Integrand size = 18, antiderivative size = 53 \[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^2} \, dx=-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x}+\frac {1}{2} b \sqrt {\pi } \text {erfc}(b x)^2-\frac {b \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\sqrt {\pi }} \] Output:
-erfc(b*x)/exp(b^2*x^2)/x+1/2*b*Pi^(1/2)*erfc(b*x)^2-b*Ei(-2*b^2*x^2)/Pi^( 1/2)
Time = 0.01 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^2} \, dx=-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x}+\frac {1}{2} b \sqrt {\pi } \text {erfc}(b x)^2-\frac {b \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\sqrt {\pi }} \] Input:
Integrate[Erfc[b*x]/(E^(b^2*x^2)*x^2),x]
Output:
-(Erfc[b*x]/(E^(b^2*x^2)*x)) + (b*Sqrt[Pi]*Erfc[b*x]^2)/2 - (b*ExpIntegral Ei[-2*b^2*x^2])/Sqrt[Pi]
Time = 0.33 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {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.
\(\displaystyle \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^2} \, dx\) |
\(\Big \downarrow \) 6946 |
\(\displaystyle -2 b^2 \int e^{-b^2 x^2} \text {erfc}(b x)dx-\frac {2 b \int \frac {e^{-2 b^2 x^2}}{x}dx}{\sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x}\) |
\(\Big \downarrow \) 2639 |
\(\displaystyle -2 b^2 \int e^{-b^2 x^2} \text {erfc}(b x)dx-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x}-\frac {b \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\sqrt {\pi }}\) |
\(\Big \downarrow \) 6928 |
\(\displaystyle \sqrt {\pi } b \int \text {erfc}(b x)d\text {erfc}(b x)-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x}-\frac {b \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\sqrt {\pi }}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x}-\frac {b \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\sqrt {\pi }}+\frac {1}{2} \sqrt {\pi } b \text {erfc}(b x)^2\) |
Input:
Int[Erfc[b*x]/(E^(b^2*x^2)*x^2),x]
Output:
-(Erfc[b*x]/(E^(b^2*x^2)*x)) + (b*Sqrt[Pi]*Erfc[b*x]^2)/2 - (b*ExpIntegral Ei[-2*b^2*x^2])/Sqrt[Pi]
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[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]
\[\int \frac {\operatorname {erfc}\left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}}{x^{2}}d x\]
Input:
int(erfc(b*x)/exp(b^2*x^2)/x^2,x)
Output:
int(erfc(b*x)/exp(b^2*x^2)/x^2,x)
Time = 0.09 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.45 \[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^2} \, dx=-\frac {2 \, \pi ^{\frac {3}{2}} \sqrt {b^{2}} x \operatorname {erf}\left (\sqrt {b^{2}} x\right ) + 2 \, {\left (\pi - \pi \operatorname {erf}\left (b x\right )\right )} e^{\left (-b^{2} x^{2}\right )} - \sqrt {\pi } {\left (\pi b x \operatorname {erf}\left (b x\right )^{2} - 2 \, b x {\rm Ei}\left (-2 \, b^{2} x^{2}\right )\right )}}{2 \, \pi x} \] Input:
integrate(erfc(b*x)/exp(b^2*x^2)/x^2,x, algorithm="fricas")
Output:
-1/2*(2*pi^(3/2)*sqrt(b^2)*x*erf(sqrt(b^2)*x) + 2*(pi - pi*erf(b*x))*e^(-b ^2*x^2) - sqrt(pi)*(pi*b*x*erf(b*x)^2 - 2*b*x*Ei(-2*b^2*x^2)))/(pi*x)
\[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^2} \, dx=\int \frac {e^{- b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}}{x^{2}}\, dx \] Input:
integrate(erfc(b*x)/exp(b**2*x**2)/x**2,x)
Output:
Integral(exp(-b**2*x**2)*erfc(b*x)/x**2, x)
\[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^2} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{2}} \,d x } \] Input:
integrate(erfc(b*x)/exp(b^2*x^2)/x^2,x, algorithm="maxima")
Output:
integrate(erfc(b*x)*e^(-b^2*x^2)/x^2, x)
\[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^2} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{2}} \,d x } \] Input:
integrate(erfc(b*x)/exp(b^2*x^2)/x^2,x, algorithm="giac")
Output:
integrate(erfc(b*x)*e^(-b^2*x^2)/x^2, x)
Timed out. \[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^2} \, dx=\int \frac {{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right )}{x^2} \,d x \] Input:
int((exp(-b^2*x^2)*erfc(b*x))/x^2,x)
Output:
int((exp(-b^2*x^2)*erfc(b*x))/x^2, x)
\[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^2} \, dx=\frac {-\sqrt {\pi }\, e^{b^{2} x^{2}} \mathrm {erf}\left (b x \right ) b x -e^{b^{2} x^{2}} \left (\int \frac {\mathrm {erf}\left (b x \right )}{e^{b^{2} x^{2}} x^{2}}d x \right ) x -1}{e^{b^{2} x^{2}} x} \] Input:
int(erfc(b*x)/exp(b^2*x^2)/x^2,x)
Output:
( - (sqrt(pi)*e**(b**2*x**2)*erf(b*x)*b*x + e**(b**2*x**2)*int(erf(b*x)/(e **(b**2*x**2)*x**2),x)*x + 1))/(e**(b**2*x**2)*x)