\(\int e^{-b^2 x^2} x^5 \text {erfc}(b x) \, dx\) [178]

Optimal result

Integrand size = 18, antiderivative size = 135 \[ \int e^{-b^2 x^2} x^5 \text {erfc}(b x) \, dx=\frac {11 e^{-2 b^2 x^2} x}{16 b^5 \sqrt {\pi }}+\frac {e^{-2 b^2 x^2} x^3}{4 b^3 \sqrt {\pi }}-\frac {43 \text {erf}\left (\sqrt {2} b x\right )}{32 \sqrt {2} b^6}-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{b^6}-\frac {e^{-b^2 x^2} x^2 \text {erfc}(b x)}{b^4}-\frac {e^{-b^2 x^2} x^4 \text {erfc}(b x)}{2 b^2} \] Output:

11/16*x/b^5/exp(2*b^2*x^2)/Pi^(1/2)+1/4*x^3/b^3/exp(2*b^2*x^2)/Pi^(1/2)-43 
/64*erf(2^(1/2)*b*x)*2^(1/2)/b^6-erfc(b*x)/b^6/exp(b^2*x^2)-x^2*erfc(b*x)/ 
b^4/exp(b^2*x^2)-1/2*x^4*erfc(b*x)/b^2/exp(b^2*x^2)
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.64 \[ \int e^{-b^2 x^2} x^5 \text {erfc}(b x) \, dx=\frac {-43 \sqrt {2} \text {erf}\left (\sqrt {2} b x\right )+4 e^{-2 b^2 x^2} \left (\frac {b x \left (11+4 b^2 x^2\right )}{\sqrt {\pi }}-8 e^{b^2 x^2} \left (2+2 b^2 x^2+b^4 x^4\right ) \text {erfc}(b x)\right )}{64 b^6} \] Input:

Integrate[(x^5*Erfc[b*x])/E^(b^2*x^2),x]
 

Output:

(-43*Sqrt[2]*Erf[Sqrt[2]*b*x] + (4*((b*x*(11 + 4*b^2*x^2))/Sqrt[Pi] - 8*E^ 
(b^2*x^2)*(2 + 2*b^2*x^2 + b^4*x^4)*Erfc[b*x]))/E^(2*b^2*x^2))/(64*b^6)
 

Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.76, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6940, 2641, 2641, 2634, 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.

\(\displaystyle \int x^5 e^{-b^2 x^2} \text {erfc}(b x) \, dx\)

\(\Big \downarrow \) 6940

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

\(\Big \downarrow \) 2641

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

\(\Big \downarrow \) 2641

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

\(\Big \downarrow \) 2634

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

\(\Big \downarrow \) 6940

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

\(\Big \downarrow \) 2641

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

\(\Big \downarrow \) 2634

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

\(\Big \downarrow \) 6937

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

\(\Big \downarrow \) 2634

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

Input:

Int[(x^5*Erfc[b*x])/E^(b^2*x^2),x]
 

Output:

-((-1/4*x^3/(b^2*E^(2*b^2*x^2)) + (3*(-1/4*x/(b^2*E^(2*b^2*x^2)) + (Sqrt[P 
i/2]*Erf[Sqrt[2]*b*x])/(8*b^3)))/(4*b^2))/(b*Sqrt[Pi])) - (x^4*Erfc[b*x])/ 
(2*b^2*E^(b^2*x^2)) + (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])) - (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))/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 = 2.24 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.27

method	result	size
default	\(\frac {\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}} x^{4} b^{4}}{2}-{\mathrm e}^{-b^{2} x^{2}} x^{2} b^{2}-{\mathrm e}^{-b^{2} x^{2}}}{b^{5}}-\frac {\operatorname {erf}\left (b x \right ) \left (-\frac {{\mathrm e}^{-b^{2} x^{2}} x^{4} b^{4}}{2}-{\mathrm e}^{-b^{2} x^{2}} x^{2} b^{2}-{\mathrm e}^{-b^{2} x^{2}}\right )}{b^{5}}+\frac {-\frac {43 \sqrt {2}\, \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {2}\, b x \right )}{64}+\frac {11 \,{\mathrm e}^{-2 b^{2} x^{2}} b x}{16}+\frac {{\mathrm e}^{-2 b^{2} x^{2}} b^{3} x^{3}}{4}}{\sqrt {\pi }\, b^{5}}}{b}\)	\(172\)

Input:

int(x^5*erfc(b*x)/exp(b^2*x^2),x,method=_RETURNVERBOSE)
 

Output:

(1/b^5*(-1/2/exp(b^2*x^2)*b^4*x^4-b^2*x^2/exp(b^2*x^2)-1/exp(b^2*x^2))-erf 
(b*x)/b^5*(-1/2/exp(b^2*x^2)*b^4*x^4-b^2*x^2/exp(b^2*x^2)-1/exp(b^2*x^2))+ 
1/Pi^(1/2)/b^5*(-43/64*2^(1/2)*Pi^(1/2)*erf(2^(1/2)*b*x)+1/4/exp(b^2*x^2)^ 
2*b^3*x^3+11/16/exp(b^2*x^2)^2*b*x))/b
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.90 \[ \int e^{-b^2 x^2} x^5 \text {erfc}(b x) \, dx=-\frac {43 \, \sqrt {2} \pi \sqrt {b^{2}} \operatorname {erf}\left (\sqrt {2} \sqrt {b^{2}} x\right ) - 4 \, \sqrt {\pi } {\left (4 \, b^{4} x^{3} + 11 \, b^{2} x\right )} e^{\left (-2 \, b^{2} x^{2}\right )} + 32 \, {\left (\pi b^{5} x^{4} + 2 \, \pi b^{3} x^{2} + 2 \, \pi b - {\left (\pi b^{5} x^{4} + 2 \, \pi b^{3} x^{2} + 2 \, \pi b\right )} \operatorname {erf}\left (b x\right )\right )} e^{\left (-b^{2} x^{2}\right )}}{64 \, \pi b^{7}} \] Input:

integrate(x^5*erfc(b*x)/exp(b^2*x^2),x, algorithm="fricas")
 

Output:

-1/64*(43*sqrt(2)*pi*sqrt(b^2)*erf(sqrt(2)*sqrt(b^2)*x) - 4*sqrt(pi)*(4*b^ 
4*x^3 + 11*b^2*x)*e^(-2*b^2*x^2) + 32*(pi*b^5*x^4 + 2*pi*b^3*x^2 + 2*pi*b 
- (pi*b^5*x^4 + 2*pi*b^3*x^2 + 2*pi*b)*erf(b*x))*e^(-b^2*x^2))/(pi*b^7)
 

Sympy [F]

\[ \int e^{-b^2 x^2} x^5 \text {erfc}(b x) \, dx=\int x^{5} e^{- b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}\, dx \] Input:

integrate(x**5*erfc(b*x)/exp(b**2*x**2),x)
 

Output:

Integral(x**5*exp(-b**2*x**2)*erfc(b*x), x)
 

Maxima [F]

\[ \int e^{-b^2 x^2} x^5 \text {erfc}(b x) \, dx=\int { x^{5} \operatorname {erfc}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} \,d x } \] Input:

integrate(x^5*erfc(b*x)/exp(b^2*x^2),x, algorithm="maxima")
 

Output:

integrate(x^5*erfc(b*x)*e^(-b^2*x^2), x)
 

Giac [F]

\[ \int e^{-b^2 x^2} x^5 \text {erfc}(b x) \, dx=\int { x^{5} \operatorname {erfc}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} \,d x } \] Input:

integrate(x^5*erfc(b*x)/exp(b^2*x^2),x, algorithm="giac")
 

Output:

integrate(x^5*erfc(b*x)*e^(-b^2*x^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int e^{-b^2 x^2} x^5 \text {erfc}(b x) \, dx=\int x^5\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right ) \,d x \] Input:

int(x^5*exp(-b^2*x^2)*erfc(b*x),x)
 

Output:

int(x^5*exp(-b^2*x^2)*erfc(b*x), x)
 

Reduce [F]

\[ \int e^{-b^2 x^2} x^5 \text {erfc}(b x) \, dx=\frac {-2 e^{b^{2} x^{2}} \left (\int \frac {\mathrm {erf}\left (b x \right ) x^{5}}{e^{b^{2} x^{2}}}d x \right ) b^{6}-b^{4} x^{4}-2 b^{2} x^{2}-2}{2 e^{b^{2} x^{2}} b^{6}} \] Input:

int(x^5*erfc(b*x)/exp(b^2*x^2),x)
 

Output:

( - 2*e**(b**2*x**2)*int((erf(b*x)*x**5)/e**(b**2*x**2),x)*b**6 - b**4*x** 
4 - 2*b**2*x**2 - 2)/(2*e**(b**2*x**2)*b**6)