Integrand size = 17, antiderivative size = 86 \[ \int e^{c+d x^2} x \text {erfc}(a+b x) \, dx=\frac {b e^{c+\frac {a^2 d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )}{2 \sqrt {b^2-d} d}+\frac {e^{c+d x^2} \text {erfc}(a+b x)}{2 d} \] Output:
1/2*b*exp(c+a^2*d/(b^2-d))*erf((a*b+(b^2-d)*x)/(b^2-d)^(1/2))/(b^2-d)^(1/2 )/d+1/2*exp(d*x^2+c)*erfc(b*x+a)/d
Time = 0.09 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.94 \[ \int e^{c+d x^2} x \text {erfc}(a+b x) \, dx=\frac {e^c \left (e^{d x^2} \text {erfc}(a+b x)+\frac {b e^{\frac {a^2 d}{b^2-d}} \text {erfi}\left (\frac {-a b+\left (-b^2+d\right ) x}{\sqrt {-b^2+d}}\right )}{\sqrt {-b^2+d}}\right )}{2 d} \] Input:
Integrate[E^(c + d*x^2)*x*Erfc[a + b*x],x]
Output:
(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)
Time = 0.33 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.10, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {6937, 2664, 2634}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x e^{c+d x^2} \text {erfc}(a+b x) \, dx\) |
\(\Big \downarrow \) 6937 |
\(\displaystyle \frac {b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c}dx}{\sqrt {\pi } d}+\frac {e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\) |
\(\Big \downarrow \) 2664 |
\(\displaystyle \frac {b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \int e^{-\frac {\left (a b+\left (b^2-d\right ) x\right )^2}{b^2-d}}dx}{\sqrt {\pi } d}+\frac {e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\) |
\(\Big \downarrow \) 2634 |
\(\displaystyle \frac {b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 d \sqrt {b^2-d}}+\frac {e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\) |
Input:
Int[E^(c + d*x^2)*x*Erfc[a + b*x],x]
Output:
(b*E^((b^2*c + a^2*d - c*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)
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] /; Fr eeQ[{F, a, b, c, d}, x] && NegQ[b]
Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[F^(a - b^2/ (4*c)) Int[F^((b + 2*c*x)^2/(4*c)), x], x] /; FreeQ[{F, a, b, c}, x]
Int[E^((c_.) + (d_.)*(x_)^2)*Erfc[(a_.) + (b_.)*(x_)]*(x_), x_Symbol] :> Si mp[E^(c + d*x^2)*(Erfc[a + b*x]/(2*d)), x] + Simp[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]
Leaf count of result is larger than twice the leaf count of optimal. \(178\) vs. \(2(76)=152\).
Time = 0.83 (sec) , antiderivative size = 179, normalized size of antiderivative = 2.08
method | result | size |
default | \(\frac {\frac {b \,{\mathrm e}^{\frac {d \,a^{2}-2 d a \left (b x +a \right )+b^{2} c +d \left (b x +a \right )^{2}}{b^{2}}}}{2 d}-\frac {\operatorname {erf}\left (b x +a \right ) b \,{\mathrm e}^{\frac {d \,a^{2}-2 d a \left (b x +a \right )+b^{2} c +d \left (b x +a \right )^{2}}{b^{2}}}}{2 d}+\frac {b \,{\mathrm e}^{\frac {d \,a^{2}+b^{2} c}{b^{2}}-\frac {a^{2} d^{2}}{b^{4} \left (-1+\frac {d}{b^{2}}\right )}} \operatorname {erf}\left (\sqrt {1-\frac {d}{b^{2}}}\, \left (b x +a \right )+\frac {a d}{b^{2} \sqrt {1-\frac {d}{b^{2}}}}\right )}{2 d \sqrt {1-\frac {d}{b^{2}}}}}{b}\) | \(179\) |
Input:
int(exp(d*x^2+c)*x*erfc(b*x+a),x,method=_RETURNVERBOSE)
Output:
(1/2*b*exp((d*a^2-2*d*a*(b*x+a)+b^2*c+d*(b*x+a)^2)/b^2)/d-1/2*erf(b*x+a)*b *exp((d*a^2-2*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
Time = 0.10 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.26 \[ \int e^{c+d x^2} x \text {erfc}(a+b x) \, dx=\frac {\sqrt {b^{2} - d} b \operatorname {erf}\left (\frac {a b + {\left (b^{2} - d\right )} x}{\sqrt {b^{2} - d}}\right ) e^{\left (\frac {b^{2} c + {\left (a^{2} - c\right )} d}{b^{2} - d}\right )} + {\left (b^{2} - {\left (b^{2} - d\right )} \operatorname {erf}\left (b x + a\right ) - d\right )} e^{\left (d x^{2} + c\right )}}{2 \, {\left (b^{2} d - d^{2}\right )}} \] Input:
integrate(exp(d*x^2+c)*x*erfc(b*x+a),x, algorithm="fricas")
Output:
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)
\[ \int e^{c+d x^2} x \text {erfc}(a+b x) \, dx=e^{c} \int x e^{d x^{2}} \operatorname {erfc}{\left (a + b x \right )}\, dx \] Input:
integrate(exp(d*x**2+c)*x*erfc(b*x+a),x)
Output:
exp(c)*Integral(x*exp(d*x**2)*erfc(a + b*x), x)
\[ \int e^{c+d x^2} x \text {erfc}(a+b x) \, dx=\int { x \operatorname {erfc}\left (b x + a\right ) e^{\left (d x^{2} + c\right )} \,d x } \] Input:
integrate(exp(d*x^2+c)*x*erfc(b*x+a),x, algorithm="maxima")
Output:
integrate(x*erfc(b*x + a)*e^(d*x^2 + c), x)
\[ \int e^{c+d x^2} x \text {erfc}(a+b x) \, dx=\int { x \operatorname {erfc}\left (b x + a\right ) e^{\left (d x^{2} + c\right )} \,d x } \] Input:
integrate(exp(d*x^2+c)*x*erfc(b*x+a),x, algorithm="giac")
Output:
integrate(x*erfc(b*x + a)*e^(d*x^2 + c), x)
Timed out. \[ \int e^{c+d x^2} x \text {erfc}(a+b x) \, dx=\int x\,\mathrm {erfc}\left (a+b\,x\right )\,{\mathrm {e}}^{d\,x^2+c} \,d x \] Input:
int(x*erfc(a + b*x)*exp(c + d*x^2),x)
Output:
int(x*erfc(a + b*x)*exp(c + d*x^2), x)
\[ \int e^{c+d x^2} x \text {erfc}(a+b x) \, dx=\frac {e^{c} \left (e^{d \,x^{2}}-2 \left (\int e^{d \,x^{2}} \mathrm {erf}\left (b x +a \right ) x d x \right ) d \right )}{2 d} \] Input:
int(exp(d*x^2+c)*x*erfc(b*x+a),x)
Output:
(e**c*(e**(d*x**2) - 2*int(e**(d*x**2)*erf(a + b*x)*x,x)*d))/(2*d)