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]

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

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Rubi [A]  time = 0.03, antiderivative size = 37, normalized size of antiderivative = 1.00, 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 8

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

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]

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*}

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Mathematica [A]  time = 0.03, 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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fricas [A]  time = 0.58, size = 35, normalized size = 0.95 \[ -\frac {2 \, \sqrt {\pi } b x e^{c} - \pi \operatorname {erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{2 \, \pi b^{2}} \]

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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giac [A]  time = 0.28, size = 31, normalized size = 0.84 \[ -\frac {x e^{c}}{\sqrt {\pi } b} + \frac {\operatorname {erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{2 \, b^{2}} \]

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

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maple [A]  time = 0.10, size = 51, normalized size = 1.38 \[ \frac {-2 \,{\mathrm e}^{b^{2} x^{2}+c} {\mathrm e}^{-b^{2} x^{2}} x b +{\mathrm e}^{b^{2} x^{2}+c} \erf \left (b x \right ) \sqrt {\pi }}{2 \sqrt {\pi }\, b^{2}} \]

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))/Pi^(1/2)/b^2

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maxima [A]  time = 0.32, size = 34, normalized size = 0.92 \[ -\frac {2 \, b x e^{c} - \sqrt {\pi } \operatorname {erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{2 \, \sqrt {\pi } b^{2}} \]

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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mupad [B]  time = 0.10, size = 31, normalized size = 0.84 \[ \frac {{\mathrm {e}}^{b^2\,x^2}\,{\mathrm {e}}^c\,\mathrm {erf}\left (b\,x\right )}{2\,b^2}-\frac {x\,{\mathrm {e}}^c}{b\,\sqrt {\pi }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*exp(c + b^2*x^2)*erf(b*x),x)

[Out]

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

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sympy [A]  time = 14.84, size = 34, normalized size = 0.92 \[ \begin {cases} - \frac {x e^{c}}{\sqrt {\pi } b} + \frac {e^{c} e^{b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{2 b^{2}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]

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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