Integrand size = 40, antiderivative size = 60 \[ \int \left (\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^3}+\frac {b^2 e^{-b^2 x^2} \text {erfc}(b x)}{x}\right ) \, dx=\frac {b e^{-2 b^2 x^2}}{\sqrt {\pi } x}+\sqrt {2} b^2 \text {erf}\left (\sqrt {2} b x\right )-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{2 x^2} \] Output:
b/exp(2*b^2*x^2)/Pi^(1/2)/x+2^(1/2)*b^2*erf(2^(1/2)*b*x)-1/2*erfc(b*x)/exp (b^2*x^2)/x^2
Time = 0.06 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00 \[ \int \left (\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^3}+\frac {b^2 e^{-b^2 x^2} \text {erfc}(b x)}{x}\right ) \, dx=\frac {b e^{-2 b^2 x^2}}{\sqrt {\pi } x}+\sqrt {2} b^2 \text {erf}\left (\sqrt {2} b x\right )-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{2 x^2} \] Input:
Integrate[Erfc[b*x]/(E^(b^2*x^2)*x^3) + (b^2*Erfc[b*x])/(E^(b^2*x^2)*x),x]
Output:
b/(E^(2*b^2*x^2)*Sqrt[Pi]*x) + Sqrt[2]*b^2*Erf[Sqrt[2]*b*x] - Erfc[b*x]/(2 *E^(b^2*x^2)*x^2)
Time = 0.27 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {2009}
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.
\(\displaystyle \int \left (\frac {b^2 e^{-b^2 x^2} \text {erfc}(b x)}{x}+\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^3}\right ) \, dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \sqrt {2} b^2 \text {erf}\left (\sqrt {2} b x\right )-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{2 x^2}+\frac {b e^{-2 b^2 x^2}}{\sqrt {\pi } x}\) |
Input:
Int[Erfc[b*x]/(E^(b^2*x^2)*x^3) + (b^2*Erfc[b*x])/(E^(b^2*x^2)*x),x]
Output:
b/(E^(2*b^2*x^2)*Sqrt[Pi]*x) + Sqrt[2]*b^2*Erf[Sqrt[2]*b*x] - Erfc[b*x]/(2 *E^(b^2*x^2)*x^2)
Time = 1.74 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.40
method | result | size |
default | \(\frac {-\frac {b \,{\mathrm e}^{-b^{2} x^{2}}}{2 x^{2}}+\frac {\operatorname {erf}\left (b x \right ) b \,{\mathrm e}^{-b^{2} x^{2}}}{2 x^{2}}-\frac {b^{3} \left (-\frac {{\mathrm e}^{-2 b^{2} x^{2}}}{b x}-\sqrt {2}\, \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {2}\, b x \right )\right )}{\sqrt {\pi }}}{b}\) | \(84\) |
Input:
int(erfc(b*x)/exp(b^2*x^2)/x^3+b^2*erfc(b*x)/exp(b^2*x^2)/x,x,method=_RETU RNVERBOSE)
Output:
(-1/2*b/exp(b^2*x^2)/x^2+1/2*erf(b*x)*b/exp(b^2*x^2)/x^2-1/Pi^(1/2)*b^3*(- 1/exp(b^2*x^2)^2/b/x-2^(1/2)*Pi^(1/2)*erf(2^(1/2)*b*x)))/b
Time = 0.09 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.18 \[ \int \left (\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^3}+\frac {b^2 e^{-b^2 x^2} \text {erfc}(b x)}{x}\right ) \, dx=\frac {2 \, \sqrt {2} \pi \sqrt {b^{2}} b x^{2} \operatorname {erf}\left (\sqrt {2} \sqrt {b^{2}} x\right ) + 2 \, \sqrt {\pi } b x e^{\left (-2 \, b^{2} x^{2}\right )} - {\left (\pi - \pi \operatorname {erf}\left (b x\right )\right )} e^{\left (-b^{2} x^{2}\right )}}{2 \, \pi x^{2}} \] Input:
integrate(erfc(b*x)/exp(b^2*x^2)/x^3+b^2*erfc(b*x)/exp(b^2*x^2)/x,x, algor ithm="fricas")
Output:
1/2*(2*sqrt(2)*pi*sqrt(b^2)*b*x^2*erf(sqrt(2)*sqrt(b^2)*x) + 2*sqrt(pi)*b* x*e^(-2*b^2*x^2) - (pi - pi*erf(b*x))*e^(-b^2*x^2))/(pi*x^2)
\[ \int \left (\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^3}+\frac {b^2 e^{-b^2 x^2} \text {erfc}(b x)}{x}\right ) \, dx=\int \frac {\left (b^{2} x^{2} + 1\right ) e^{- b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}}{x^{3}}\, dx \] Input:
integrate(erfc(b*x)/exp(b**2*x**2)/x**3+b**2*erfc(b*x)/exp(b**2*x**2)/x,x)
Output:
Integral((b**2*x**2 + 1)*exp(-b**2*x**2)*erfc(b*x)/x**3, x)
\[ \int \left (\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^3}+\frac {b^2 e^{-b^2 x^2} \text {erfc}(b x)}{x}\right ) \, dx=\int { \frac {b^{2} \operatorname {erfc}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x} + \frac {\operatorname {erfc}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{3}} \,d x } \] Input:
integrate(erfc(b*x)/exp(b^2*x^2)/x^3+b^2*erfc(b*x)/exp(b^2*x^2)/x,x, algor ithm="maxima")
Output:
integrate(b^2*erfc(b*x)*e^(-b^2*x^2)/x + erfc(b*x)*e^(-b^2*x^2)/x^3, x)
\[ \int \left (\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^3}+\frac {b^2 e^{-b^2 x^2} \text {erfc}(b x)}{x}\right ) \, dx=\int { \frac {b^{2} \operatorname {erfc}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x} + \frac {\operatorname {erfc}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{3}} \,d x } \] Input:
integrate(erfc(b*x)/exp(b^2*x^2)/x^3+b^2*erfc(b*x)/exp(b^2*x^2)/x,x, algor ithm="giac")
Output:
integrate(b^2*erfc(b*x)*e^(-b^2*x^2)/x + erfc(b*x)*e^(-b^2*x^2)/x^3, x)
Timed out. \[ \int \left (\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^3}+\frac {b^2 e^{-b^2 x^2} \text {erfc}(b x)}{x}\right ) \, dx=\int \frac {{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right )}{x^3}+\frac {b^2\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right )}{x} \,d x \] Input:
int((exp(-b^2*x^2)*erfc(b*x))/x^3 + (b^2*exp(-b^2*x^2)*erfc(b*x))/x,x)
Output:
int((exp(-b^2*x^2)*erfc(b*x))/x^3 + (b^2*exp(-b^2*x^2)*erfc(b*x))/x, x)
\[ \int \left (\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^3}+\frac {b^2 e^{-b^2 x^2} \text {erfc}(b x)}{x}\right ) \, dx=\frac {-2 e^{b^{2} x^{2}} \left (\int \frac {\mathrm {erf}\left (b x \right )}{e^{b^{2} x^{2}} x^{3}}d x \right ) x^{2}-2 e^{b^{2} x^{2}} \left (\int \frac {\mathrm {erf}\left (b x \right )}{e^{b^{2} x^{2}} x}d x \right ) b^{2} x^{2}-1}{2 e^{b^{2} x^{2}} x^{2}} \] Input:
int(erfc(b*x)/exp(b^2*x^2)/x^3+b^2*erfc(b*x)/exp(b^2*x^2)/x,x)
Output:
( - 2*e**(b**2*x**2)*int(erf(b*x)/(e**(b**2*x**2)*x**3),x)*x**2 - 2*e**(b* *2*x**2)*int(erf(b*x)/(e**(b**2*x**2)*x),x)*b**2*x**2 - 1)/(2*e**(b**2*x** 2)*x**2)