3.75 \(\int e^{-b^2 x^2} x^5 \text {erf}(b x) \, dx\)

Optimal. Leaf size=135 \[ \frac {43 \text {erf}\left (\sqrt {2} b x\right )}{32 \sqrt {2} b^6}-\frac {x^4 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{b^6}-\frac {11 x e^{-2 b^2 x^2}}{16 \sqrt {\pi } b^5}-\frac {x^2 e^{-b^2 x^2} \text {erf}(b x)}{b^4}-\frac {x^3 e^{-2 b^2 x^2}}{4 \sqrt {\pi } b^3} \]

[Out]

-erf(b*x)/b^6/exp(b^2*x^2)-x^2*erf(b*x)/b^4/exp(b^2*x^2)-1/2*x^4*erf(b*x)/b^2/exp(b^2*x^2)+43/64*erf(b*x*2^(1/
2))/b^6*2^(1/2)-11/16*x/b^5/exp(2*b^2*x^2)/Pi^(1/2)-1/4*x^3/b^3/exp(2*b^2*x^2)/Pi^(1/2)

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Rubi [A]  time = 0.20, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6385, 6382, 2205, 2212} \[ -\frac {x^4 e^{-b^2 x^2} \text {Erf}(b x)}{2 b^2}-\frac {x^2 e^{-b^2 x^2} \text {Erf}(b x)}{b^4}-\frac {e^{-b^2 x^2} \text {Erf}(b x)}{b^6}+\frac {43 \text {Erf}\left (\sqrt {2} b x\right )}{32 \sqrt {2} b^6}-\frac {x^3 e^{-2 b^2 x^2}}{4 \sqrt {\pi } b^3}-\frac {11 x e^{-2 b^2 x^2}}{16 \sqrt {\pi } b^5} \]

Antiderivative was successfully verified.

[In]

Int[(x^5*Erf[b*x])/E^(b^2*x^2),x]

[Out]

(-11*x)/(16*b^5*E^(2*b^2*x^2)*Sqrt[Pi]) - x^3/(4*b^3*E^(2*b^2*x^2)*Sqrt[Pi]) - Erf[b*x]/(b^6*E^(b^2*x^2)) - (x
^2*Erf[b*x])/(b^4*E^(b^2*x^2)) - (x^4*Erf[b*x])/(2*b^2*E^(b^2*x^2)) + (43*Erf[Sqrt[2]*b*x])/(32*Sqrt[2]*b^6)

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 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 e^{-b^2 x^2} x^5 \text {erf}(b x) \, dx &=-\frac {e^{-b^2 x^2} x^4 \text {erf}(b x)}{2 b^2}+\frac {2 \int e^{-b^2 x^2} x^3 \text {erf}(b x) \, dx}{b^2}+\frac {\int e^{-2 b^2 x^2} x^4 \, dx}{b \sqrt {\pi }}\\ &=-\frac {e^{-2 b^2 x^2} x^3}{4 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} x^2 \text {erf}(b x)}{b^4}-\frac {e^{-b^2 x^2} x^4 \text {erf}(b x)}{2 b^2}+\frac {2 \int e^{-b^2 x^2} x \text {erf}(b x) \, dx}{b^4}+\frac {3 \int e^{-2 b^2 x^2} x^2 \, dx}{4 b^3 \sqrt {\pi }}+\frac {2 \int e^{-2 b^2 x^2} x^2 \, dx}{b^3 \sqrt {\pi }}\\ &=-\frac {11 e^{-2 b^2 x^2} x}{16 b^5 \sqrt {\pi }}-\frac {e^{-2 b^2 x^2} x^3}{4 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{b^6}-\frac {e^{-b^2 x^2} x^2 \text {erf}(b x)}{b^4}-\frac {e^{-b^2 x^2} x^4 \text {erf}(b x)}{2 b^2}+\frac {3 \int e^{-2 b^2 x^2} \, dx}{16 b^5 \sqrt {\pi }}+\frac {\int e^{-2 b^2 x^2} \, dx}{2 b^5 \sqrt {\pi }}+\frac {2 \int e^{-2 b^2 x^2} \, dx}{b^5 \sqrt {\pi }}\\ &=-\frac {11 e^{-2 b^2 x^2} x}{16 b^5 \sqrt {\pi }}-\frac {e^{-2 b^2 x^2} x^3}{4 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{b^6}-\frac {e^{-b^2 x^2} x^2 \text {erf}(b x)}{b^4}-\frac {e^{-b^2 x^2} x^4 \text {erf}(b x)}{2 b^2}+\frac {43 \text {erf}\left (\sqrt {2} b x\right )}{32 \sqrt {2} b^6}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 86, normalized size = 0.64 \[ \frac {-\frac {4 b x e^{-2 b^2 x^2} \left (4 b^2 x^2+11\right )}{\sqrt {\pi }}-32 e^{-b^2 x^2} \left (b^4 x^4+2 b^2 x^2+2\right ) \text {erf}(b x)+43 \sqrt {2} \text {erf}\left (\sqrt {2} b x\right )}{64 b^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^5*Erf[b*x])/E^(b^2*x^2),x]

[Out]

((-4*b*x*(11 + 4*b^2*x^2))/(E^(2*b^2*x^2)*Sqrt[Pi]) - (32*(2 + 2*b^2*x^2 + b^4*x^4)*Erf[b*x])/E^(b^2*x^2) + 43
*Sqrt[2]*Erf[Sqrt[2]*b*x])/(64*b^6)

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fricas [A]  time = 0.57, size = 97, normalized size = 0.72 \[ \frac {43 \, \sqrt {2} \pi \sqrt {b^{2}} \operatorname {erf}\left (\sqrt {2} \sqrt {b^{2}} x\right ) - 32 \, {\left (\pi b^{5} x^{4} + 2 \, \pi b^{3} x^{2} + 2 \, \pi b\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} - 4 \, \sqrt {\pi } {\left (4 \, b^{4} x^{3} + 11 \, b^{2} x\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{64 \, \pi b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(43*sqrt(2)*pi*sqrt(b^2)*erf(sqrt(2)*sqrt(b^2)*x) - 32*(pi*b^5*x^4 + 2*pi*b^3*x^2 + 2*pi*b)*erf(b*x)*e^(-
b^2*x^2) - 4*sqrt(pi)*(4*b^4*x^3 + 11*b^2*x)*e^(-2*b^2*x^2))/(pi*b^7)

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giac [A]  time = 0.37, size = 153, normalized size = 1.13 \[ -\frac {{\left (b^{4} x^{4} + 2 \, b^{2} x^{2} + 2\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{2 \, b^{6}} - \frac {b^{4} {\left (\frac {4 \, {\left (4 \, b^{2} x^{3} + 3 \, x\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{b^{4}} + \frac {3 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {2} b x\right )}{b^{5}}\right )} + 8 \, b^{2} {\left (\frac {4 \, x e^{\left (-2 \, b^{2} x^{2}\right )}}{b^{2}} + \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {2} b x\right )}{b^{3}}\right )} + \frac {32 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {2} b x\right )}{b}}{64 \, \sqrt {\pi } b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(b^4*x^4 + 2*b^2*x^2 + 2)*erf(b*x)*e^(-b^2*x^2)/b^6 - 1/64*(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)

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maple [A]  time = 0.08, size = 119, normalized size = 0.88 \[ \frac {\frac {\erf \left (b x \right ) \left (-\frac {{\mathrm e}^{-b^{2} x^{2}} b^{4} x^{4}}{2}-{\mathrm e}^{-b^{2} x^{2}} b^{2} x^{2}-{\mathrm e}^{-b^{2} x^{2}}\right )}{b^{5}}-\frac {-\frac {43 \sqrt {2}\, \sqrt {\pi }\, \erf \left (b x \sqrt {2}\right )}{64}+\frac {11 \,{\mathrm e}^{-2 b^{2} x^{2}} b x}{16}+\frac {{\mathrm e}^{-2 b^{2} x^{2}} b^{3} x^{3}}{4}}{\sqrt {\pi }\, b^{5}}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(erf(b*x)/b^5*(-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/Pi^(1/2)/b^5*(-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))/b

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {{\left (b^{4} x^{4} + 2 \, b^{2} x^{2} + 2\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{2 \, b^{6}} + \frac {-\frac {1}{64} \, b^{4} {\left (\frac {4 \, {\left (4 \, b^{2} x^{3} + 3 \, x\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{b^{4}} - \frac {3 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\sqrt {2} b x\right )}{b^{5}}\right )} - \frac {1}{8} \, b^{2} {\left (\frac {4 \, x e^{\left (-2 \, b^{2} x^{2}\right )}}{b^{2}} - \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\sqrt {2} b x\right )}{b^{3}}\right )} + \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\sqrt {2} b x\right )}{2 \, b}}{\sqrt {\pi } b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(b^4*x^4 + 2*b^2*x^2 + 2)*erf(b*x)*e^(-b^2*x^2)/b^6 + integrate((b^4*x^4 + 2*b^2*x^2 + 2)*e^(-2*b^2*x^2),
x)/(sqrt(pi)*b^5)

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mupad [B]  time = 0.45, size = 192, normalized size = 1.42 \[ \frac {\sqrt {2}\,\mathrm {erf}\left (\sqrt {2}\,x\,\sqrt {b^2}\right )}{2\,b\,{\left (b^2\right )}^{5/2}}-\frac {\mathrm {erfi}\left (x\,\sqrt {-2\,b^2}\right )}{2\,b^3\,{\left (-2\,b^2\right )}^{3/2}}-\frac {x^3\,{\mathrm {e}}^{-2\,b^2\,x^2}}{4\,b^3\,\sqrt {\pi }}-\mathrm {erf}\left (b\,x\right )\,\left (\frac {{\mathrm {e}}^{-b^2\,x^2}}{b^6}+\frac {x^4\,{\mathrm {e}}^{-b^2\,x^2}}{2\,b^2}+\frac {x^2\,{\mathrm {e}}^{-b^2\,x^2}}{b^4}\right )-\frac {11\,x\,{\mathrm {e}}^{-2\,b^2\,x^2}}{16\,b^5\,\sqrt {\pi }}+\frac {3\,\sqrt {2}\,x^5}{64\,b\,{\left (b^2\,x^2\right )}^{5/2}}-\frac {3\,\sqrt {2}\,x^5\,\mathrm {erfc}\left (\sqrt {2\,b^2\,x^2}\right )}{64\,b\,{\left (b^2\,x^2\right )}^{5/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2^(1/2)*erf(2^(1/2)*x*(b^2)^(1/2)))/(2*b*(b^2)^(5/2)) - erfi(x*(-2*b^2)^(1/2))/(2*b^3*(-2*b^2)^(3/2)) - (x^3*
exp(-2*b^2*x^2))/(4*b^3*pi^(1/2)) - erf(b*x)*(exp(-b^2*x^2)/b^6 + (x^4*exp(-b^2*x^2))/(2*b^2) + (x^2*exp(-b^2*
x^2))/b^4) - (11*x*exp(-2*b^2*x^2))/(16*b^5*pi^(1/2)) + (3*2^(1/2)*x^5)/(64*b*(b^2*x^2)^(5/2)) - (3*2^(1/2)*x^
5*erfc((2*b^2*x^2)^(1/2)))/(64*b*(b^2*x^2)^(5/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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