Integrand size = 18, antiderivative size = 112 \[ \int e^{-b^2 x^2} x^4 \text {erf}(b x) \, dx=-\frac {e^{-2 b^2 x^2}}{2 b^5 \sqrt {\pi }}-\frac {e^{-2 b^2 x^2} x^2}{4 b^3 \sqrt {\pi }}-\frac {3 e^{-b^2 x^2} x \text {erf}(b x)}{4 b^4}-\frac {e^{-b^2 x^2} x^3 \text {erf}(b x)}{2 b^2}+\frac {3 \sqrt {\pi } \text {erf}(b x)^2}{16 b^5} \]
-3/4*x*erf(b*x)/b^4/exp(b^2*x^2)-1/2*x^3*erf(b*x)/b^2/exp(b^2*x^2)-1/2/b^5 /exp(2*b^2*x^2)/Pi^(1/2)-1/4*x^2/b^3/exp(2*b^2*x^2)/Pi^(1/2)+3/16*erf(b*x) ^2*Pi^(1/2)/b^5
Time = 0.02 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.76 \[ \int e^{-b^2 x^2} x^4 \text {erf}(b x) \, dx=\frac {e^{-2 b^2 x^2} \left (-4 \left (2+b^2 x^2\right )-4 b e^{b^2 x^2} \sqrt {\pi } x \left (3+2 b^2 x^2\right ) \text {erf}(b x)+3 e^{2 b^2 x^2} \pi \text {erf}(b x)^2\right )}{16 b^5 \sqrt {\pi }} \]
(-4*(2 + b^2*x^2) - 4*b*E^(b^2*x^2)*Sqrt[Pi]*x*(3 + 2*b^2*x^2)*Erf[b*x] + 3*E^(2*b^2*x^2)*Pi*Erf[b*x]^2)/(16*b^5*E^(2*b^2*x^2)*Sqrt[Pi])
Time = 0.62 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.27, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {6939, 2641, 2638, 6939, 2638, 6927, 15}
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.
\(\displaystyle \int x^4 e^{-b^2 x^2} \text {erf}(b x) \, dx\) |
\(\Big \downarrow \) 6939 |
\(\displaystyle \frac {3 \int e^{-b^2 x^2} x^2 \text {erf}(b x)dx}{2 b^2}+\frac {\int e^{-2 b^2 x^2} x^3dx}{\sqrt {\pi } b}-\frac {x^3 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}\) |
\(\Big \downarrow \) 2641 |
\(\displaystyle \frac {3 \int e^{-b^2 x^2} x^2 \text {erf}(b x)dx}{2 b^2}+\frac {\frac {\int e^{-2 b^2 x^2} xdx}{2 b^2}-\frac {x^2 e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}-\frac {x^3 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}\) |
\(\Big \downarrow \) 2638 |
\(\displaystyle \frac {3 \int e^{-b^2 x^2} x^2 \text {erf}(b x)dx}{2 b^2}-\frac {x^3 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}+\frac {-\frac {x^2 e^{-2 b^2 x^2}}{4 b^2}-\frac {e^{-2 b^2 x^2}}{8 b^4}}{\sqrt {\pi } b}\) |
\(\Big \downarrow \) 6939 |
\(\displaystyle \frac {3 \left (\frac {\int e^{-b^2 x^2} \text {erf}(b x)dx}{2 b^2}+\frac {\int e^{-2 b^2 x^2} xdx}{\sqrt {\pi } b}-\frac {x e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}\right )}{2 b^2}-\frac {x^3 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}+\frac {-\frac {x^2 e^{-2 b^2 x^2}}{4 b^2}-\frac {e^{-2 b^2 x^2}}{8 b^4}}{\sqrt {\pi } b}\) |
\(\Big \downarrow \) 2638 |
\(\displaystyle \frac {3 \left (\frac {\int e^{-b^2 x^2} \text {erf}(b x)dx}{2 b^2}-\frac {x e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}-\frac {e^{-2 b^2 x^2}}{4 \sqrt {\pi } b^3}\right )}{2 b^2}-\frac {x^3 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}+\frac {-\frac {x^2 e^{-2 b^2 x^2}}{4 b^2}-\frac {e^{-2 b^2 x^2}}{8 b^4}}{\sqrt {\pi } b}\) |
\(\Big \downarrow \) 6927 |
\(\displaystyle \frac {3 \left (\frac {\sqrt {\pi } \int \text {erf}(b x)d\text {erf}(b x)}{4 b^3}-\frac {x e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}-\frac {e^{-2 b^2 x^2}}{4 \sqrt {\pi } b^3}\right )}{2 b^2}-\frac {x^3 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}+\frac {-\frac {x^2 e^{-2 b^2 x^2}}{4 b^2}-\frac {e^{-2 b^2 x^2}}{8 b^4}}{\sqrt {\pi } b}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {x^3 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}+\frac {-\frac {x^2 e^{-2 b^2 x^2}}{4 b^2}-\frac {e^{-2 b^2 x^2}}{8 b^4}}{\sqrt {\pi } b}+\frac {3 \left (\frac {\sqrt {\pi } \text {erf}(b x)^2}{8 b^3}-\frac {x e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}-\frac {e^{-2 b^2 x^2}}{4 \sqrt {\pi } b^3}\right )}{2 b^2}\) |
(-1/8*1/(b^4*E^(2*b^2*x^2)) - x^2/(4*b^2*E^(2*b^2*x^2)))/(b*Sqrt[Pi]) - (x ^3*Erf[b*x])/(2*b^2*E^(b^2*x^2)) + (3*(-1/4*1/(b^3*E^(2*b^2*x^2)*Sqrt[Pi]) - (x*Erf[b*x])/(2*b^2*E^(b^2*x^2)) + (Sqrt[Pi]*Erf[b*x]^2)/(8*b^3)))/(2*b ^2)
3.1.81.3.1 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[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_ .), x_Symbol] :> Simp[(e + f*x)^n*(F^(a + b*(c + d*x)^n)/(b*f*n*(c + d*x)^n *Log[F])), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] && EqQ [d*e - c*f, 0]
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[(b_.)*(x_)]^(n_.), x_Symbol] :> Simp[E^c*( Sqrt[Pi]/(2*b)) Subst[Int[x^n, x], x, Erf[b*x]], x] /; FreeQ[{b, c, d, n} , x] && EqQ[d, -b^2]
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]
\[\int x^{4} \operatorname {erf}\left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}d x\]
Time = 0.25 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.66 \[ \int e^{-b^2 x^2} x^4 \text {erf}(b x) \, dx=-\frac {4 \, {\left (2 \, \pi b^{3} x^{3} + 3 \, \pi b x\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} - \sqrt {\pi } {\left (3 \, \pi \operatorname {erf}\left (b x\right )^{2} - 4 \, {\left (b^{2} x^{2} + 2\right )} e^{\left (-2 \, b^{2} x^{2}\right )}\right )}}{16 \, \pi b^{5}} \]
-1/16*(4*(2*pi*b^3*x^3 + 3*pi*b*x)*erf(b*x)*e^(-b^2*x^2) - sqrt(pi)*(3*pi* erf(b*x)^2 - 4*(b^2*x^2 + 2)*e^(-2*b^2*x^2)))/(pi*b^5)
Time = 6.61 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.97 \[ \int e^{-b^2 x^2} x^4 \text {erf}(b x) \, dx=\begin {cases} - \frac {x^{3} e^{- b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{2 b^{2}} - \frac {x^{2} e^{- 2 b^{2} x^{2}}}{4 \sqrt {\pi } b^{3}} - \frac {3 x e^{- b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{4 b^{4}} + \frac {3 \sqrt {\pi } \operatorname {erf}^{2}{\left (b x \right )}}{16 b^{5}} - \frac {e^{- 2 b^{2} x^{2}}}{2 \sqrt {\pi } b^{5}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]
Piecewise((-x**3*exp(-b**2*x**2)*erf(b*x)/(2*b**2) - x**2*exp(-2*b**2*x**2 )/(4*sqrt(pi)*b**3) - 3*x*exp(-b**2*x**2)*erf(b*x)/(4*b**4) + 3*sqrt(pi)*e rf(b*x)**2/(16*b**5) - exp(-2*b**2*x**2)/(2*sqrt(pi)*b**5), Ne(b, 0)), (0, True))
\[ \int e^{-b^2 x^2} x^4 \text {erf}(b x) \, dx=\int { x^{4} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} \,d x } \]
1/2*integrate((2*b^2*x^3 + 3*x)*e^(-2*b^2*x^2), x)/(sqrt(pi)*b^3) - 1/16*( 4*(2*b^3*x^3 + 3*b*x)*erf(b*x)*e^(-b^2*x^2) - 3*sqrt(pi)*erf(b*x)^2)/b^5
\[ \int e^{-b^2 x^2} x^4 \text {erf}(b x) \, dx=\int { x^{4} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} \,d x } \]
Time = 5.70 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.80 \[ \int e^{-b^2 x^2} x^4 \text {erf}(b x) \, dx=-\frac {8\,{\mathrm {e}}^{-2\,b^2\,x^2}-3\,\pi \,{\mathrm {erf}\left (b\,x\right )}^2}{16\,b^5\,\sqrt {\pi }}-\frac {x^2\,{\mathrm {e}}^{-2\,b^2\,x^2}}{4\,b^3\,\sqrt {\pi }}-\frac {3\,x\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erf}\left (b\,x\right )}{4\,b^4}-\frac {x^3\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erf}\left (b\,x\right )}{2\,b^2} \]