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

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

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 57, normalized size of antiderivative = 1.00, 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 = {6383, 2205} \[ \frac {b e^c \text {Erf}\left (x \sqrt {b^2-d}\right )}{2 d \sqrt {b^2-d}}+\frac {\text {Erfc}(b x) e^{c+d x^2}}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[E^(c + d*x^2)*x*Erfc[b*x],x]

[Out]

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

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]

Rule 6383

Int[E^((c_.) + (d_.)*(x_)^2)*Erfc[(a_.) + (b_.)*(x_)]*(x_), x_Symbol] :> Simp[(E^(c + d*x^2)*Erfc[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+d x^2} x \text {erfc}(b x) \, dx &=\frac {e^{c+d x^2} \text {erfc}(b x)}{2 d}+\frac {b \int e^{c-\left (b^2-d\right ) x^2} \, dx}{d \sqrt {\pi }}\\ &=\frac {b e^c \text {erf}\left (\sqrt {b^2-d} x\right )}{2 \sqrt {b^2-d} d}+\frac {e^{c+d x^2} \text {erfc}(b x)}{2 d}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 50, normalized size = 0.88 \[ \frac {e^c \left (\frac {b \text {erfi}\left (x \sqrt {d-b^2}\right )}{\sqrt {d-b^2}}+e^{d x^2} \text {erfc}(b x)\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.46, size = 70, normalized size = 1.23 \[ \frac {\sqrt {b^{2} - d} b \operatorname {erf}\left (\sqrt {b^{2} - d} x\right ) e^{c} + {\left (b^{2} - {\left (b^{2} - d\right )} \operatorname {erf}\left (b x\right ) - d\right )} e^{\left (d x^{2} + c\right )}}{2 \, {\left (b^{2} d - d^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \operatorname {erfc}\left (b x\right ) e^{\left (d x^{2} + c\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.21, size = 92, normalized size = 1.61 \[ \frac {\frac {b \,{\mathrm e}^{\frac {b^{2} d \,x^{2}+c \,b^{2}}{b^{2}}}}{2 d}-\frac {\erf \left (b x \right ) b \,{\mathrm e}^{\frac {b^{2} d \,x^{2}+c \,b^{2}}{b^{2}}}}{2 d}+\frac {b \,{\mathrm e}^{c} \erf \left (\sqrt {1-\frac {d}{b^{2}}}\, b x \right )}{2 d \sqrt {1-\frac {d}{b^{2}}}}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1/2*b*exp((b^2*d*x^2+b^2*c)/b^2)/d-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 [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \operatorname {erfc}\left (b x\right ) e^{\left (d x^{2} + c\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int x\,{\mathrm {e}}^{d\,x^2+c}\,\mathrm {erfc}\left (b\,x\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*exp(c + d*x^2)*erfc(b*x),x)

[Out]

int(x*exp(c + d*x^2)*erfc(b*x), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ e^{c} \int x e^{d x^{2}} \operatorname {erfc}{\left (b x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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