Integrand size = 18, antiderivative size = 135 \[ \int e^{-b^2 x^2} x^5 \text {erf}(b x) \, dx=-\frac {11 e^{-2 b^2 x^2} x}{16 b^5 \sqrt {\pi }}-\frac {e^{-2 b^2 x^2} x^3}{4 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{b^6}-\frac {e^{-b^2 x^2} x^2 \text {erf}(b x)}{b^4}-\frac {e^{-b^2 x^2} x^4 \text {erf}(b x)}{2 b^2}+\frac {43 \text {erf}\left (\sqrt {2} b x\right )}{32 \sqrt {2} b^6} \]
-erf(b*x)/b^6/exp(b^2*x^2)-x^2*erf(b*x)/b^4/exp(b^2*x^2)-1/2*x^4*erf(b*x)/ b^2/exp(b^2*x^2)+43/64*erf(b*x*2^(1/2))/b^6*2^(1/2)-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)
Time = 0.05 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.64 \[ \int e^{-b^2 x^2} x^5 \text {erf}(b x) \, dx=\frac {-\frac {4 b e^{-2 b^2 x^2} x \left (11+4 b^2 x^2\right )}{\sqrt {\pi }}-32 e^{-b^2 x^2} \left (2+2 b^2 x^2+b^4 x^4\right ) \text {erf}(b x)+43 \sqrt {2} \text {erf}\left (\sqrt {2} b x\right )}{64 b^6} \]
((-4*b*x*(11 + 4*b^2*x^2))/(E^(2*b^2*x^2)*Sqrt[Pi]) - (32*(2 + 2*b^2*x^2 + b^4*x^4)*Erf[b*x])/E^(b^2*x^2) + 43*Sqrt[2]*Erf[Sqrt[2]*b*x])/(64*b^6)
Time = 0.92 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.75, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6939, 2641, 2641, 2634, 6939, 2641, 2634, 6936, 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.
| \(\displaystyle \int x^5 e^{-b^2 x^2} \text {erf}(b x) \, dx\) |
| \(\Big \downarrow \) 6939 |
| \(\displaystyle \frac {2 \int e^{-b^2 x^2} x^3 \text {erf}(b x)dx}{b^2}+\frac {\int e^{-2 b^2 x^2} x^4dx}{\sqrt {\pi } b}-\frac {x^4 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}\) |
| \(\Big \downarrow \) 2641 |
| \(\displaystyle \frac {2 \int e^{-b^2 x^2} x^3 \text {erf}(b x)dx}{b^2}+\frac {\frac {3 \int e^{-2 b^2 x^2} x^2dx}{4 b^2}-\frac {x^3 e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}-\frac {x^4 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}\) |
| \(\Big \downarrow \) 2641 |
| \(\displaystyle \frac {2 \int e^{-b^2 x^2} x^3 \text {erf}(b x)dx}{b^2}+\frac {\frac {3 \left (\frac {\int e^{-2 b^2 x^2}dx}{4 b^2}-\frac {x e^{-2 b^2 x^2}}{4 b^2}\right )}{4 b^2}-\frac {x^3 e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}-\frac {x^4 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle \frac {2 \int e^{-b^2 x^2} x^3 \text {erf}(b x)dx}{b^2}-\frac {x^4 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}+\frac {\frac {3 \left (\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} b x\right )}{8 b^3}-\frac {x e^{-2 b^2 x^2}}{4 b^2}\right )}{4 b^2}-\frac {x^3 e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}\) |
| \(\Big \downarrow \) 6939 |
| \(\displaystyle \frac {2 \left (\frac {\int e^{-b^2 x^2} x \text {erf}(b x)dx}{b^2}+\frac {\int e^{-2 b^2 x^2} x^2dx}{\sqrt {\pi } b}-\frac {x^2 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}\right )}{b^2}-\frac {x^4 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}+\frac {\frac {3 \left (\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} b x\right )}{8 b^3}-\frac {x e^{-2 b^2 x^2}}{4 b^2}\right )}{4 b^2}-\frac {x^3 e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}\) |
| \(\Big \downarrow \) 2641 |
| \(\displaystyle \frac {2 \left (\frac {\int e^{-b^2 x^2} x \text {erf}(b x)dx}{b^2}+\frac {\frac {\int e^{-2 b^2 x^2}dx}{4 b^2}-\frac {x e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}-\frac {x^2 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}\right )}{b^2}-\frac {x^4 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}+\frac {\frac {3 \left (\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} b x\right )}{8 b^3}-\frac {x e^{-2 b^2 x^2}}{4 b^2}\right )}{4 b^2}-\frac {x^3 e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle \frac {2 \left (\frac {\int e^{-b^2 x^2} x \text {erf}(b x)dx}{b^2}-\frac {x^2 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}+\frac {\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} b x\right )}{8 b^3}-\frac {x e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}\right )}{b^2}-\frac {x^4 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}+\frac {\frac {3 \left (\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} b x\right )}{8 b^3}-\frac {x e^{-2 b^2 x^2}}{4 b^2}\right )}{4 b^2}-\frac {x^3 e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}\) |
| \(\Big \downarrow \) 6936 |
| \(\displaystyle \frac {2 \left (\frac {\frac {\int e^{-2 b^2 x^2}dx}{\sqrt {\pi } b}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}}{b^2}-\frac {x^2 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}+\frac {\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} b x\right )}{8 b^3}-\frac {x e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}\right )}{b^2}-\frac {x^4 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}+\frac {\frac {3 \left (\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} b x\right )}{8 b^3}-\frac {x e^{-2 b^2 x^2}}{4 b^2}\right )}{4 b^2}-\frac {x^3 e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle -\frac {x^4 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}+\frac {2 \left (-\frac {x^2 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}+\frac {\frac {\text {erf}\left (\sqrt {2} b x\right )}{2 \sqrt {2} b^2}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}}{b^2}+\frac {\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} b x\right )}{8 b^3}-\frac {x e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}\right )}{b^2}+\frac {\frac {3 \left (\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} b x\right )}{8 b^3}-\frac {x e^{-2 b^2 x^2}}{4 b^2}\right )}{4 b^2}-\frac {x^3 e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}\) |
-1/2*(x^4*Erf[b*x])/(b^2*E^(b^2*x^2)) + (-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*Sqrt[Pi]) + (2*(-1/2*(x^2*Erf[b*x])/(b^2*E^(b^2*x^2)) + (-1/2*Erf [b*x]/(b^2*E^(b^2*x^2)) + Erf[Sqrt[2]*b*x]/(2*Sqrt[2]*b^2))/b^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])) )/b^2
3.1.75.3.1 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)*Erf[(a_.) + (b_.)*(x_)]*(x_), x_Symbol] :> Sim p[E^(c + d*x^2)*(Erf[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)*Erf[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] : > Simp[x^(m - 1)*E^(c + d*x^2)*(Erf[a + b*x]/(2*d)), x] + (-Simp[(m - 1)/(2 *d) Int[x^(m - 2)*E^(c + d*x^2)*Erf[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]) /; FreeQ[ {a, b, c, d}, x] && IGtQ[m, 1]
Time = 2.31 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.88
| method | result | size |
| default | \(\frac {\frac {\operatorname {erf}\left (b x \right ) \left (-\frac {{\mathrm e}^{-b^{2} x^{2}} x^{4} b^{4}}{2}-x^{2} {\mathrm e}^{-b^{2} x^{2}} b^{2}-{\mathrm e}^{-b^{2} x^{2}}\right )}{b^{5}}-\frac {-\frac {43 \sqrt {2}\, \sqrt {\pi }\, \operatorname {erf}\left (b x \sqrt {2}\right )}{64}+\frac {{\mathrm e}^{-2 b^{2} x^{2}} b^{3} x^{3}}{4}+\frac {11 \,{\mathrm e}^{-2 b^{2} x^{2}} b x}{16}}{\sqrt {\pi }\, b^{5}}}{b}\) | \(119\) |
(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(b*x*2^(1/2))+1/4/exp(b^2*x ^2)^2*b^3*x^3+11/16/exp(b^2*x^2)^2*b*x))/b
Time = 0.25 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.72 \[ \int e^{-b^2 x^2} x^5 \text {erf}(b x) \, dx=\frac {43 \, \sqrt {2} \pi \sqrt {b^{2}} \operatorname {erf}\left (\sqrt {2} \sqrt {b^{2}} x\right ) - 32 \, {\left (\pi b^{5} x^{4} + 2 \, \pi b^{3} x^{2} + 2 \, \pi b\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} - 4 \, \sqrt {\pi } {\left (4 \, b^{4} x^{3} + 11 \, b^{2} x\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{64 \, \pi b^{7}} \]
1/64*(43*sqrt(2)*pi*sqrt(b^2)*erf(sqrt(2)*sqrt(b^2)*x) - 32*(pi*b^5*x^4 + 2*pi*b^3*x^2 + 2*pi*b)*erf(b*x)*e^(-b^2*x^2) - 4*sqrt(pi)*(4*b^4*x^3 + 11* b^2*x)*e^(-2*b^2*x^2))/(pi*b^7)
\[ \int e^{-b^2 x^2} x^5 \text {erf}(b x) \, dx=\int x^{5} e^{- b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}\, dx \]
\[ \int e^{-b^2 x^2} x^5 \text {erf}(b x) \, dx=\int { x^{5} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} \,d x } \]
-1/2*(b^4*x^4 + 2*b^2*x^2 + 2)*erf(b*x)*e^(-b^2*x^2)/b^6 + integrate((b^4* x^4 + 2*b^2*x^2 + 2)*e^(-2*b^2*x^2), x)/(sqrt(pi)*b^5)
Time = 0.28 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.13 \[ \int e^{-b^2 x^2} x^5 \text {erf}(b x) \, dx=-\frac {{\left (b^{4} x^{4} + 2 \, b^{2} x^{2} + 2\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{2 \, b^{6}} - \frac {b^{4} {\left (\frac {4 \, {\left (4 \, b^{2} x^{3} + 3 \, x\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{b^{4}} + \frac {3 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {2} b x\right )}{b^{5}}\right )} + 8 \, b^{2} {\left (\frac {4 \, x e^{\left (-2 \, b^{2} x^{2}\right )}}{b^{2}} + \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {2} b x\right )}{b^{3}}\right )} + \frac {32 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {2} b x\right )}{b}}{64 \, \sqrt {\pi } b^{5}} \]
-1/2*(b^4*x^4 + 2*b^2*x^2 + 2)*erf(b*x)*e^(-b^2*x^2)/b^6 - 1/64*(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)
Time = 0.48 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.42 \[ \int e^{-b^2 x^2} x^5 \text {erf}(b x) \, dx=\frac {\sqrt {2}\,\mathrm {erf}\left (\sqrt {2}\,x\,\sqrt {b^2}\right )}{2\,b\,{\left (b^2\right )}^{5/2}}-\frac {\mathrm {erfi}\left (x\,\sqrt {-2\,b^2}\right )}{2\,b^3\,{\left (-2\,b^2\right )}^{3/2}}-\frac {x^3\,{\mathrm {e}}^{-2\,b^2\,x^2}}{4\,b^3\,\sqrt {\pi }}-\mathrm {erf}\left (b\,x\right )\,\left (\frac {{\mathrm {e}}^{-b^2\,x^2}}{b^6}+\frac {x^4\,{\mathrm {e}}^{-b^2\,x^2}}{2\,b^2}+\frac {x^2\,{\mathrm {e}}^{-b^2\,x^2}}{b^4}\right )-\frac {11\,x\,{\mathrm {e}}^{-2\,b^2\,x^2}}{16\,b^5\,\sqrt {\pi }}+\frac {3\,\sqrt {2}\,x^5}{64\,b\,{\left (b^2\,x^2\right )}^{5/2}}-\frac {3\,\sqrt {2}\,x^5\,\mathrm {erfc}\left (\sqrt {2\,b^2\,x^2}\right )}{64\,b\,{\left (b^2\,x^2\right )}^{5/2}} \]
(2^(1/2)*erf(2^(1/2)*x*(b^2)^(1/2)))/(2*b*(b^2)^(5/2)) - erfi(x*(-2*b^2)^( 1/2))/(2*b^3*(-2*b^2)^(3/2)) - (x^3*exp(-2*b^2*x^2))/(4*b^3*pi^(1/2)) - er f(b*x)*(exp(-b^2*x^2)/b^6 + (x^4*exp(-b^2*x^2))/(2*b^2) + (x^2*exp(-b^2*x^ 2))/b^4) - (11*x*exp(-2*b^2*x^2))/(16*b^5*pi^(1/2)) + (3*2^(1/2)*x^5)/(64* b*(b^2*x^2)^(5/2)) - (3*2^(1/2)*x^5*erfc((2*b^2*x^2)^(1/2)))/(64*b*(b^2*x^ 2)^(5/2))