3.285 \(\int e^{c+b^2 x^2} x \text{Erfi}(b x) \, dx\)

Optimal. Leaf size=47 \[ \frac{e^{b^2 x^2+c} \text{Erfi}(b x)}{2 b^2}-\frac{e^c \text{Erfi}\left (\sqrt{2} b x\right )}{2 \sqrt{2} b^2} \]

[Out]

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

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Rubi [A]  time = 0.0380216, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {6384, 2204} \[ \frac{e^{b^2 x^2+c} \text{Erfi}(b x)}{2 b^2}-\frac{e^c \text{Erfi}\left (\sqrt{2} b x\right )}{2 \sqrt{2} b^2} \]

Antiderivative was successfully verified.

[In]

Int[E^(c + b^2*x^2)*x*Erfi[b*x],x]

[Out]

(E^(c + b^2*x^2)*Erfi[b*x])/(2*b^2) - (E^c*Erfi[Sqrt[2]*b*x])/(2*Sqrt[2]*b^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 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rubi steps

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

Mathematica [A]  time = 0.009964, size = 42, normalized size = 0.89 \[ \frac{e^c \left (2 e^{b^2 x^2} \text{Erfi}(b x)-\sqrt{2} \text{Erfi}\left (\sqrt{2} b x\right )\right )}{4 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(E^c*(2*E^(b^2*x^2)*Erfi[b*x] - Sqrt[2]*Erfi[Sqrt[2]*b*x]))/(4*b^2)

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Maple [F]  time = 0.227, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{{b}^{2}{x}^{2}+c}}x{\it erfi} \left ( bx \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(b^2*x^2+c)*x*erfi(b*x),x)

[Out]

int(exp(b^2*x^2+c)*x*erfi(b*x),x)

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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} + c\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b^2*x^2+c)*x*erfi(b*x),x, algorithm="maxima")

[Out]

integrate(x*erfi(b*x)*e^(b^2*x^2 + c), x)

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Fricas [A]  time = 2.69679, size = 123, normalized size = 2.62 \begin{align*} \frac{2 \, b \operatorname{erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} - \sqrt{2} \sqrt{b^{2}} \operatorname{erfi}\left (\sqrt{2} \sqrt{b^{2}} x\right ) e^{c}}{4 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b**2*x**2+c)*x*erfi(b*x),x)

[Out]

exp(c)*Integral(x*exp(b**2*x**2)*erfi(b*x), x)

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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} + c\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b^2*x^2+c)*x*erfi(b*x),x, algorithm="giac")

[Out]

integrate(x*erfi(b*x)*e^(b^2*x^2 + c), x)