\(\int e^{c+d x^2} x \text {erfc}(a+b x) \, dx\) [190]

Optimal result

Integrand size = 17, antiderivative size = 86 \[ \int e^{c+d x^2} x \text {erfc}(a+b x) \, dx=\frac {b e^{c+\frac {a^2 d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )}{2 \sqrt {b^2-d} d}+\frac {e^{c+d x^2} \text {erfc}(a+b x)}{2 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {6518, 2266, 2236} \[ \int e^{c+d x^2} x \text {erfc}(a+b x) \, dx=\frac {b e^{\frac {a^2 d}{b^2-d}+c} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 d \sqrt {b^2-d}}+\frac {e^{c+d x^2} \text {erfc}(a+b x)}{2 d} \]

[In]

Int[E^(c + d*x^2)*x*Erfc[a + b*x],x]

[Out]

(b*E^(c + (a^2*d)/(b^2 - d))*Erf[(a*b + (b^2 - d)*x)/Sqrt[b^2 - d]])/(2*Sqrt[b^2 - d]*d) + (E^(c + d*x^2)*Erfc
[a + 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 2266

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[F^(a - b^2/(4*c)), Int[F^((b + 2*c*x)^2/(4*c))
, x], x] /; FreeQ[{F, a, b, c}, x]

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}(a+b x)}{2 d}+\frac {b \int e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2} \, dx}{d \sqrt {\pi }} \\ & = \frac {e^{c+d x^2} \text {erfc}(a+b x)}{2 d}+\frac {\left (b e^{\frac {b^2 c+a^2 d-c d}{b^2-d}}\right ) \int \exp \left (\frac {\left (-2 a b+2 \left (-b^2+d\right ) x\right )^2}{4 \left (-b^2+d\right )}\right ) \, dx}{d \sqrt {\pi }} \\ & = \frac {b e^{\frac {b^2 c+a^2 d-c d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )}{2 \sqrt {b^2-d} d}+\frac {e^{c+d x^2} \text {erfc}(a+b x)}{2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.94 \[ \int e^{c+d x^2} x \text {erfc}(a+b x) \, dx=\frac {e^c \left (e^{d x^2} \text {erfc}(a+b x)+\frac {b e^{\frac {a^2 d}{b^2-d}} \text {erfi}\left (\frac {-a b+\left (-b^2+d\right ) x}{\sqrt {-b^2+d}}\right )}{\sqrt {-b^2+d}}\right )}{2 d} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(178\) vs. \(2(76)=152\).

Time = 0.90 (sec) , antiderivative size = 179, normalized size of antiderivative = 2.08

method	result	size
default	\(\frac {\frac {b \,{\mathrm e}^{\frac {a^{2} d -2 d a \left (b x +a \right )+b^{2} c +d \left (b x +a \right )^{2}}{b^{2}}}}{2 d}-\frac {\operatorname {erf}\left (b x +a \right ) b \,{\mathrm e}^{\frac {a^{2} d -2 d a \left (b x +a \right )+b^{2} c +d \left (b x +a \right )^{2}}{b^{2}}}}{2 d}+\frac {b \,{\mathrm e}^{\frac {a^{2} d +b^{2} c}{b^{2}}-\frac {a^{2} d^{2}}{b^{4} \left (-1+\frac {d}{b^{2}}\right )}} \operatorname {erf}\left (\sqrt {1-\frac {d}{b^{2}}}\, \left (b x +a \right )+\frac {a d}{b^{2} \sqrt {1-\frac {d}{b^{2}}}}\right )}{2 d \sqrt {1-\frac {d}{b^{2}}}}}{b}\)	\(179\)

[In]

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

[Out]

(1/2*b*exp((a^2*d-2*d*a*(b*x+a)+b^2*c+d*(b*x+a)^2)/b^2)/d-1/2*erf(b*x+a)*b*exp((a^2*d-2*d*a*(b*x+a)+b^2*c+d*(b
*x+a)^2)/b^2)/d+1/2*b/d*exp((a^2*d+b^2*c)/b^2-1/b^4*a^2*d^2/(-1+d/b^2))/(1-d/b^2)^(1/2)*erf((1-d/b^2)^(1/2)*(b
*x+a)+1/b^2*a*d/(1-d/b^2)^(1/2)))/b

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.26 \[ \int e^{c+d x^2} x \text {erfc}(a+b x) \, dx=\frac {\sqrt {b^{2} - d} b \operatorname {erf}\left (\frac {a b + {\left (b^{2} - d\right )} x}{\sqrt {b^{2} - d}}\right ) e^{\left (\frac {b^{2} c + {\left (a^{2} - c\right )} d}{b^{2} - d}\right )} + {\left (b^{2} - {\left (b^{2} - d\right )} \operatorname {erf}\left (b x + a\right ) - d\right )} e^{\left (d x^{2} + c\right )}}{2 \, {\left (b^{2} d - d^{2}\right )}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(x*erfc(a + b*x)*exp(c + d*x^2),x)

[Out]

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