Integrand size = 8, antiderivative size = 96 \[ \int \frac {\text {erfc}(b x)}{x^7} \, dx=\frac {b e^{-b^2 x^2}}{15 \sqrt {\pi } x^5}-\frac {2 b^3 e^{-b^2 x^2}}{45 \sqrt {\pi } x^3}+\frac {4 b^5 e^{-b^2 x^2}}{45 \sqrt {\pi } x}+\frac {4}{45} b^6 \text {erf}(b x)-\frac {\text {erfc}(b x)}{6 x^6} \]
4/45*b^6*erf(b*x)-1/6*erfc(b*x)/x^6+1/15*b/exp(b^2*x^2)/x^5/Pi^(1/2)-2/45* b^3/exp(b^2*x^2)/x^3/Pi^(1/2)+4/45*b^5/exp(b^2*x^2)/x/Pi^(1/2)
Time = 0.04 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.65 \[ \int \frac {\text {erfc}(b x)}{x^7} \, dx=\frac {1}{90} \left (\frac {2 b e^{-b^2 x^2} \left (3-2 b^2 x^2+4 b^4 x^4\right )}{\sqrt {\pi } x^5}+8 b^6 \text {erf}(b x)-\frac {15 \text {erfc}(b x)}{x^6}\right ) \]
((2*b*(3 - 2*b^2*x^2 + 4*b^4*x^4))/(E^(b^2*x^2)*Sqrt[Pi]*x^5) + 8*b^6*Erf[ b*x] - (15*Erfc[b*x])/x^6)/90
Time = 0.36 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6916, 2643, 2643, 2643, 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.
\(\displaystyle \int \frac {\text {erfc}(b x)}{x^7} \, dx\) |
\(\Big \downarrow \) 6916 |
\(\displaystyle -\frac {b \int \frac {e^{-b^2 x^2}}{x^6}dx}{3 \sqrt {\pi }}-\frac {\text {erfc}(b x)}{6 x^6}\) |
\(\Big \downarrow \) 2643 |
\(\displaystyle -\frac {b \left (-\frac {2}{5} b^2 \int \frac {e^{-b^2 x^2}}{x^4}dx-\frac {e^{-b^2 x^2}}{5 x^5}\right )}{3 \sqrt {\pi }}-\frac {\text {erfc}(b x)}{6 x^6}\) |
\(\Big \downarrow \) 2643 |
\(\displaystyle -\frac {b \left (-\frac {2}{5} b^2 \left (-\frac {2}{3} b^2 \int \frac {e^{-b^2 x^2}}{x^2}dx-\frac {e^{-b^2 x^2}}{3 x^3}\right )-\frac {e^{-b^2 x^2}}{5 x^5}\right )}{3 \sqrt {\pi }}-\frac {\text {erfc}(b x)}{6 x^6}\) |
\(\Big \downarrow \) 2643 |
\(\displaystyle -\frac {b \left (-\frac {2}{5} b^2 \left (-\frac {2}{3} b^2 \left (-2 b^2 \int e^{-b^2 x^2}dx-\frac {e^{-b^2 x^2}}{x}\right )-\frac {e^{-b^2 x^2}}{3 x^3}\right )-\frac {e^{-b^2 x^2}}{5 x^5}\right )}{3 \sqrt {\pi }}-\frac {\text {erfc}(b x)}{6 x^6}\) |
\(\Big \downarrow \) 2634 |
\(\displaystyle -\frac {b \left (-\frac {2}{5} b^2 \left (-\frac {2}{3} b^2 \left (\sqrt {\pi } (-b) \text {erf}(b x)-\frac {e^{-b^2 x^2}}{x}\right )-\frac {e^{-b^2 x^2}}{3 x^3}\right )-\frac {e^{-b^2 x^2}}{5 x^5}\right )}{3 \sqrt {\pi }}-\frac {\text {erfc}(b x)}{6 x^6}\) |
-1/3*(b*(-1/5*1/(E^(b^2*x^2)*x^5) - (2*b^2*(-1/3*1/(E^(b^2*x^2)*x^3) - (2* b^2*(-(1/(E^(b^2*x^2)*x)) - b*Sqrt[Pi]*Erf[b*x]))/3))/5))/Sqrt[Pi] - Erfc[ b*x]/(6*x^6)
3.2.10.3.1 Defintions of rubi rules used
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 + 1)*(F^(a + b*(c + d*x)^n)/(d*(m + 1))) , x] - Simp[b*n*(Log[F]/(m + 1)) 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[ -4, (m + 1)/n, 5] && IntegerQ[n] && ((GtQ[n, 0] && LtQ[m, -1]) || (GtQ[-n, 0] && LeQ[-n, m + 1]))
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 = 0.47 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.84
method | result | size |
parallelrisch | \(-\frac {8 \,\operatorname {erfc}\left (b x \right ) x^{6} b^{6} \sqrt {\pi }-8 \,{\mathrm e}^{-b^{2} x^{2}} x^{5} b^{5}+4 x^{3} {\mathrm e}^{-b^{2} x^{2}} b^{3}-6 \,{\mathrm e}^{-b^{2} x^{2}} b x +15 \,\operatorname {erfc}\left (b x \right ) \sqrt {\pi }}{90 \sqrt {\pi }\, x^{6}}\) | \(81\) |
parts | \(-\frac {\operatorname {erfc}\left (b x \right )}{6 x^{6}}-\frac {b \left (-\frac {{\mathrm e}^{-b^{2} x^{2}}}{5 x^{5}}-\frac {2 b^{2} \left (-\frac {{\mathrm e}^{-b^{2} x^{2}}}{3 x^{3}}-\frac {2 b^{2} \left (-\frac {{\mathrm e}^{-b^{2} x^{2}}}{x}-b \sqrt {\pi }\, \operatorname {erf}\left (b x \right )\right )}{3}\right )}{5}\right )}{3 \sqrt {\pi }}\) | \(82\) |
derivativedivides | \(b^{6} \left (-\frac {\operatorname {erfc}\left (b x \right )}{6 b^{6} x^{6}}-\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}}}{5 b^{5} x^{5}}+\frac {2 \,{\mathrm e}^{-b^{2} x^{2}}}{15 b^{3} x^{3}}-\frac {4 \,{\mathrm e}^{-b^{2} x^{2}}}{15 b x}-\frac {4 \,\operatorname {erf}\left (b x \right ) \sqrt {\pi }}{15}}{3 \sqrt {\pi }}\right )\) | \(87\) |
default | \(b^{6} \left (-\frac {\operatorname {erfc}\left (b x \right )}{6 b^{6} x^{6}}-\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}}}{5 b^{5} x^{5}}+\frac {2 \,{\mathrm e}^{-b^{2} x^{2}}}{15 b^{3} x^{3}}-\frac {4 \,{\mathrm e}^{-b^{2} x^{2}}}{15 b x}-\frac {4 \,\operatorname {erf}\left (b x \right ) \sqrt {\pi }}{15}}{3 \sqrt {\pi }}\right )\) | \(87\) |
-1/90*(8*erfc(b*x)*x^6*b^6*Pi^(1/2)-8*exp(-b^2*x^2)*x^5*b^5+4*x^3*exp(-b^2 *x^2)*b^3-6*exp(-b^2*x^2)*b*x+15*erfc(b*x)*Pi^(1/2))/Pi^(1/2)/x^6
Time = 0.25 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.69 \[ \int \frac {\text {erfc}(b x)}{x^7} \, dx=-\frac {15 \, \pi - 2 \, \sqrt {\pi } {\left (4 \, b^{5} x^{5} - 2 \, b^{3} x^{3} + 3 \, b x\right )} e^{\left (-b^{2} x^{2}\right )} - {\left (15 \, \pi + 8 \, \pi b^{6} x^{6}\right )} \operatorname {erf}\left (b x\right )}{90 \, \pi x^{6}} \]
-1/90*(15*pi - 2*sqrt(pi)*(4*b^5*x^5 - 2*b^3*x^3 + 3*b*x)*e^(-b^2*x^2) - ( 15*pi + 8*pi*b^6*x^6)*erf(b*x))/(pi*x^6)
Time = 0.48 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.91 \[ \int \frac {\text {erfc}(b x)}{x^7} \, dx=- \frac {4 b^{6} \operatorname {erfc}{\left (b x \right )}}{45} + \frac {4 b^{5} e^{- b^{2} x^{2}}}{45 \sqrt {\pi } x} - \frac {2 b^{3} e^{- b^{2} x^{2}}}{45 \sqrt {\pi } x^{3}} + \frac {b e^{- b^{2} x^{2}}}{15 \sqrt {\pi } x^{5}} - \frac {\operatorname {erfc}{\left (b x \right )}}{6 x^{6}} \]
-4*b**6*erfc(b*x)/45 + 4*b**5*exp(-b**2*x**2)/(45*sqrt(pi)*x) - 2*b**3*exp (-b**2*x**2)/(45*sqrt(pi)*x**3) + b*exp(-b**2*x**2)/(15*sqrt(pi)*x**5) - e rfc(b*x)/(6*x**6)
Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.36 \[ \int \frac {\text {erfc}(b x)}{x^7} \, dx=\frac {b^{6} {\left (x^{2}\right )}^{\frac {5}{2}} \Gamma \left (-\frac {5}{2}, b^{2} x^{2}\right )}{6 \, \sqrt {\pi } x^{5}} - \frac {\operatorname {erfc}\left (b x\right )}{6 \, x^{6}} \]
\[ \int \frac {\text {erfc}(b x)}{x^7} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right )}{x^{7}} \,d x } \]
Timed out. \[ \int \frac {\text {erfc}(b x)}{x^7} \, dx=\int \frac {\mathrm {erfc}\left (b\,x\right )}{x^7} \,d x \]