Integrand size = 19, antiderivative size = 95 \[ \int e^{c+b^2 x^2} x^2 \text {erfc}(b x) \, dx=\frac {e^c x^2}{2 b \sqrt {\pi }}+\frac {e^{c+b^2 x^2} x \text {erfc}(b x)}{2 b^2}-\frac {e^c \sqrt {\pi } \text {erfi}(b x)}{4 b^3}+\frac {e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 b \sqrt {\pi }} \] Output:
1/2*exp(c)*x^2/b/Pi^(1/2)+1/2*exp(b^2*x^2+c)*x*erfc(b*x)/b^2-1/4*exp(c)*Pi ^(1/2)*erfi(b*x)/b^3+1/2*exp(c)*x^2*hypergeom([1, 1],[3/2, 2],b^2*x^2)/b/P i^(1/2)
Time = 0.23 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.09 \[ \int e^{c+b^2 x^2} x^2 \text {erfc}(b x) \, dx=-\frac {e^c \left (-2 b e^{b^2 x^2} \sqrt {\pi } x-2 b^2 x^2+\pi \text {erfi}(b x)+\text {erf}(b x) \left (2 b e^{b^2 x^2} \sqrt {\pi } x-\pi \text {erfi}(b x)\right )+2 b^2 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )\right )}{4 b^3 \sqrt {\pi }} \] Input:
Integrate[E^(c + b^2*x^2)*x^2*Erfc[b*x],x]
Output:
-1/4*(E^c*(-2*b*E^(b^2*x^2)*Sqrt[Pi]*x - 2*b^2*x^2 + Pi*Erfi[b*x] + Erf[b* x]*(2*b*E^(b^2*x^2)*Sqrt[Pi]*x - Pi*Erfi[b*x]) + 2*b^2*x^2*HypergeometricP FQ[{1, 1}, {3/2, 2}, -(b^2*x^2)]))/(b^3*Sqrt[Pi])
Time = 0.47 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {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.
\(\displaystyle \int x^2 e^{b^2 x^2+c} \text {erfc}(b x) \, dx\) |
\(\Big \downarrow \) 6940 |
\(\displaystyle -\frac {\int e^{b^2 x^2+c} \text {erfc}(b x)dx}{2 b^2}+\frac {\int e^c xdx}{\sqrt {\pi } b}+\frac {x e^{b^2 x^2+c} \text {erfc}(b x)}{2 b^2}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {\int e^{b^2 x^2+c} \text {erfc}(b x)dx}{2 b^2}+\frac {x e^{b^2 x^2+c} \text {erfc}(b x)}{2 b^2}+\frac {e^c x^2}{2 \sqrt {\pi } b}\) |
\(\Big \downarrow \) 6931 |
\(\displaystyle -\frac {\int e^{b^2 x^2+c}dx-\int e^{b^2 x^2+c} \text {erf}(b x)dx}{2 b^2}+\frac {x e^{b^2 x^2+c} \text {erfc}(b x)}{2 b^2}+\frac {e^c x^2}{2 \sqrt {\pi } b}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle -\frac {\frac {\sqrt {\pi } e^c \text {erfi}(b x)}{2 b}-\int e^{b^2 x^2+c} \text {erf}(b x)dx}{2 b^2}+\frac {x e^{b^2 x^2+c} \text {erfc}(b x)}{2 b^2}+\frac {e^c x^2}{2 \sqrt {\pi } b}\) |
\(\Big \downarrow \) 6930 |
\(\displaystyle -\frac {\frac {\sqrt {\pi } e^c \text {erfi}(b x)}{2 b}-\frac {b e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{\sqrt {\pi }}}{2 b^2}+\frac {x e^{b^2 x^2+c} \text {erfc}(b x)}{2 b^2}+\frac {e^c x^2}{2 \sqrt {\pi } b}\) |
Input:
Int[E^(c + b^2*x^2)*x^2*Erfc[b*x],x]
Output:
(E^c*x^2)/(2*b*Sqrt[Pi]) + (E^(c + b^2*x^2)*x*Erfc[b*x])/(2*b^2) - ((E^c*S qrt[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)
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_))^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]
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]
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]
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]
\[\int {\mathrm e}^{b^{2} x^{2}+c} x^{2} \operatorname {erfc}\left (b x \right )d x\]
Input:
int(exp(b^2*x^2+c)*x^2*erfc(b*x),x)
Output:
int(exp(b^2*x^2+c)*x^2*erfc(b*x),x)
\[ \int e^{c+b^2 x^2} x^2 \text {erfc}(b x) \, dx=\int { x^{2} \operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} \,d x } \] Input:
integrate(exp(b^2*x^2+c)*x^2*erfc(b*x),x, algorithm="fricas")
Output:
integral(-(x^2*erf(b*x) - x^2)*e^(b^2*x^2 + c), x)
Result contains complex when optimal does not.
Time = 28.46 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.68 \[ \int e^{c+b^2 x^2} x^2 \text {erfc}(b x) \, dx=- \frac {b x^{4} e^{c} {{}_{2}F_{2}\left (\begin {matrix} 1, 2 \\ \frac {3}{2}, 3 \end {matrix}\middle | {b^{2} x^{2}} \right )}}{2 \sqrt {\pi }} + \frac {x e^{c} e^{b^{2} x^{2}}}{2 b^{2}} + \frac {i \sqrt {\pi } e^{c} \operatorname {erf}{\left (i b x \right )}}{4 b^{3}} \] Input:
integrate(exp(b**2*x**2+c)*x**2*erfc(b*x),x)
Output:
-b*x**4*exp(c)*hyper((1, 2), (3/2, 3), b**2*x**2)/(2*sqrt(pi)) + x*exp(c)* exp(b**2*x**2)/(2*b**2) + I*sqrt(pi)*exp(c)*erf(I*b*x)/(4*b**3)
\[ \int e^{c+b^2 x^2} x^2 \text {erfc}(b x) \, dx=\int { x^{2} \operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} \,d x } \] Input:
integrate(exp(b^2*x^2+c)*x^2*erfc(b*x),x, algorithm="maxima")
Output:
integrate(x^2*erfc(b*x)*e^(b^2*x^2 + c), x)
\[ \int e^{c+b^2 x^2} x^2 \text {erfc}(b x) \, dx=\int { x^{2} \operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} \,d x } \] Input:
integrate(exp(b^2*x^2+c)*x^2*erfc(b*x),x, algorithm="giac")
Output:
integrate(x^2*erfc(b*x)*e^(b^2*x^2 + c), x)
Timed out. \[ \int e^{c+b^2 x^2} x^2 \text {erfc}(b x) \, dx=\int x^2\,{\mathrm {e}}^{b^2\,x^2+c}\,\mathrm {erfc}\left (b\,x\right ) \,d x \] Input:
int(x^2*exp(c + b^2*x^2)*erfc(b*x),x)
Output:
int(x^2*exp(c + b^2*x^2)*erfc(b*x), x)
\[ \int e^{c+b^2 x^2} x^2 \text {erfc}(b x) \, dx=\frac {e^{c} \left (\sqrt {\pi }\, \mathrm {erf}\left (b i x \right ) i +2 e^{b^{2} x^{2}} b x -4 \left (\int e^{b^{2} x^{2}} \mathrm {erf}\left (b x \right ) x^{2}d x \right ) b^{3}\right )}{4 b^{3}} \] Input:
int(exp(b^2*x^2+c)*x^2*erfc(b*x),x)
Output:
(e**c*(sqrt(pi)*erf(b*i*x)*i + 2*e**(b**2*x**2)*b*x - 4*int(e**(b**2*x**2) *erf(b*x)*x**2,x)*b**3))/(4*b**3)