Integrand size = 19, antiderivative size = 118 \[ \int e^{c+b^2 x^2} x^5 \text {erfc}(b x) \, dx=\frac {2 e^c x}{b^5 \sqrt {\pi }}-\frac {2 e^c x^3}{3 b^3 \sqrt {\pi }}+\frac {e^c x^5}{5 b \sqrt {\pi }}+\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{b^6}-\frac {e^{c+b^2 x^2} x^2 \text {erfc}(b x)}{b^4}+\frac {e^{c+b^2 x^2} x^4 \text {erfc}(b x)}{2 b^2} \]
exp(b^2*x^2+c)*erfc(b*x)/b^6-exp(b^2*x^2+c)*x^2*erfc(b*x)/b^4+1/2*exp(b^2* x^2+c)*x^4*erfc(b*x)/b^2+2*exp(c)*x/b^5/Pi^(1/2)-2/3*exp(c)*x^3/b^3/Pi^(1/ 2)+1/5*exp(c)*x^5/b/Pi^(1/2)
Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.62 \[ \int e^{c+b^2 x^2} x^5 \text {erfc}(b x) \, dx=\frac {e^c \left (60 b x-20 b^3 x^3+6 b^5 x^5+15 e^{b^2 x^2} \sqrt {\pi } \left (2-2 b^2 x^2+b^4 x^4\right ) \text {erfc}(b x)\right )}{30 b^6 \sqrt {\pi }} \]
(E^c*(60*b*x - 20*b^3*x^3 + 6*b^5*x^5 + 15*E^(b^2*x^2)*Sqrt[Pi]*(2 - 2*b^2 *x^2 + b^4*x^4)*Erfc[b*x]))/(30*b^6*Sqrt[Pi])
Time = 0.48 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.14, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {6940, 15, 6940, 15, 6937, 24}
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.
\(\displaystyle \int x^5 e^{b^2 x^2+c} \text {erfc}(b x) \, dx\) |
\(\Big \downarrow \) 6940 |
\(\displaystyle -\frac {2 \int e^{b^2 x^2+c} x^3 \text {erfc}(b x)dx}{b^2}+\frac {\int e^c x^4dx}{\sqrt {\pi } b}+\frac {x^4 e^{b^2 x^2+c} \text {erfc}(b x)}{2 b^2}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {2 \int e^{b^2 x^2+c} x^3 \text {erfc}(b x)dx}{b^2}+\frac {x^4 e^{b^2 x^2+c} \text {erfc}(b x)}{2 b^2}+\frac {e^c x^5}{5 \sqrt {\pi } b}\) |
\(\Big \downarrow \) 6940 |
\(\displaystyle -\frac {2 \left (-\frac {\int e^{b^2 x^2+c} x \text {erfc}(b x)dx}{b^2}+\frac {\int e^c x^2dx}{\sqrt {\pi } b}+\frac {x^2 e^{b^2 x^2+c} \text {erfc}(b x)}{2 b^2}\right )}{b^2}+\frac {x^4 e^{b^2 x^2+c} \text {erfc}(b x)}{2 b^2}+\frac {e^c x^5}{5 \sqrt {\pi } b}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {2 \left (-\frac {\int e^{b^2 x^2+c} x \text {erfc}(b x)dx}{b^2}+\frac {x^2 e^{b^2 x^2+c} \text {erfc}(b x)}{2 b^2}+\frac {e^c x^3}{3 \sqrt {\pi } b}\right )}{b^2}+\frac {x^4 e^{b^2 x^2+c} \text {erfc}(b x)}{2 b^2}+\frac {e^c x^5}{5 \sqrt {\pi } b}\) |
\(\Big \downarrow \) 6937 |
\(\displaystyle -\frac {2 \left (-\frac {\frac {\int e^cdx}{\sqrt {\pi } b}+\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{2 b^2}}{b^2}+\frac {x^2 e^{b^2 x^2+c} \text {erfc}(b x)}{2 b^2}+\frac {e^c x^3}{3 \sqrt {\pi } b}\right )}{b^2}+\frac {x^4 e^{b^2 x^2+c} \text {erfc}(b x)}{2 b^2}+\frac {e^c x^5}{5 \sqrt {\pi } b}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {x^4 e^{b^2 x^2+c} \text {erfc}(b x)}{2 b^2}-\frac {2 \left (\frac {x^2 e^{b^2 x^2+c} \text {erfc}(b x)}{2 b^2}-\frac {\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{2 b^2}+\frac {e^c x}{\sqrt {\pi } b}}{b^2}+\frac {e^c x^3}{3 \sqrt {\pi } b}\right )}{b^2}+\frac {e^c x^5}{5 \sqrt {\pi } b}\) |
(E^c*x^5)/(5*b*Sqrt[Pi]) + (E^(c + b^2*x^2)*x^4*Erfc[b*x])/(2*b^2) - (2*(( E^c*x^3)/(3*b*Sqrt[Pi]) + (E^(c + b^2*x^2)*x^2*Erfc[b*x])/(2*b^2) - ((E^c* x)/(b*Sqrt[Pi]) + (E^(c + b^2*x^2)*Erfc[b*x])/(2*b^2))/b^2))/b^2
3.2.67.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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 = 2.60 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.14
method | result | size |
default | \(\frac {\frac {{\mathrm e}^{c} \left (\frac {{\mathrm e}^{b^{2} x^{2}} b^{4} x^{4}}{2}-b^{2} x^{2} {\mathrm e}^{b^{2} x^{2}}+{\mathrm e}^{b^{2} x^{2}}\right )}{b^{5}}-\frac {\operatorname {erf}\left (b x \right ) {\mathrm e}^{c} \left (\frac {{\mathrm e}^{b^{2} x^{2}} b^{4} x^{4}}{2}-b^{2} x^{2} {\mathrm e}^{b^{2} x^{2}}+{\mathrm e}^{b^{2} x^{2}}\right )}{b^{5}}+\frac {{\mathrm e}^{c} \left (\frac {1}{5} b^{5} x^{5}-\frac {2}{3} b^{3} x^{3}+2 b x \right )}{\sqrt {\pi }\, b^{5}}}{b}\) | \(135\) |
parallelrisch | \(\frac {6 \,{\mathrm e}^{b^{2} x^{2}+c} {\mathrm e}^{-b^{2} x^{2}} x^{5} b^{5}+15 \,{\mathrm e}^{b^{2} x^{2}+c} x^{4} \operatorname {erfc}\left (b x \right ) b^{4} \sqrt {\pi }-20 \,{\mathrm e}^{b^{2} x^{2}+c} x^{3} {\mathrm e}^{-b^{2} x^{2}} b^{3}-30 \,{\mathrm e}^{b^{2} x^{2}+c} x^{2} \operatorname {erfc}\left (b x \right ) b^{2} \sqrt {\pi }+60 \,{\mathrm e}^{b^{2} x^{2}+c} x \,{\mathrm e}^{-b^{2} x^{2}} b +30 \,{\mathrm e}^{b^{2} x^{2}+c} \operatorname {erfc}\left (b x \right ) \sqrt {\pi }}{30 b^{6} \sqrt {\pi }}\) | \(156\) |
(1/b^5*exp(c)*(1/2*exp(b^2*x^2)*b^4*x^4-b^2*x^2*exp(b^2*x^2)+exp(b^2*x^2)) -erf(b*x)/b^5*exp(c)*(1/2*exp(b^2*x^2)*b^4*x^4-b^2*x^2*exp(b^2*x^2)+exp(b^ 2*x^2))+1/Pi^(1/2)/b^5*exp(c)*(1/5*b^5*x^5-2/3*b^3*x^3+2*b*x))/b
Time = 0.25 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.82 \[ \int e^{c+b^2 x^2} x^5 \text {erfc}(b x) \, dx=\frac {2 \, \sqrt {\pi } {\left (3 \, b^{5} x^{5} - 10 \, b^{3} x^{3} + 30 \, b x\right )} e^{c} + 15 \, {\left (2 \, \pi + \pi b^{4} x^{4} - 2 \, \pi b^{2} x^{2} - {\left (2 \, \pi + \pi b^{4} x^{4} - 2 \, \pi b^{2} x^{2}\right )} \operatorname {erf}\left (b x\right )\right )} e^{\left (b^{2} x^{2} + c\right )}}{30 \, \pi b^{6}} \]
1/30*(2*sqrt(pi)*(3*b^5*x^5 - 10*b^3*x^3 + 30*b*x)*e^c + 15*(2*pi + pi*b^4 *x^4 - 2*pi*b^2*x^2 - (2*pi + pi*b^4*x^4 - 2*pi*b^2*x^2)*erf(b*x))*e^(b^2* x^2 + c))/(pi*b^6)
Time = 33.43 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.07 \[ \int e^{c+b^2 x^2} x^5 \text {erfc}(b x) \, dx=\begin {cases} \frac {x^{5} e^{c}}{5 \sqrt {\pi } b} + \frac {x^{4} e^{c} e^{b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}}{2 b^{2}} - \frac {2 x^{3} e^{c}}{3 \sqrt {\pi } b^{3}} - \frac {x^{2} e^{c} e^{b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}}{b^{4}} + \frac {2 x e^{c}}{\sqrt {\pi } b^{5}} + \frac {e^{c} e^{b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}}{b^{6}} & \text {for}\: b \neq 0 \\\frac {x^{6} e^{c}}{6} & \text {otherwise} \end {cases} \]
Piecewise((x**5*exp(c)/(5*sqrt(pi)*b) + x**4*exp(c)*exp(b**2*x**2)*erfc(b* x)/(2*b**2) - 2*x**3*exp(c)/(3*sqrt(pi)*b**3) - x**2*exp(c)*exp(b**2*x**2) *erfc(b*x)/b**4 + 2*x*exp(c)/(sqrt(pi)*b**5) + exp(c)*exp(b**2*x**2)*erfc( b*x)/b**6, Ne(b, 0)), (x**6*exp(c)/6, True))
\[ \int e^{c+b^2 x^2} x^5 \text {erfc}(b x) \, dx=\int { x^{5} \operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} \,d x } \]
\[ \int e^{c+b^2 x^2} x^5 \text {erfc}(b x) \, dx=\int { x^{5} \operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} \,d x } \]
Time = 4.97 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.80 \[ \int e^{c+b^2 x^2} x^5 \text {erfc}(b x) \, dx=\frac {{\mathrm {e}}^c\,\left (60\,b\,x-20\,b^3\,x^3+6\,b^5\,x^5+30\,\sqrt {\pi }\,{\mathrm {e}}^{b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right )-30\,b^2\,x^2\,\sqrt {\pi }\,{\mathrm {e}}^{b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right )+15\,b^4\,x^4\,\sqrt {\pi }\,{\mathrm {e}}^{b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right )\right )}{30\,b^6\,\sqrt {\pi }} \]