Integrand size = 10, antiderivative size = 126 \[ \int x^3 \text {erf}(b x)^2 \, dx=\frac {e^{-2 b^2 x^2}}{2 b^4 \pi }+\frac {e^{-2 b^2 x^2} x^2}{4 b^2 \pi }+\frac {3 e^{-b^2 x^2} x \text {erf}(b x)}{4 b^3 \sqrt {\pi }}+\frac {e^{-b^2 x^2} x^3 \text {erf}(b x)}{2 b \sqrt {\pi }}-\frac {3 \text {erf}(b x)^2}{16 b^4}+\frac {1}{4} x^4 \text {erf}(b x)^2 \]
1/2/b^4/exp(2*b^2*x^2)/Pi+1/4*x^2/b^2/exp(2*b^2*x^2)/Pi-3/16*erf(b*x)^2/b^ 4+1/4*x^4*erf(b*x)^2+3/4*x*erf(b*x)/b^3/exp(b^2*x^2)/Pi^(1/2)+1/2*x^3*erf( b*x)/b/exp(b^2*x^2)/Pi^(1/2)
Time = 0.03 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.71 \[ \int x^3 \text {erf}(b x)^2 \, dx=\frac {e^{-2 b^2 x^2} \left (8+4 b^2 x^2+4 b e^{b^2 x^2} \sqrt {\pi } x \left (3+2 b^2 x^2\right ) \text {erf}(b x)+e^{2 b^2 x^2} \pi \left (-3+4 b^4 x^4\right ) \text {erf}(b x)^2\right )}{16 b^4 \pi } \]
(8 + 4*b^2*x^2 + 4*b*E^(b^2*x^2)*Sqrt[Pi]*x*(3 + 2*b^2*x^2)*Erf[b*x] + E^( 2*b^2*x^2)*Pi*(-3 + 4*b^4*x^4)*Erf[b*x]^2)/(16*b^4*E^(2*b^2*x^2)*Pi)
Time = 0.75 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.30, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {6918, 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.
\(\displaystyle \int x^3 \text {erf}(b x)^2 \, dx\) |
\(\Big \downarrow \) 6918 |
\(\displaystyle \frac {1}{4} x^4 \text {erf}(b x)^2-\frac {b \int e^{-b^2 x^2} x^4 \text {erf}(b x)dx}{\sqrt {\pi }}\) |
\(\Big \downarrow \) 6939 |
\(\displaystyle \frac {1}{4} x^4 \text {erf}(b x)^2-\frac {b \left (\frac {3 \int e^{-b^2 x^2} x^2 \text {erf}(b x)dx}{2 b^2}+\frac {\int e^{-2 b^2 x^2} x^3dx}{\sqrt {\pi } b}-\frac {x^3 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}\right )}{\sqrt {\pi }}\) |
\(\Big \downarrow \) 2641 |
\(\displaystyle \frac {1}{4} x^4 \text {erf}(b x)^2-\frac {b \left (\frac {3 \int e^{-b^2 x^2} x^2 \text {erf}(b x)dx}{2 b^2}+\frac {\frac {\int e^{-2 b^2 x^2} xdx}{2 b^2}-\frac {x^2 e^{-2 b^2 x^2}}{4 b^2}}{\sqrt {\pi } b}-\frac {x^3 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}\right )}{\sqrt {\pi }}\) |
\(\Big \downarrow \) 2638 |
\(\displaystyle \frac {1}{4} x^4 \text {erf}(b x)^2-\frac {b \left (\frac {3 \int e^{-b^2 x^2} x^2 \text {erf}(b x)dx}{2 b^2}-\frac {x^3 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}+\frac {-\frac {x^2 e^{-2 b^2 x^2}}{4 b^2}-\frac {e^{-2 b^2 x^2}}{8 b^4}}{\sqrt {\pi } b}\right )}{\sqrt {\pi }}\) |
\(\Big \downarrow \) 6939 |
\(\displaystyle \frac {1}{4} x^4 \text {erf}(b x)^2-\frac {b \left (\frac {3 \left (\frac {\int e^{-b^2 x^2} \text {erf}(b x)dx}{2 b^2}+\frac {\int e^{-2 b^2 x^2} xdx}{\sqrt {\pi } b}-\frac {x e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}\right )}{2 b^2}-\frac {x^3 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}+\frac {-\frac {x^2 e^{-2 b^2 x^2}}{4 b^2}-\frac {e^{-2 b^2 x^2}}{8 b^4}}{\sqrt {\pi } b}\right )}{\sqrt {\pi }}\) |
\(\Big \downarrow \) 2638 |
\(\displaystyle \frac {1}{4} x^4 \text {erf}(b x)^2-\frac {b \left (\frac {3 \left (\frac {\int e^{-b^2 x^2} \text {erf}(b x)dx}{2 b^2}-\frac {x e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}-\frac {e^{-2 b^2 x^2}}{4 \sqrt {\pi } b^3}\right )}{2 b^2}-\frac {x^3 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}+\frac {-\frac {x^2 e^{-2 b^2 x^2}}{4 b^2}-\frac {e^{-2 b^2 x^2}}{8 b^4}}{\sqrt {\pi } b}\right )}{\sqrt {\pi }}\) |
\(\Big \downarrow \) 6927 |
\(\displaystyle \frac {1}{4} x^4 \text {erf}(b x)^2-\frac {b \left (\frac {3 \left (\frac {\sqrt {\pi } \int \text {erf}(b x)d\text {erf}(b x)}{4 b^3}-\frac {x e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}-\frac {e^{-2 b^2 x^2}}{4 \sqrt {\pi } b^3}\right )}{2 b^2}-\frac {x^3 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}+\frac {-\frac {x^2 e^{-2 b^2 x^2}}{4 b^2}-\frac {e^{-2 b^2 x^2}}{8 b^4}}{\sqrt {\pi } b}\right )}{\sqrt {\pi }}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {1}{4} x^4 \text {erf}(b x)^2-\frac {b \left (-\frac {x^3 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}+\frac {-\frac {x^2 e^{-2 b^2 x^2}}{4 b^2}-\frac {e^{-2 b^2 x^2}}{8 b^4}}{\sqrt {\pi } b}+\frac {3 \left (\frac {\sqrt {\pi } \text {erf}(b x)^2}{8 b^3}-\frac {x e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}-\frac {e^{-2 b^2 x^2}}{4 \sqrt {\pi } b^3}\right )}{2 b^2}\right )}{\sqrt {\pi }}\) |
(x^4*Erf[b*x]^2)/4 - (b*((-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]*E rf[b*x]^2)/(8*b^3)))/(2*b^2)))/Sqrt[Pi]
3.1.23.3.1 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[Erf[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(Erf[b*x]^2/( m + 1)), x] - Simp[4*(b/(Sqrt[Pi]*(m + 1))) Int[(x^(m + 1)*Erf[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)*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]
Time = 0.31 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.92
method | result | size |
parallelrisch | \(\frac {4 \operatorname {erf}\left (b x \right )^{2} x^{4} \pi ^{\frac {3}{2}} b^{4}+8 \,{\mathrm e}^{-b^{2} x^{2}} \operatorname {erf}\left (b x \right ) x^{3} b^{3} \pi +4 x^{2} {\mathrm e}^{-2 b^{2} x^{2}} b^{2} \sqrt {\pi }+12 \,{\mathrm e}^{-b^{2} x^{2}} x \,\operatorname {erf}\left (b x \right ) b \pi -3 \operatorname {erf}\left (b x \right )^{2} \pi ^{\frac {3}{2}}+8 \,{\mathrm e}^{-2 b^{2} x^{2}} \sqrt {\pi }}{16 \pi ^{\frac {3}{2}} b^{4}}\) | \(116\) |
1/16*(4*erf(b*x)^2*x^4*Pi^(3/2)*b^4+8*exp(-b^2*x^2)*erf(b*x)*x^3*b^3*Pi+4* x^2*exp(-b^2*x^2)^2*b^2*Pi^(1/2)+12*exp(-b^2*x^2)*x*erf(b*x)*b*Pi-3*erf(b* x)^2*Pi^(3/2)+8*exp(-b^2*x^2)^2*Pi^(1/2))/Pi^(3/2)/b^4
Time = 0.25 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.64 \[ \int x^3 \text {erf}(b x)^2 \, dx=\frac {4 \, \sqrt {\pi } {\left (2 \, b^{3} x^{3} + 3 \, b x\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} - {\left (3 \, \pi - 4 \, \pi b^{4} x^{4}\right )} \operatorname {erf}\left (b x\right )^{2} + 4 \, {\left (b^{2} x^{2} + 2\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{16 \, \pi b^{4}} \]
1/16*(4*sqrt(pi)*(2*b^3*x^3 + 3*b*x)*erf(b*x)*e^(-b^2*x^2) - (3*pi - 4*pi* b^4*x^4)*erf(b*x)^2 + 4*(b^2*x^2 + 2)*e^(-2*b^2*x^2))/(pi*b^4)
Time = 0.33 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.93 \[ \int x^3 \text {erf}(b x)^2 \, dx=\begin {cases} \frac {x^{4} \operatorname {erf}^{2}{\left (b x \right )}}{4} + \frac {x^{3} e^{- b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{2 \sqrt {\pi } b} + \frac {x^{2} e^{- 2 b^{2} x^{2}}}{4 \pi b^{2}} + \frac {3 x e^{- b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{4 \sqrt {\pi } b^{3}} - \frac {3 \operatorname {erf}^{2}{\left (b x \right )}}{16 b^{4}} + \frac {e^{- 2 b^{2} x^{2}}}{2 \pi b^{4}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]
Piecewise((x**4*erf(b*x)**2/4 + x**3*exp(-b**2*x**2)*erf(b*x)/(2*sqrt(pi)* b) + x**2*exp(-2*b**2*x**2)/(4*pi*b**2) + 3*x*exp(-b**2*x**2)*erf(b*x)/(4* sqrt(pi)*b**3) - 3*erf(b*x)**2/(16*b**4) + exp(-2*b**2*x**2)/(2*pi*b**4), Ne(b, 0)), (0, True))
\[ \int x^3 \text {erf}(b x)^2 \, dx=\int { x^{3} \operatorname {erf}\left (b x\right )^{2} \,d x } \]
-1/2*integrate((2*b^2*x^3 + 3*x)*e^(-2*b^2*x^2), x)/(pi*b^2) - 1/16*((3*pi - 4*pi*b^4*x^4)*erf(b*x)^2 - 4*(2*sqrt(pi)*b^3*x^3 + 3*sqrt(pi)*b*x)*erf( b*x)*e^(-b^2*x^2))/(pi*b^4)
\[ \int x^3 \text {erf}(b x)^2 \, dx=\int { x^{3} \operatorname {erf}\left (b x\right )^{2} \,d x } \]
Time = 5.15 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.80 \[ \int x^3 \text {erf}(b x)^2 \, dx=\frac {x^4\,{\mathrm {erf}\left (b\,x\right )}^2}{4}+\frac {\frac {{\mathrm {e}}^{-2\,b^2\,x^2}}{2}-\frac {3\,\pi \,{\mathrm {erf}\left (b\,x\right )}^2}{16}+\frac {b^2\,x^2\,{\mathrm {e}}^{-2\,b^2\,x^2}}{4}+\frac {b^3\,x^3\,\sqrt {\pi }\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erf}\left (b\,x\right )}{2}+\frac {3\,b\,x\,\sqrt {\pi }\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erf}\left (b\,x\right )}{4}}{b^4\,\pi } \]