Integrand size = 8, antiderivative size = 81 \[ \int \frac {\text {erfc}(b x)}{x^6} \, dx=\frac {b e^{-b^2 x^2}}{10 \sqrt {\pi } x^4}-\frac {b^3 e^{-b^2 x^2}}{10 \sqrt {\pi } x^2}-\frac {\text {erfc}(b x)}{5 x^5}-\frac {b^5 \operatorname {ExpIntegralEi}\left (-b^2 x^2\right )}{10 \sqrt {\pi }} \] Output:
1/10*b/exp(b^2*x^2)/Pi^(1/2)/x^4-1/10*b^3/exp(b^2*x^2)/Pi^(1/2)/x^2-1/5*er fc(b*x)/x^5-1/10*b^5*Ei(-b^2*x^2)/Pi^(1/2)
Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.90 \[ \int \frac {\text {erfc}(b x)}{x^6} \, dx=e^{-b^2 x^2} \left (\frac {b}{10 \sqrt {\pi } x^4}-\frac {b^3}{10 \sqrt {\pi } x^2}\right )-\frac {\text {erfc}(b x)}{5 x^5}-\frac {b^5 \operatorname {ExpIntegralEi}\left (-b^2 x^2\right )}{10 \sqrt {\pi }} \] Input:
Integrate[Erfc[b*x]/x^6,x]
Output:
(b/(10*Sqrt[Pi]*x^4) - b^3/(10*Sqrt[Pi]*x^2))/E^(b^2*x^2) - Erfc[b*x]/(5*x ^5) - (b^5*ExpIntegralEi[-(b^2*x^2)])/(10*Sqrt[Pi])
Time = 0.32 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6916, 2643, 2643, 2639}
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.
\(\displaystyle \int \frac {\text {erfc}(b x)}{x^6} \, dx\) |
\(\Big \downarrow \) 6916 |
\(\displaystyle -\frac {2 b \int \frac {e^{-b^2 x^2}}{x^5}dx}{5 \sqrt {\pi }}-\frac {\text {erfc}(b x)}{5 x^5}\) |
\(\Big \downarrow \) 2643 |
\(\displaystyle -\frac {2 b \left (-\frac {1}{2} b^2 \int \frac {e^{-b^2 x^2}}{x^3}dx-\frac {e^{-b^2 x^2}}{4 x^4}\right )}{5 \sqrt {\pi }}-\frac {\text {erfc}(b x)}{5 x^5}\) |
\(\Big \downarrow \) 2643 |
\(\displaystyle -\frac {2 b \left (-\frac {1}{2} b^2 \left (b^2 \left (-\int \frac {e^{-b^2 x^2}}{x}dx\right )-\frac {e^{-b^2 x^2}}{2 x^2}\right )-\frac {e^{-b^2 x^2}}{4 x^4}\right )}{5 \sqrt {\pi }}-\frac {\text {erfc}(b x)}{5 x^5}\) |
\(\Big \downarrow \) 2639 |
\(\displaystyle -\frac {2 b \left (-\frac {1}{2} b^2 \left (-\frac {1}{2} b^2 \operatorname {ExpIntegralEi}\left (-b^2 x^2\right )-\frac {e^{-b^2 x^2}}{2 x^2}\right )-\frac {e^{-b^2 x^2}}{4 x^4}\right )}{5 \sqrt {\pi }}-\frac {\text {erfc}(b x)}{5 x^5}\) |
Input:
Int[Erfc[b*x]/x^6,x]
Output:
-1/5*Erfc[b*x]/x^5 - (2*b*(-1/4*1/(E^(b^2*x^2)*x^4) - (b^2*(-1/2*1/(E^(b^2 *x^2)*x^2) - (b^2*ExpIntegralEi[-(b^2*x^2)])/2))/2))/(5*Sqrt[Pi])
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_ Symbol] :> Simp[F^a*(ExpIntegralEi[b*(c + d*x)^n*Log[F]]/(f*n)), x] /; Free Q[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]
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 = 0.61 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.81
method | result | size |
parts | \(-\frac {\operatorname {erfc}\left (b x \right )}{5 x^{5}}-\frac {2 b \left (-\frac {{\mathrm e}^{-b^{2} x^{2}}}{4 x^{4}}-\frac {b^{2} \left (-\frac {{\mathrm e}^{-b^{2} x^{2}}}{2 x^{2}}+\frac {b^{2} \operatorname {expIntegral}_{1}\left (b^{2} x^{2}\right )}{2}\right )}{2}\right )}{5 \sqrt {\pi }}\) | \(66\) |
derivativedivides | \(b^{5} \left (-\frac {\operatorname {erfc}\left (b x \right )}{5 b^{5} x^{5}}-\frac {2 \left (-\frac {{\mathrm e}^{-b^{2} x^{2}}}{4 b^{4} x^{4}}+\frac {{\mathrm e}^{-b^{2} x^{2}}}{4 x^{2} b^{2}}-\frac {\operatorname {expIntegral}_{1}\left (b^{2} x^{2}\right )}{4}\right )}{5 \sqrt {\pi }}\right )\) | \(71\) |
default | \(b^{5} \left (-\frac {\operatorname {erfc}\left (b x \right )}{5 b^{5} x^{5}}-\frac {2 \left (-\frac {{\mathrm e}^{-b^{2} x^{2}}}{4 b^{4} x^{4}}+\frac {{\mathrm e}^{-b^{2} x^{2}}}{4 x^{2} b^{2}}-\frac {\operatorname {expIntegral}_{1}\left (b^{2} x^{2}\right )}{4}\right )}{5 \sqrt {\pi }}\right )\) | \(71\) |
Input:
int(erfc(b*x)/x^6,x,method=_RETURNVERBOSE)
Output:
-1/5*erfc(b*x)/x^5-2/5/Pi^(1/2)*b*(-1/4/x^4*exp(-b^2*x^2)-1/2*b^2*(-1/2/x^ 2*exp(-b^2*x^2)+1/2*b^2*Ei(1,b^2*x^2)))
Time = 0.10 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.77 \[ \int \frac {\text {erfc}(b x)}{x^6} \, dx=-\frac {2 \, \pi - 2 \, \pi \operatorname {erf}\left (b x\right ) + \sqrt {\pi } {\left (b^{5} x^{5} {\rm Ei}\left (-b^{2} x^{2}\right ) + {\left (b^{3} x^{3} - b x\right )} e^{\left (-b^{2} x^{2}\right )}\right )}}{10 \, \pi x^{5}} \] Input:
integrate(erfc(b*x)/x^6,x, algorithm="fricas")
Output:
-1/10*(2*pi - 2*pi*erf(b*x) + sqrt(pi)*(b^5*x^5*Ei(-b^2*x^2) + (b^3*x^3 - b*x)*e^(-b^2*x^2)))/(pi*x^5)
Time = 1.99 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.86 \[ \int \frac {\text {erfc}(b x)}{x^6} \, dx=\frac {b^{5} \operatorname {E}_{1}\left (b^{2} x^{2}\right )}{10 \sqrt {\pi }} - \frac {b^{3} e^{- b^{2} x^{2}}}{10 \sqrt {\pi } x^{2}} + \frac {b e^{- b^{2} x^{2}}}{10 \sqrt {\pi } x^{4}} - \frac {\operatorname {erfc}{\left (b x \right )}}{5 x^{5}} \] Input:
integrate(erfc(b*x)/x**6,x)
Output:
b**5*expint(1, b**2*x**2)/(10*sqrt(pi)) - b**3*exp(-b**2*x**2)/(10*sqrt(pi )*x**2) + b*exp(-b**2*x**2)/(10*sqrt(pi)*x**4) - erfc(b*x)/(5*x**5)
Time = 0.07 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.33 \[ \int \frac {\text {erfc}(b x)}{x^6} \, dx=\frac {b^{5} \Gamma \left (-2, b^{2} x^{2}\right )}{5 \, \sqrt {\pi }} - \frac {\operatorname {erfc}\left (b x\right )}{5 \, x^{5}} \] Input:
integrate(erfc(b*x)/x^6,x, algorithm="maxima")
Output:
1/5*b^5*gamma(-2, b^2*x^2)/sqrt(pi) - 1/5*erfc(b*x)/x^5
\[ \int \frac {\text {erfc}(b x)}{x^6} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right )}{x^{6}} \,d x } \] Input:
integrate(erfc(b*x)/x^6,x, algorithm="giac")
Output:
integrate(erfc(b*x)/x^6, x)
Time = 3.93 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.81 \[ \int \frac {\text {erfc}(b x)}{x^6} \, dx=-\frac {\frac {\mathrm {erfc}\left (b\,x\right )}{5}+\frac {b^3\,x^3\,{\mathrm {e}}^{-b^2\,x^2}}{10\,\sqrt {\pi }}-\frac {b\,x\,{\mathrm {e}}^{-b^2\,x^2}}{10\,\sqrt {\pi }}}{x^5}-\frac {b^5\,\mathrm {ei}\left (-b^2\,x^2\right )}{10\,\sqrt {\pi }} \] Input:
int(erfc(b*x)/x^6,x)
Output:
- (erfc(b*x)/5 + (b^3*x^3*exp(-b^2*x^2))/(10*pi^(1/2)) - (b*x*exp(-b^2*x^2 ))/(10*pi^(1/2)))/x^5 - (b^5*ei(-b^2*x^2))/(10*pi^(1/2))
Time = 0.15 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.12 \[ \int \frac {\text {erfc}(b x)}{x^6} \, dx=\frac {-\sqrt {\pi }\, e^{b^{2} x^{2}} \mathit {ei} \left (-b^{2} x^{2}\right ) b^{5} x^{5}+2 e^{b^{2} x^{2}} \mathrm {erf}\left (b x \right ) \pi -2 e^{b^{2} x^{2}} \pi -\sqrt {\pi }\, b^{3} x^{3}+\sqrt {\pi }\, b x}{10 e^{b^{2} x^{2}} \pi \,x^{5}} \] Input:
int(erfc(b*x)/x^6,x)
Output:
( - sqrt(pi)*e**(b**2*x**2)*ei( - b**2*x**2)*b**5*x**5 + 2*e**(b**2*x**2)* erf(b*x)*pi - 2*e**(b**2*x**2)*pi - sqrt(pi)*b**3*x**3 + sqrt(pi)*b*x)/(10 *e**(b**2*x**2)*pi*x**5)