Integrand size = 18, antiderivative size = 63 \[ \int e^{-b^2 x^2} x^2 \text {erfc}(b x) \, dx=\frac {e^{-2 b^2 x^2}}{4 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} x \text {erfc}(b x)}{2 b^2}-\frac {\sqrt {\pi } \text {erfc}(b x)^2}{8 b^3} \] Output:
1/4/b^3/exp(2*b^2*x^2)/Pi^(1/2)-1/2*x*erfc(b*x)/b^2/exp(b^2*x^2)-1/8*Pi^(1 /2)*erfc(b*x)^2/b^3
Time = 0.09 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.25 \[ \int e^{-b^2 x^2} x^2 \text {erfc}(b x) \, dx=\frac {2 e^{-2 b^2 x^2} \left (\frac {1}{\sqrt {\pi }}-2 b e^{b^2 x^2} x\right )+\left (2 \sqrt {\pi }+4 b e^{-b^2 x^2} x\right ) \text {erf}(b x)-\sqrt {\pi } \text {erf}(b x)^2}{8 b^3} \] Input:
Integrate[(x^2*Erfc[b*x])/E^(b^2*x^2),x]
Output:
((2*(1/Sqrt[Pi] - 2*b*E^(b^2*x^2)*x))/E^(2*b^2*x^2) + (2*Sqrt[Pi] + (4*b*x )/E^(b^2*x^2))*Erf[b*x] - Sqrt[Pi]*Erf[b*x]^2)/(8*b^3)
Time = 0.34 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6940, 2638, 6928, 15}
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.
\(\displaystyle \int x^2 e^{-b^2 x^2} \text {erfc}(b x) \, dx\) |
\(\Big \downarrow \) 6940 |
\(\displaystyle \frac {\int e^{-b^2 x^2} \text {erfc}(b x)dx}{2 b^2}-\frac {\int e^{-2 b^2 x^2} xdx}{\sqrt {\pi } b}-\frac {x e^{-b^2 x^2} \text {erfc}(b x)}{2 b^2}\) |
\(\Big \downarrow \) 2638 |
\(\displaystyle \frac {\int e^{-b^2 x^2} \text {erfc}(b x)dx}{2 b^2}-\frac {x e^{-b^2 x^2} \text {erfc}(b x)}{2 b^2}+\frac {e^{-2 b^2 x^2}}{4 \sqrt {\pi } b^3}\) |
\(\Big \downarrow \) 6928 |
\(\displaystyle -\frac {\sqrt {\pi } \int \text {erfc}(b x)d\text {erfc}(b x)}{4 b^3}-\frac {x e^{-b^2 x^2} \text {erfc}(b x)}{2 b^2}+\frac {e^{-2 b^2 x^2}}{4 \sqrt {\pi } b^3}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {\sqrt {\pi } \text {erfc}(b x)^2}{8 b^3}-\frac {x e^{-b^2 x^2} \text {erfc}(b x)}{2 b^2}+\frac {e^{-2 b^2 x^2}}{4 \sqrt {\pi } b^3}\) |
Input:
Int[(x^2*Erfc[b*x])/E^(b^2*x^2),x]
Output:
1/(4*b^3*E^(2*b^2*x^2)*Sqrt[Pi]) - (x*Erfc[b*x])/(2*b^2*E^(b^2*x^2)) - (Sq rt[Pi]*Erfc[b*x]^2)/(8*b^3)
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_ .), x_Symbol] :> Simp[(e + f*x)^n*(F^(a + b*(c + d*x)^n)/(b*f*n*(c + d*x)^n *Log[F])), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] && EqQ [d*e - c*f, 0]
Int[E^((c_.) + (d_.)*(x_)^2)*Erfc[(b_.)*(x_)]^(n_.), x_Symbol] :> Simp[(-E^ c)*(Sqrt[Pi]/(2*b)) Subst[Int[x^n, x], x, Erfc[b*x]], x] /; FreeQ[{b, c, d, n}, x] && EqQ[d, -b^2]
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]
\[\int x^{2} \operatorname {erfc}\left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}d x\]
Input:
int(x^2*erfc(b*x)/exp(b^2*x^2),x)
Output:
int(x^2*erfc(b*x)/exp(b^2*x^2),x)
Time = 0.11 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.05 \[ \int e^{-b^2 x^2} x^2 \text {erfc}(b x) \, dx=\frac {4 \, {\left (\pi b x \operatorname {erf}\left (b x\right ) - \pi b x\right )} e^{\left (-b^{2} x^{2}\right )} - \sqrt {\pi } {\left (\pi \operatorname {erf}\left (b x\right )^{2} - 2 \, \pi \operatorname {erf}\left (b x\right ) - 2 \, e^{\left (-2 \, b^{2} x^{2}\right )}\right )}}{8 \, \pi b^{3}} \] Input:
integrate(x^2*erfc(b*x)/exp(b^2*x^2),x, algorithm="fricas")
Output:
1/8*(4*(pi*b*x*erf(b*x) - pi*b*x)*e^(-b^2*x^2) - sqrt(pi)*(pi*erf(b*x)^2 - 2*pi*erf(b*x) - 2*e^(-2*b^2*x^2)))/(pi*b^3)
Time = 1.00 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00 \[ \int e^{-b^2 x^2} x^2 \text {erfc}(b x) \, dx=\begin {cases} - \frac {x e^{- b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}}{2 b^{2}} - \frac {\sqrt {\pi } \operatorname {erfc}^{2}{\left (b x \right )}}{8 b^{3}} + \frac {e^{- 2 b^{2} x^{2}}}{4 \sqrt {\pi } b^{3}} & \text {for}\: b \neq 0 \\\frac {x^{3}}{3} & \text {otherwise} \end {cases} \] Input:
integrate(x**2*erfc(b*x)/exp(b**2*x**2),x)
Output:
Piecewise((-x*exp(-b**2*x**2)*erfc(b*x)/(2*b**2) - sqrt(pi)*erfc(b*x)**2/( 8*b**3) + exp(-2*b**2*x**2)/(4*sqrt(pi)*b**3), Ne(b, 0)), (x**3/3, True))
\[ \int e^{-b^2 x^2} x^2 \text {erfc}(b x) \, dx=\int { x^{2} \operatorname {erfc}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} \,d x } \] Input:
integrate(x^2*erfc(b*x)/exp(b^2*x^2),x, algorithm="maxima")
Output:
integrate(x^2*erfc(b*x)*e^(-b^2*x^2), x)
\[ \int e^{-b^2 x^2} x^2 \text {erfc}(b x) \, dx=\int { x^{2} \operatorname {erfc}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} \,d x } \] Input:
integrate(x^2*erfc(b*x)/exp(b^2*x^2),x, algorithm="giac")
Output:
integrate(x^2*erfc(b*x)*e^(-b^2*x^2), x)
Time = 4.36 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.78 \[ \int e^{-b^2 x^2} x^2 \text {erfc}(b x) \, dx=\frac {2\,{\mathrm {e}}^{-2\,b^2\,x^2}-\pi \,{\mathrm {erfc}\left (b\,x\right )}^2}{8\,b^3\,\sqrt {\pi }}-\frac {x\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right )}{2\,b^2} \] Input:
int(x^2*exp(-b^2*x^2)*erfc(b*x),x)
Output:
(2*exp(-2*b^2*x^2) - pi*erfc(b*x)^2)/(8*b^3*pi^(1/2)) - (x*exp(-b^2*x^2)*e rfc(b*x))/(2*b^2)
\[ \int e^{-b^2 x^2} x^2 \text {erfc}(b x) \, dx=\frac {\sqrt {\pi }\, e^{b^{2} x^{2}} \mathrm {erf}\left (b x \right )-4 e^{b^{2} x^{2}} \left (\int \frac {\mathrm {erf}\left (b x \right ) x^{2}}{e^{b^{2} x^{2}}}d x \right ) b^{3}-2 b x}{4 e^{b^{2} x^{2}} b^{3}} \] Input:
int(x^2*erfc(b*x)/exp(b^2*x^2),x)
Output:
(sqrt(pi)*e**(b**2*x**2)*erf(b*x) - 4*e**(b**2*x**2)*int((erf(b*x)*x**2)/e **(b**2*x**2),x)*b**3 - 2*b*x)/(4*e**(b**2*x**2)*b**3)