Integrand size = 10, antiderivative size = 67 \[ \int \frac {\text {erfc}(b x)^2}{x^3} \, dx=\frac {2 b e^{-b^2 x^2} \text {erfc}(b x)}{\sqrt {\pi } x}-b^2 \text {erfc}(b x)^2-\frac {\text {erfc}(b x)^2}{2 x^2}+\frac {2 b^2 \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\pi } \]
2*b^2*Ei(-2*b^2*x^2)/Pi-b^2*erfc(b*x)^2-1/2*erfc(b*x)^2/x^2+2*b*erfc(b*x)/ exp(b^2*x^2)/x/Pi^(1/2)
Time = 0.02 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.94 \[ \int \frac {\text {erfc}(b x)^2}{x^3} \, dx=\frac {2 b e^{-b^2 x^2} \text {erfc}(b x)}{\sqrt {\pi } x}+\left (-b^2-\frac {1}{2 x^2}\right ) \text {erfc}(b x)^2+\frac {2 b^2 \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\pi } \]
(2*b*Erfc[b*x])/(E^(b^2*x^2)*Sqrt[Pi]*x) + (-b^2 - 1/(2*x^2))*Erfc[b*x]^2 + (2*b^2*ExpIntegralEi[-2*b^2*x^2])/Pi
Time = 0.41 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.12, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6919, 6946, 2639, 6928, 15}
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.
\(\displaystyle \int \frac {\text {erfc}(b x)^2}{x^3} \, dx\) |
\(\Big \downarrow \) 6919 |
\(\displaystyle -\frac {2 b \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^2}dx}{\sqrt {\pi }}-\frac {\text {erfc}(b x)^2}{2 x^2}\) |
\(\Big \downarrow \) 6946 |
\(\displaystyle -\frac {2 b \left (-2 b^2 \int e^{-b^2 x^2} \text {erfc}(b x)dx-\frac {2 b \int \frac {e^{-2 b^2 x^2}}{x}dx}{\sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x}\right )}{\sqrt {\pi }}-\frac {\text {erfc}(b x)^2}{2 x^2}\) |
\(\Big \downarrow \) 2639 |
\(\displaystyle -\frac {2 b \left (-2 b^2 \int e^{-b^2 x^2} \text {erfc}(b x)dx-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x}-\frac {b \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\sqrt {\pi }}\right )}{\sqrt {\pi }}-\frac {\text {erfc}(b x)^2}{2 x^2}\) |
\(\Big \downarrow \) 6928 |
\(\displaystyle -\frac {2 b \left (\sqrt {\pi } b \int \text {erfc}(b x)d\text {erfc}(b x)-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x}-\frac {b \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\sqrt {\pi }}\right )}{\sqrt {\pi }}-\frac {\text {erfc}(b x)^2}{2 x^2}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {2 b \left (-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x}-\frac {b \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\sqrt {\pi }}+\frac {1}{2} \sqrt {\pi } b \text {erfc}(b x)^2\right )}{\sqrt {\pi }}-\frac {\text {erfc}(b x)^2}{2 x^2}\) |
-1/2*Erfc[b*x]^2/x^2 - (2*b*(-(Erfc[b*x]/(E^(b^2*x^2)*x)) + (b*Sqrt[Pi]*Er fc[b*x]^2)/2 - (b*ExpIntegralEi[-2*b^2*x^2])/Sqrt[Pi]))/Sqrt[Pi]
3.2.29.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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[Erfc[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(Erfc[b*x]^2 /(m + 1)), x] + Simp[4*(b/(Sqrt[Pi]*(m + 1))) Int[(x^(m + 1)*Erfc[b*x])/E ^(b^2*x^2), x], x] /; FreeQ[b, x] && (IGtQ[m, 0] || ILtQ[(m + 1)/2, 0])
Int[E^((c_.) + (d_.)*(x_)^2)*Erfc[(b_.)*(x_)]^(n_.), x_Symbol] :> Simp[(-E^ c)*(Sqrt[Pi]/(2*b)) Subst[Int[x^n, x], x, Erfc[b*x]], x] /; FreeQ[{b, c, d, n}, x] && EqQ[d, -b^2]
Int[E^((c_.) + (d_.)*(x_)^2)*Erfc[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[x^(m + 1)*E^(c + d*x^2)*(Erfc[a + b*x]/(m + 1)), x] + (-Simp[2*(d/( m + 1)) Int[x^(m + 2)*E^(c + d*x^2)*Erfc[a + b*x], x], x] + Simp[2*(b/((m + 1)*Sqrt[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] && ILtQ[m, -1]
\[\int \frac {\operatorname {erfc}\left (b x \right )^{2}}{x^{3}}d x\]
Time = 0.26 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.46 \[ \int \frac {\text {erfc}(b x)^2}{x^3} \, dx=-\frac {\pi - 4 \, \pi \sqrt {b^{2}} b x^{2} \operatorname {erf}\left (\sqrt {b^{2}} x\right ) - 4 \, b^{2} x^{2} {\rm Ei}\left (-2 \, b^{2} x^{2}\right ) + {\left (\pi + 2 \, \pi b^{2} x^{2}\right )} \operatorname {erf}\left (b x\right )^{2} + 4 \, \sqrt {\pi } {\left (b x \operatorname {erf}\left (b x\right ) - b x\right )} e^{\left (-b^{2} x^{2}\right )} - 2 \, \pi \operatorname {erf}\left (b x\right )}{2 \, \pi x^{2}} \]
-1/2*(pi - 4*pi*sqrt(b^2)*b*x^2*erf(sqrt(b^2)*x) - 4*b^2*x^2*Ei(-2*b^2*x^2 ) + (pi + 2*pi*b^2*x^2)*erf(b*x)^2 + 4*sqrt(pi)*(b*x*erf(b*x) - b*x)*e^(-b ^2*x^2) - 2*pi*erf(b*x))/(pi*x^2)
\[ \int \frac {\text {erfc}(b x)^2}{x^3} \, dx=\int \frac {\operatorname {erfc}^{2}{\left (b x \right )}}{x^{3}}\, dx \]
\[ \int \frac {\text {erfc}(b x)^2}{x^3} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right )^{2}}{x^{3}} \,d x } \]
\[ \int \frac {\text {erfc}(b x)^2}{x^3} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right )^{2}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\text {erfc}(b x)^2}{x^3} \, dx=\int \frac {{\mathrm {erfc}\left (b\,x\right )}^2}{x^3} \,d x \]