Integrand size = 18, antiderivative size = 90 \[ \int e^{-b^2 x^2} x^3 \text {erfc}(b x) \, dx=\frac {e^{-2 b^2 x^2} x}{4 b^3 \sqrt {\pi }}-\frac {5 \text {erf}\left (\sqrt {2} b x\right )}{8 \sqrt {2} b^4}-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{2 b^4}-\frac {e^{-b^2 x^2} x^2 \text {erfc}(b x)}{2 b^2} \]
-1/2*erfc(b*x)/b^4/exp(b^2*x^2)-1/2*x^2*erfc(b*x)/b^2/exp(b^2*x^2)-5/16*er f(b*x*2^(1/2))/b^4*2^(1/2)+1/4*x/b^3/exp(2*b^2*x^2)/Pi^(1/2)
Time = 0.06 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.77 \[ \int e^{-b^2 x^2} x^3 \text {erfc}(b x) \, dx=\frac {-5 \sqrt {2} \text {erf}\left (\sqrt {2} b x\right )+4 e^{-2 b^2 x^2} \left (\frac {b x}{\sqrt {\pi }}-2 e^{b^2 x^2} \left (1+b^2 x^2\right ) \text {erfc}(b x)\right )}{16 b^4} \]
(-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)
Time = 0.46 (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 |
\(\displaystyle \frac {-\frac {\text {erf}\left (\sqrt {2} b x\right )}{2 \sqrt {2} b^2}-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{2 b^2}}{b^2}-\frac {x^2 e^{-b^2 x^2} \text {erfc}(b x)}{2 b^2}-\frac {\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} b x\right )}{8 b^3}-\frac {x e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}\) |
-((-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
3.2.79.3.1 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 = 0.97 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.31
method | result | size |
default | \(\frac {\frac {-\frac {x^{2} {\mathrm e}^{-b^{2} x^{2}} b^{2}}{2}-\frac {{\mathrm e}^{-b^{2} x^{2}}}{2}}{b^{3}}-\frac {\operatorname {erf}\left (b x \right ) \left (-\frac {x^{2} {\mathrm e}^{-b^{2} x^{2}} b^{2}}{2}-\frac {{\mathrm e}^{-b^{2} x^{2}}}{2}\right )}{b^{3}}+\frac {-\frac {5 \sqrt {2}\, \sqrt {\pi }\, \operatorname {erf}\left (b x \sqrt {2}\right )}{16}+\frac {{\mathrm e}^{-2 b^{2} x^{2}} b x}{4}}{\sqrt {\pi }\, b^{3}}}{b}\) | \(118\) |
(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(b*x*2^(1/2))+1/4/exp(b^2*x^2)^2*b*x))/b
Time = 0.26 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00 \[ \int e^{-b^2 x^2} x^3 \text {erfc}(b x) \, dx=\frac {4 \, \sqrt {\pi } b^{2} x e^{\left (-2 \, b^{2} x^{2}\right )} - 5 \, \sqrt {2} \pi \sqrt {b^{2}} \operatorname {erf}\left (\sqrt {2} \sqrt {b^{2}} x\right ) - 8 \, {\left (\pi b^{3} x^{2} + \pi b - {\left (\pi b^{3} x^{2} + \pi b\right )} \operatorname {erf}\left (b x\right )\right )} e^{\left (-b^{2} x^{2}\right )}}{16 \, \pi b^{5}} \]
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)
\[ \int e^{-b^2 x^2} x^3 \text {erfc}(b x) \, dx=\int x^{3} e^{- b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}\, dx \]
\[ \int e^{-b^2 x^2} x^3 \text {erfc}(b x) \, dx=\int { x^{3} \operatorname {erfc}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} \,d x } \]
\[ \int e^{-b^2 x^2} x^3 \text {erfc}(b x) \, dx=\int { x^{3} \operatorname {erfc}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} \,d x } \]
Timed out. \[ \int e^{-b^2 x^2} x^3 \text {erfc}(b x) \, dx=\int x^3\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right ) \,d x \]