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

Optimal. Leaf size=37 \[ \frac{e^{b^2 x^2+c} \text{Erf}(b x)}{2 b^2}-\frac{e^c x}{\sqrt{\pi } b} \]

[Out]

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

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Rubi [A]  time = 0.031491, antiderivative size = 37, 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 = {6382, 8} \[ \frac{e^{b^2 x^2+c} \text{Erf}(b x)}{2 b^2}-\frac{e^c x}{\sqrt{\pi } b} \]

Antiderivative was successfully verified.

[In]

Int[E^(c + b^2*x^2)*x*Erf[b*x],x]

[Out]

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

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 8

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

Rubi steps

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

Mathematica [A]  time = 0.0201227, size = 34, normalized size = 0.92 \[ \frac{e^c \left (e^{b^2 x^2} \text{Erf}(b x)-\frac{2 b x}{\sqrt{\pi }}\right )}{2 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.136, size = 51, normalized size = 1.4 \begin{align*}{\frac{-2\,{{\rm e}^{{b}^{2}{x}^{2}+c}}{{\rm e}^{-{b}^{2}{x}^{2}}}xb+{{\rm e}^{{b}^{2}{x}^{2}+c}}{\it Erf} \left ( bx \right ) \sqrt{\pi }}{2\,{b}^{2}\sqrt{\pi }}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.04374, size = 46, normalized size = 1.24 \begin{align*} -\frac{2 \, b x e^{c} - \sqrt{\pi } \operatorname{erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{2 \, \sqrt{\pi } b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 3.00517, size = 89, normalized size = 2.41 \begin{align*} -\frac{2 \, \sqrt{\pi } b x e^{c} - \pi \operatorname{erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{2 \, \pi b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 26.7039, size = 34, normalized size = 0.92 \begin{align*} \begin{cases} - \frac{x e^{c}}{\sqrt{\pi } b} + \frac{e^{c} e^{b^{2} x^{2}} \operatorname{erf}{\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(exp(b**2*x**2+c)*x*erf(b*x),x)

[Out]

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

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Giac [A]  time = 1.30347, size = 42, normalized size = 1.14 \begin{align*} -\frac{x e^{c}}{\sqrt{\pi } b} + \frac{\operatorname{erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{2 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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