\(\int e^{c+d x^2} x^3 \text {erfc}(b x) \, dx\) [157]

Optimal result

Integrand size = 17, antiderivative size = 155 \[ \int e^{c+d x^2} x^3 \text {erfc}(b x) \, dx=-\frac {b e^{c-\left (b^2-d\right ) x^2} x}{2 \left (b^2-d\right ) d \sqrt {\pi }}-\frac {b e^c \text {erf}\left (\sqrt {b^2-d} x\right )}{2 \sqrt {b^2-d} d^2}+\frac {b e^c \text {erf}\left (\sqrt {b^2-d} x\right )}{4 \left (b^2-d\right )^{3/2} d}-\frac {e^{c+d x^2} \text {erfc}(b x)}{2 d^2}+\frac {e^{c+d x^2} x^2 \text {erfc}(b x)}{2 d} \] Output:

-1/2*b*exp(c-(b^2-d)*x^2)*x/(b^2-d)/d/Pi^(1/2)-1/2*b*exp(c)*erf((b^2-d)^(1 
/2)*x)/(b^2-d)^(1/2)/d^2+1/4*b*exp(c)*erf((b^2-d)^(1/2)*x)/(b^2-d)^(3/2)/d 
-1/2*exp(d*x^2+c)*erfc(b*x)/d^2+1/2*exp(d*x^2+c)*x^2*erfc(b*x)/d
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.64 \[ \int e^{c+d x^2} x^3 \text {erfc}(b x) \, dx=\frac {e^c \left (\frac {2 b d e^{\left (-b^2+d\right ) x^2} x}{\left (-b^2+d\right ) \sqrt {\pi }}+2 e^{d x^2} \left (-1+d x^2\right ) \text {erfc}(b x)+\frac {\left (2 b^3-3 b d\right ) \text {erfi}\left (\sqrt {-b^2+d} x\right )}{\left (-b^2+d\right )^{3/2}}\right )}{4 d^2} \] Input:

Integrate[E^(c + d*x^2)*x^3*Erfc[b*x],x]
 

Output:

(E^c*((2*b*d*E^((-b^2 + d)*x^2)*x)/((-b^2 + d)*Sqrt[Pi]) + 2*E^(d*x^2)*(-1 
 + d*x^2)*Erfc[b*x] + ((2*b^3 - 3*b*d)*Erfi[Sqrt[-b^2 + d]*x])/(-b^2 + d)^ 
(3/2)))/(4*d^2)
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {6940, 2641, 2634, 6937, 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.

\(\displaystyle \int x^3 \text {erfc}(b x) e^{c+d x^2} \, dx\)

\(\Big \downarrow \) 6940

\(\displaystyle \frac {b \int e^{c-\left (b^2-d\right ) x^2} x^2dx}{\sqrt {\pi } d}-\frac {\int e^{d x^2+c} x \text {erfc}(b x)dx}{d}+\frac {x^2 \text {erfc}(b x) e^{c+d x^2}}{2 d}\)

\(\Big \downarrow \) 2641

\(\displaystyle \frac {b \left (\frac {\int e^{c-\left (b^2-d\right ) x^2}dx}{2 \left (b^2-d\right )}-\frac {x e^{c-x^2 \left (b^2-d\right )}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {\int e^{d x^2+c} x \text {erfc}(b x)dx}{d}+\frac {x^2 \text {erfc}(b x) e^{c+d x^2}}{2 d}\)

\(\Big \downarrow \) 2634

\(\displaystyle -\frac {\int e^{d x^2+c} x \text {erfc}(b x)dx}{d}+\frac {b \left (\frac {\sqrt {\pi } e^c \text {erf}\left (x \sqrt {b^2-d}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{c-x^2 \left (b^2-d\right )}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}+\frac {x^2 \text {erfc}(b x) e^{c+d x^2}}{2 d}\)

\(\Big \downarrow \) 6937

\(\displaystyle -\frac {\frac {b \int e^{c-\left (b^2-d\right ) x^2}dx}{\sqrt {\pi } d}+\frac {\text {erfc}(b x) e^{c+d x^2}}{2 d}}{d}+\frac {b \left (\frac {\sqrt {\pi } e^c \text {erf}\left (x \sqrt {b^2-d}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{c-x^2 \left (b^2-d\right )}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}+\frac {x^2 \text {erfc}(b x) e^{c+d x^2}}{2 d}\)

\(\Big \downarrow \) 2634

\(\displaystyle -\frac {\frac {b e^c \text {erf}\left (x \sqrt {b^2-d}\right )}{2 d \sqrt {b^2-d}}+\frac {\text {erfc}(b x) e^{c+d x^2}}{2 d}}{d}+\frac {b \left (\frac {\sqrt {\pi } e^c \text {erf}\left (x \sqrt {b^2-d}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{c-x^2 \left (b^2-d\right )}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}+\frac {x^2 \text {erfc}(b x) e^{c+d x^2}}{2 d}\)

Input:

Int[E^(c + d*x^2)*x^3*Erfc[b*x],x]
 

Output:

(b*(-1/2*(E^(c - (b^2 - d)*x^2)*x)/(b^2 - d) + (E^c*Sqrt[Pi]*Erf[Sqrt[b^2 
- d]*x])/(4*(b^2 - d)^(3/2))))/(d*Sqrt[Pi]) + (E^(c + d*x^2)*x^2*Erfc[b*x] 
)/(2*d) - ((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))/d
 

Defintions of rubi rules used

rule 2634

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]
 

rule 2641

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_ 
.), x_Symbol] :> Simp[(c + d*x)^(m - n + 1)*(F^(a + b*(c + d*x)^n)/(b*d*n*L 
og[F])), x] - Simp[(m - n + 1)/(b*n*Log[F])   Int[(c + d*x)^(m - n)*F^(a + 
b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2*((m + 1)/ 
n)] && LtQ[0, (m + 1)/n, 5] && IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n 
, 0])
 

rule 6937

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]
 

rule 6940

Int[E^((c_.) + (d_.)*(x_)^2)*Erfc[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] 
:> Simp[x^(m - 1)*E^(c + d*x^2)*(Erfc[a + b*x]/(2*d)), x] + (-Simp[(m - 1)/ 
(2*d)   Int[x^(m - 2)*E^(c + d*x^2)*Erfc[a + b*x], x], x] + Simp[b/(d*Sqrt[ 
Pi])   Int[x^(m - 1)*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2), x], x]) /; Fre 
eQ[{a, b, c, d}, x] && IGtQ[m, 1]
 

Maple [A] (verified)

Time = 1.38 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.33

method	result	size
default	\(\frac {\frac {{\mathrm e}^{c} \left (\frac {b^{4} x^{2} {\mathrm e}^{d \,x^{2}}}{2 d}-\frac {b^{4} {\mathrm e}^{d \,x^{2}}}{2 d^{2}}\right )}{b^{3}}-\frac {\operatorname {erf}\left (b x \right ) {\mathrm e}^{c} \left (\frac {b^{4} x^{2} {\mathrm e}^{d \,x^{2}}}{2 d}-\frac {b^{4} {\mathrm e}^{d \,x^{2}}}{2 d^{2}}\right )}{b^{3}}+\frac {{\mathrm e}^{c} \left (\frac {b^{2} \left (\frac {b x \,{\mathrm e}^{\left (-1+\frac {d}{b^{2}}\right ) b^{2} x^{2}}}{-2+\frac {2 d}{b^{2}}}-\frac {\sqrt {\pi }\, \operatorname {erf}\left (\sqrt {1-\frac {d}{b^{2}}}\, b x \right )}{4 \left (-1+\frac {d}{b^{2}}\right ) \sqrt {1-\frac {d}{b^{2}}}}\right )}{d}-\frac {b^{4} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {1-\frac {d}{b^{2}}}\, b x \right )}{2 d^{2} \sqrt {1-\frac {d}{b^{2}}}}\right )}{\sqrt {\pi }\, b^{3}}}{b}\)	\(206\)

Input:

int(exp(d*x^2+c)*x^3*erfc(b*x),x,method=_RETURNVERBOSE)
 

Output:

(1/b^3*exp(c)*(1/2/d*b^4*x^2*exp(d*x^2)-1/2/d^2*b^4*exp(d*x^2))-erf(b*x)/b 
^3*exp(c)*(1/2/d*b^4*x^2*exp(d*x^2)-1/2/d^2*b^4*exp(d*x^2))+1/Pi^(1/2)/b^3 
*exp(c)*(1/d*b^2*(1/2/(-1+d/b^2)*b*x*exp((-1+d/b^2)*b^2*x^2)-1/4/(-1+d/b^2 
)*Pi^(1/2)/(1-d/b^2)^(1/2)*erf((1-d/b^2)^(1/2)*b*x))-1/2/d^2*b^4*Pi^(1/2)/ 
(1-d/b^2)^(1/2)*erf((1-d/b^2)^(1/2)*b*x)))/b
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.23 \[ \int e^{c+d x^2} x^3 \text {erfc}(b x) \, dx=-\frac {\pi {\left (2 \, b^{3} - 3 \, b d\right )} \sqrt {b^{2} - d} \operatorname {erf}\left (\sqrt {b^{2} - d} x\right ) e^{c} + 2 \, \sqrt {\pi } {\left (b^{3} d - b d^{2}\right )} x e^{\left (-b^{2} x^{2} + d x^{2} + c\right )} - 2 \, {\left (\pi {\left (b^{4} d - 2 \, b^{2} d^{2} + d^{3}\right )} x^{2} - \pi {\left (b^{4} - 2 \, b^{2} d + d^{2}\right )} - {\left (\pi {\left (b^{4} d - 2 \, b^{2} d^{2} + d^{3}\right )} x^{2} - \pi {\left (b^{4} - 2 \, b^{2} d + d^{2}\right )}\right )} \operatorname {erf}\left (b x\right )\right )} e^{\left (d x^{2} + c\right )}}{4 \, \pi {\left (b^{4} d^{2} - 2 \, b^{2} d^{3} + d^{4}\right )}} \] Input:

integrate(exp(d*x^2+c)*x^3*erfc(b*x),x, algorithm="fricas")
 

Output:

-1/4*(pi*(2*b^3 - 3*b*d)*sqrt(b^2 - d)*erf(sqrt(b^2 - d)*x)*e^c + 2*sqrt(p 
i)*(b^3*d - b*d^2)*x*e^(-b^2*x^2 + d*x^2 + c) - 2*(pi*(b^4*d - 2*b^2*d^2 + 
 d^3)*x^2 - pi*(b^4 - 2*b^2*d + d^2) - (pi*(b^4*d - 2*b^2*d^2 + d^3)*x^2 - 
 pi*(b^4 - 2*b^2*d + d^2))*erf(b*x))*e^(d*x^2 + c))/(pi*(b^4*d^2 - 2*b^2*d 
^3 + d^4))
 

Sympy [F]

\[ \int e^{c+d x^2} x^3 \text {erfc}(b x) \, dx=e^{c} \int x^{3} e^{d x^{2}} \operatorname {erfc}{\left (b x \right )}\, dx \] Input:

integrate(exp(d*x**2+c)*x**3*erfc(b*x),x)
 

Output:

exp(c)*Integral(x**3*exp(d*x**2)*erfc(b*x), x)
 

Maxima [F]

\[ \int e^{c+d x^2} x^3 \text {erfc}(b x) \, dx=\int { x^{3} \operatorname {erfc}\left (b x\right ) e^{\left (d x^{2} + c\right )} \,d x } \] Input:

integrate(exp(d*x^2+c)*x^3*erfc(b*x),x, algorithm="maxima")
 

Output:

integrate(x^3*erfc(b*x)*e^(d*x^2 + c), x)
 

Giac [F]

\[ \int e^{c+d x^2} x^3 \text {erfc}(b x) \, dx=\int { x^{3} \operatorname {erfc}\left (b x\right ) e^{\left (d x^{2} + c\right )} \,d x } \] Input:

integrate(exp(d*x^2+c)*x^3*erfc(b*x),x, algorithm="giac")
 

Output:

integrate(x^3*erfc(b*x)*e^(d*x^2 + c), x)
 

Mupad [F(-1)]

Timed out. \[ \int e^{c+d x^2} x^3 \text {erfc}(b x) \, dx=\int x^3\,{\mathrm {e}}^{d\,x^2+c}\,\mathrm {erfc}\left (b\,x\right ) \,d x \] Input:

int(x^3*exp(c + d*x^2)*erfc(b*x),x)
 

Output:

int(x^3*exp(c + d*x^2)*erfc(b*x), x)
 

Reduce [F]

\[ \int e^{c+d x^2} x^3 \text {erfc}(b x) \, dx=\frac {e^{c} \left (-e^{b^{2} x^{2}+d \,x^{2}} \mathrm {erf}\left (b x \right ) b^{2} d \pi \,x^{2}+e^{b^{2} x^{2}+d \,x^{2}} \mathrm {erf}\left (b x \right ) b^{2} \pi +e^{b^{2} x^{2}+d \,x^{2}} \mathrm {erf}\left (b x \right ) d^{2} \pi \,x^{2}-e^{b^{2} x^{2}+d \,x^{2}} \mathrm {erf}\left (b x \right ) d \pi +e^{b^{2} x^{2}+d \,x^{2}} b^{2} d \pi \,x^{2}-e^{b^{2} x^{2}+d \,x^{2}} b^{2} \pi -e^{b^{2} x^{2}+d \,x^{2}} d^{2} \pi \,x^{2}+e^{b^{2} x^{2}+d \,x^{2}} d \pi -2 \sqrt {\pi }\, e^{b^{2} x^{2}} \left (\int \frac {e^{d \,x^{2}}}{e^{b^{2} x^{2}}}d x \right ) b^{3}+3 \sqrt {\pi }\, e^{b^{2} x^{2}} \left (\int \frac {e^{d \,x^{2}}}{e^{b^{2} x^{2}}}d x \right ) b d -\sqrt {\pi }\, e^{d \,x^{2}} b d x \right )}{2 e^{b^{2} x^{2}} d^{2} \pi \left (b^{2}-d \right )} \] Input:

int(exp(d*x^2+c)*x^3*erfc(b*x),x)
 

Output:

(e**c*( - e**(b**2*x**2 + d*x**2)*erf(b*x)*b**2*d*pi*x**2 + e**(b**2*x**2 
+ d*x**2)*erf(b*x)*b**2*pi + e**(b**2*x**2 + d*x**2)*erf(b*x)*d**2*pi*x**2 
 - e**(b**2*x**2 + d*x**2)*erf(b*x)*d*pi + e**(b**2*x**2 + d*x**2)*b**2*d* 
pi*x**2 - e**(b**2*x**2 + d*x**2)*b**2*pi - e**(b**2*x**2 + d*x**2)*d**2*p 
i*x**2 + e**(b**2*x**2 + d*x**2)*d*pi - 2*sqrt(pi)*e**(b**2*x**2)*int(e**( 
d*x**2)/e**(b**2*x**2),x)*b**3 + 3*sqrt(pi)*e**(b**2*x**2)*int(e**(d*x**2) 
/e**(b**2*x**2),x)*b*d - sqrt(pi)*e**(d*x**2)*b*d*x))/(2*e**(b**2*x**2)*d* 
*2*pi*(b**2 - d))