Integrand size = 17, antiderivative size = 283 \[ \int e^{c+d x^2} x^5 \text {erfc}(b x) \, dx=\frac {b e^{c-\left (b^2-d\right ) x^2} x}{\left (b^2-d\right ) d^2 \sqrt {\pi }}-\frac {3 b e^{c-\left (b^2-d\right ) x^2} x}{4 \left (b^2-d\right )^2 d \sqrt {\pi }}-\frac {b e^{c-\left (b^2-d\right ) x^2} x^3}{2 \left (b^2-d\right ) d \sqrt {\pi }}+\frac {b e^c \text {erf}\left (\sqrt {b^2-d} x\right )}{\sqrt {b^2-d} d^3}-\frac {b e^c \text {erf}\left (\sqrt {b^2-d} x\right )}{2 \left (b^2-d\right )^{3/2} d^2}+\frac {3 b e^c \text {erf}\left (\sqrt {b^2-d} x\right )}{8 \left (b^2-d\right )^{5/2} d}+\frac {e^{c+d x^2} \text {erfc}(b x)}{d^3}-\frac {e^{c+d x^2} x^2 \text {erfc}(b x)}{d^2}+\frac {e^{c+d x^2} x^4 \text {erfc}(b x)}{2 d} \]
-1/2*b*exp(c)*erf(x*(b^2-d)^(1/2))/(b^2-d)^(3/2)/d^2+3/8*b*exp(c)*erf(x*(b ^2-d)^(1/2))/(b^2-d)^(5/2)/d+exp(d*x^2+c)*erfc(b*x)/d^3-exp(d*x^2+c)*x^2*e rfc(b*x)/d^2+1/2*exp(d*x^2+c)*x^4*erfc(b*x)/d+b*exp(c)*erf(x*(b^2-d)^(1/2) )/d^3/(b^2-d)^(1/2)+b*exp(c-(b^2-d)*x^2)*x/(b^2-d)/d^2/Pi^(1/2)-3/4*b*exp( c-(b^2-d)*x^2)*x/(b^2-d)^2/d/Pi^(1/2)-1/2*b*exp(c-(b^2-d)*x^2)*x^3/(b^2-d) /d/Pi^(1/2)
Time = 0.28 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.49 \[ \int e^{c+d x^2} x^5 \text {erfc}(b x) \, dx=\frac {e^c \left (\frac {2 b d e^{\left (-b^2+d\right ) x^2} x \left (b^2 \left (4-2 d x^2\right )+d \left (-7+2 d x^2\right )\right )}{\left (b^2-d\right )^2 \sqrt {\pi }}+4 e^{d x^2} \left (2-2 d x^2+d^2 x^4\right ) \text {erfc}(b x)+\frac {b \left (8 b^4-20 b^2 d+15 d^2\right ) \text {erfi}\left (\sqrt {-b^2+d} x\right )}{\left (-b^2+d\right )^{5/2}}\right )}{8 d^3} \]
(E^c*((2*b*d*E^((-b^2 + d)*x^2)*x*(b^2*(4 - 2*d*x^2) + d*(-7 + 2*d*x^2)))/ ((b^2 - d)^2*Sqrt[Pi]) + 4*E^(d*x^2)*(2 - 2*d*x^2 + d^2*x^4)*Erfc[b*x] + ( b*(8*b^4 - 20*b^2*d + 15*d^2)*Erfi[Sqrt[-b^2 + d]*x])/(-b^2 + d)^(5/2)))/( 8*d^3)
Time = 1.28 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.12, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, 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.
| \(\displaystyle \int x^5 \text {erfc}(b x) e^{c+d x^2} \, dx\) |
| \(\Big \downarrow \) 6940 |
| \(\displaystyle \frac {b \int e^{c-\left (b^2-d\right ) x^2} x^4dx}{\sqrt {\pi } d}-\frac {2 \int e^{d x^2+c} x^3 \text {erfc}(b x)dx}{d}+\frac {x^4 \text {erfc}(b x) e^{c+d x^2}}{2 d}\) |
| \(\Big \downarrow \) 2641 |
| \(\displaystyle \frac {b \left (\frac {3 \int e^{c-\left (b^2-d\right ) x^2} x^2dx}{2 \left (b^2-d\right )}-\frac {x^3 e^{c-x^2 \left (b^2-d\right )}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {2 \int e^{d x^2+c} x^3 \text {erfc}(b x)dx}{d}+\frac {x^4 \text {erfc}(b x) e^{c+d x^2}}{2 d}\) |
| \(\Big \downarrow \) 2641 |
| \(\displaystyle \frac {b \left (\frac {3 \left (\frac {\int e^{c-\left (b^2-d\right ) x^2}dx}{2 \left (b^2-d\right )}-\frac {x e^{c-x^2 \left (b^2-d\right )}}{2 \left (b^2-d\right )}\right )}{2 \left (b^2-d\right )}-\frac {x^3 e^{c-x^2 \left (b^2-d\right )}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {2 \int e^{d x^2+c} x^3 \text {erfc}(b x)dx}{d}+\frac {x^4 \text {erfc}(b x) e^{c+d x^2}}{2 d}\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle -\frac {2 \int e^{d x^2+c} x^3 \text {erfc}(b x)dx}{d}+\frac {b \left (\frac {3 \left (\frac {\sqrt {\pi } e^c \text {erf}\left (x \sqrt {b^2-d}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{c-x^2 \left (b^2-d\right )}}{2 \left (b^2-d\right )}\right )}{2 \left (b^2-d\right )}-\frac {x^3 e^{c-x^2 \left (b^2-d\right )}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}+\frac {x^4 \text {erfc}(b x) e^{c+d x^2}}{2 d}\) |
| \(\Big \downarrow \) 6940 |
| \(\displaystyle -\frac {2 \left (\frac {b \int e^{c-\left (b^2-d\right ) x^2} x^2dx}{\sqrt {\pi } d}-\frac {\int e^{d x^2+c} x \text {erfc}(b x)dx}{d}+\frac {x^2 \text {erfc}(b x) e^{c+d x^2}}{2 d}\right )}{d}+\frac {b \left (\frac {3 \left (\frac {\sqrt {\pi } e^c \text {erf}\left (x \sqrt {b^2-d}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{c-x^2 \left (b^2-d\right )}}{2 \left (b^2-d\right )}\right )}{2 \left (b^2-d\right )}-\frac {x^3 e^{c-x^2 \left (b^2-d\right )}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}+\frac {x^4 \text {erfc}(b x) e^{c+d x^2}}{2 d}\) |
| \(\Big \downarrow \) 2641 |
| \(\displaystyle -\frac {2 \left (\frac {b \left (\frac {\int e^{c-\left (b^2-d\right ) x^2}dx}{2 \left (b^2-d\right )}-\frac {x e^{c-x^2 \left (b^2-d\right )}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {\int e^{d x^2+c} x \text {erfc}(b x)dx}{d}+\frac {x^2 \text {erfc}(b x) e^{c+d x^2}}{2 d}\right )}{d}+\frac {b \left (\frac {3 \left (\frac {\sqrt {\pi } e^c \text {erf}\left (x \sqrt {b^2-d}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{c-x^2 \left (b^2-d\right )}}{2 \left (b^2-d\right )}\right )}{2 \left (b^2-d\right )}-\frac {x^3 e^{c-x^2 \left (b^2-d\right )}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}+\frac {x^4 \text {erfc}(b x) e^{c+d x^2}}{2 d}\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle -\frac {2 \left (-\frac {\int e^{d x^2+c} x \text {erfc}(b x)dx}{d}+\frac {b \left (\frac {\sqrt {\pi } e^c \text {erf}\left (x \sqrt {b^2-d}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{c-x^2 \left (b^2-d\right )}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}+\frac {x^2 \text {erfc}(b x) e^{c+d x^2}}{2 d}\right )}{d}+\frac {b \left (\frac {3 \left (\frac {\sqrt {\pi } e^c \text {erf}\left (x \sqrt {b^2-d}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{c-x^2 \left (b^2-d\right )}}{2 \left (b^2-d\right )}\right )}{2 \left (b^2-d\right )}-\frac {x^3 e^{c-x^2 \left (b^2-d\right )}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}+\frac {x^4 \text {erfc}(b x) e^{c+d x^2}}{2 d}\) |
| \(\Big \downarrow \) 6937 |
| \(\displaystyle -\frac {2 \left (-\frac {\frac {b \int e^{c-\left (b^2-d\right ) x^2}dx}{\sqrt {\pi } d}+\frac {\text {erfc}(b x) e^{c+d x^2}}{2 d}}{d}+\frac {b \left (\frac {\sqrt {\pi } e^c \text {erf}\left (x \sqrt {b^2-d}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{c-x^2 \left (b^2-d\right )}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}+\frac {x^2 \text {erfc}(b x) e^{c+d x^2}}{2 d}\right )}{d}+\frac {b \left (\frac {3 \left (\frac {\sqrt {\pi } e^c \text {erf}\left (x \sqrt {b^2-d}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{c-x^2 \left (b^2-d\right )}}{2 \left (b^2-d\right )}\right )}{2 \left (b^2-d\right )}-\frac {x^3 e^{c-x^2 \left (b^2-d\right )}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}+\frac {x^4 \text {erfc}(b x) e^{c+d x^2}}{2 d}\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle -\frac {2 \left (-\frac {\frac {b e^c \text {erf}\left (x \sqrt {b^2-d}\right )}{2 d \sqrt {b^2-d}}+\frac {\text {erfc}(b x) e^{c+d x^2}}{2 d}}{d}+\frac {b \left (\frac {\sqrt {\pi } e^c \text {erf}\left (x \sqrt {b^2-d}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{c-x^2 \left (b^2-d\right )}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}+\frac {x^2 \text {erfc}(b x) e^{c+d x^2}}{2 d}\right )}{d}+\frac {b \left (\frac {3 \left (\frac {\sqrt {\pi } e^c \text {erf}\left (x \sqrt {b^2-d}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{c-x^2 \left (b^2-d\right )}}{2 \left (b^2-d\right )}\right )}{2 \left (b^2-d\right )}-\frac {x^3 e^{c-x^2 \left (b^2-d\right )}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}+\frac {x^4 \text {erfc}(b x) e^{c+d x^2}}{2 d}\) |
(b*(-1/2*(E^(c - (b^2 - d)*x^2)*x^3)/(b^2 - d) + (3*(-1/2*(E^(c - (b^2 - d )*x^2)*x)/(b^2 - d) + (E^c*Sqrt[Pi]*Erf[Sqrt[b^2 - d]*x])/(4*(b^2 - d)^(3/ 2))))/(2*(b^2 - d))))/(d*Sqrt[Pi]) + (E^(c + d*x^2)*x^4*Erfc[b*x])/(2*d) - (2*((b*(-1/2*(E^(c - (b^2 - d)*x^2)*x)/(b^2 - d) + (E^c*Sqrt[Pi]*Erf[Sqrt [b^2 - d]*x])/(4*(b^2 - d)^(3/2))))/(d*Sqrt[Pi]) + (E^(c + d*x^2)*x^2*Erfc [b*x])/(2*d) - ((b*E^c*Erf[Sqrt[b^2 - d]*x])/(2*Sqrt[b^2 - d]*d) + (E^(c + d*x^2)*Erfc[b*x])/(2*d))/d))/d
3.2.56.3.1 Defintions of rubi rules used
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]
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])
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]
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]
Time = 5.43 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.33
| method | result | size |
| default | \(\frac {\frac {{\mathrm e}^{c} \left (\frac {{\mathrm e}^{d \,x^{2}} b^{6} x^{4}}{2 d}-\frac {2 b^{2} \left (\frac {b^{4} x^{2} {\mathrm e}^{d \,x^{2}}}{2 d}-\frac {b^{4} {\mathrm e}^{d \,x^{2}}}{2 d^{2}}\right )}{d}\right )}{b^{5}}-\frac {\operatorname {erf}\left (b x \right ) {\mathrm e}^{c} \left (\frac {{\mathrm e}^{d \,x^{2}} b^{6} x^{4}}{2 d}-\frac {2 b^{2} \left (\frac {b^{4} x^{2} {\mathrm e}^{d \,x^{2}}}{2 d}-\frac {b^{4} {\mathrm e}^{d \,x^{2}}}{2 d^{2}}\right )}{d}\right )}{b^{5}}+\frac {{\mathrm e}^{c} \left (\frac {b^{2} \left (\frac {b^{3} x^{3} {\mathrm e}^{\left (-1+\frac {d}{b^{2}}\right ) b^{2} x^{2}}}{-2+\frac {2 d}{b^{2}}}-\frac {3 \left (\frac {b x \,{\mathrm e}^{\left (-1+\frac {d}{b^{2}}\right ) b^{2} x^{2}}}{-2+\frac {2 d}{b^{2}}}-\frac {\sqrt {\pi }\, \operatorname {erf}\left (\sqrt {1-\frac {d}{b^{2}}}\, b x \right )}{4 \left (-1+\frac {d}{b^{2}}\right ) \sqrt {1-\frac {d}{b^{2}}}}\right )}{2 \left (-1+\frac {d}{b^{2}}\right )}\right )}{d}+\frac {b^{6} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {1-\frac {d}{b^{2}}}\, b x \right )}{d^{3} \sqrt {1-\frac {d}{b^{2}}}}-\frac {2 b^{4} \left (\frac {b x \,{\mathrm e}^{\left (-1+\frac {d}{b^{2}}\right ) b^{2} x^{2}}}{-2+\frac {2 d}{b^{2}}}-\frac {\sqrt {\pi }\, \operatorname {erf}\left (\sqrt {1-\frac {d}{b^{2}}}\, b x \right )}{4 \left (-1+\frac {d}{b^{2}}\right ) \sqrt {1-\frac {d}{b^{2}}}}\right )}{d^{2}}\right )}{\sqrt {\pi }\, b^{5}}}{b}\) | \(376\) |
(1/b^5*exp(c)*(1/2*exp(d*x^2)*b^6*x^4/d-2/d*b^2*(1/2/d*b^4*x^2*exp(d*x^2)- 1/2/d^2*b^4*exp(d*x^2)))-erf(b*x)/b^5*exp(c)*(1/2*exp(d*x^2)*b^6*x^4/d-2/d *b^2*(1/2/d*b^4*x^2*exp(d*x^2)-1/2/d^2*b^4*exp(d*x^2)))+1/Pi^(1/2)/b^5*exp (c)*(1/d*b^2*(1/2/(-1+d/b^2)*b^3*x^3*exp((-1+d/b^2)*b^2*x^2)-3/2/(-1+d/b^2 )*(1/2/(-1+d/b^2)*b*x*exp((-1+d/b^2)*b^2*x^2)-1/4/(-1+d/b^2)*Pi^(1/2)/(1-d /b^2)^(1/2)*erf((1-d/b^2)^(1/2)*b*x)))+1/d^3*b^6*Pi^(1/2)/(1-d/b^2)^(1/2)* erf((1-d/b^2)^(1/2)*b*x)-2/d^2*b^4*(1/2/(-1+d/b^2)*b*x*exp((-1+d/b^2)*b^2* x^2)-1/4/(-1+d/b^2)*Pi^(1/2)/(1-d/b^2)^(1/2)*erf((1-d/b^2)^(1/2)*b*x))))/b
Time = 0.28 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.26 \[ \int e^{c+d x^2} x^5 \text {erfc}(b x) \, dx=\frac {\pi {\left (8 \, b^{5} - 20 \, b^{3} d + 15 \, b d^{2}\right )} \sqrt {b^{2} - d} \operatorname {erf}\left (\sqrt {b^{2} - d} x\right ) e^{c} - 2 \, \sqrt {\pi } {\left (2 \, {\left (b^{5} d^{2} - 2 \, b^{3} d^{3} + b d^{4}\right )} x^{3} - {\left (4 \, b^{5} d - 11 \, b^{3} d^{2} + 7 \, b d^{3}\right )} x\right )} e^{\left (-b^{2} x^{2} + d x^{2} + c\right )} + 4 \, {\left (\pi {\left (b^{6} d^{2} - 3 \, b^{4} d^{3} + 3 \, b^{2} d^{4} - d^{5}\right )} x^{4} - 2 \, \pi {\left (b^{6} d - 3 \, b^{4} d^{2} + 3 \, b^{2} d^{3} - d^{4}\right )} x^{2} + 2 \, \pi {\left (b^{6} - 3 \, b^{4} d + 3 \, b^{2} d^{2} - d^{3}\right )} - {\left (\pi {\left (b^{6} d^{2} - 3 \, b^{4} d^{3} + 3 \, b^{2} d^{4} - d^{5}\right )} x^{4} - 2 \, \pi {\left (b^{6} d - 3 \, b^{4} d^{2} + 3 \, b^{2} d^{3} - d^{4}\right )} x^{2} + 2 \, \pi {\left (b^{6} - 3 \, b^{4} d + 3 \, b^{2} d^{2} - d^{3}\right )}\right )} \operatorname {erf}\left (b x\right )\right )} e^{\left (d x^{2} + c\right )}}{8 \, \pi {\left (b^{6} d^{3} - 3 \, b^{4} d^{4} + 3 \, b^{2} d^{5} - d^{6}\right )}} \]
1/8*(pi*(8*b^5 - 20*b^3*d + 15*b*d^2)*sqrt(b^2 - d)*erf(sqrt(b^2 - d)*x)*e ^c - 2*sqrt(pi)*(2*(b^5*d^2 - 2*b^3*d^3 + b*d^4)*x^3 - (4*b^5*d - 11*b^3*d ^2 + 7*b*d^3)*x)*e^(-b^2*x^2 + d*x^2 + c) + 4*(pi*(b^6*d^2 - 3*b^4*d^3 + 3 *b^2*d^4 - d^5)*x^4 - 2*pi*(b^6*d - 3*b^4*d^2 + 3*b^2*d^3 - d^4)*x^2 + 2*p i*(b^6 - 3*b^4*d + 3*b^2*d^2 - d^3) - (pi*(b^6*d^2 - 3*b^4*d^3 + 3*b^2*d^4 - d^5)*x^4 - 2*pi*(b^6*d - 3*b^4*d^2 + 3*b^2*d^3 - d^4)*x^2 + 2*pi*(b^6 - 3*b^4*d + 3*b^2*d^2 - d^3))*erf(b*x))*e^(d*x^2 + c))/(pi*(b^6*d^3 - 3*b^4 *d^4 + 3*b^2*d^5 - d^6))
\[ \int e^{c+d x^2} x^5 \text {erfc}(b x) \, dx=e^{c} \int x^{5} e^{d x^{2}} \operatorname {erfc}{\left (b x \right )}\, dx \]
\[ \int e^{c+d x^2} x^5 \text {erfc}(b x) \, dx=\int { x^{5} \operatorname {erfc}\left (b x\right ) e^{\left (d x^{2} + c\right )} \,d x } \]
\[ \int e^{c+d x^2} x^5 \text {erfc}(b x) \, dx=\int { x^{5} \operatorname {erfc}\left (b x\right ) e^{\left (d x^{2} + c\right )} \,d x } \]
Timed out. \[ \int e^{c+d x^2} x^5 \text {erfc}(b x) \, dx=\int x^5\,{\mathrm {e}}^{d\,x^2+c}\,\mathrm {erfc}\left (b\,x\right ) \,d x \]