3.22 \(\int x^5 \text{Erf}(b x)^2 \, dx\)

Optimal. Leaf size=178 \[ \frac{x^5 e^{-b^2 x^2} \text{Erf}(b x)}{3 \sqrt{\pi } b}+\frac{5 x^3 e^{-b^2 x^2} \text{Erf}(b x)}{6 \sqrt{\pi } b^3}+\frac{5 x e^{-b^2 x^2} \text{Erf}(b x)}{4 \sqrt{\pi } b^5}-\frac{5 \text{Erf}(b x)^2}{16 b^6}+\frac{x^4 e^{-2 b^2 x^2}}{6 \pi b^2}+\frac{7 x^2 e^{-2 b^2 x^2}}{12 \pi b^4}+\frac{11 e^{-2 b^2 x^2}}{12 \pi b^6}+\frac{1}{6} x^6 \text{Erf}(b x)^2 \]

[Out]

11/(12*b^6*E^(2*b^2*x^2)*Pi) + (7*x^2)/(12*b^4*E^(2*b^2*x^2)*Pi) + x^4/(6*b^2*E^(2*b^2*x^2)*Pi) + (5*x*Erf[b*x
])/(4*b^5*E^(b^2*x^2)*Sqrt[Pi]) + (5*x^3*Erf[b*x])/(6*b^3*E^(b^2*x^2)*Sqrt[Pi]) + (x^5*Erf[b*x])/(3*b*E^(b^2*x
^2)*Sqrt[Pi]) - (5*Erf[b*x]^2)/(16*b^6) + (x^6*Erf[b*x]^2)/6

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Rubi [A]  time = 0.294206, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {6364, 6385, 6373, 30, 2209, 2212} \[ \frac{x^5 e^{-b^2 x^2} \text{Erf}(b x)}{3 \sqrt{\pi } b}+\frac{5 x^3 e^{-b^2 x^2} \text{Erf}(b x)}{6 \sqrt{\pi } b^3}+\frac{5 x e^{-b^2 x^2} \text{Erf}(b x)}{4 \sqrt{\pi } b^5}-\frac{5 \text{Erf}(b x)^2}{16 b^6}+\frac{x^4 e^{-2 b^2 x^2}}{6 \pi b^2}+\frac{7 x^2 e^{-2 b^2 x^2}}{12 \pi b^4}+\frac{11 e^{-2 b^2 x^2}}{12 \pi b^6}+\frac{1}{6} x^6 \text{Erf}(b x)^2 \]

Antiderivative was successfully verified.

[In]

Int[x^5*Erf[b*x]^2,x]

[Out]

11/(12*b^6*E^(2*b^2*x^2)*Pi) + (7*x^2)/(12*b^4*E^(2*b^2*x^2)*Pi) + x^4/(6*b^2*E^(2*b^2*x^2)*Pi) + (5*x*Erf[b*x
])/(4*b^5*E^(b^2*x^2)*Sqrt[Pi]) + (5*x^3*Erf[b*x])/(6*b^3*E^(b^2*x^2)*Sqrt[Pi]) + (x^5*Erf[b*x])/(3*b*E^(b^2*x
^2)*Sqrt[Pi]) - (5*Erf[b*x]^2)/(16*b^6) + (x^6*Erf[b*x]^2)/6

Rule 6364

Int[Erf[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[(x^(m + 1)*Erf[b*x]^2)/(m + 1), x] - Dist[(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])

Rule 6385

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] + (-Dist[(m - 1)/(2*d), Int[x^(m - 2)*E^(c + d*x^2)*Erf[a + b*x], x], x] - Dist[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]

Rule 6373

Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(b_.)*(x_)]^(n_.), x_Symbol] :> Dist[(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]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2209

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]

Rule 2212

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*Log[F]), x] - Dist[(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])

Rubi steps

\begin{align*} \int x^5 \text{erf}(b x)^2 \, dx &=\frac{1}{6} x^6 \text{erf}(b x)^2-\frac{(2 b) \int e^{-b^2 x^2} x^6 \text{erf}(b x) \, dx}{3 \sqrt{\pi }}\\ &=\frac{e^{-b^2 x^2} x^5 \text{erf}(b x)}{3 b \sqrt{\pi }}+\frac{1}{6} x^6 \text{erf}(b x)^2-\frac{2 \int e^{-2 b^2 x^2} x^5 \, dx}{3 \pi }-\frac{5 \int e^{-b^2 x^2} x^4 \text{erf}(b x) \, dx}{3 b \sqrt{\pi }}\\ &=\frac{e^{-2 b^2 x^2} x^4}{6 b^2 \pi }+\frac{5 e^{-b^2 x^2} x^3 \text{erf}(b x)}{6 b^3 \sqrt{\pi }}+\frac{e^{-b^2 x^2} x^5 \text{erf}(b x)}{3 b \sqrt{\pi }}+\frac{1}{6} x^6 \text{erf}(b x)^2-\frac{2 \int e^{-2 b^2 x^2} x^3 \, dx}{3 b^2 \pi }-\frac{5 \int e^{-2 b^2 x^2} x^3 \, dx}{3 b^2 \pi }-\frac{5 \int e^{-b^2 x^2} x^2 \text{erf}(b x) \, dx}{2 b^3 \sqrt{\pi }}\\ &=\frac{7 e^{-2 b^2 x^2} x^2}{12 b^4 \pi }+\frac{e^{-2 b^2 x^2} x^4}{6 b^2 \pi }+\frac{5 e^{-b^2 x^2} x \text{erf}(b x)}{4 b^5 \sqrt{\pi }}+\frac{5 e^{-b^2 x^2} x^3 \text{erf}(b x)}{6 b^3 \sqrt{\pi }}+\frac{e^{-b^2 x^2} x^5 \text{erf}(b x)}{3 b \sqrt{\pi }}+\frac{1}{6} x^6 \text{erf}(b x)^2-\frac{\int e^{-2 b^2 x^2} x \, dx}{3 b^4 \pi }-\frac{5 \int e^{-2 b^2 x^2} x \, dx}{6 b^4 \pi }-\frac{5 \int e^{-2 b^2 x^2} x \, dx}{2 b^4 \pi }-\frac{5 \int e^{-b^2 x^2} \text{erf}(b x) \, dx}{4 b^5 \sqrt{\pi }}\\ &=\frac{11 e^{-2 b^2 x^2}}{12 b^6 \pi }+\frac{7 e^{-2 b^2 x^2} x^2}{12 b^4 \pi }+\frac{e^{-2 b^2 x^2} x^4}{6 b^2 \pi }+\frac{5 e^{-b^2 x^2} x \text{erf}(b x)}{4 b^5 \sqrt{\pi }}+\frac{5 e^{-b^2 x^2} x^3 \text{erf}(b x)}{6 b^3 \sqrt{\pi }}+\frac{e^{-b^2 x^2} x^5 \text{erf}(b x)}{3 b \sqrt{\pi }}+\frac{1}{6} x^6 \text{erf}(b x)^2-\frac{5 \operatorname{Subst}(\int x \, dx,x,\text{erf}(b x))}{8 b^6}\\ &=\frac{11 e^{-2 b^2 x^2}}{12 b^6 \pi }+\frac{7 e^{-2 b^2 x^2} x^2}{12 b^4 \pi }+\frac{e^{-2 b^2 x^2} x^4}{6 b^2 \pi }+\frac{5 e^{-b^2 x^2} x \text{erf}(b x)}{4 b^5 \sqrt{\pi }}+\frac{5 e^{-b^2 x^2} x^3 \text{erf}(b x)}{6 b^3 \sqrt{\pi }}+\frac{e^{-b^2 x^2} x^5 \text{erf}(b x)}{3 b \sqrt{\pi }}-\frac{5 \text{erf}(b x)^2}{16 b^6}+\frac{1}{6} x^6 \text{erf}(b x)^2\\ \end{align*}

Mathematica [A]  time = 0.0451967, size = 106, normalized size = 0.6 \[ \frac{e^{-2 b^2 x^2} \left (4 \sqrt{\pi } b x e^{b^2 x^2} \left (4 b^4 x^4+10 b^2 x^2+15\right ) \text{Erf}(b x)+\pi e^{2 b^2 x^2} \left (8 b^6 x^6-15\right ) \text{Erf}(b x)^2+8 b^4 x^4+28 b^2 x^2+44\right )}{48 \pi b^6} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*Erf[b*x]^2,x]

[Out]

(44 + 28*b^2*x^2 + 8*b^4*x^4 + 4*b*E^(b^2*x^2)*Sqrt[Pi]*x*(15 + 10*b^2*x^2 + 4*b^4*x^4)*Erf[b*x] + E^(2*b^2*x^
2)*Pi*(-15 + 8*b^6*x^6)*Erf[b*x]^2)/(48*b^6*E^(2*b^2*x^2)*Pi)

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Maple [F]  time = 0.053, size = 0, normalized size = 0. \begin{align*} \int{x}^{5} \left ({\it Erf} \left ( bx \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*erf(b*x)^2,x)

[Out]

int(x^5*erf(b*x)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{-\frac{{\left (2 \, b^{4} x^{4} + 2 \, b^{2} x^{2} + 1\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{2 \, b^{2}} - \frac{5 \,{\left (2 \, b^{2} x^{2} + 1\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{4 \, b^{2}} - \frac{15 \, e^{\left (-2 \, b^{2} x^{2}\right )}}{4 \, b^{2}}}{6 \, \pi b^{4}} + \frac{{\left (8 \, \sqrt{\pi } b^{6} x^{6} - 15 \, \sqrt{\pi }\right )} \operatorname{erf}\left (b x\right )^{2} + 4 \,{\left (4 \, b^{5} x^{5} + 10 \, b^{3} x^{3} + 15 \, b x\right )} \operatorname{erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{48 \, \sqrt{\pi } b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*erf(b*x)^2,x, algorithm="maxima")

[Out]

-1/6*integrate((4*b^4*x^5 + 10*b^2*x^3 + 15*x)*e^(-2*b^2*x^2), x)/(pi*b^4) + 1/48*((8*sqrt(pi)*b^6*x^6 - 15*sq
rt(pi))*erf(b*x)^2 + 4*(4*b^5*x^5 + 10*b^3*x^3 + 15*b*x)*erf(b*x)*e^(-b^2*x^2))/(sqrt(pi)*b^6)

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Fricas [A]  time = 2.57921, size = 227, normalized size = 1.28 \begin{align*} \frac{4 \, \sqrt{\pi }{\left (4 \, b^{5} x^{5} + 10 \, b^{3} x^{3} + 15 \, b x\right )} \operatorname{erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} -{\left (15 \, \pi - 8 \, \pi b^{6} x^{6}\right )} \operatorname{erf}\left (b x\right )^{2} + 4 \,{\left (2 \, b^{4} x^{4} + 7 \, b^{2} x^{2} + 11\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{48 \, \pi b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*erf(b*x)^2,x, algorithm="fricas")

[Out]

1/48*(4*sqrt(pi)*(4*b^5*x^5 + 10*b^3*x^3 + 15*b*x)*erf(b*x)*e^(-b^2*x^2) - (15*pi - 8*pi*b^6*x^6)*erf(b*x)^2 +
4*(2*b^4*x^4 + 7*b^2*x^2 + 11)*e^(-2*b^2*x^2))/(pi*b^6)

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Sympy [A]  time = 10.3349, size = 168, normalized size = 0.94 \begin{align*} \begin{cases} \frac{x^{6} \operatorname{erf}^{2}{\left (b x \right )}}{6} + \frac{x^{5} e^{- b^{2} x^{2}} \operatorname{erf}{\left (b x \right )}}{3 \sqrt{\pi } b} + \frac{x^{4} e^{- 2 b^{2} x^{2}}}{6 \pi b^{2}} + \frac{5 x^{3} e^{- b^{2} x^{2}} \operatorname{erf}{\left (b x \right )}}{6 \sqrt{\pi } b^{3}} + \frac{7 x^{2} e^{- 2 b^{2} x^{2}}}{12 \pi b^{4}} + \frac{5 x e^{- b^{2} x^{2}} \operatorname{erf}{\left (b x \right )}}{4 \sqrt{\pi } b^{5}} - \frac{5 \operatorname{erf}^{2}{\left (b x \right )}}{16 b^{6}} + \frac{11 e^{- 2 b^{2} x^{2}}}{12 \pi b^{6}} & \text{for}\: b \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*erf(b*x)**2,x)

[Out]

Piecewise((x**6*erf(b*x)**2/6 + x**5*exp(-b**2*x**2)*erf(b*x)/(3*sqrt(pi)*b) + x**4*exp(-2*b**2*x**2)/(6*pi*b*
*2) + 5*x**3*exp(-b**2*x**2)*erf(b*x)/(6*sqrt(pi)*b**3) + 7*x**2*exp(-2*b**2*x**2)/(12*pi*b**4) + 5*x*exp(-b**
2*x**2)*erf(b*x)/(4*sqrt(pi)*b**5) - 5*erf(b*x)**2/(16*b**6) + 11*exp(-2*b**2*x**2)/(12*pi*b**6), Ne(b, 0)), (
0, True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{5} \operatorname{erf}\left (b x\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*erf(b*x)^2,x, algorithm="giac")

[Out]

integrate(x^5*erf(b*x)^2, x)