3.271 \(\int e^{-b^2 x^2} x^3 \text{Erfi}(b x) \, dx\)

Optimal. Leaf size=71 \[ -\frac{x^2 e^{-b^2 x^2} \text{Erfi}(b x)}{2 b^2}-\frac{e^{-b^2 x^2} \text{Erfi}(b x)}{2 b^4}+\frac{x}{\sqrt{\pi } b^3}+\frac{x^3}{3 \sqrt{\pi } b} \]

[Out]

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

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Rubi [A]  time = 0.0685207, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6387, 6384, 8, 30} \[ -\frac{x^2 e^{-b^2 x^2} \text{Erfi}(b x)}{2 b^2}-\frac{e^{-b^2 x^2} \text{Erfi}(b x)}{2 b^4}+\frac{x}{\sqrt{\pi } b^3}+\frac{x^3}{3 \sqrt{\pi } b} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*Erfi[b*x])/E^(b^2*x^2),x]

[Out]

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

Rule 6387

Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[(x^(m - 1)*E^(c + d*x^2)*Er
fi[a + b*x])/(2*d), x] + (-Dist[(m - 1)/(2*d), Int[x^(m - 2)*E^(c + d*x^2)*Erfi[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]

Rule 6384

Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(a_.) + (b_.)*(x_)]*(x_), x_Symbol] :> Simp[(E^(c + d*x^2)*Erfi[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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.026179, size = 51, normalized size = 0.72 \[ \frac{\frac{2 b x \left (b^2 x^2+3\right )}{\sqrt{\pi }}-3 e^{-b^2 x^2} \left (b^2 x^2+1\right ) \text{Erfi}(b x)}{6 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^3*Erfi[b*x])/E^(b^2*x^2),x]

[Out]

((2*b*x*(3 + b^2*x^2))/Sqrt[Pi] - (3*(1 + b^2*x^2)*Erfi[b*x])/E^(b^2*x^2))/(6*b^4)

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Maple [A]  time = 0.115, size = 72, normalized size = 1. \begin{align*}{\frac{2\,{{\rm e}^{{b}^{2}{x}^{2}}}{b}^{3}{x}^{3}-3\,\sqrt{\pi }{\it erfi} \left ( bx \right ){b}^{2}{x}^{2}+6\,{{\rm e}^{{b}^{2}{x}^{2}}}bx-3\,\sqrt{\pi }{\it erfi} \left ( bx \right ) }{6\,\sqrt{\pi }{b}^{4}{{\rm e}^{{b}^{2}{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*erfi(b*x)/exp(b^2*x^2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \operatorname{erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*erfi(b*x)/exp(b^2*x^2),x, algorithm="maxima")

[Out]

integrate(x^3*erfi(b*x)*e^(-b^2*x^2), x)

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Fricas [A]  time = 2.57583, size = 138, normalized size = 1.94 \begin{align*} \frac{{\left (2 \, \sqrt{\pi }{\left (b^{3} x^{3} + 3 \, b x\right )} e^{\left (b^{2} x^{2}\right )} - 3 \,{\left (\pi + \pi b^{2} x^{2}\right )} \operatorname{erfi}\left (b x\right )\right )} e^{\left (-b^{2} x^{2}\right )}}{6 \, \pi b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*erfi(b*x)/exp(b^2*x^2),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*erfi(b*x)/exp(b**2*x**2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \operatorname{erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*erfi(b*x)/exp(b^2*x^2),x, algorithm="giac")

[Out]

integrate(x^3*erfi(b*x)*e^(-b^2*x^2), x)