3.8 \(\int x^6 \text{Erf}(b x) \, dx\)

Optimal. Leaf size=109 \[ \frac{x^6 e^{-b^2 x^2}}{7 \sqrt{\pi } b}+\frac{3 x^4 e^{-b^2 x^2}}{7 \sqrt{\pi } b^3}+\frac{6 x^2 e^{-b^2 x^2}}{7 \sqrt{\pi } b^5}+\frac{6 e^{-b^2 x^2}}{7 \sqrt{\pi } b^7}+\frac{1}{7} x^7 \text{Erf}(b x) \]

[Out]

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

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Rubi [A]  time = 0.0989079, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6361, 2212, 2209} \[ \frac{x^6 e^{-b^2 x^2}}{7 \sqrt{\pi } b}+\frac{3 x^4 e^{-b^2 x^2}}{7 \sqrt{\pi } b^3}+\frac{6 x^2 e^{-b^2 x^2}}{7 \sqrt{\pi } b^5}+\frac{6 e^{-b^2 x^2}}{7 \sqrt{\pi } b^7}+\frac{1}{7} x^7 \text{Erf}(b x) \]

Antiderivative was successfully verified.

[In]

Int[x^6*Erf[b*x],x]

[Out]

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

Rule 6361

Int[Erf[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m + 1)*Erf[a + b*x])/(d*
(m + 1)), x] - Dist[(2*b)/(Sqrt[Pi]*d*(m + 1)), Int[(c + d*x)^(m + 1)/E^(a + b*x)^2, x], x] /; FreeQ[{a, b, c,
d, m}, x] && NeQ[m, -1]

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 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]

Rubi steps

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

Mathematica [A]  time = 0.0177674, size = 72, normalized size = 0.66 \[ \frac{e^{-b^2 x^2} \left (\sqrt{\pi } b^7 x^7 e^{b^2 x^2} \text{Erf}(b x)+b^6 x^6+3 b^4 x^4+6 b^2 x^2+6\right )}{7 \sqrt{\pi } b^7} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6*Erf[b*x],x]

[Out]

(6 + 6*b^2*x^2 + 3*b^4*x^4 + b^6*x^6 + b^7*E^(b^2*x^2)*Sqrt[Pi]*x^7*Erf[b*x])/(7*b^7*E^(b^2*x^2)*Sqrt[Pi])

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Maple [A]  time = 0.044, size = 90, normalized size = 0.8 \begin{align*}{\frac{1}{{b}^{7}} \left ({\frac{{\it Erf} \left ( bx \right ){b}^{7}{x}^{7}}{7}}-{\frac{2}{7\,\sqrt{\pi }} \left ( -{\frac{{b}^{6}{x}^{6}}{2\,{{\rm e}^{{b}^{2}{x}^{2}}}}}-{\frac{3\,{b}^{4}{x}^{4}}{2\,{{\rm e}^{{b}^{2}{x}^{2}}}}}-3\,{\frac{{b}^{2}{x}^{2}}{{{\rm e}^{{b}^{2}{x}^{2}}}}}-3\, \left ({{\rm e}^{{b}^{2}{x}^{2}}} \right ) ^{-1} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*erf(b*x),x)

[Out]

1/b^7*(1/7*erf(b*x)*b^7*x^7-2/7/Pi^(1/2)*(-1/2/exp(b^2*x^2)*b^6*x^6-3/2*b^4*x^4/exp(b^2*x^2)-3*b^2*x^2/exp(b^2
*x^2)-3/exp(b^2*x^2)))

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Maxima [A]  time = 1.23354, size = 70, normalized size = 0.64 \begin{align*} \frac{1}{7} \, x^{7} \operatorname{erf}\left (b x\right ) + \frac{{\left (b^{6} x^{6} + 3 \, b^{4} x^{4} + 6 \, b^{2} x^{2} + 6\right )} e^{\left (-b^{2} x^{2}\right )}}{7 \, \sqrt{\pi } b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*erf(b*x),x, algorithm="maxima")

[Out]

1/7*x^7*erf(b*x) + 1/7*(b^6*x^6 + 3*b^4*x^4 + 6*b^2*x^2 + 6)*e^(-b^2*x^2)/(sqrt(pi)*b^7)

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Fricas [A]  time = 2.52048, size = 132, normalized size = 1.21 \begin{align*} \frac{\pi b^{7} x^{7} \operatorname{erf}\left (b x\right ) + \sqrt{\pi }{\left (b^{6} x^{6} + 3 \, b^{4} x^{4} + 6 \, b^{2} x^{2} + 6\right )} e^{\left (-b^{2} x^{2}\right )}}{7 \, \pi b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*erf(b*x),x, algorithm="fricas")

[Out]

1/7*(pi*b^7*x^7*erf(b*x) + sqrt(pi)*(b^6*x^6 + 3*b^4*x^4 + 6*b^2*x^2 + 6)*e^(-b^2*x^2))/(pi*b^7)

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Sympy [A]  time = 7.56104, size = 99, normalized size = 0.91 \begin{align*} \begin{cases} \frac{x^{7} \operatorname{erf}{\left (b x \right )}}{7} + \frac{x^{6} e^{- b^{2} x^{2}}}{7 \sqrt{\pi } b} + \frac{3 x^{4} e^{- b^{2} x^{2}}}{7 \sqrt{\pi } b^{3}} + \frac{6 x^{2} e^{- b^{2} x^{2}}}{7 \sqrt{\pi } b^{5}} + \frac{6 e^{- b^{2} x^{2}}}{7 \sqrt{\pi } b^{7}} & \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**6*erf(b*x),x)

[Out]

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

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Giac [A]  time = 1.32549, size = 70, normalized size = 0.64 \begin{align*} \frac{1}{7} \, x^{7} \operatorname{erf}\left (b x\right ) + \frac{{\left (b^{6} x^{6} + 3 \, b^{4} x^{4} + 6 \, b^{2} x^{2} + 6\right )} e^{\left (-b^{2} x^{2}\right )}}{7 \, \sqrt{\pi } b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*erf(b*x),x, algorithm="giac")

[Out]

1/7*x^7*erf(b*x) + 1/7*(b^6*x^6 + 3*b^4*x^4 + 6*b^2*x^2 + 6)*e^(-b^2*x^2)/(sqrt(pi)*b^7)