Optimal. Leaf size=86 \[ -b^2 \text {Int}\left (\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x},x\right )+\sqrt {2} b^2 \text {erf}\left (\sqrt {2} b x\right )-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{2 x^2}+\frac {b e^{-2 b^2 x^2}}{\sqrt {\pi } x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {e^{-b^2 x^2} \text {Erfc}(b x)}{x^3} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^3} \, dx &=-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{2 x^2}-b^2 \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x} \, dx-\frac {b \int \frac {e^{-2 b^2 x^2}}{x^2} \, dx}{\sqrt {\pi }}\\ &=\frac {b e^{-2 b^2 x^2}}{\sqrt {\pi } x}-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{2 x^2}-b^2 \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x} \, dx+\frac {\left (4 b^3\right ) \int e^{-2 b^2 x^2} \, dx}{\sqrt {\pi }}\\ &=\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}-b^2 \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.30, size = 0, normalized size = 0.00 \[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^3} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (\operatorname {erf}\left (b x\right ) - 1\right )} e^{\left (-b^{2} x^{2}\right )}}{x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erfc}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.26, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {erfc}\left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erfc}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right )}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{- b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________