Integrand size = 8, antiderivative size = 71 \[ \int x^3 \text {erfc}(b x) \, dx=-\frac {3 e^{-b^2 x^2} x}{8 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} x^3}{4 b \sqrt {\pi }}+\frac {3 \text {erf}(b x)}{16 b^4}+\frac {1}{4} x^4 \text {erfc}(b x) \] Output:
-3/8*x/b^3/exp(b^2*x^2)/Pi^(1/2)-1/4*x^3/b/exp(b^2*x^2)/Pi^(1/2)+3/16*erf( b*x)/b^4+1/4*x^4*erfc(b*x)
Time = 0.04 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.76 \[ \int x^3 \text {erfc}(b x) \, dx=\frac {1}{16} \left (-\frac {2 e^{-b^2 x^2} x \left (3+2 b^2 x^2\right )}{b^3 \sqrt {\pi }}+\frac {3 \text {erf}(b x)}{b^4}+4 x^4 \text {erfc}(b x)\right ) \] Input:
Integrate[x^3*Erfc[b*x],x]
Output:
((-2*x*(3 + 2*b^2*x^2))/(b^3*E^(b^2*x^2)*Sqrt[Pi]) + (3*Erf[b*x])/b^4 + 4* x^4*Erfc[b*x])/16
Time = 0.31 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.20, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6916, 2641, 2641, 2634}
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.
\(\displaystyle \int x^3 \text {erfc}(b x) \, dx\) |
\(\Big \downarrow \) 6916 |
\(\displaystyle \frac {b \int e^{-b^2 x^2} x^4dx}{2 \sqrt {\pi }}+\frac {1}{4} x^4 \text {erfc}(b x)\) |
\(\Big \downarrow \) 2641 |
\(\displaystyle \frac {b \left (\frac {3 \int e^{-b^2 x^2} x^2dx}{2 b^2}-\frac {x^3 e^{-b^2 x^2}}{2 b^2}\right )}{2 \sqrt {\pi }}+\frac {1}{4} x^4 \text {erfc}(b x)\) |
\(\Big \downarrow \) 2641 |
\(\displaystyle \frac {b \left (\frac {3 \left (\frac {\int e^{-b^2 x^2}dx}{2 b^2}-\frac {x e^{-b^2 x^2}}{2 b^2}\right )}{2 b^2}-\frac {x^3 e^{-b^2 x^2}}{2 b^2}\right )}{2 \sqrt {\pi }}+\frac {1}{4} x^4 \text {erfc}(b x)\) |
\(\Big \downarrow \) 2634 |
\(\displaystyle \frac {b \left (\frac {3 \left (\frac {\sqrt {\pi } \text {erf}(b x)}{4 b^3}-\frac {x e^{-b^2 x^2}}{2 b^2}\right )}{2 b^2}-\frac {x^3 e^{-b^2 x^2}}{2 b^2}\right )}{2 \sqrt {\pi }}+\frac {1}{4} x^4 \text {erfc}(b x)\) |
Input:
Int[x^3*Erfc[b*x],x]
Output:
(b*(-1/2*x^3/(b^2*E^(b^2*x^2)) + (3*(-1/2*x/(b^2*E^(b^2*x^2)) + (Sqrt[Pi]* Erf[b*x])/(4*b^3)))/(2*b^2)))/(2*Sqrt[Pi]) + (x^4*Erfc[b*x])/4
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F], 2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; Fr eeQ[{F, a, b, c, d}, x] && NegQ[b]
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*L og[F])), x] - Simp[(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])
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] + Simp[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]
Time = 0.16 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.90
method | result | size |
parallelrisch | \(\frac {4 \,\operatorname {erfc}\left (b x \right ) x^{4} \sqrt {\pi }\, b^{4}-4 \,{\mathrm e}^{-b^{2} x^{2}} x^{3} b^{3}-6 \,{\mathrm e}^{-b^{2} x^{2}} b x -3 \,\operatorname {erfc}\left (b x \right ) \sqrt {\pi }}{16 \sqrt {\pi }\, b^{4}}\) | \(64\) |
derivativedivides | \(\frac {\frac {b^{4} x^{4} \operatorname {erfc}\left (b x \right )}{4}+\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}} x^{3} b^{3}}{2}-\frac {3 \,{\mathrm e}^{-b^{2} x^{2}} b x}{4}+\frac {3 \,\operatorname {erf}\left (b x \right ) \sqrt {\pi }}{8}}{2 \sqrt {\pi }}}{b^{4}}\) | \(65\) |
default | \(\frac {\frac {b^{4} x^{4} \operatorname {erfc}\left (b x \right )}{4}+\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}} x^{3} b^{3}}{2}-\frac {3 \,{\mathrm e}^{-b^{2} x^{2}} b x}{4}+\frac {3 \,\operatorname {erf}\left (b x \right ) \sqrt {\pi }}{8}}{2 \sqrt {\pi }}}{b^{4}}\) | \(65\) |
parts | \(\frac {x^{4} \operatorname {erfc}\left (b x \right )}{4}+\frac {b \left (-\frac {x^{3} {\mathrm e}^{-b^{2} x^{2}}}{2 b^{2}}+\frac {-\frac {3 x \,{\mathrm e}^{-b^{2} x^{2}}}{4 b^{2}}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (b x \right )}{8 b^{3}}}{b^{2}}\right )}{2 \sqrt {\pi }}\) | \(68\) |
Input:
int(x^3*erfc(b*x),x,method=_RETURNVERBOSE)
Output:
1/16*(4*erfc(b*x)*x^4*Pi^(1/2)*b^4-4*exp(-b^2*x^2)*x^3*b^3-6*exp(-b^2*x^2) *b*x-3*erfc(b*x)*Pi^(1/2))/Pi^(1/2)/b^4
Time = 0.08 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.89 \[ \int x^3 \text {erfc}(b x) \, dx=\frac {4 \, \pi b^{4} x^{4} - 2 \, \sqrt {\pi } {\left (2 \, b^{3} x^{3} + 3 \, b x\right )} e^{\left (-b^{2} x^{2}\right )} + {\left (3 \, \pi - 4 \, \pi b^{4} x^{4}\right )} \operatorname {erf}\left (b x\right )}{16 \, \pi b^{4}} \] Input:
integrate(x^3*erfc(b*x),x, algorithm="fricas")
Output:
1/16*(4*pi*b^4*x^4 - 2*sqrt(pi)*(2*b^3*x^3 + 3*b*x)*e^(-b^2*x^2) + (3*pi - 4*pi*b^4*x^4)*erf(b*x))/(pi*b^4)
Time = 0.26 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.96 \[ \int x^3 \text {erfc}(b x) \, dx=\begin {cases} \frac {x^{4} \operatorname {erfc}{\left (b x \right )}}{4} - \frac {x^{3} e^{- b^{2} x^{2}}}{4 \sqrt {\pi } b} - \frac {3 x e^{- b^{2} x^{2}}}{8 \sqrt {\pi } b^{3}} - \frac {3 \operatorname {erfc}{\left (b x \right )}}{16 b^{4}} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases} \] Input:
integrate(x**3*erfc(b*x),x)
Output:
Piecewise((x**4*erfc(b*x)/4 - x**3*exp(-b**2*x**2)/(4*sqrt(pi)*b) - 3*x*ex p(-b**2*x**2)/(8*sqrt(pi)*b**3) - 3*erfc(b*x)/(16*b**4), Ne(b, 0)), (x**4/ 4, True))
Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.77 \[ \int x^3 \text {erfc}(b x) \, dx=\frac {1}{4} \, x^{4} \operatorname {erfc}\left (b x\right ) - \frac {b {\left (\frac {2 \, {\left (2 \, b^{2} x^{3} + 3 \, x\right )} e^{\left (-b^{2} x^{2}\right )}}{b^{4}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (b x\right )}{b^{5}}\right )}}{16 \, \sqrt {\pi }} \] Input:
integrate(x^3*erfc(b*x),x, algorithm="maxima")
Output:
1/4*x^4*erfc(b*x) - 1/16*b*(2*(2*b^2*x^3 + 3*x)*e^(-b^2*x^2)/b^4 - 3*sqrt( pi)*erf(b*x)/b^5)/sqrt(pi)
Time = 0.13 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.86 \[ \int x^3 \text {erfc}(b x) \, dx=-\frac {1}{4} \, x^{4} \operatorname {erf}\left (b x\right ) + \frac {1}{4} \, x^{4} - \frac {b {\left (\frac {2 \, {\left (2 \, b^{2} x^{3} + 3 \, x\right )} e^{\left (-b^{2} x^{2}\right )}}{b^{4}} + \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-b x\right )}{b^{5}}\right )}}{16 \, \sqrt {\pi }} \] Input:
integrate(x^3*erfc(b*x),x, algorithm="giac")
Output:
-1/4*x^4*erf(b*x) + 1/4*x^4 - 1/16*b*(2*(2*b^2*x^3 + 3*x)*e^(-b^2*x^2)/b^4 + 3*sqrt(pi)*erf(-b*x)/b^5)/sqrt(pi)
Time = 3.88 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.82 \[ \int x^3 \text {erfc}(b x) \, dx=\frac {x^4\,\mathrm {erfc}\left (b\,x\right )}{4}-\frac {\frac {3\,\mathrm {erfc}\left (b\,x\right )}{16}+\frac {b^3\,x^3\,{\mathrm {e}}^{-b^2\,x^2}}{4\,\sqrt {\pi }}+\frac {3\,b\,x\,{\mathrm {e}}^{-b^2\,x^2}}{8\,\sqrt {\pi }}}{b^4} \] Input:
int(x^3*erfc(b*x),x)
Output:
(x^4*erfc(b*x))/4 - ((3*erfc(b*x))/16 + (b^3*x^3*exp(-b^2*x^2))/(4*pi^(1/2 )) + (3*b*x*exp(-b^2*x^2))/(8*pi^(1/2)))/b^4
Time = 0.16 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.30 \[ \int x^3 \text {erfc}(b x) \, dx=\frac {-4 e^{b^{2} x^{2}} \mathrm {erf}\left (b x \right ) b^{4} \pi \,x^{4}+3 e^{b^{2} x^{2}} \mathrm {erf}\left (b x \right ) \pi +4 e^{b^{2} x^{2}} b^{4} \pi \,x^{4}-4 \sqrt {\pi }\, b^{3} x^{3}-6 \sqrt {\pi }\, b x}{16 e^{b^{2} x^{2}} b^{4} \pi } \] Input:
int(x^3*erfc(b*x),x)
Output:
( - 4*e**(b**2*x**2)*erf(b*x)*b**4*pi*x**4 + 3*e**(b**2*x**2)*erf(b*x)*pi + 4*e**(b**2*x**2)*b**4*pi*x**4 - 4*sqrt(pi)*b**3*x**3 - 6*sqrt(pi)*b*x)/( 16*e**(b**2*x**2)*b**4*pi)