\(\int e^{c+b^2 x^2} x^2 \text {erfc}(b x) \, dx\) [174]

Optimal result

Integrand size = 19, antiderivative size = 95 \[ \int e^{c+b^2 x^2} x^2 \text {erfc}(b x) \, dx=\frac {e^c x^2}{2 b \sqrt {\pi }}+\frac {e^{c+b^2 x^2} x \text {erfc}(b x)}{2 b^2}-\frac {e^c \sqrt {\pi } \text {erfi}(b x)}{4 b^3}+\frac {e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 b \sqrt {\pi }} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {6521, 6512, 2235, 6511, 12, 30} \[ \int e^{c+b^2 x^2} x^2 \text {erfc}(b x) \, dx=\frac {e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 \sqrt {\pi } b}-\frac {\sqrt {\pi } e^c \text {erfi}(b x)}{4 b^3}+\frac {x e^{b^2 x^2+c} \text {erfc}(b x)}{2 b^2}+\frac {e^c x^2}{2 \sqrt {\pi } b} \]

[In]

Int[E^(c + b^2*x^2)*x^2*Erfc[b*x],x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

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

Rule 2235

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 6511

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 6512

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 6521

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

Rubi steps \begin{align*} \text {integral}& = \frac {e^{c+b^2 x^2} x \text {erfc}(b x)}{2 b^2}-\frac {\int e^{c+b^2 x^2} \text {erfc}(b x) \, dx}{2 b^2}+\frac {\int e^c x \, dx}{b \sqrt {\pi }} \\ & = \frac {e^{c+b^2 x^2} x \text {erfc}(b x)}{2 b^2}-\frac {\int e^{c+b^2 x^2} \, dx}{2 b^2}+\frac {\int e^{c+b^2 x^2} \text {erf}(b x) \, dx}{2 b^2}+\frac {e^c \int x \, dx}{b \sqrt {\pi }} \\ & = \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 }} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.09 \[ \int e^{c+b^2 x^2} x^2 \text {erfc}(b x) \, dx=-\frac {e^c \left (-2 b e^{b^2 x^2} \sqrt {\pi } x-2 b^2 x^2+\pi \text {erfi}(b x)+\text {erf}(b x) \left (2 b e^{b^2 x^2} \sqrt {\pi } x-\pi \text {erfi}(b x)\right )+2 b^2 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )\right )}{4 b^3 \sqrt {\pi }} \]

[In]

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

[Out]

-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*HypergeometricPFQ[{1, 1}, {3/2, 2}, -(b^2*x^2)]))/(b^3*Sqrt[Pi])

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 28.60 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.68 \[ \int e^{c+b^2 x^2} x^2 \text {erfc}(b x) \, dx=- \frac {b x^{4} e^{c} {{}_{2}F_{2}\left (\begin {matrix} 1, 2 \\ \frac {3}{2}, 3 \end {matrix}\middle | {b^{2} x^{2}} \right )}}{2 \sqrt {\pi }} + \frac {x e^{c} e^{b^{2} x^{2}}}{2 b^{2}} + \frac {i \sqrt {\pi } e^{c} \operatorname {erf}{\left (i b x \right )}}{4 b^{3}} \]

[In]

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

[Out]

-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)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(x^2*exp(c + b^2*x^2)*erfc(b*x),x)

[Out]

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