\(\int x^3 \text {erf}(b x)^2 \, dx\) [23]

Optimal result

Integrand size = 10, antiderivative size = 126 \[ \int x^3 \text {erf}(b x)^2 \, dx=\frac {e^{-2 b^2 x^2}}{2 b^4 \pi }+\frac {e^{-2 b^2 x^2} x^2}{4 b^2 \pi }+\frac {3 e^{-b^2 x^2} x \text {erf}(b x)}{4 b^3 \sqrt {\pi }}+\frac {e^{-b^2 x^2} x^3 \text {erf}(b x)}{2 b \sqrt {\pi }}-\frac {3 \text {erf}(b x)^2}{16 b^4}+\frac {1}{4} x^4 \text {erf}(b x)^2 \] Output:

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.71 \[ \int x^3 \text {erf}(b x)^2 \, dx=\frac {e^{-2 b^2 x^2} \left (8+4 b^2 x^2+4 b e^{b^2 x^2} \sqrt {\pi } x \left (3+2 b^2 x^2\right ) \text {erf}(b x)+e^{2 b^2 x^2} \pi \left (-3+4 b^4 x^4\right ) \text {erf}(b x)^2\right )}{16 b^4 \pi } \] Input:

Integrate[x^3*Erf[b*x]^2,x]
 

Output:

(8 + 4*b^2*x^2 + 4*b*E^(b^2*x^2)*Sqrt[Pi]*x*(3 + 2*b^2*x^2)*Erf[b*x] + E^( 
2*b^2*x^2)*Pi*(-3 + 4*b^4*x^4)*Erf[b*x]^2)/(16*b^4*E^(2*b^2*x^2)*Pi)
 

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.30, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {6918, 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.

\(\displaystyle \int x^3 \text {erf}(b x)^2 \, dx\)

\(\Big \downarrow \) 6918

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

\(\Big \downarrow \) 6939

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

\(\Big \downarrow \) 2641

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

\(\Big \downarrow \) 2638

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

\(\Big \downarrow \) 6939

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

\(\Big \downarrow \) 2638

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

\(\Big \downarrow \) 6927

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

\(\Big \downarrow \) 15

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

Input:

Int[x^3*Erf[b*x]^2,x]
 

Output:

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

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]
 

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.92

method	result	size
parallelrisch	\(\frac {4 \operatorname {erf}\left (b x \right )^{2} x^{4} \pi ^{\frac {3}{2}} b^{4}+8 \,{\mathrm e}^{-b^{2} x^{2}} x^{3} \operatorname {erf}\left (b x \right ) b^{3} \pi +4 \,{\mathrm e}^{-2 b^{2} x^{2}} x^{2} b^{2} \sqrt {\pi }+12 \,{\mathrm e}^{-b^{2} x^{2}} x \,\operatorname {erf}\left (b x \right ) b \pi -3 \operatorname {erf}\left (b x \right )^{2} \pi ^{\frac {3}{2}}+8 \,{\mathrm e}^{-2 b^{2} x^{2}} \sqrt {\pi }}{16 \pi ^{\frac {3}{2}} b^{4}}\)	\(116\)
orering	\(\frac {\left (4 b^{6} x^{6}-9 b^{4} x^{4}-12 b^{2} x^{2}+12\right ) \operatorname {erf}\left (b x \right )^{2}}{16 b^{6} x^{2}}+\frac {\left (3 b^{4} x^{4}+2 b^{2} x^{2}-6\right ) \left (3 x^{2} \operatorname {erf}\left (b x \right )^{2}+\frac {4 x^{3} \operatorname {erf}\left (b x \right ) {\mathrm e}^{-b^{2} x^{2}} b}{\sqrt {\pi }}\right )}{16 b^{6} x^{4}}+\frac {\left (b^{2} x^{2}+2\right ) \left (6 x \operatorname {erf}\left (b x \right )^{2}+\frac {24 x^{2} \operatorname {erf}\left (b x \right ) {\mathrm e}^{-b^{2} x^{2}} b}{\sqrt {\pi }}+\frac {8 x^{3} {\mathrm e}^{-2 b^{2} x^{2}} b^{2}}{\pi }-\frac {8 x^{4} \operatorname {erf}\left (b x \right ) b^{3} {\mathrm e}^{-b^{2} x^{2}}}{\sqrt {\pi }}\right )}{32 b^{6} x^{3}}\)	\(197\)

Input:

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

Output:

1/16*(4*erf(b*x)^2*x^4*Pi^(3/2)*b^4+8*exp(-b^2*x^2)*x^3*erf(b*x)*b^3*Pi+4* 
exp(-b^2*x^2)^2*x^2*b^2*Pi^(1/2)+12*exp(-b^2*x^2)*x*erf(b*x)*b*Pi-3*erf(b* 
x)^2*Pi^(3/2)+8*exp(-b^2*x^2)^2*Pi^(1/2))/Pi^(3/2)/b^4
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.64 \[ \int x^3 \text {erf}(b x)^2 \, dx=\frac {4 \, \sqrt {\pi } {\left (2 \, b^{3} x^{3} + 3 \, b x\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} - {\left (3 \, \pi - 4 \, \pi b^{4} x^{4}\right )} \operatorname {erf}\left (b x\right )^{2} + 4 \, {\left (b^{2} x^{2} + 2\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{16 \, \pi b^{4}} \] Input:

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

Output:

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

Sympy [A] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.93 \[ \int x^3 \text {erf}(b x)^2 \, dx=\begin {cases} \frac {x^{4} \operatorname {erf}^{2}{\left (b x \right )}}{4} + \frac {x^{3} e^{- b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{2 \sqrt {\pi } b} + \frac {x^{2} e^{- 2 b^{2} x^{2}}}{4 \pi b^{2}} + \frac {3 x e^{- b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{4 \sqrt {\pi } b^{3}} - \frac {3 \operatorname {erf}^{2}{\left (b x \right )}}{16 b^{4}} + \frac {e^{- 2 b^{2} x^{2}}}{2 \pi b^{4}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

integrate(x^3*erf(b*x)^2, x)
 

Mupad [B] (verification not implemented)

Time = 4.06 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.80 \[ \int x^3 \text {erf}(b x)^2 \, dx=\frac {x^4\,{\mathrm {erf}\left (b\,x\right )}^2}{4}+\frac {\frac {{\mathrm {e}}^{-2\,b^2\,x^2}}{2}-\frac {3\,\pi \,{\mathrm {erf}\left (b\,x\right )}^2}{16}+\frac {b^2\,x^2\,{\mathrm {e}}^{-2\,b^2\,x^2}}{4}+\frac {b^3\,x^3\,\sqrt {\pi }\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erf}\left (b\,x\right )}{2}+\frac {3\,b\,x\,\sqrt {\pi }\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erf}\left (b\,x\right )}{4}}{b^4\,\pi } \] Input:

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

Output:

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

Reduce [F]

\[ \int x^3 \text {erf}(b x)^2 \, dx=\int \mathrm {erf}\left (b x \right )^{2} x^{3}d x \] Input:

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

Output:

int(erf(b*x)**2*x**3,x)