\(\int x^2 \text {erfc}(b x)^2 \, dx\) [133]

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.78 \[ \int x^2 \text {erfc}(b x)^2 \, dx=\frac {4 b e^{-2 b^2 x^2} x-5 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} b x\right )-8 e^{-b^2 x^2} \sqrt {\pi } \left (1+b^2 x^2\right ) \text {erfc}(b x)+4 b^3 \pi x^3 \text {erfc}(b x)^2}{12 b^3 \pi } \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.33, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6919, 6940, 2641, 2634, 6937, 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.

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

\(\Big \downarrow \) 6919

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

\(\Big \downarrow \) 6940

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

\(\Big \downarrow \) 2641

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

\(\Big \downarrow \) 2634

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

\(\Big \downarrow \) 6937

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

\(\Big \downarrow \) 2634

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

Input:

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

Output:

(x^3*Erfc[b*x]^2)/3 + (4*b*(-((-1/4*x/(b^2*E^(2*b^2*x^2)) + (Sqrt[Pi/2]*Er 
f[Sqrt[2]*b*x])/(8*b^3))/(b*Sqrt[Pi])) - (x^2*Erfc[b*x])/(2*b^2*E^(b^2*x^2 
)) + (-1/2*Erf[Sqrt[2]*b*x]/(Sqrt[2]*b^2) - Erfc[b*x]/(2*b^2*E^(b^2*x^2))) 
/b^2))/(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 6919

Int[Erfc[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(Erfc[b*x]^2 
/(m + 1)), x] + Simp[4*(b/(Sqrt[Pi]*(m + 1)))   Int[(x^(m + 1)*Erfc[b*x])/E 
^(b^2*x^2), x], x] /; FreeQ[b, x] && (IGtQ[m, 0] || ILtQ[(m + 1)/2, 0])
 

rule 6937

Int[E^((c_.) + (d_.)*(x_)^2)*Erfc[(a_.) + (b_.)*(x_)]*(x_), x_Symbol] :> Si 
mp[E^(c + d*x^2)*(Erfc[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 6940

Int[E^((c_.) + (d_.)*(x_)^2)*Erfc[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] 
:> Simp[x^(m - 1)*E^(c + d*x^2)*(Erfc[a + b*x]/(2*d)), x] + (-Simp[(m - 1)/ 
(2*d)   Int[x^(m - 2)*E^(c + d*x^2)*Erfc[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]) /; Fre 
eQ[{a, b, c, d}, x] && IGtQ[m, 1]
 

Maple [A] (verified)

Time = 0.69 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.34

method	result	size
derivativedivides	\(\frac {\frac {b^{3} x^{3}}{3}-\frac {2 \,\operatorname {erf}\left (b x \right ) b^{3} x^{3}}{3}+\frac {-\frac {2 \,{\mathrm e}^{-b^{2} x^{2}} x^{2} b^{2}}{3}-\frac {2 \,{\mathrm e}^{-b^{2} x^{2}}}{3}}{\sqrt {\pi }}+\frac {\operatorname {erf}\left (b x \right )^{2} b^{3} x^{3}}{3}-\frac {4 \,\operatorname {erf}\left (b x \right ) \left (-\frac {{\mathrm e}^{-b^{2} x^{2}} x^{2} b^{2}}{2}-\frac {{\mathrm e}^{-b^{2} x^{2}}}{2}\right )}{3 \sqrt {\pi }}+\frac {-\frac {5 \sqrt {2}\, \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {2}\, b x \right )}{12}+\frac {{\mathrm e}^{-2 b^{2} x^{2}} b x}{3}}{\pi }}{b^{3}}\)	\(151\)
default	\(\frac {\frac {b^{3} x^{3}}{3}-\frac {2 \,\operatorname {erf}\left (b x \right ) b^{3} x^{3}}{3}+\frac {-\frac {2 \,{\mathrm e}^{-b^{2} x^{2}} x^{2} b^{2}}{3}-\frac {2 \,{\mathrm e}^{-b^{2} x^{2}}}{3}}{\sqrt {\pi }}+\frac {\operatorname {erf}\left (b x \right )^{2} b^{3} x^{3}}{3}-\frac {4 \,\operatorname {erf}\left (b x \right ) \left (-\frac {{\mathrm e}^{-b^{2} x^{2}} x^{2} b^{2}}{2}-\frac {{\mathrm e}^{-b^{2} x^{2}}}{2}\right )}{3 \sqrt {\pi }}+\frac {-\frac {5 \sqrt {2}\, \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {2}\, b x \right )}{12}+\frac {{\mathrm e}^{-2 b^{2} x^{2}} b x}{3}}{\pi }}{b^{3}}\)	\(151\)

Input:

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

Output:

1/b^3*(1/3*b^3*x^3-2/3*erf(b*x)*b^3*x^3+4/3/Pi^(1/2)*(-1/2*b^2*x^2/exp(b^2 
*x^2)-1/2/exp(b^2*x^2))+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 = 0.09 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.09 \[ \int x^2 \text {erfc}(b x)^2 \, dx=\frac {4 \, \pi b^{4} x^{3} \operatorname {erf}\left (b x\right )^{2} - 8 \, \pi b^{4} x^{3} \operatorname {erf}\left (b x\right ) + 4 \, \pi b^{4} x^{3} + 4 \, b^{2} x e^{\left (-2 \, b^{2} x^{2}\right )} - 5 \, \sqrt {2} \sqrt {\pi } \sqrt {b^{2}} \operatorname {erf}\left (\sqrt {2} \sqrt {b^{2}} x\right ) - 8 \, \sqrt {\pi } {\left (b^{3} x^{2} - {\left (b^{3} x^{2} + b\right )} \operatorname {erf}\left (b x\right ) + b\right )} e^{\left (-b^{2} x^{2}\right )}}{12 \, \pi b^{4}} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^2 \text {erfc}(b x)^2 \, dx=\frac {-2 e^{b^{2} x^{2}} \mathrm {erf}\left (b x \right ) b^{3} \pi \,x^{3}+3 e^{b^{2} x^{2}} \left (\int \mathrm {erf}\left (b x \right )^{2} x^{2}d x \right ) b^{3} \pi +e^{b^{2} x^{2}} b^{3} \pi \,x^{3}-2 \sqrt {\pi }\, b^{2} x^{2}-2 \sqrt {\pi }}{3 e^{b^{2} x^{2}} b^{3} \pi } \] Input:

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

Output:

( - 2*e**(b**2*x**2)*erf(b*x)*b**3*pi*x**3 + 3*e**(b**2*x**2)*int(erf(b*x) 
**2*x**2,x)*b**3*pi + e**(b**2*x**2)*b**3*pi*x**3 - 2*sqrt(pi)*b**2*x**2 - 
 2*sqrt(pi))/(3*e**(b**2*x**2)*b**3*pi)