\(\int e^{c+d x^2} x \text {erfc}(b x) \, dx\) [158]

Optimal result

Integrand size = 15, antiderivative size = 57 \[ \int e^{c+d x^2} x \text {erfc}(b x) \, dx=\frac {b e^c \text {erf}\left (\sqrt {b^2-d} x\right )}{2 \sqrt {b^2-d} d}+\frac {e^{c+d x^2} \text {erfc}(b x)}{2 d} \]

[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)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {6518, 2236} \[ \int e^{c+d x^2} x \text {erfc}(b x) \, dx=\frac {b e^c \text {erf}\left (x \sqrt {b^2-d}\right )}{2 d \sqrt {b^2-d}}+\frac {\text {erfc}(b x) e^{c+d x^2}}{2 d} \]

[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 2236

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 6518

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*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.88 \[ \int e^{c+d x^2} x \text {erfc}(b x) \, dx=\frac {e^c \left (e^{d x^2} \text {erfc}(b x)+\frac {b \text {erfi}\left (\sqrt {-b^2+d} x\right )}{\sqrt {-b^2+d}}\right )}{2 d} \]

[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)

Maple [A] (verified)

Time = 0.62 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.61

method	result	size
default	\(\frac {\frac {b \,{\mathrm e}^{\frac {b^{2} d \,x^{2}+b^{2} c}{b^{2}}}}{2 d}-\frac {\operatorname {erf}\left (b x \right ) b \,{\mathrm e}^{\frac {b^{2} d \,x^{2}+b^{2} c}{b^{2}}}}{2 d}+\frac {b \,{\mathrm e}^{c} \operatorname {erf}\left (\sqrt {1-\frac {d}{b^{2}}}\, b x \right )}{2 d \sqrt {1-\frac {d}{b^{2}}}}}{b}\)	\(92\)

[In]

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

[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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.23 \[ \int e^{c+d x^2} x \text {erfc}(b x) \, dx=\frac {\sqrt {b^{2} - d} b \operatorname {erf}\left (\sqrt {b^{2} - d} x\right ) e^{c} + {\left (b^{2} - {\left (b^{2} - d\right )} \operatorname {erf}\left (b x\right ) - d\right )} e^{\left (d x^{2} + c\right )}}{2 \, {\left (b^{2} d - d^{2}\right )}} \]

[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)

Sympy [F]

\[ \int e^{c+d x^2} x \text {erfc}(b x) \, dx=e^{c} \int x e^{d x^{2}} \operatorname {erfc}{\left (b x \right )}\, dx \]

[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)

Maxima [F]

\[ \int e^{c+d x^2} x \text {erfc}(b x) \, dx=\int { x \operatorname {erfc}\left (b x\right ) e^{\left (d x^{2} + c\right )} \,d x } \]

[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)

Giac [F]

\[ \int e^{c+d x^2} x \text {erfc}(b x) \, dx=\int { x \operatorname {erfc}\left (b x\right ) e^{\left (d x^{2} + c\right )} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int e^{c+d x^2} x \text {erfc}(b x) \, dx=\int x\,{\mathrm {e}}^{d\,x^2+c}\,\mathrm {erfc}\left (b\,x\right ) \,d x \]

[In]

int(x*exp(c + d*x^2)*erfc(b*x),x)

[Out]

int(x*exp(c + d*x^2)*erfc(b*x), x)