3.132 \(\int x^4 \text {erfc}(b x)^2 \, dx\)

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

[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*erfc(b*x)^2-4/5*erfc(b*x)/b^5/exp(b^2*x^2)
/Pi^(1/2)-4/5*x^2*erfc(b*x)/b^3/exp(b^2*x^2)/Pi^(1/2)-2/5*x^4*erfc(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 = 0.23, 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 = {6365, 6386, 6383, 2205, 2212} \[ -\frac {43 \text {Erf}\left (\sqrt {2} b x\right )}{40 \sqrt {2 \pi } b^5}-\frac {2 x^4 e^{-b^2 x^2} \text {Erfc}(b x)}{5 \sqrt {\pi } b}-\frac {4 x^2 e^{-b^2 x^2} \text {Erfc}(b x)}{5 \sqrt {\pi } b^3}-\frac {4 e^{-b^2 x^2} \text {Erfc}(b x)}{5 \sqrt {\pi } b^5}+\frac {x^3 e^{-2 b^2 x^2}}{5 \pi b^2}+\frac {11 x e^{-2 b^2 x^2}}{20 \pi b^4}+\frac {1}{5} x^5 \text {Erfc}(b x)^2 \]

Antiderivative was successfully verified.

[In]

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

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 6365

Int[Erfc[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[(x^(m + 1)*Erfc[b*x]^2)/(m + 1), x] + Dist[(4*b)/(Sqrt[Pi
]*(m + 1)), Int[(x^(m + 1)*Erfc[b*x])/E^(b^2*x^2), x], x] /; FreeQ[b, x] && (IGtQ[m, 0] || ILtQ[(m + 1)/2, 0])

Rule 6383

Int[E^((c_.) + (d_.)*(x_)^2)*Erfc[(a_.) + (b_.)*(x_)]*(x_), x_Symbol] :> Simp[(E^(c + d*x^2)*Erfc[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 6386

Int[E^((c_.) + (d_.)*(x_)^2)*Erfc[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[(x^(m - 1)*E^(c + d*x^2)*Er
fc[a + b*x])/(2*d), x] + (-Dist[(m - 1)/(2*d), Int[x^(m - 2)*E^(c + d*x^2)*Erfc[a + b*x], x], x] + Dist[b/(d*S
qrt[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 {erfc}(b x)^2 \, dx &=\frac {1}{5} x^5 \text {erfc}(b x)^2+\frac {(4 b) \int e^{-b^2 x^2} x^5 \text {erfc}(b x) \, dx}{5 \sqrt {\pi }}\\ &=-\frac {2 e^{-b^2 x^2} x^4 \text {erfc}(b x)}{5 b \sqrt {\pi }}+\frac {1}{5} x^5 \text {erfc}(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 {erfc}(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 {erfc}(b x)}{5 b^3 \sqrt {\pi }}-\frac {2 e^{-b^2 x^2} x^4 \text {erfc}(b x)}{5 b \sqrt {\pi }}+\frac {1}{5} x^5 \text {erfc}(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 {erfc}(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 {erfc}(b x)}{5 b^5 \sqrt {\pi }}-\frac {4 e^{-b^2 x^2} x^2 \text {erfc}(b x)}{5 b^3 \sqrt {\pi }}-\frac {2 e^{-b^2 x^2} x^4 \text {erfc}(b x)}{5 b \sqrt {\pi }}+\frac {1}{5} x^5 \text {erfc}(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 {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 }}-\frac {4 e^{-b^2 x^2} \text {erfc}(b x)}{5 b^5 \sqrt {\pi }}-\frac {4 e^{-b^2 x^2} x^2 \text {erfc}(b x)}{5 b^3 \sqrt {\pi }}-\frac {2 e^{-b^2 x^2} x^4 \text {erfc}(b x)}{5 b \sqrt {\pi }}+\frac {1}{5} x^5 \text {erfc}(b x)^2\\ \end {align*}

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Mathematica [A]  time = 0.16, size = 108, normalized size = 0.65 \[ \frac {4 \left (4 \pi b^5 x^5 \text {erfc}(b x)^2+b x e^{-2 b^2 x^2} \left (4 b^2 x^2+11\right )-8 \sqrt {\pi } e^{-b^2 x^2} \left (b^4 x^4+2 b^2 x^2+2\right ) \text {erfc}(b x)\right )-43 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} b x\right )}{80 \pi b^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-43*Sqrt[2*Pi]*Erf[Sqrt[2]*b*x] + 4*((b*x*(11 + 4*b^2*x^2))/E^(2*b^2*x^2) - (8*Sqrt[Pi]*(2 + 2*b^2*x^2 + b^4*
x^4)*Erfc[b*x])/E^(b^2*x^2) + 4*b^5*Pi*x^5*Erfc[b*x]^2))/(80*b^5*Pi)

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fricas [A]  time = 0.40, size = 154, normalized size = 0.93 \[ \frac {16 \, \pi b^{6} x^{5} \operatorname {erf}\left (b x\right )^{2} - 32 \, \pi b^{6} x^{5} \operatorname {erf}\left (b x\right ) + 16 \, \pi b^{6} x^{5} - 43 \, \sqrt {2} \sqrt {\pi } \sqrt {b^{2}} \operatorname {erf}\left (\sqrt {2} \sqrt {b^{2}} x\right ) - 32 \, \sqrt {\pi } {\left (b^{5} x^{4} + 2 \, b^{3} x^{2} - {\left (b^{5} x^{4} + 2 \, b^{3} x^{2} + 2 \, b\right )} \operatorname {erf}\left (b x\right ) + 2 \, b\right )} e^{\left (-b^{2} x^{2}\right )} + 4 \, {\left (4 \, b^{4} x^{3} + 11 \, b^{2} x\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{80 \, \pi b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/80*(16*pi*b^6*x^5*erf(b*x)^2 - 32*pi*b^6*x^5*erf(b*x) + 16*pi*b^6*x^5 - 43*sqrt(2)*sqrt(pi)*sqrt(b^2)*erf(sq
rt(2)*sqrt(b^2)*x) - 32*sqrt(pi)*(b^5*x^4 + 2*b^3*x^2 - (b^5*x^4 + 2*b^3*x^2 + 2*b)*erf(b*x) + 2*b)*e^(-b^2*x^
2) + 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 = 0.86, size = 218, normalized size = 1.32 \[ \frac {1}{5} \, x^{5} \operatorname {erf}\left (b x\right )^{2} - \frac {2}{5} \, x^{5} \operatorname {erf}\left (b x\right ) + \frac {1}{5} \, x^{5} + \frac {b {\left (\frac {32 \, {\left (b^{4} x^{4} + 2 \, b^{2} x^{2} + 2\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{b^{6}} + \frac {b^{4} {\left (\frac {4 \, {\left (4 \, b^{2} x^{3} + 3 \, x\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{b^{4}} + \frac {3 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {2} b x\right )}{b^{5}}\right )} + 8 \, b^{2} {\left (\frac {4 \, x e^{\left (-2 \, b^{2} x^{2}\right )}}{b^{2}} + \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {2} b x\right )}{b^{3}}\right )} + \frac {32 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {2} b x\right )}{b}}{\sqrt {\pi } b^{5}}\right )}}{80 \, \sqrt {\pi }} - \frac {2 \, {\left (b^{4} x^{4} + 2 \, b^{2} x^{2} + 2\right )} e^{\left (-b^{2} x^{2}\right )}}{5 \, \sqrt {\pi } b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 205, normalized size = 1.24 \[ \frac {\frac {b^{5} x^{5}}{5}-\frac {2 b^{5} x^{5} \erf \left (b x \right )}{5}+\frac {-\frac {2 \,{\mathrm e}^{-b^{2} x^{2}} b^{4} x^{4}}{5}-\frac {4 \,{\mathrm e}^{-b^{2} x^{2}} b^{2} x^{2}}{5}-\frac {4 \,{\mathrm e}^{-b^{2} x^{2}}}{5}}{\sqrt {\pi }}+\frac {b^{5} x^{5} \erf \left (b x \right )^{2}}{5}-\frac {4 \erf \left (b x \right ) \left (-\frac {{\mathrm e}^{-b^{2} x^{2}} b^{4} x^{4}}{2}-{\mathrm e}^{-b^{2} x^{2}} b^{2} x^{2}-{\mathrm e}^{-b^{2} x^{2}}\right )}{5 \sqrt {\pi }}+\frac {-\frac {43 \sqrt {2}\, \sqrt {\pi }\, \erf \left (b x \sqrt {2}\right )}{80}+\frac {11 \,{\mathrm e}^{-2 b^{2} x^{2}} b x}{20}+\frac {{\mathrm e}^{-2 b^{2} x^{2}} b^{3} x^{3}}{5}}{\pi }}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b^5*(1/5*b^5*x^5-2/5*b^5*x^5*erf(b*x)+4/5/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))+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 \[ \int x^{4} \operatorname {erfc}\left (b x\right )^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^4*erfc(b*x)^2, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^4\,{\mathrm {erfc}\left (b\,x\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^4*erfc(b*x)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**4*erfc(b*x)**2, x)

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