3.214 \(\int x^6 \text{Erfi}(b x) \, dx\)

Optimal. Leaf size=105 \[ -\frac{x^6 e^{b^2 x^2}}{7 \sqrt{\pi } b}+\frac{3 x^4 e^{b^2 x^2}}{7 \sqrt{\pi } b^3}-\frac{6 x^2 e^{b^2 x^2}}{7 \sqrt{\pi } b^5}+\frac{6 e^{b^2 x^2}}{7 \sqrt{\pi } b^7}+\frac{1}{7} x^7 \text{Erfi}(b x) \]

[Out]

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

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Rubi [A]  time = 0.0859314, antiderivative size = 105, 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 = {6363, 2212, 2209} \[ -\frac{x^6 e^{b^2 x^2}}{7 \sqrt{\pi } b}+\frac{3 x^4 e^{b^2 x^2}}{7 \sqrt{\pi } b^3}-\frac{6 x^2 e^{b^2 x^2}}{7 \sqrt{\pi } b^5}+\frac{6 e^{b^2 x^2}}{7 \sqrt{\pi } b^7}+\frac{1}{7} x^7 \text{Erfi}(b x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6363

Int[Erfi[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m + 1)*Erfi[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{erfi}(b x) \, dx &=\frac{1}{7} x^7 \text{erfi}(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{erfi}(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{erfi}(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{erfi}(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{erfi}(b x)\\ \end{align*}

Mathematica [A]  time = 0.0363018, size = 57, normalized size = 0.54 \[ \frac{1}{7} \left (\frac{e^{b^2 x^2} \left (-b^6 x^6+3 b^4 x^4-6 b^2 x^2+6\right )}{\sqrt{\pi } b^7}+x^7 \text{Erfi}(b x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.041, size = 82, normalized size = 0.8 \begin{align*}{\frac{1}{{b}^{7}} \left ({\frac{{b}^{7}{x}^{7}{\it erfi} \left ( bx \right ) }{7}}-{\frac{2}{7\,\sqrt{\pi }} \left ({\frac{{b}^{6}{x}^{6}{{\rm e}^{{b}^{2}{x}^{2}}}}{2}}-{\frac{3\,{{\rm e}^{{b}^{2}{x}^{2}}}{b}^{4}{x}^{4}}{2}}+3\,{b}^{2}{x}^{2}{{\rm e}^{{b}^{2}{x}^{2}}}-3\,{{\rm e}^{{b}^{2}{x}^{2}}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.00238, size = 69, normalized size = 0.66 \begin{align*} \frac{1}{7} \, x^{7} \operatorname{erfi}\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*erfi(b*x),x, algorithm="maxima")

[Out]

1/7*x^7*erfi(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 = 2.34847, size = 132, normalized size = 1.26 \begin{align*} \frac{\pi b^{7} x^{7} \operatorname{erfi}\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*erfi(b*x),x, algorithm="fricas")

[Out]

1/7*(pi*b^7*x^7*erfi(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 = 6.57559, size = 99, normalized size = 0.94 \begin{align*} \begin{cases} \frac{x^{7} \operatorname{erfi}{\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*erfi(b*x),x)

[Out]

Piecewise((x**7*erfi(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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{6} \operatorname{erfi}\left (b x\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^6*erfi(b*x), x)