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

Optimal. Leaf size=53 \[ \frac{\text{Erfi}(b x) e^{c+d x^2}}{2 d}-\frac{b e^c \text{Erfi}\left (x \sqrt{b^2+d}\right )}{2 d \sqrt{b^2+d}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0391289, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {6384, 2204} \[ \frac{\text{Erfi}(b x) e^{c+d x^2}}{2 d}-\frac{b e^c \text{Erfi}\left (x \sqrt{b^2+d}\right )}{2 d \sqrt{b^2+d}} \]

Antiderivative was successfully verified.

[In]

Int[E^(c + d*x^2)*x*Erfi[b*x],x]

[Out]

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

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

Mathematica [A]  time = 0.017985, size = 47, normalized size = 0.89 \[ \frac{e^c \left (e^{d x^2} \text{Erfi}(b x)-\frac{b \text{Erfi}\left (x \sqrt{b^2+d}\right )}{\sqrt{b^2+d}}\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.237, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{d{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(d*x^2+c)*x*erfi(b*x),x)

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.70942, size = 135, normalized size = 2.55 \begin{align*} \frac{\sqrt{-b^{2} - d} b \operatorname{erf}\left (\sqrt{-b^{2} - d} x\right ) e^{c} +{\left (b^{2} + d\right )} \operatorname{erfi}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{2 \,{\left (b^{2} d + d^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(sqrt(-b^2 - d)*b*erf(sqrt(-b^2 - d)*x)*e^c + (b^2 + d)*erfi(b*x)*e^(d*x^2 + c))/(b^2*d + d^2)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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