3.29 \(\int x^4 \text {erf}(b x)^2 \, dx\)

Optimal. Leaf size=165 \[ -\frac {43 \text {erf}\left (\sqrt {2} b x\right )}{40 \sqrt {2 \pi } b^5}+\frac {2 x^4 e^{-b^2 x^2} \text {erf}(b x)}{5 \sqrt {\pi } b}+\frac {x^3 e^{-2 b^2 x^2}}{5 \pi b^2}+\frac {4 e^{-b^2 x^2} \text {erf}(b x)}{5 \sqrt {\pi } b^5}+\frac {11 x e^{-2 b^2 x^2}}{20 \pi b^4}+\frac {4 x^2 e^{-b^2 x^2} \text {erf}(b x)}{5 \sqrt {\pi } b^3}+\frac {1}{5} x^5 \text {erf}(b x)^2 \]

[Out]

11/20*x/b^4/exp(2*b^2*x^2)/Pi+1/5*x^3/b^2/exp(2*b^2*x^2)/Pi+1/5*x^5*erf(b*x)^2+4/5*erf(b*x)/b^5/exp(b^2*x^2)/P
i^(1/2)+4/5*x^2*erf(b*x)/b^3/exp(b^2*x^2)/Pi^(1/2)+2/5*x^4*erf(b*x)/b/exp(b^2*x^2)/Pi^(1/2)-43/80*erf(b*x*2^(1
/2))/b^5*2^(1/2)/Pi^(1/2)

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Rubi [A]  time = 0.25, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6364, 6385, 6382, 2205, 2212} \[ \frac {2 x^4 e^{-b^2 x^2} \text {Erf}(b x)}{5 \sqrt {\pi } b}+\frac {4 x^2 e^{-b^2 x^2} \text {Erf}(b x)}{5 \sqrt {\pi } b^3}+\frac {4 e^{-b^2 x^2} \text {Erf}(b x)}{5 \sqrt {\pi } b^5}-\frac {43 \text {Erf}\left (\sqrt {2} b x\right )}{40 \sqrt {2 \pi } b^5}+\frac {x^3 e^{-2 b^2 x^2}}{5 \pi b^2}+\frac {11 x e^{-2 b^2 x^2}}{20 \pi b^4}+\frac {1}{5} x^5 \text {Erf}(b x)^2 \]

Antiderivative was successfully verified.

[In]

Int[x^4*Erf[b*x]^2,x]

[Out]

(11*x)/(20*b^4*E^(2*b^2*x^2)*Pi) + x^3/(5*b^2*E^(2*b^2*x^2)*Pi) + (4*Erf[b*x])/(5*b^5*E^(b^2*x^2)*Sqrt[Pi]) +
(4*x^2*Erf[b*x])/(5*b^3*E^(b^2*x^2)*Sqrt[Pi]) + (2*x^4*Erf[b*x])/(5*b*E^(b^2*x^2)*Sqrt[Pi]) + (x^5*Erf[b*x]^2)
/5 - (43*Erf[Sqrt[2]*b*x])/(40*b^5*Sqrt[2*Pi])

Rule 2205

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] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

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 6382

Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(a_.) + (b_.)*(x_)]*(x_), x_Symbol] :> Simp[(E^(c + d*x^2)*Erf[a + b*x])/(2*d
), x] - Dist[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]

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^4 \text {erf}(b x)^2 \, dx &=\frac {1}{5} x^5 \text {erf}(b x)^2-\frac {(4 b) \int e^{-b^2 x^2} x^5 \text {erf}(b x) \, dx}{5 \sqrt {\pi }}\\ &=\frac {2 e^{-b^2 x^2} x^4 \text {erf}(b x)}{5 b \sqrt {\pi }}+\frac {1}{5} x^5 \text {erf}(b x)^2-\frac {4 \int e^{-2 b^2 x^2} x^4 \, dx}{5 \pi }-\frac {8 \int e^{-b^2 x^2} x^3 \text {erf}(b x) \, dx}{5 b \sqrt {\pi }}\\ &=\frac {e^{-2 b^2 x^2} x^3}{5 b^2 \pi }+\frac {4 e^{-b^2 x^2} x^2 \text {erf}(b x)}{5 b^3 \sqrt {\pi }}+\frac {2 e^{-b^2 x^2} x^4 \text {erf}(b x)}{5 b \sqrt {\pi }}+\frac {1}{5} x^5 \text {erf}(b x)^2-\frac {3 \int e^{-2 b^2 x^2} x^2 \, dx}{5 b^2 \pi }-\frac {8 \int e^{-2 b^2 x^2} x^2 \, dx}{5 b^2 \pi }-\frac {8 \int e^{-b^2 x^2} x \text {erf}(b x) \, dx}{5 b^3 \sqrt {\pi }}\\ &=\frac {11 e^{-2 b^2 x^2} x}{20 b^4 \pi }+\frac {e^{-2 b^2 x^2} x^3}{5 b^2 \pi }+\frac {4 e^{-b^2 x^2} \text {erf}(b x)}{5 b^5 \sqrt {\pi }}+\frac {4 e^{-b^2 x^2} x^2 \text {erf}(b x)}{5 b^3 \sqrt {\pi }}+\frac {2 e^{-b^2 x^2} x^4 \text {erf}(b x)}{5 b \sqrt {\pi }}+\frac {1}{5} x^5 \text {erf}(b x)^2-\frac {3 \int e^{-2 b^2 x^2} \, dx}{20 b^4 \pi }-\frac {2 \int e^{-2 b^2 x^2} \, dx}{5 b^4 \pi }-\frac {8 \int e^{-2 b^2 x^2} \, dx}{5 b^4 \pi }\\ &=\frac {11 e^{-2 b^2 x^2} x}{20 b^4 \pi }+\frac {e^{-2 b^2 x^2} x^3}{5 b^2 \pi }+\frac {4 e^{-b^2 x^2} \text {erf}(b x)}{5 b^5 \sqrt {\pi }}+\frac {4 e^{-b^2 x^2} x^2 \text {erf}(b x)}{5 b^3 \sqrt {\pi }}+\frac {2 e^{-b^2 x^2} x^4 \text {erf}(b x)}{5 b \sqrt {\pi }}+\frac {1}{5} x^5 \text {erf}(b x)^2-\frac {2 \sqrt {\frac {2}{\pi }} \text {erf}\left (\sqrt {2} b x\right )}{5 b^5}-\frac {11 \text {erf}\left (\sqrt {2} b x\right )}{40 b^5 \sqrt {2 \pi }}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 106, normalized size = 0.64 \[ \frac {16 \pi b^5 x^5 \text {erf}(b x)^2+4 b x e^{-2 b^2 x^2} \left (4 b^2 x^2+11\right )+32 \sqrt {\pi } e^{-b^2 x^2} \left (b^4 x^4+2 b^2 x^2+2\right ) \text {erf}(b x)-43 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} b x\right )}{80 \pi b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*Erf[b*x]^2,x]

[Out]

((4*b*x*(11 + 4*b^2*x^2))/E^(2*b^2*x^2) + (32*Sqrt[Pi]*(2 + 2*b^2*x^2 + b^4*x^4)*Erf[b*x])/E^(b^2*x^2) + 16*b^
5*Pi*x^5*Erf[b*x]^2 - 43*Sqrt[2*Pi]*Erf[Sqrt[2]*b*x])/(80*b^5*Pi)

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fricas [A]  time = 0.85, size = 111, normalized size = 0.67 \[ \frac {16 \, \pi b^{6} x^{5} \operatorname {erf}\left (b x\right )^{2} + 32 \, \sqrt {\pi } {\left (b^{5} x^{4} + 2 \, b^{3} x^{2} + 2 \, b\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} - 43 \, \sqrt {2} \sqrt {\pi } \sqrt {b^{2}} \operatorname {erf}\left (\sqrt {2} \sqrt {b^{2}} x\right ) + 4 \, {\left (4 \, b^{4} x^{3} + 11 \, b^{2} x\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{80 \, \pi b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*erf(b*x)^2,x, algorithm="fricas")

[Out]

1/80*(16*pi*b^6*x^5*erf(b*x)^2 + 32*sqrt(pi)*(b^5*x^4 + 2*b^3*x^2 + 2*b)*erf(b*x)*e^(-b^2*x^2) - 43*sqrt(2)*sq
rt(pi)*sqrt(b^2)*erf(sqrt(2)*sqrt(b^2)*x) + 4*(4*b^4*x^3 + 11*b^2*x)*e^(-2*b^2*x^2))/(pi*b^6)

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giac [A]  time = 0.38, size = 170, normalized size = 1.03 \[ \frac {1}{5} \, x^{5} \operatorname {erf}\left (b x\right )^{2} + \frac {b {\left (\frac {32 \, {\left (b^{4} x^{4} + 2 \, b^{2} x^{2} + 2\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{b^{6}} + \frac {b^{4} {\left (\frac {4 \, {\left (4 \, b^{2} x^{3} + 3 \, x\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{b^{4}} + \frac {3 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {2} b x\right )}{b^{5}}\right )} + 8 \, b^{2} {\left (\frac {4 \, x e^{\left (-2 \, b^{2} x^{2}\right )}}{b^{2}} + \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {2} b x\right )}{b^{3}}\right )} + \frac {32 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {2} b x\right )}{b}}{\sqrt {\pi } b^{5}}\right )}}{80 \, \sqrt {\pi }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*erf(b*x)^2,x, algorithm="giac")

[Out]

1/5*x^5*erf(b*x)^2 + 1/80*b*(32*(b^4*x^4 + 2*b^2*x^2 + 2)*erf(b*x)*e^(-b^2*x^2)/b^6 + (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)*sqr
t(pi)*erf(-sqrt(2)*b*x)/b^3) + 32*sqrt(2)*sqrt(pi)*erf(-sqrt(2)*b*x)/b)/(sqrt(pi)*b^5))/sqrt(pi)

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maple [A]  time = 0.01, size = 131, normalized size = 0.79 \[ \frac {\frac {b^{5} x^{5} \erf \left (b x \right )^{2}}{5}-\frac {4 \erf \left (b x \right ) \left (-\frac {{\mathrm e}^{-b^{2} x^{2}} b^{4} x^{4}}{2}-{\mathrm e}^{-b^{2} x^{2}} b^{2} x^{2}-{\mathrm e}^{-b^{2} x^{2}}\right )}{5 \sqrt {\pi }}+\frac {-\frac {43 \sqrt {2}\, \sqrt {\pi }\, \erf \left (b x \sqrt {2}\right )}{80}+\frac {11 \,{\mathrm e}^{-2 b^{2} x^{2}} b x}{20}+\frac {{\mathrm e}^{-2 b^{2} x^{2}} b^{3} x^{3}}{5}}{\pi }}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*erf(b*x)^2,x)

[Out]

1/b^5*(1/5*b^5*x^5*erf(b*x)^2-4/5*erf(b*x)/Pi^(1/2)*(-1/2/exp(b^2*x^2)*b^4*x^4-1/exp(b^2*x^2)*b^2*x^2-1/exp(b^
2*x^2))+4/5/Pi*(-43/64*2^(1/2)*Pi^(1/2)*erf(b*x*2^(1/2))+11/16/exp(b^2*x^2)^2*b*x+1/4/exp(b^2*x^2)^2*b^3*x^3))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {-\frac {1}{16} \, b^{4} {\left (\frac {4 \, {\left (4 \, b^{2} x^{3} + 3 \, x\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{b^{4}} - \frac {3 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\sqrt {2} b x\right )}{b^{5}}\right )} - \frac {1}{2} \, b^{2} {\left (\frac {4 \, x e^{\left (-2 \, b^{2} x^{2}\right )}}{b^{2}} - \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\sqrt {2} b x\right )}{b^{3}}\right )} + \frac {2 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\sqrt {2} b x\right )}{b}}{5 \, \pi b^{4}} + \frac {\sqrt {\pi } b^{5} x^{5} \operatorname {erf}\left (b x\right )^{2} + 2 \, {\left (b^{4} x^{4} + 2 \, b^{2} x^{2} + 2\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{5 \, \sqrt {\pi } b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*erf(b*x)^2,x, algorithm="maxima")

[Out]

-1/5*integrate(4*(b^4*x^4 + 2*b^2*x^2 + 2)*e^(-2*b^2*x^2), x)/(pi*b^4) + 1/5*(sqrt(pi)*b^5*x^5*erf(b*x)^2 + 2*
(b^4*x^4 + 2*b^2*x^2 + 2)*erf(b*x)*e^(-b^2*x^2))/(sqrt(pi)*b^5)

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mupad [B]  time = 0.21, size = 131, normalized size = 0.79 \[ \frac {x^5\,{\mathrm {erf}\left (b\,x\right )}^2}{5}+\frac {\frac {4\,\sqrt {\pi }\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erf}\left (b\,x\right )}{5}+\frac {b^3\,x^3\,{\mathrm {e}}^{-2\,b^2\,x^2}}{5}-\frac {43\,\sqrt {2}\,\sqrt {\pi }\,\mathrm {erf}\left (\sqrt {2}\,b\,x\right )}{80}+\frac {11\,b\,x\,{\mathrm {e}}^{-2\,b^2\,x^2}}{20}+\frac {4\,b^2\,x^2\,\sqrt {\pi }\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erf}\left (b\,x\right )}{5}+\frac {2\,b^4\,x^4\,\sqrt {\pi }\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erf}\left (b\,x\right )}{5}}{b^5\,\pi } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*erf(b*x)^2,x)

[Out]

(x^5*erf(b*x)^2)/5 + ((4*pi^(1/2)*exp(-b^2*x^2)*erf(b*x))/5 + (b^3*x^3*exp(-2*b^2*x^2))/5 - (43*2^(1/2)*pi^(1/
2)*erf(2^(1/2)*b*x))/80 + (11*b*x*exp(-2*b^2*x^2))/20 + (4*b^2*x^2*pi^(1/2)*exp(-b^2*x^2)*erf(b*x))/5 + (2*b^4
*x^4*pi^(1/2)*exp(-b^2*x^2)*erf(b*x))/5)/(b^5*pi)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \operatorname {erf}^{2}{\left (b x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*erf(b*x)**2,x)

[Out]

Integral(x**4*erf(b*x)**2, x)

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