3.168 \(\int e^{c+b^2 x^2} x^3 \text {erfc}(b x) \, dx\)

Optimal. Leaf size=80 \[ -\frac {e^c x}{\sqrt {\pi } b^3}+\frac {x^2 e^{b^2 x^2+c} \text {erfc}(b x)}{2 b^2}-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{2 b^4}+\frac {e^c x^3}{3 \sqrt {\pi } b} \]

[Out]

-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-exp(c)*x/b^3/Pi^(1/2)+1/3*exp(c)*x^3/b/
Pi^(1/2)

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Rubi [A]  time = 0.09, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6386, 6383, 8, 12, 30} \[ \frac {x^2 e^{b^2 x^2+c} \text {Erfc}(b x)}{2 b^2}-\frac {e^{b^2 x^2+c} \text {Erfc}(b x)}{2 b^4}-\frac {e^c x}{\sqrt {\pi } b^3}+\frac {e^c x^3}{3 \sqrt {\pi } b} \]

Antiderivative was successfully verified.

[In]

Int[E^(c + b^2*x^2)*x^3*Erfc[b*x],x]

[Out]

-((E^c*x)/(b^3*Sqrt[Pi])) + (E^c*x^3)/(3*b*Sqrt[Pi]) - (E^(c + b^2*x^2)*Erfc[b*x])/(2*b^4) + (E^(c + b^2*x^2)*
x^2*Erfc[b*x])/(2*b^2)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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 6383

Int[E^((c_.) + (d_.)*(x_)^2)*Erfc[(a_.) + (b_.)*(x_)]*(x_), x_Symbol] :> Simp[(E^(c + d*x^2)*Erfc[a + b*x])/(2
*d), x] + Dist[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]

Rule 6386

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*} \int e^{c+b^2 x^2} x^3 \text {erfc}(b x) \, dx &=\frac {e^{c+b^2 x^2} x^2 \text {erfc}(b x)}{2 b^2}-\frac {\int e^{c+b^2 x^2} x \text {erfc}(b x) \, dx}{b^2}+\frac {\int e^c x^2 \, dx}{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}-\frac {\int e^c \, dx}{b^3 \sqrt {\pi }}+\frac {e^c \int x^2 \, dx}{b \sqrt {\pi }}\\ &=-\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}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 58, normalized size = 0.72 \[ \frac {e^c \left (3 \sqrt {\pi } e^{b^2 x^2} \left (b^2 x^2-1\right ) \text {erfc}(b x)+2 b x \left (b^2 x^2-3\right )\right )}{6 \sqrt {\pi } b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(c + b^2*x^2)*x^3*Erfc[b*x],x]

[Out]

(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])

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fricas [A]  time = 0.41, size = 68, normalized size = 0.85 \[ \frac {2 \, \sqrt {\pi } {\left (b^{3} x^{3} - 3 \, b x\right )} e^{c} - 3 \, {\left (\pi - \pi b^{2} x^{2} - {\left (\pi - \pi b^{2} x^{2}\right )} \operatorname {erf}\left (b x\right )\right )} e^{\left (b^{2} x^{2} + c\right )}}{6 \, \pi b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b^2*x^2+c)*x^3*erfc(b*x),x, algorithm="fricas")

[Out]

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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b^2*x^2+c)*x^3*erfc(b*x),x, algorithm="giac")

[Out]

integrate(x^3*erfc(b*x)*e^(b^2*x^2 + c), x)

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maple [A]  time = 0.18, size = 99, normalized size = 1.24 \[ \frac {\frac {{\mathrm e}^{c} \left (\frac {b^{2} x^{2} {\mathrm e}^{b^{2} x^{2}}}{2}-\frac {{\mathrm e}^{b^{2} x^{2}}}{2}\right )}{b^{3}}-\frac {\erf \left (b x \right ) {\mathrm e}^{c} \left (\frac {b^{2} x^{2} {\mathrm e}^{b^{2} x^{2}}}{2}-\frac {{\mathrm e}^{b^{2} x^{2}}}{2}\right )}{b^{3}}+\frac {{\mathrm e}^{c} \left (\frac {1}{3} b^{3} x^{3}-b x \right )}{\sqrt {\pi }\, b^{3}}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(b^2*x^2+c)*x^3*erfc(b*x),x)

[Out]

(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*ex
p(b^2*x^2))+1/Pi^(1/2)/b^3*exp(c)*(1/3*b^3*x^3-b*x))/b

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b^2*x^2+c)*x^3*erfc(b*x),x, algorithm="maxima")

[Out]

integrate(x^3*erfc(b*x)*e^(b^2*x^2 + c), x)

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mupad [B]  time = 0.16, size = 63, normalized size = 0.79 \[ -\frac {{\mathrm {e}}^c\,\left (6\,b\,x-2\,b^3\,x^3+3\,\sqrt {\pi }\,{\mathrm {e}}^{b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right )-3\,b^2\,x^2\,\sqrt {\pi }\,{\mathrm {e}}^{b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right )\right )}{6\,b^4\,\sqrt {\pi }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*exp(c + b^2*x^2)*erfc(b*x),x)

[Out]

-(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))

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sympy [A]  time = 120.01, size = 83, normalized size = 1.04 \[ \begin {cases} \frac {x^{3} e^{c}}{3 \sqrt {\pi } b} + \frac {x^{2} e^{c} e^{b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}}{2 b^{2}} - \frac {x e^{c}}{\sqrt {\pi } b^{3}} - \frac {e^{c} e^{b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}}{2 b^{4}} & \text {for}\: b \neq 0 \\\frac {x^{4} e^{c}}{4} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b**2*x**2+c)*x**3*erfc(b*x),x)

[Out]

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))

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