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

Optimal. Leaf size=32 \[ \frac{x}{\sqrt{\pi } b}-\frac{e^{-b^2 x^2} \text{Erfi}(b x)}{2 b^2} \]

[Out]

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

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Rubi [A]  time = 0.027782, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {6384, 8} \[ \frac{x}{\sqrt{\pi } b}-\frac{e^{-b^2 x^2} \text{Erfi}(b x)}{2 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

Rubi steps

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

Mathematica [A]  time = 0.0140244, size = 32, normalized size = 1. \[ \frac{x}{\sqrt{\pi } b}-\frac{e^{-b^2 x^2} \text{Erfi}(b x)}{2 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.063, size = 41, normalized size = 1.3 \begin{align*}{\frac{2\,{{\rm e}^{{b}^{2}{x}^{2}}}bx-\sqrt{\pi }{\it erfi} \left ( bx \right ) }{2\,{b}^{2}\sqrt{\pi }{{\rm e}^{{b}^{2}{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*exp(b^2*x^2)*b*x-Pi^(1/2)*erfi(b*x))/Pi^(1/2)/b^2/exp(b^2*x^2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \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*erfi(b*x)/exp(b^2*x^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.43269, size = 96, normalized size = 3. \begin{align*} \frac{{\left (2 \, \sqrt{\pi } b x e^{\left (b^{2} x^{2}\right )} - \pi \operatorname{erfi}\left (b x\right )\right )} e^{\left (-b^{2} x^{2}\right )}}{2 \, \pi b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*sqrt(pi)*b*x*e^(b^2*x^2) - pi*erfi(b*x))*e^(-b^2*x^2)/(pi*b^2)

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Sympy [A]  time = 32.3085, size = 27, normalized size = 0.84 \begin{align*} \begin{cases} \frac{x}{\sqrt{\pi } b} - \frac{e^{- b^{2} x^{2}} \operatorname{erfi}{\left (b x \right )}}{2 b^{2}} & \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*erfi(b*x)/exp(b**2*x**2),x)

[Out]

Piecewise((x/(sqrt(pi)*b) - exp(-b**2*x**2)*erfi(b*x)/(2*b**2), Ne(b, 0)), (0, True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \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*erfi(b*x)/exp(b^2*x^2),x, algorithm="giac")

[Out]

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