Integrand size = 10, antiderivative size = 165 \[ \int x^4 \text {erfc}(b x)^2 \, dx=\frac {11 e^{-2 b^2 x^2} x}{20 b^4 \pi }+\frac {e^{-2 b^2 x^2} x^3}{5 b^2 \pi }-\frac {43 \text {erf}\left (\sqrt {2} b x\right )}{40 b^5 \sqrt {2 \pi }}-\frac {4 e^{-b^2 x^2} \text {erfc}(b x)}{5 b^5 \sqrt {\pi }}-\frac {4 e^{-b^2 x^2} x^2 \text {erfc}(b x)}{5 b^3 \sqrt {\pi }}-\frac {2 e^{-b^2 x^2} x^4 \text {erfc}(b x)}{5 b \sqrt {\pi }}+\frac {1}{5} x^5 \text {erfc}(b x)^2 \]
11/20*x/b^4/exp(2*b^2*x^2)/Pi+1/5*x^3/b^2/exp(2*b^2*x^2)/Pi+1/5*x^5*erfc(b *x)^2-4/5*erfc(b*x)/b^5/exp(b^2*x^2)/Pi^(1/2)-4/5*x^2*erfc(b*x)/b^3/exp(b^ 2*x^2)/Pi^(1/2)-2/5*x^4*erfc(b*x)/b/exp(b^2*x^2)/Pi^(1/2)-43/80*erf(b*x*2^ (1/2))/b^5*2^(1/2)/Pi^(1/2)
Time = 0.10 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.65 \[ \int x^4 \text {erfc}(b x)^2 \, dx=\frac {-43 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} b x\right )+4 \left (b e^{-2 b^2 x^2} x \left (11+4 b^2 x^2\right )-8 e^{-b^2 x^2} \sqrt {\pi } \left (2+2 b^2 x^2+b^4 x^4\right ) \text {erfc}(b x)+4 b^5 \pi x^5 \text {erfc}(b x)^2\right )}{80 b^5 \pi } \]
(-43*Sqrt[2*Pi]*Erf[Sqrt[2]*b*x] + 4*((b*x*(11 + 4*b^2*x^2))/E^(2*b^2*x^2) - (8*Sqrt[Pi]*(2 + 2*b^2*x^2 + b^4*x^4)*Erfc[b*x])/E^(b^2*x^2) + 4*b^5*Pi *x^5*Erfc[b*x]^2))/(80*b^5*Pi)
Time = 1.05 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.59, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6919, 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.
| \(\displaystyle \int x^4 \text {erfc}(b x)^2 \, dx\) |
| \(\Big \downarrow \) 6919 |
| \(\displaystyle \frac {4 b \int e^{-b^2 x^2} x^5 \text {erfc}(b x)dx}{5 \sqrt {\pi }}+\frac {1}{5} x^5 \text {erfc}(b x)^2\) |
| \(\Big \downarrow \) 6940 |
| \(\displaystyle \frac {4 b \left (\frac {2 \int e^{-b^2 x^2} x^3 \text {erfc}(b x)dx}{b^2}-\frac {\int e^{-2 b^2 x^2} x^4dx}{\sqrt {\pi } b}-\frac {x^4 e^{-b^2 x^2} \text {erfc}(b x)}{2 b^2}\right )}{5 \sqrt {\pi }}+\frac {1}{5} x^5 \text {erfc}(b x)^2\) |
| \(\Big \downarrow \) 2641 |
| \(\displaystyle \frac {4 b \left (\frac {2 \int e^{-b^2 x^2} x^3 \text {erfc}(b x)dx}{b^2}-\frac {\frac {3 \int e^{-2 b^2 x^2} x^2dx}{4 b^2}-\frac {x^3 e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}-\frac {x^4 e^{-b^2 x^2} \text {erfc}(b x)}{2 b^2}\right )}{5 \sqrt {\pi }}+\frac {1}{5} x^5 \text {erfc}(b x)^2\) |
| \(\Big \downarrow \) 2641 |
| \(\displaystyle \frac {4 b \left (\frac {2 \int e^{-b^2 x^2} x^3 \text {erfc}(b x)dx}{b^2}-\frac {\frac {3 \left (\frac {\int e^{-2 b^2 x^2}dx}{4 b^2}-\frac {x e^{-2 b^2 x^2}}{4 b^2}\right )}{4 b^2}-\frac {x^3 e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}-\frac {x^4 e^{-b^2 x^2} \text {erfc}(b x)}{2 b^2}\right )}{5 \sqrt {\pi }}+\frac {1}{5} x^5 \text {erfc}(b x)^2\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle \frac {4 b \left (\frac {2 \int e^{-b^2 x^2} x^3 \text {erfc}(b x)dx}{b^2}-\frac {x^4 e^{-b^2 x^2} \text {erfc}(b x)}{2 b^2}-\frac {\frac {3 \left (\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} b x\right )}{8 b^3}-\frac {x e^{-2 b^2 x^2}}{4 b^2}\right )}{4 b^2}-\frac {x^3 e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}\right )}{5 \sqrt {\pi }}+\frac {1}{5} x^5 \text {erfc}(b x)^2\) |
| \(\Big \downarrow \) 6940 |
| \(\displaystyle \frac {4 b \left (\frac {2 \left (\frac {\int e^{-b^2 x^2} x \text {erfc}(b x)dx}{b^2}-\frac {\int e^{-2 b^2 x^2} x^2dx}{\sqrt {\pi } b}-\frac {x^2 e^{-b^2 x^2} \text {erfc}(b x)}{2 b^2}\right )}{b^2}-\frac {x^4 e^{-b^2 x^2} \text {erfc}(b x)}{2 b^2}-\frac {\frac {3 \left (\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} b x\right )}{8 b^3}-\frac {x e^{-2 b^2 x^2}}{4 b^2}\right )}{4 b^2}-\frac {x^3 e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}\right )}{5 \sqrt {\pi }}+\frac {1}{5} x^5 \text {erfc}(b x)^2\) |
| \(\Big \downarrow \) 2641 |
| \(\displaystyle \frac {4 b \left (\frac {2 \left (\frac {\int e^{-b^2 x^2} x \text {erfc}(b x)dx}{b^2}-\frac {\frac {\int e^{-2 b^2 x^2}dx}{4 b^2}-\frac {x e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}-\frac {x^2 e^{-b^2 x^2} \text {erfc}(b x)}{2 b^2}\right )}{b^2}-\frac {x^4 e^{-b^2 x^2} \text {erfc}(b x)}{2 b^2}-\frac {\frac {3 \left (\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} b x\right )}{8 b^3}-\frac {x e^{-2 b^2 x^2}}{4 b^2}\right )}{4 b^2}-\frac {x^3 e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}\right )}{5 \sqrt {\pi }}+\frac {1}{5} x^5 \text {erfc}(b x)^2\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle \frac {4 b \left (\frac {2 \left (\frac {\int e^{-b^2 x^2} x \text {erfc}(b x)dx}{b^2}-\frac {x^2 e^{-b^2 x^2} \text {erfc}(b x)}{2 b^2}-\frac {\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} b x\right )}{8 b^3}-\frac {x e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}\right )}{b^2}-\frac {x^4 e^{-b^2 x^2} \text {erfc}(b x)}{2 b^2}-\frac {\frac {3 \left (\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} b x\right )}{8 b^3}-\frac {x e^{-2 b^2 x^2}}{4 b^2}\right )}{4 b^2}-\frac {x^3 e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}\right )}{5 \sqrt {\pi }}+\frac {1}{5} x^5 \text {erfc}(b x)^2\) |
| \(\Big \downarrow \) 6937 |
| \(\displaystyle \frac {4 b \left (\frac {2 \left (\frac {-\frac {\int e^{-2 b^2 x^2}dx}{\sqrt {\pi } b}-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{2 b^2}}{b^2}-\frac {x^2 e^{-b^2 x^2} \text {erfc}(b x)}{2 b^2}-\frac {\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} b x\right )}{8 b^3}-\frac {x e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}\right )}{b^2}-\frac {x^4 e^{-b^2 x^2} \text {erfc}(b x)}{2 b^2}-\frac {\frac {3 \left (\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} b x\right )}{8 b^3}-\frac {x e^{-2 b^2 x^2}}{4 b^2}\right )}{4 b^2}-\frac {x^3 e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}\right )}{5 \sqrt {\pi }}+\frac {1}{5} x^5 \text {erfc}(b x)^2\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle \frac {4 b \left (-\frac {x^4 e^{-b^2 x^2} \text {erfc}(b x)}{2 b^2}+\frac {2 \left (\frac {-\frac {\text {erf}\left (\sqrt {2} b x\right )}{2 \sqrt {2} b^2}-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{2 b^2}}{b^2}-\frac {x^2 e^{-b^2 x^2} \text {erfc}(b x)}{2 b^2}-\frac {\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} b x\right )}{8 b^3}-\frac {x e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}\right )}{b^2}-\frac {\frac {3 \left (\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} b x\right )}{8 b^3}-\frac {x e^{-2 b^2 x^2}}{4 b^2}\right )}{4 b^2}-\frac {x^3 e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}\right )}{5 \sqrt {\pi }}+\frac {1}{5} x^5 \text {erfc}(b x)^2\) |
(x^5*Erfc[b*x]^2)/5 + (4*b*(-((-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[Pi/2]*Erf[Sqrt[2]*b*x])/(8*b^3)))/(4*b^2))/(b*S qrt[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*Er fc[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))/(5*Sqrt[Pi])
3.2.32.3.1 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[Erfc[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(Erfc[b*x]^2 /(m + 1)), x] + Simp[4*(b/(Sqrt[Pi]*(m + 1))) Int[(x^(m + 1)*Erfc[b*x])/E ^(b^2*x^2), x], x] /; FreeQ[b, x] && (IGtQ[m, 0] || ILtQ[(m + 1)/2, 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 = 0.54 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.24
| method | result | size |
| derivativedivides | \(\frac {\frac {b^{5} x^{5}}{5}-\frac {2 \,\operatorname {erf}\left (b x \right ) b^{5} x^{5}}{5}+\frac {-\frac {2 \,{\mathrm e}^{-b^{2} x^{2}} x^{4} b^{4}}{5}-\frac {4 x^{2} {\mathrm e}^{-b^{2} x^{2}} b^{2}}{5}-\frac {4 \,{\mathrm e}^{-b^{2} x^{2}}}{5}}{\sqrt {\pi }}+\frac {\operatorname {erf}\left (b x \right )^{2} b^{5} x^{5}}{5}-\frac {4 \,\operatorname {erf}\left (b x \right ) \left (-\frac {{\mathrm e}^{-b^{2} x^{2}} x^{4} b^{4}}{2}-x^{2} {\mathrm e}^{-b^{2} x^{2}} b^{2}-{\mathrm e}^{-b^{2} x^{2}}\right )}{5 \sqrt {\pi }}+\frac {-\frac {43 \sqrt {2}\, \sqrt {\pi }\, \operatorname {erf}\left (b x \sqrt {2}\right )}{80}+\frac {11 \,{\mathrm e}^{-2 b^{2} x^{2}} b x}{20}+\frac {{\mathrm e}^{-2 b^{2} x^{2}} b^{3} x^{3}}{5}}{\pi }}{b^{5}}\) | \(205\) |
| default | \(\frac {\frac {b^{5} x^{5}}{5}-\frac {2 \,\operatorname {erf}\left (b x \right ) b^{5} x^{5}}{5}+\frac {-\frac {2 \,{\mathrm e}^{-b^{2} x^{2}} x^{4} b^{4}}{5}-\frac {4 x^{2} {\mathrm e}^{-b^{2} x^{2}} b^{2}}{5}-\frac {4 \,{\mathrm e}^{-b^{2} x^{2}}}{5}}{\sqrt {\pi }}+\frac {\operatorname {erf}\left (b x \right )^{2} b^{5} x^{5}}{5}-\frac {4 \,\operatorname {erf}\left (b x \right ) \left (-\frac {{\mathrm e}^{-b^{2} x^{2}} x^{4} b^{4}}{2}-x^{2} {\mathrm e}^{-b^{2} x^{2}} b^{2}-{\mathrm e}^{-b^{2} x^{2}}\right )}{5 \sqrt {\pi }}+\frac {-\frac {43 \sqrt {2}\, \sqrt {\pi }\, \operatorname {erf}\left (b x \sqrt {2}\right )}{80}+\frac {11 \,{\mathrm e}^{-2 b^{2} x^{2}} b x}{20}+\frac {{\mathrm e}^{-2 b^{2} x^{2}} b^{3} x^{3}}{5}}{\pi }}{b^{5}}\) | \(205\) |
1/b^5*(1/5*b^5*x^5-2/5*erf(b*x)*b^5*x^5+4/5/Pi^(1/2)*(-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))+1/5*erf(b*x)^2*b^5*x^5-4/5*erf( b*x)/Pi^(1/2)*(-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))+4/5/Pi*(-43/64*2^(1/2)*Pi^(1/2)*erf(b*x*2^(1/2))+11/16/exp(b^2*x^2)^2* b*x+1/4/exp(b^2*x^2)^2*b^3*x^3))
Time = 0.27 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.93 \[ \int x^4 \text {erfc}(b x)^2 \, dx=\frac {16 \, \pi b^{6} x^{5} \operatorname {erf}\left (b x\right )^{2} - 32 \, \pi b^{6} x^{5} \operatorname {erf}\left (b x\right ) + 16 \, \pi b^{6} x^{5} - 43 \, \sqrt {2} \sqrt {\pi } \sqrt {b^{2}} \operatorname {erf}\left (\sqrt {2} \sqrt {b^{2}} x\right ) - 32 \, \sqrt {\pi } {\left (b^{5} x^{4} + 2 \, b^{3} x^{2} - {\left (b^{5} x^{4} + 2 \, b^{3} x^{2} + 2 \, b\right )} \operatorname {erf}\left (b x\right ) + 2 \, b\right )} e^{\left (-b^{2} x^{2}\right )} + 4 \, {\left (4 \, b^{4} x^{3} + 11 \, b^{2} x\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{80 \, \pi b^{6}} \]
1/80*(16*pi*b^6*x^5*erf(b*x)^2 - 32*pi*b^6*x^5*erf(b*x) + 16*pi*b^6*x^5 - 43*sqrt(2)*sqrt(pi)*sqrt(b^2)*erf(sqrt(2)*sqrt(b^2)*x) - 32*sqrt(pi)*(b^5* x^4 + 2*b^3*x^2 - (b^5*x^4 + 2*b^3*x^2 + 2*b)*erf(b*x) + 2*b)*e^(-b^2*x^2) + 4*(4*b^4*x^3 + 11*b^2*x)*e^(-2*b^2*x^2))/(pi*b^6)
\[ \int x^4 \text {erfc}(b x)^2 \, dx=\int x^{4} \operatorname {erfc}^{2}{\left (b x \right )}\, dx \]
\[ \int x^4 \text {erfc}(b x)^2 \, dx=\int { x^{4} \operatorname {erfc}\left (b x\right )^{2} \,d x } \]
Time = 0.34 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.32 \[ \int x^4 \text {erfc}(b x)^2 \, dx=\frac {1}{5} \, x^{5} \operatorname {erf}\left (b x\right )^{2} - \frac {2}{5} \, x^{5} \operatorname {erf}\left (b x\right ) + \frac {1}{5} \, x^{5} + \frac {b {\left (\frac {32 \, {\left (b^{4} x^{4} + 2 \, b^{2} x^{2} + 2\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{b^{6}} + \frac {b^{4} {\left (\frac {4 \, {\left (4 \, b^{2} x^{3} + 3 \, x\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{b^{4}} + \frac {3 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {2} b x\right )}{b^{5}}\right )} + 8 \, b^{2} {\left (\frac {4 \, x e^{\left (-2 \, b^{2} x^{2}\right )}}{b^{2}} + \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {2} b x\right )}{b^{3}}\right )} + \frac {32 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {2} b x\right )}{b}}{\sqrt {\pi } b^{5}}\right )}}{80 \, \sqrt {\pi }} - \frac {2 \, {\left (b^{4} x^{4} + 2 \, b^{2} x^{2} + 2\right )} e^{\left (-b^{2} x^{2}\right )}}{5 \, \sqrt {\pi } b^{5}} \]
1/5*x^5*erf(b*x)^2 - 2/5*x^5*erf(b*x) + 1/5*x^5 + 1/80*b*(32*(b^4*x^4 + 2* b^2*x^2 + 2)*erf(b*x)*e^(-b^2*x^2)/b^6 + (b^4*(4*(4*b^2*x^3 + 3*x)*e^(-2*b ^2*x^2)/b^4 + 3*sqrt(2)*sqrt(pi)*erf(-sqrt(2)*b*x)/b^5) + 8*b^2*(4*x*e^(-2 *b^2*x^2)/b^2 + sqrt(2)*sqrt(pi)*erf(-sqrt(2)*b*x)/b^3) + 32*sqrt(2)*sqrt( pi)*erf(-sqrt(2)*b*x)/b)/(sqrt(pi)*b^5))/sqrt(pi) - 2/5*(b^4*x^4 + 2*b^2*x ^2 + 2)*e^(-b^2*x^2)/(sqrt(pi)*b^5)
Timed out. \[ \int x^4 \text {erfc}(b x)^2 \, dx=\int x^4\,{\mathrm {erfc}\left (b\,x\right )}^2 \,d x \]