\(\int x^2 \text {erfc}(b x) \, dx\) [113]

Optimal result

Integrand size = 8, antiderivative size = 59 \[ \int x^2 \text {erfc}(b x) \, dx=-\frac {e^{-b^2 x^2}}{3 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} x^2}{3 b \sqrt {\pi }}+\frac {1}{3} x^3 \text {erfc}(b x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6497, 2243, 2240} \[ \int x^2 \text {erfc}(b x) \, dx=-\frac {x^2 e^{-b^2 x^2}}{3 \sqrt {\pi } b}-\frac {e^{-b^2 x^2}}{3 \sqrt {\pi } b^3}+\frac {1}{3} x^3 \text {erfc}(b x) \]

[In]

Int[x^2*Erfc[b*x],x]

[Out]

-1/3*1/(b^3*E^(b^2*x^2)*Sqrt[Pi]) - x^2/(3*b*E^(b^2*x^2)*Sqrt[Pi]) + (x^3*Erfc[b*x])/3

Rule 2240

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]

Rule 2243

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m
- n + 1)*(F^(a + b*(c + d*x)^n)/(b*d*n*Log[F])), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^(m - n)*F^(
a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2*((m + 1)/n)] && LtQ[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

Rule 6497

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 \text {erfc}(b x)+\frac {(2 b) \int e^{-b^2 x^2} x^3 \, dx}{3 \sqrt {\pi }} \\ & = -\frac {e^{-b^2 x^2} x^2}{3 b \sqrt {\pi }}+\frac {1}{3} x^3 \text {erfc}(b x)+\frac {2 \int e^{-b^2 x^2} x \, dx}{3 b \sqrt {\pi }} \\ & = -\frac {e^{-b^2 x^2}}{3 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} x^2}{3 b \sqrt {\pi }}+\frac {1}{3} x^3 \text {erfc}(b x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.71 \[ \int x^2 \text {erfc}(b x) \, dx=\frac {1}{3} \left (-\frac {e^{-b^2 x^2} \left (1+b^2 x^2\right )}{b^3 \sqrt {\pi }}+x^3 \text {erfc}(b x)\right ) \]

[In]

Integrate[x^2*Erfc[b*x],x]

[Out]

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

Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.83

method	result	size
parts	\(\frac {x^{3} \operatorname {erfc}\left (b x \right )}{3}+\frac {2 b \left (-\frac {x^{2} {\mathrm e}^{-b^{2} x^{2}}}{2 b^{2}}-\frac {{\mathrm e}^{-b^{2} x^{2}}}{2 b^{4}}\right )}{3 \sqrt {\pi }}\)	\(49\)
parallelrisch	\(\frac {x^{3} \operatorname {erfc}\left (b x \right ) b^{3} \sqrt {\pi }-x^{2} {\mathrm e}^{-b^{2} x^{2}} b^{2}-{\mathrm e}^{-b^{2} x^{2}}}{3 b^{3} \sqrt {\pi }}\)	\(52\)
derivativedivides	\(\frac {\frac {b^{3} x^{3} \operatorname {erfc}\left (b x \right )}{3}+\frac {-\frac {x^{2} {\mathrm e}^{-b^{2} x^{2}} b^{2}}{3}-\frac {{\mathrm e}^{-b^{2} x^{2}}}{3}}{\sqrt {\pi }}}{b^{3}}\)	\(54\)
default	\(\frac {\frac {b^{3} x^{3} \operatorname {erfc}\left (b x \right )}{3}+\frac {-\frac {x^{2} {\mathrm e}^{-b^{2} x^{2}} b^{2}}{3}-\frac {{\mathrm e}^{-b^{2} x^{2}}}{3}}{\sqrt {\pi }}}{b^{3}}\)	\(54\)

[In]

int(x^2*erfc(b*x),x,method=_RETURNVERBOSE)

[Out]

1/3*x^3*erfc(b*x)+2/3/Pi^(1/2)*b*(-1/2/b^2*x^2*exp(-b^2*x^2)-1/2/b^4*exp(-b^2*x^2))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.88 \[ \int x^2 \text {erfc}(b x) \, dx=-\frac {\pi b^{3} x^{3} \operatorname {erf}\left (b x\right ) - \pi b^{3} x^{3} + \sqrt {\pi } {\left (b^{2} x^{2} + 1\right )} e^{\left (-b^{2} x^{2}\right )}}{3 \, \pi b^{3}} \]

[In]

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

[Out]

-1/3*(pi*b^3*x^3*erf(b*x) - pi*b^3*x^3 + sqrt(pi)*(b^2*x^2 + 1)*e^(-b^2*x^2))/(pi*b^3)

Sympy [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.92 \[ \int x^2 \text {erfc}(b x) \, dx=\begin {cases} \frac {x^{3} \operatorname {erfc}{\left (b x \right )}}{3} - \frac {x^{2} e^{- b^{2} x^{2}}}{3 \sqrt {\pi } b} - \frac {e^{- b^{2} x^{2}}}{3 \sqrt {\pi } b^{3}} & \text {for}\: b \neq 0 \\\frac {x^{3}}{3} & \text {otherwise} \end {cases} \]

[In]

integrate(x**2*erfc(b*x),x)

[Out]

Piecewise((x**3*erfc(b*x)/3 - x**2*exp(-b**2*x**2)/(3*sqrt(pi)*b) - exp(-b**2*x**2)/(3*sqrt(pi)*b**3), Ne(b, 0
)), (x**3/3, True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.61 \[ \int x^2 \text {erfc}(b x) \, dx=\frac {1}{3} \, x^{3} \operatorname {erfc}\left (b x\right ) - \frac {{\left (b^{2} x^{2} + 1\right )} e^{\left (-b^{2} x^{2}\right )}}{3 \, \sqrt {\pi } b^{3}} \]

[In]

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

[Out]

1/3*x^3*erfc(b*x) - 1/3*(b^2*x^2 + 1)*e^(-b^2*x^2)/(sqrt(pi)*b^3)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.69 \[ \int x^2 \text {erfc}(b x) \, dx=-\frac {1}{3} \, x^{3} \operatorname {erf}\left (b x\right ) + \frac {1}{3} \, x^{3} - \frac {{\left (b^{2} x^{2} + 1\right )} e^{\left (-b^{2} x^{2}\right )}}{3 \, \sqrt {\pi } b^{3}} \]

[In]

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

[Out]

-1/3*x^3*erf(b*x) + 1/3*x^3 - 1/3*(b^2*x^2 + 1)*e^(-b^2*x^2)/(sqrt(pi)*b^3)

Mupad [B] (verification not implemented)

Time = 5.37 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.85 \[ \int x^2 \text {erfc}(b x) \, dx=\frac {x^3\,\mathrm {erfc}\left (b\,x\right )}{3}-\frac {\frac {{\mathrm {e}}^{-b^2\,x^2}}{3\,\sqrt {\pi }}+\frac {b^2\,x^2\,{\mathrm {e}}^{-b^2\,x^2}}{3\,\sqrt {\pi }}}{b^3} \]

[In]

int(x^2*erfc(b*x),x)

[Out]

(x^3*erfc(b*x))/3 - (exp(-b^2*x^2)/(3*pi^(1/2)) + (b^2*x^2*exp(-b^2*x^2))/(3*pi^(1/2)))/b^3