\(\int e^{c+b^2 x^2} x^4 \text {erfc}(b x) \, dx\) [173]

Optimal result

Integrand size = 19, antiderivative size = 138 \[ \int e^{c+b^2 x^2} x^4 \text {erfc}(b x) \, dx=-\frac {3 e^c x^2}{4 b^3 \sqrt {\pi }}+\frac {e^c x^4}{4 b \sqrt {\pi }}-\frac {3 e^{c+b^2 x^2} x \text {erfc}(b x)}{4 b^4}+\frac {e^{c+b^2 x^2} x^3 \text {erfc}(b x)}{2 b^2}+\frac {3 e^c \sqrt {\pi } \text {erfi}(b x)}{8 b^5}-\frac {3 e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{4 b^3 \sqrt {\pi }} \] Output:

-3/4*exp(c)*x^2/b^3/Pi^(1/2)+1/4*exp(c)*x^4/b/Pi^(1/2)-3/4*exp(b^2*x^2+c)* 
x*erfc(b*x)/b^4+1/2*exp(b^2*x^2+c)*x^3*erfc(b*x)/b^2+3/8*exp(c)*Pi^(1/2)*e 
rfi(b*x)/b^5-3/4*exp(c)*x^2*hypergeom([1, 1],[3/2, 2],b^2*x^2)/b^3/Pi^(1/2 
)
 

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.07 \[ \int e^{c+b^2 x^2} x^4 \text {erfc}(b x) \, dx=-\frac {e^c \left (6 b e^{b^2 x^2} \sqrt {\pi } x+6 b^2 x^2-4 b^3 e^{b^2 x^2} \sqrt {\pi } x^3-2 b^4 x^4+2 b e^{b^2 x^2} \sqrt {\pi } x \left (-3+2 b^2 x^2\right ) \text {erf}(b x)-3 \pi \text {erfi}(b x)+3 \pi \text {erf}(b x) \text {erfi}(b x)-6 b^2 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )\right )}{8 b^5 \sqrt {\pi }} \] Input:

Integrate[E^(c + b^2*x^2)*x^4*Erfc[b*x],x]
 

Output:

-1/8*(E^c*(6*b*E^(b^2*x^2)*Sqrt[Pi]*x + 6*b^2*x^2 - 4*b^3*E^(b^2*x^2)*Sqrt 
[Pi]*x^3 - 2*b^4*x^4 + 2*b*E^(b^2*x^2)*Sqrt[Pi]*x*(-3 + 2*b^2*x^2)*Erf[b*x 
] - 3*Pi*Erfi[b*x] + 3*Pi*Erf[b*x]*Erfi[b*x] - 6*b^2*x^2*HypergeometricPFQ 
[{1, 1}, {3/2, 2}, -(b^2*x^2)]))/(b^5*Sqrt[Pi])
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {6940, 15, 6940, 15, 6931, 2633, 6930}

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.

\(\displaystyle \int x^4 e^{b^2 x^2+c} \text {erfc}(b x) \, dx\)

\(\Big \downarrow \) 6940

\(\displaystyle -\frac {3 \int e^{b^2 x^2+c} x^2 \text {erfc}(b x)dx}{2 b^2}+\frac {\int e^c x^3dx}{\sqrt {\pi } b}+\frac {x^3 e^{b^2 x^2+c} \text {erfc}(b x)}{2 b^2}\)

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 6940

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 6931

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

\(\Big \downarrow \) 2633

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

\(\Big \downarrow \) 6930

\(\displaystyle -\frac {3 \left (-\frac {\frac {\sqrt {\pi } e^c \text {erfi}(b x)}{2 b}-\frac {b e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{\sqrt {\pi }}}{2 b^2}+\frac {x e^{b^2 x^2+c} \text {erfc}(b x)}{2 b^2}+\frac {e^c x^2}{2 \sqrt {\pi } b}\right )}{2 b^2}+\frac {x^3 e^{b^2 x^2+c} \text {erfc}(b x)}{2 b^2}+\frac {e^c x^4}{4 \sqrt {\pi } b}\)

Input:

Int[E^(c + b^2*x^2)*x^4*Erfc[b*x],x]
 

Output:

(E^c*x^4)/(4*b*Sqrt[Pi]) + (E^(c + b^2*x^2)*x^3*Erfc[b*x])/(2*b^2) - (3*(( 
E^c*x^2)/(2*b*Sqrt[Pi]) + (E^(c + b^2*x^2)*x*Erfc[b*x])/(2*b^2) - ((E^c*Sq 
rt[Pi]*Erfi[b*x])/(2*b) - (b*E^c*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b 
^2*x^2])/Sqrt[Pi])/(2*b^2)))/(2*b^2)
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 2633

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt 
[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ 
F, a, b, c, d}, x] && PosQ[b]
 

rule 6930

Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(b_.)*(x_)], x_Symbol] :> Simp[b*E^c*(x^2/ 
Sqrt[Pi])*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2], x] /; FreeQ[{b, c, 
d}, x] && EqQ[d, b^2]
 

rule 6931

Int[E^((c_.) + (d_.)*(x_)^2)*Erfc[(b_.)*(x_)], x_Symbol] :> Int[E^(c + d*x^ 
2), x] - Int[E^(c + d*x^2)*Erf[b*x], x] /; FreeQ[{b, c, d}, x] && EqQ[d, b^ 
2]
 

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 [F]

\[\int {\mathrm e}^{b^{2} x^{2}+c} x^{4} \operatorname {erfc}\left (b x \right )d x\]

Input:

int(exp(b^2*x^2+c)*x^4*erfc(b*x),x)
 

Output:

int(exp(b^2*x^2+c)*x^4*erfc(b*x),x)
 

Fricas [F]

\[ \int e^{c+b^2 x^2} x^4 \text {erfc}(b x) \, dx=\int { x^{4} \operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} \,d x } \] Input:

integrate(exp(b^2*x^2+c)*x^4*erfc(b*x),x, algorithm="fricas")
 

Output:

integral(-(x^4*erf(b*x) - x^4)*e^(b^2*x^2 + c), x)
 

Sympy [F(-1)]

Timed out. \[ \int e^{c+b^2 x^2} x^4 \text {erfc}(b x) \, dx=\text {Timed out} \] Input:

integrate(exp(b**2*x**2+c)*x**4*erfc(b*x),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int e^{c+b^2 x^2} x^4 \text {erfc}(b x) \, dx=\int { x^{4} \operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} \,d x } \] Input:

integrate(exp(b^2*x^2+c)*x^4*erfc(b*x),x, algorithm="maxima")
 

Output:

integrate(x^4*erfc(b*x)*e^(b^2*x^2 + c), x)
 

Giac [F]

\[ \int e^{c+b^2 x^2} x^4 \text {erfc}(b x) \, dx=\int { x^{4} \operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} \,d x } \] Input:

integrate(exp(b^2*x^2+c)*x^4*erfc(b*x),x, algorithm="giac")
 

Output:

integrate(x^4*erfc(b*x)*e^(b^2*x^2 + c), x)
 

Mupad [F(-1)]

Timed out. \[ \int e^{c+b^2 x^2} x^4 \text {erfc}(b x) \, dx=\int x^4\,{\mathrm {e}}^{b^2\,x^2+c}\,\mathrm {erfc}\left (b\,x\right ) \,d x \] Input:

int(x^4*exp(c + b^2*x^2)*erfc(b*x),x)
 

Output:

int(x^4*exp(c + b^2*x^2)*erfc(b*x), x)
 

Reduce [F]

\[ \int e^{c+b^2 x^2} x^4 \text {erfc}(b x) \, dx=\frac {e^{c} \left (-3 \sqrt {\pi }\, \mathrm {erf}\left (b i x \right ) i +4 e^{b^{2} x^{2}} b^{3} x^{3}-6 e^{b^{2} x^{2}} b x -8 \left (\int e^{b^{2} x^{2}} \mathrm {erf}\left (b x \right ) x^{4}d x \right ) b^{5}\right )}{8 b^{5}} \] Input:

int(exp(b^2*x^2+c)*x^4*erfc(b*x),x)
 

Output:

(e**c*( - 3*sqrt(pi)*erf(b*i*x)*i + 4*e**(b**2*x**2)*b**3*x**3 - 6*e**(b** 
2*x**2)*b*x - 8*int(e**(b**2*x**2)*erf(b*x)*x**4,x)*b**5))/(8*b**5)