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

Optimal. Leaf size=57 \[ \frac{\text{Erf}(b x) e^{c+d x^2}}{2 d}-\frac{b e^c \text{Erf}\left (x \sqrt{b^2-d}\right )}{2 d \sqrt{b^2-d}} \]

[Out]

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

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Rubi [A]  time = 0.0387532, antiderivative size = 57, 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 = {6382, 2205} \[ \frac{\text{Erf}(b x) e^{c+d x^2}}{2 d}-\frac{b e^c \text{Erf}\left (x \sqrt{b^2-d}\right )}{2 d \sqrt{b^2-d}} \]

Antiderivative was successfully verified.

[In]

Int[E^(c + d*x^2)*x*Erf[b*x],x]

[Out]

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

Rule 6382

Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(a_.) + (b_.)*(x_)]*(x_), x_Symbol] :> Simp[(E^(c + d*x^2)*Erf[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 2205

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

Rubi steps

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

Mathematica [A]  time = 0.0381589, size = 51, normalized size = 0.89 \[ \frac{e^c \left (e^{d x^2} \text{Erf}(b x)-\frac{b \text{Erfi}\left (x \sqrt{d-b^2}\right )}{\sqrt{d-b^2}}\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(c + d*x^2)*x*Erf[b*x],x]

[Out]

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

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Maple [A]  time = 0.333, size = 67, normalized size = 1.2 \begin{align*}{\frac{1}{b} \left ({\frac{{\it Erf} \left ( bx \right ) b}{2\,d}{{\rm e}^{{\frac{{b}^{2}d{x}^{2}+{b}^{2}c}{{b}^{2}}}}}}-{\frac{b{{\rm e}^{c}}}{2\,d}{\it Erf} \left ( \sqrt{1-{\frac{d}{{b}^{2}}}}bx \right ){\frac{1}{\sqrt{1-{\frac{d}{{b}^{2}}}}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1/2*erf(b*x)*b*exp((b^2*d*x^2+b^2*c)/b^2)/d-1/2*b/d*exp(c)/(1-d/b^2)^(1/2)*erf((1-d/b^2)^(1/2)*b*x))/b

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Maxima [A]  time = 1.0571, size = 63, normalized size = 1.11 \begin{align*} -\frac{b \operatorname{erf}\left (\sqrt{b^{2} - d} x\right ) e^{c}}{2 \, \sqrt{b^{2} - d} d} + \frac{\operatorname{erf}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.9042, size = 132, normalized size = 2.32 \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{erf}\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*erf(b*x),x, algorithm="fricas")

[Out]

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

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

[Out]

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

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Giac [A]  time = 1.22563, size = 65, normalized size = 1.14 \begin{align*} \frac{b \operatorname{erf}\left (-\sqrt{b^{2} - d} x\right ) e^{c}}{2 \, \sqrt{b^{2} - d} d} + \frac{\operatorname{erf}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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