Integrand size = 19, antiderivative size = 80 \[ \int e^{c+b^2 x^2} x^3 \text {erfc}(b x) \, dx=-\frac {e^c x}{b^3 \sqrt {\pi }}+\frac {e^c x^3}{3 b \sqrt {\pi }}-\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{2 b^4}+\frac {e^{c+b^2 x^2} x^2 \text {erfc}(b x)}{2 b^2} \] Output:
-exp(c)*x/b^3/Pi^(1/2)+1/3*exp(c)*x^3/b/Pi^(1/2)-1/2*exp(b^2*x^2+c)*erfc(b *x)/b^4+1/2*exp(b^2*x^2+c)*x^2*erfc(b*x)/b^2
Time = 0.03 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.72 \[ \int e^{c+b^2 x^2} x^3 \text {erfc}(b x) \, dx=\frac {e^c \left (2 b x \left (-3+b^2 x^2\right )+3 e^{b^2 x^2} \sqrt {\pi } \left (-1+b^2 x^2\right ) \text {erfc}(b x)\right )}{6 b^4 \sqrt {\pi }} \] Input:
Integrate[E^(c + b^2*x^2)*x^3*Erfc[b*x],x]
Output:
(E^c*(2*b*x*(-3 + b^2*x^2) + 3*E^(b^2*x^2)*Sqrt[Pi]*(-1 + b^2*x^2)*Erfc[b* x]))/(6*b^4*Sqrt[Pi])
Time = 0.36 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {6940, 15, 6937, 24}
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.
\(\displaystyle \int x^3 e^{b^2 x^2+c} \text {erfc}(b x) \, dx\) |
\(\Big \downarrow \) 6940 |
\(\displaystyle -\frac {\int e^{b^2 x^2+c} x \text {erfc}(b x)dx}{b^2}+\frac {\int e^c x^2dx}{\sqrt {\pi } b}+\frac {x^2 e^{b^2 x^2+c} \text {erfc}(b x)}{2 b^2}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {\int e^{b^2 x^2+c} x \text {erfc}(b x)dx}{b^2}+\frac {x^2 e^{b^2 x^2+c} \text {erfc}(b x)}{2 b^2}+\frac {e^c x^3}{3 \sqrt {\pi } b}\) |
\(\Big \downarrow \) 6937 |
\(\displaystyle -\frac {\frac {\int e^cdx}{\sqrt {\pi } b}+\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{2 b^2}}{b^2}+\frac {x^2 e^{b^2 x^2+c} \text {erfc}(b x)}{2 b^2}+\frac {e^c x^3}{3 \sqrt {\pi } b}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {x^2 e^{b^2 x^2+c} \text {erfc}(b x)}{2 b^2}-\frac {\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{2 b^2}+\frac {e^c x}{\sqrt {\pi } b}}{b^2}+\frac {e^c x^3}{3 \sqrt {\pi } b}\) |
Input:
Int[E^(c + b^2*x^2)*x^3*Erfc[b*x],x]
Output:
(E^c*x^3)/(3*b*Sqrt[Pi]) + (E^(c + b^2*x^2)*x^2*Erfc[b*x])/(2*b^2) - ((E^c *x)/(b*Sqrt[Pi]) + (E^(c + b^2*x^2)*Erfc[b*x])/(2*b^2))/b^2
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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]
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]
Time = 1.19 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.24
method | result | size |
default | \(\frac {\frac {{\mathrm e}^{c} \left (\frac {b^{2} x^{2} {\mathrm e}^{b^{2} x^{2}}}{2}-\frac {{\mathrm e}^{b^{2} x^{2}}}{2}\right )}{b^{3}}-\frac {\operatorname {erf}\left (b x \right ) {\mathrm e}^{c} \left (\frac {b^{2} x^{2} {\mathrm e}^{b^{2} x^{2}}}{2}-\frac {{\mathrm e}^{b^{2} x^{2}}}{2}\right )}{b^{3}}+\frac {{\mathrm e}^{c} \left (\frac {1}{3} b^{3} x^{3}-b x \right )}{\sqrt {\pi }\, b^{3}}}{b}\) | \(99\) |
parallelrisch | \(\frac {2 \,{\mathrm e}^{b^{2} x^{2}+c} {\mathrm e}^{-b^{2} x^{2}} x^{3} b^{3}+3 \,{\mathrm e}^{b^{2} x^{2}+c} x^{2} \operatorname {erfc}\left (b x \right ) b^{2} \sqrt {\pi }-6 \,{\mathrm e}^{b^{2} x^{2}+c} x \,{\mathrm e}^{-b^{2} x^{2}} b -3 \,\operatorname {erfc}\left (b x \right ) {\mathrm e}^{b^{2} x^{2}+c} \sqrt {\pi }}{6 \sqrt {\pi }\, b^{4}}\) | \(104\) |
Input:
int(exp(b^2*x^2+c)*x^3*erfc(b*x),x,method=_RETURNVERBOSE)
Output:
(1/b^3*exp(c)*(1/2*b^2*x^2*exp(b^2*x^2)-1/2*exp(b^2*x^2))-erf(b*x)/b^3*exp (c)*(1/2*b^2*x^2*exp(b^2*x^2)-1/2*exp(b^2*x^2))+1/Pi^(1/2)/b^3*exp(c)*(1/3 *b^3*x^3-b*x))/b
Time = 0.10 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.85 \[ \int e^{c+b^2 x^2} x^3 \text {erfc}(b x) \, dx=\frac {2 \, \sqrt {\pi } {\left (b^{3} x^{3} - 3 \, b x\right )} e^{c} - 3 \, {\left (\pi - \pi b^{2} x^{2} - {\left (\pi - \pi b^{2} x^{2}\right )} \operatorname {erf}\left (b x\right )\right )} e^{\left (b^{2} x^{2} + c\right )}}{6 \, \pi b^{4}} \] Input:
integrate(exp(b^2*x^2+c)*x^3*erfc(b*x),x, algorithm="fricas")
Output:
1/6*(2*sqrt(pi)*(b^3*x^3 - 3*b*x)*e^c - 3*(pi - pi*b^2*x^2 - (pi - pi*b^2* x^2)*erf(b*x))*e^(b^2*x^2 + c))/(pi*b^4)
Time = 6.97 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.04 \[ \int e^{c+b^2 x^2} x^3 \text {erfc}(b x) \, dx=\begin {cases} \frac {x^{3} e^{c}}{3 \sqrt {\pi } b} + \frac {x^{2} e^{c} e^{b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}}{2 b^{2}} - \frac {x e^{c}}{\sqrt {\pi } b^{3}} - \frac {e^{c} e^{b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}}{2 b^{4}} & \text {for}\: b \neq 0 \\\frac {x^{4} e^{c}}{4} & \text {otherwise} \end {cases} \] Input:
integrate(exp(b**2*x**2+c)*x**3*erfc(b*x),x)
Output:
Piecewise((x**3*exp(c)/(3*sqrt(pi)*b) + x**2*exp(c)*exp(b**2*x**2)*erfc(b* x)/(2*b**2) - x*exp(c)/(sqrt(pi)*b**3) - exp(c)*exp(b**2*x**2)*erfc(b*x)/( 2*b**4), Ne(b, 0)), (x**4*exp(c)/4, True))
\[ \int e^{c+b^2 x^2} x^3 \text {erfc}(b x) \, dx=\int { x^{3} \operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} \,d x } \] Input:
integrate(exp(b^2*x^2+c)*x^3*erfc(b*x),x, algorithm="maxima")
Output:
integrate(x^3*erfc(b*x)*e^(b^2*x^2 + c), x)
\[ \int e^{c+b^2 x^2} x^3 \text {erfc}(b x) \, dx=\int { x^{3} \operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} \,d x } \] Input:
integrate(exp(b^2*x^2+c)*x^3*erfc(b*x),x, algorithm="giac")
Output:
integrate(x^3*erfc(b*x)*e^(b^2*x^2 + c), x)
Time = 4.21 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.79 \[ \int e^{c+b^2 x^2} x^3 \text {erfc}(b x) \, dx=-\frac {{\mathrm {e}}^c\,\left (6\,b\,x-2\,b^3\,x^3+3\,\sqrt {\pi }\,{\mathrm {e}}^{b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right )-3\,b^2\,x^2\,\sqrt {\pi }\,{\mathrm {e}}^{b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right )\right )}{6\,b^4\,\sqrt {\pi }} \] Input:
int(x^3*exp(c + b^2*x^2)*erfc(b*x),x)
Output:
-(exp(c)*(6*b*x - 2*b^3*x^3 + 3*pi^(1/2)*exp(b^2*x^2)*erfc(b*x) - 3*b^2*x^ 2*pi^(1/2)*exp(b^2*x^2)*erfc(b*x)))/(6*b^4*pi^(1/2))
Time = 0.15 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.20 \[ \int e^{c+b^2 x^2} x^3 \text {erfc}(b x) \, dx=\frac {e^{c} \left (-3 e^{b^{2} x^{2}} \mathrm {erf}\left (b x \right ) b^{2} \pi \,x^{2}+3 e^{b^{2} x^{2}} \mathrm {erf}\left (b x \right ) \pi +3 e^{b^{2} x^{2}} b^{2} \pi \,x^{2}-3 e^{b^{2} x^{2}} \pi +2 \sqrt {\pi }\, b^{3} x^{3}-6 \sqrt {\pi }\, b x \right )}{6 b^{4} \pi } \] Input:
int(exp(b^2*x^2+c)*x^3*erfc(b*x),x)
Output:
(e**c*( - 3*e**(b**2*x**2)*erf(b*x)*b**2*pi*x**2 + 3*e**(b**2*x**2)*erf(b* x)*pi + 3*e**(b**2*x**2)*b**2*pi*x**2 - 3*e**(b**2*x**2)*pi + 2*sqrt(pi)*b **3*x**3 - 6*sqrt(pi)*b*x))/(6*b**4*pi)