3.23 \(\int x^3 \text {erf}(b x)^2 \, dx\)

Optimal. Leaf size=126 \[ -\frac {3 \text {erf}(b x)^2}{16 b^4}+\frac {x^3 e^{-b^2 x^2} \text {erf}(b x)}{2 \sqrt {\pi } b}+\frac {x^2 e^{-2 b^2 x^2}}{4 \pi b^2}+\frac {e^{-2 b^2 x^2}}{2 \pi b^4}+\frac {3 x e^{-b^2 x^2} \text {erf}(b x)}{4 \sqrt {\pi } b^3}+\frac {1}{4} x^4 \text {erf}(b x)^2 \]

[Out]

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)

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Rubi [A]  time = 0.18, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6364, 6385, 6373, 30, 2209, 2212} \[ \frac {x^3 e^{-b^2 x^2} \text {Erf}(b x)}{2 \sqrt {\pi } b}+\frac {3 x e^{-b^2 x^2} \text {Erf}(b x)}{4 \sqrt {\pi } b^3}-\frac {3 \text {Erf}(b x)^2}{16 b^4}+\frac {x^2 e^{-2 b^2 x^2}}{4 \pi b^2}+\frac {e^{-2 b^2 x^2}}{2 \pi b^4}+\frac {1}{4} x^4 \text {Erf}(b x)^2 \]

Antiderivative was successfully verified.

[In]

Int[x^3*Erf[b*x]^2,x]

[Out]

1/(2*b^4*E^(2*b^2*x^2)*Pi) + x^2/(4*b^2*E^(2*b^2*x^2)*Pi) + (3*x*Erf[b*x])/(4*b^3*E^(b^2*x^2)*Sqrt[Pi]) + (x^3
*Erf[b*x])/(2*b*E^(b^2*x^2)*Sqrt[Pi]) - (3*Erf[b*x]^2)/(16*b^4) + (x^4*Erf[b*x]^2)/4

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])

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 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 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]

Rubi steps

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

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Mathematica [A]  time = 0.04, size = 90, normalized size = 0.71 \[ \frac {e^{-2 b^2 x^2} \left (4 \sqrt {\pi } b x e^{b^2 x^2} \left (2 b^2 x^2+3\right ) \text {erf}(b x)+4 b^2 x^2+\pi e^{2 b^2 x^2} \left (4 b^4 x^4-3\right ) \text {erf}(b x)^2+8\right )}{16 \pi b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*Erf[b*x]^2,x]

[Out]

(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)

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fricas [A]  time = 0.41, size = 81, normalized size = 0.64 \[ \frac {4 \, \sqrt {\pi } {\left (2 \, b^{3} x^{3} + 3 \, b x\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} - {\left (3 \, \pi - 4 \, \pi b^{4} x^{4}\right )} \operatorname {erf}\left (b x\right )^{2} + 4 \, {\left (b^{2} x^{2} + 2\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{16 \, \pi b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*erf(b*x)^2,x, algorithm="fricas")

[Out]

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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \operatorname {erf}\left (b x\right )^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*erf(b*x)^2,x, algorithm="giac")

[Out]

integrate(x^3*erf(b*x)^2, x)

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[ \int x^{3} \erf \left (b x \right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*erf(b*x)^2,x)

[Out]

int(x^3*erf(b*x)^2,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {-\frac {{\left (2 \, b^{2} x^{2} + 1\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{4 \, b^{2}} - \frac {3 \, e^{\left (-2 \, b^{2} x^{2}\right )}}{4 \, b^{2}}}{2 \, \pi b^{2}} - \frac {{\left (3 \, \pi - 4 \, \pi b^{4} x^{4}\right )} \operatorname {erf}\left (b x\right )^{2} - 4 \, {\left (2 \, \sqrt {\pi } b^{3} x^{3} + 3 \, \sqrt {\pi } b x\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{16 \, \pi b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*erf(b*x)^2,x, algorithm="maxima")

[Out]

-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*s
qrt(pi)*b^3*x^3 + 3*sqrt(pi)*b*x)*erf(b*x)*e^(-b^2*x^2))/(pi*b^4)

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mupad [B]  time = 0.21, size = 101, normalized size = 0.80 \[ \frac {x^4\,{\mathrm {erf}\left (b\,x\right )}^2}{4}+\frac {\frac {{\mathrm {e}}^{-2\,b^2\,x^2}}{2}-\frac {3\,\pi \,{\mathrm {erf}\left (b\,x\right )}^2}{16}+\frac {b^2\,x^2\,{\mathrm {e}}^{-2\,b^2\,x^2}}{4}+\frac {b^3\,x^3\,\sqrt {\pi }\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erf}\left (b\,x\right )}{2}+\frac {3\,b\,x\,\sqrt {\pi }\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erf}\left (b\,x\right )}{4}}{b^4\,\pi } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*erf(b*x)^2,x)

[Out]

(x^4*erf(b*x)^2)/4 + (exp(-2*b^2*x^2)/2 - (3*pi*erf(b*x)^2)/16 + (b^2*x^2*exp(-2*b^2*x^2))/4 + (b^3*x^3*pi^(1/
2)*exp(-b^2*x^2)*erf(b*x))/2 + (3*b*x*pi^(1/2)*exp(-b^2*x^2)*erf(b*x))/4)/(b^4*pi)

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sympy [A]  time = 1.97, size = 117, normalized size = 0.93 \[ \begin {cases} \frac {x^{4} \operatorname {erf}^{2}{\left (b x \right )}}{4} + \frac {x^{3} e^{- b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{2 \sqrt {\pi } b} + \frac {x^{2} e^{- 2 b^{2} x^{2}}}{4 \pi b^{2}} + \frac {3 x e^{- b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{4 \sqrt {\pi } b^{3}} - \frac {3 \operatorname {erf}^{2}{\left (b x \right )}}{16 b^{4}} + \frac {e^{- 2 b^{2} x^{2}}}{2 \pi b^{4}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*erf(b*x)**2,x)

[Out]

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))

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