\(\int e^{-b^2 x^2} x^3 \text {erfc}(b x) \, dx\) [179]

Optimal result

Integrand size = 18, antiderivative size = 90 \[ \int e^{-b^2 x^2} x^3 \text {erfc}(b x) \, dx=\frac {e^{-2 b^2 x^2} x}{4 b^3 \sqrt {\pi }}-\frac {5 \text {erf}\left (\sqrt {2} b x\right )}{8 \sqrt {2} b^4}-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{2 b^4}-\frac {e^{-b^2 x^2} x^2 \text {erfc}(b x)}{2 b^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.77 \[ \int e^{-b^2 x^2} x^3 \text {erfc}(b x) \, dx=\frac {-5 \sqrt {2} \text {erf}\left (\sqrt {2} b x\right )+4 e^{-2 b^2 x^2} \left (\frac {b x}{\sqrt {\pi }}-2 e^{b^2 x^2} \left (1+b^2 x^2\right ) \text {erfc}(b x)\right )}{16 b^4} \] Input:

Integrate[(x^3*Erfc[b*x])/E^(b^2*x^2),x]
 

Output:

(-5*Sqrt[2]*Erf[Sqrt[2]*b*x] + (4*((b*x)/Sqrt[Pi] - 2*E^(b^2*x^2)*(1 + b^2 
*x^2)*Erfc[b*x]))/E^(2*b^2*x^2))/(16*b^4)
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.40, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {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.

\(\displaystyle \int x^3 e^{-b^2 x^2} \text {erfc}(b x) \, dx\)

\(\Big \downarrow \) 6940

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

\(\Big \downarrow \) 2641

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

\(\Big \downarrow \) 2634

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

\(\Big \downarrow \) 6937

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

\(\Big \downarrow \) 2634

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

Input:

Int[(x^3*Erfc[b*x])/E^(b^2*x^2),x]
 

Output:

-((-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])) - (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
 

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 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 = 1.06 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.31

method	result	size
default	\(\frac {\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}} x^{2} b^{2}}{2}-\frac {{\mathrm e}^{-b^{2} x^{2}}}{2}}{b^{3}}-\frac {\operatorname {erf}\left (b x \right ) \left (-\frac {{\mathrm e}^{-b^{2} x^{2}} x^{2} b^{2}}{2}-\frac {{\mathrm e}^{-b^{2} x^{2}}}{2}\right )}{b^{3}}+\frac {-\frac {5 \sqrt {2}\, \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {2}\, b x \right )}{16}+\frac {{\mathrm e}^{-2 b^{2} x^{2}} b x}{4}}{\sqrt {\pi }\, b^{3}}}{b}\)	\(118\)

Input:

int(x^3*erfc(b*x)/exp(b^2*x^2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00 \[ \int e^{-b^2 x^2} x^3 \text {erfc}(b x) \, dx=\frac {4 \, \sqrt {\pi } b^{2} x e^{\left (-2 \, b^{2} x^{2}\right )} - 5 \, \sqrt {2} \pi \sqrt {b^{2}} \operatorname {erf}\left (\sqrt {2} \sqrt {b^{2}} x\right ) - 8 \, {\left (\pi b^{3} x^{2} + \pi b - {\left (\pi b^{3} x^{2} + \pi b\right )} \operatorname {erf}\left (b x\right )\right )} e^{\left (-b^{2} x^{2}\right )}}{16 \, \pi b^{5}} \] Input:

integrate(x^3*erfc(b*x)/exp(b^2*x^2),x, algorithm="fricas")
 

Output:

1/16*(4*sqrt(pi)*b^2*x*e^(-2*b^2*x^2) - 5*sqrt(2)*pi*sqrt(b^2)*erf(sqrt(2) 
*sqrt(b^2)*x) - 8*(pi*b^3*x^2 + pi*b - (pi*b^3*x^2 + pi*b)*erf(b*x))*e^(-b 
^2*x^2))/(pi*b^5)
 

Sympy [F]

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

integrate(x**3*erfc(b*x)/exp(b**2*x**2),x)
 

Output:

Integral(x**3*exp(-b**2*x**2)*erfc(b*x), x)
 

Maxima [F]

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

integrate(x^3*erfc(b*x)/exp(b^2*x^2),x, algorithm="maxima")
 

Output:

integrate(x^3*erfc(b*x)*e^(-b^2*x^2), x)
 

Giac [F]

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

integrate(x^3*erfc(b*x)/exp(b^2*x^2),x, algorithm="giac")
 

Output:

integrate(x^3*erfc(b*x)*e^(-b^2*x^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int e^{-b^2 x^2} x^3 \text {erfc}(b x) \, dx=\int x^3\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right ) \,d x \] Input:

int(x^3*exp(-b^2*x^2)*erfc(b*x),x)
 

Output:

int(x^3*exp(-b^2*x^2)*erfc(b*x), x)
 

Reduce [F]

\[ \int e^{-b^2 x^2} x^3 \text {erfc}(b x) \, dx=\frac {-2 e^{b^{2} x^{2}} \left (\int \frac {\mathrm {erf}\left (b x \right ) x^{3}}{e^{b^{2} x^{2}}}d x \right ) b^{4}-b^{2} x^{2}-1}{2 e^{b^{2} x^{2}} b^{4}} \] Input:

int(x^3*erfc(b*x)/exp(b^2*x^2),x)
 

Output:

( - 2*e**(b**2*x**2)*int((erf(b*x)*x**3)/e**(b**2*x**2),x)*b**4 - b**2*x** 
2 - 1)/(2*e**(b**2*x**2)*b**4)