b^2 x^2}}{8 b^4}}{b^2}-\frac {x^4 e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}\right )}{3 \sqrt {\pi }}\)

\(\Big \downarrow \) 2641

\(\displaystyle \frac {1}{6} x^6 \text {erf}(b x)^2-\frac {2 b \left (\frac {5 \left (\frac {3 \int e^{-b^2 x^2} x^2 \text {erf}(b x)dx}{2 b^2}+\frac {\frac {\int e^{-2 b^2 x^2} xdx}{2 b^2}-\frac {x^2 e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}-\frac {x^3 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}\right )}{2 b^2}-\frac {x^5 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}+\frac {\frac {-\frac {x^2 e^{-2 b^2 x^2}}{4 b^2}-\frac {e^{-2 b^2 x^2}}{8 b^4}}{b^2}-\frac {x^4 e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}\right )}{3 \sqrt {\pi }}\)

\(\Big \downarrow \) 2638

\(\displaystyle \frac {1}{6} x^6 \text {erf}(b x)^2-\frac {2 b \left (\frac {5 \left (\frac {3 \int e^{-b^2 x^2} x^2 \text {erf}(b x)dx}{2 b^2}-\frac {x^3 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}+\frac {-\frac {x^2 e^{-2 b^2 x^2}}{4 b^2}-\frac {e^{-2 b^2 x^2}}{8 b^4}}{\sqrt {\pi } b}\right )}{2 b^2}-\frac {x^5 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}+\frac {\frac {-\frac {x^2 e^{-2 b^2 x^2}}{4 b^2}-\frac {e^{-2 b^2 x^2}}{8 b^4}}{b^2}-\frac {x^4 e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}\right )}{3 \sqrt {\pi }}\)

\(\Big \downarrow \) 6939

\(\displaystyle \frac {1}{6} x^6 \text {erf}(b x)^2-\frac {2 b \left (\frac {5 \left (\frac {3 \left (\frac {\int e^{-b^2 x^2} \text {erf}(b x)dx}{2 b^2}+\frac {\int e^{-2 b^2 x^2} xdx}{\sqrt {\pi } b}-\frac {x e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}\right )}{2 b^2}-\frac {x^3 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}+\frac {-\frac {x^2 e^{-2 b^2 x^2}}{4 b^2}-\frac {e^{-2 b^2 x^2}}{8 b^4}}{\sqrt {\pi } b}\right )}{2 b^2}-\frac {x^5 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}+\frac {\frac {-\frac {x^2 e^{-2 b^2 x^2}}{4 b^2}-\frac {e^{-2 b^2 x^2}}{8 b^4}}{b^2}-\frac {x^4 e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}\right )}{3 \sqrt {\pi }}\)

\(\Big \downarrow \) 2638

\(\displaystyle \frac {1}{6} x^6 \text {erf}(b x)^2-\frac {2 b \left (\frac {5 \left (\frac {3 \left (\frac {\int e^{-b^2 x^2} \text {erf}(b x)dx}{2 b^2}-\frac {x e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}-\frac {e^{-2 b^2 x^2}}{4 \sqrt {\pi } b^3}\right )}{2 b^2}-\frac {x^3 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}+\frac {-\frac {x^2 e^{-2 b^2 x^2}}{4 b^2}-\frac {e^{-2 b^2 x^2}}{8 b^4}}{\sqrt {\pi } b}\right )}{2 b^2}-\frac {x^5 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}+\frac {\frac {-\frac {x^2 e^{-2 b^2 x^2}}{4 b^2}-\frac {e^{-2 b^2 x^2}}{8 b^4}}{b^2}-\frac {x^4 e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}\right )}{3 \sqrt {\pi }}\)

\(\Big \downarrow \) 6927

\(\displaystyle \frac {1}{6} x^6 \text {erf}(b x)^2-\frac {2 b \left (\frac {5 \left (\frac {3 \left (\frac {\sqrt {\pi } \int \text {erf}(b x)d\text {erf}(b x)}{4 b^3}-\frac {x e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}-\frac {e^{-2 b^2 x^2}}{4 \sqrt {\pi } b^3}\right )}{2 b^2}-\frac {x^3 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}+\frac {-\frac {x^2 e^{-2 b^2 x^2}}{4 b^2}-\frac {e^{-2 b^2 x^2}}{8 b^4}}{\sqrt {\pi } b}\right )}{2 b^2}-\frac {x^5 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}+\frac {\frac {-\frac {x^2 e^{-2 b^2 x^2}}{4 b^2}-\frac {e^{-2 b^2 x^2}}{8 b^4}}{b^2}-\frac {x^4 e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}\right )}{3 \sqrt {\pi }}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {1}{6} x^6 \text {erf}(b x)^2-\frac {2 b \left (-\frac {x^5 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}+\frac {\frac {-\frac {x^2 e^{-2 b^2 x^2}}{4 b^2}-\frac {e^{-2 b^2 x^2}}{8 b^4}}{b^2}-\frac {x^4 e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}+\frac {5 \left (-\frac {x^3 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}+\frac {-\frac {x^2 e^{-2 b^2 x^2}}{4 b^2}-\frac {e^{-2 b^2 x^2}}{8 b^4}}{\sqrt {\pi } b}+\frac {3 \left (\frac {\sqrt {\pi } \text {erf}(b x)^2}{8 b^3}-\frac {x e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}-\frac {e^{-2 b^2 x^2}}{4 \sqrt {\pi } b^3}\right )}{2 b^2}\right )}{2 b^2}\right )}{3 \sqrt {\pi }}\)

input

Int[x^5*Erf[b*x]^2,x]
 

output

(x^6*Erf[b*x]^2)/6 - (2*b*((-1/4*x^4/(b^2*E^(2*b^2*x^2)) + (-1/8*1/(b^4*E^ 
(2*b^2*x^2)) - x^2/(4*b^2*E^(2*b^2*x^2)))/b^2)/(b*Sqrt[Pi]) - (x^5*Erf[b*x 
])/(2*b^2*E^(b^2*x^2)) + (5*((-1/8*1/(b^4*E^(2*b^2*x^2)) - x^2/(4*b^2*E^(2 
*b^2*x^2)))/(b*Sqrt[Pi]) - (x^3*Erf[b*x])/(2*b^2*E^(b^2*x^2)) + (3*(-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[P 
i]*Erf[b*x]^2)/(8*b^3)))/(2*b^2)))/(2*b^2)))/(3*Sqrt[Pi])
 

3.1.22.3.1 Defintions of rubi rules used

rule 15

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

rule 2638

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 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 6927

Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(b_.)*(x_)]^(n_.), x_Symbol] :> Simp[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 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]
 

3.1.22.4 Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.90

method	result	size
parallelrisch	\(\frac {8 \operatorname {erf}\left (b x \right )^{2} x^{6} b^{6} \pi ^{\frac {3}{2}}+16 \,{\mathrm e}^{-b^{2} x^{2}} x^{5} \operatorname {erf}\left (b x \right ) b^{5} \pi +8 \,{\mathrm e}^{-2 b^{2} x^{2}} x^{4} b^{4} \sqrt {\pi }+40 \,{\mathrm e}^{-b^{2} x^{2}} \operatorname {erf}\left (b x \right ) x^{3} b^{3} \pi +28 x^{2} {\mathrm e}^{-2 b^{2} x^{2}} b^{2} \sqrt {\pi }+60 \,{\mathrm e}^{-b^{2} x^{2}} x \,\operatorname {erf}\left (b x \right ) b \pi -15 \operatorname {erf}\left (b x \right )^{2} \pi ^{\frac {3}{2}}+44 \,{\mathrm e}^{-2 b^{2} x^{2}} \sqrt {\pi }}{48 b^{6} \pi ^{\frac {3}{2}}}\)	\(160\)

input

int(x^5*erf(b*x)^2,x,method=_RETURNVERBOSE)
 

output

1/48*(8*erf(b*x)^2*x^6*b^6*Pi^(3/2)+16*exp(-b^2*x^2)*x^5*erf(b*x)*b^5*Pi+8 
*exp(-b^2*x^2)^2*x^4*b^4*Pi^(1/2)+40*exp(-b^2*x^2)*erf(b*x)*x^3*b^3*Pi+28* 
x^2*exp(-b^2*x^2)^2*b^2*Pi^(1/2)+60*exp(-b^2*x^2)*x*erf(b*x)*b*Pi-15*erf(b 
*x)^2*Pi^(3/2)+44*exp(-b^2*x^2)^2*Pi^(1/2))/b^6/Pi^(3/2)
 

3.1.22.5 Fricas [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.55 \[ \int x^5 \text {erf}(b x)^2 \, dx=\frac {4 \, \sqrt {\pi } {\left (4 \, b^{5} x^{5} + 10 \, b^{3} x^{3} + 15 \, b x\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} - {\left (15 \, \pi - 8 \, \pi b^{6} x^{6}\right )} \operatorname {erf}\left (b x\right )^{2} + 4 \, {\left (2 \, b^{4} x^{4} + 7 \, b^{2} x^{2} + 11\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{48 \, \pi b^{6}} \]

input

integrate(x^5*erf(b*x)^2,x, algorithm="fricas")
 

output

1/48*(4*sqrt(pi)*(4*b^5*x^5 + 10*b^3*x^3 + 15*b*x)*erf(b*x)*e^(-b^2*x^2) - 
 (15*pi - 8*pi*b^6*x^6)*erf(b*x)^2 + 4*(2*b^4*x^4 + 7*b^2*x^2 + 11)*e^(-2* 
b^2*x^2))/(pi*b^6)
 

3.1.22.6 Sympy [A] (verification not implemented)

Time = 0.54 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.94 \[ \int x^5 \text {erf}(b x)^2 \, dx=\begin {cases} \frac {x^{6} \operatorname {erf}^{2}{\left (b x \right )}}{6} + \frac {x^{5} e^{- b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{3 \sqrt {\pi } b} + \frac {x^{4} e^{- 2 b^{2} x^{2}}}{6 \pi b^{2}} + \frac {5 x^{3} e^{- b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{6 \sqrt {\pi } b^{3}} + \frac {7 x^{2} e^{- 2 b^{2} x^{2}}}{12 \pi b^{4}} + \frac {5 x e^{- b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{4 \sqrt {\pi } b^{5}} - \frac {5 \operatorname {erf}^{2}{\left (b x \right )}}{16 b^{6}} + \frac {11 e^{- 2 b^{2} x^{2}}}{12 \pi b^{6}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]

input

integrate(x**5*erf(b*x)**2,x)
 

output

Piecewise((x**6*erf(b*x)**2/6 + x**5*exp(-b**2*x**2)*erf(b*x)/(3*sqrt(pi)* 
b) + x**4*exp(-2*b**2*x**2)/(6*pi*b**2) + 5*x**3*exp(-b**2*x**2)*erf(b*x)/ 
(6*sqrt(pi)*b**3) + 7*x**2*exp(-2*b**2*x**2)/(12*pi*b**4) + 5*x*exp(-b**2* 
x**2)*erf(b*x)/(4*sqrt(pi)*b**5) - 5*erf(b*x)**2/(16*b**6) + 11*exp(-2*b** 
2*x**2)/(12*pi*b**6), Ne(b, 0)), (0, True))
 

3.1.22.7 Maxima [F]

\[ \int x^5 \text {erf}(b x)^2 \, dx=\int { x^{5} \operatorname {erf}\left (b x\right )^{2} \,d x } \]

input

integrate(x^5*erf(b*x)^2,x, algorithm="maxima")
 

output

-1/6*integrate((4*b^4*x^5 + 10*b^2*x^3 + 15*x)*e^(-2*b^2*x^2), x)/(pi*b^4) 
 + 1/48*((8*sqrt(pi)*b^6*x^6 - 15*sqrt(pi))*erf(b*x)^2 + 4*(4*b^5*x^5 + 10 
*b^3*x^3 + 15*b*x)*erf(b*x)*e^(-b^2*x^2))/(sqrt(pi)*b^6)
 

3.1.22.8 Giac [F]

\[ \int x^5 \text {erf}(b x)^2 \, dx=\int { x^{5} \operatorname {erf}\left (b x\right )^{2} \,d x } \]

input

integrate(x^5*erf(b*x)^2,x, algorithm="giac")
 

output

integrate(x^5*erf(b*x)^2, x)
 

3.1.22.9 Mupad [B] (verification not implemented)

Time = 5.66 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.80 \[ \int x^5 \text {erf}(b x)^2 \, dx=\frac {x^6\,{\mathrm {erf}\left (b\,x\right )}^2}{6}+\frac {\frac {11\,{\mathrm {e}}^{-2\,b^2\,x^2}}{12}-\frac {5\,\pi \,{\mathrm {erf}\left (b\,x\right )}^2}{16}+\frac {7\,b^2\,x^2\,{\mathrm {e}}^{-2\,b^2\,x^2}}{12}+\frac {b^4\,x^4\,{\mathrm {e}}^{-2\,b^2\,x^2}}{6}+\frac {5\,b^3\,x^3\,\sqrt {\pi }\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erf}\left (b\,x\right )}{6}+\frac {b^5\,x^5\,\sqrt {\pi }\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erf}\left (b\,x\right )}{3}+\frac {5\,b\,x\,\sqrt {\pi }\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erf}\left (b\,x\right )}{4}}{b^6\,\pi } \]

input

int(x^5*erf(b*x)^2,x)
 

output

(x^6*erf(b*x)^2)/6 + ((11*exp(-2*b^2*x^2))/12 - (5*pi*erf(b*x)^2)/16 + (7* 
b^2*x^2*exp(-2*b^2*x^2))/12 + (b^4*x^4*exp(-2*b^2*x^2))/6 + (5*b^3*x^3*pi^ 
(1/2)*exp(-b^2*x^2)*erf(b*x))/6 + (b^5*x^5*pi^(1/2)*exp(-b^2*x^2)*erf(b*x) 
)/3 + (5*b*x*pi^(1/2)*exp(-b^2*x^2)*erf(b*x))/4)/(b^6*pi)