Optimal. Leaf size=52 \[ -\frac {e^{-b^2 x^2} \text {erf}(b x)}{x}+\frac {b \text {Ei}\left (-2 b^2 x^2\right )}{\sqrt {\pi }}-\frac {1}{2} \sqrt {\pi } b \text {erf}(b x)^2 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6391, 6373, 30, 2210} \[ -\frac {e^{-b^2 x^2} \text {Erf}(b x)}{x}+\frac {b \text {Ei}\left (-2 b^2 x^2\right )}{\sqrt {\pi }}-\frac {1}{2} \sqrt {\pi } b \text {Erf}(b x)^2 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 2210
Rule 6373
Rule 6391
Rubi steps
\begin {align*} \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^2} \, dx &=-\frac {e^{-b^2 x^2} \text {erf}(b x)}{x}-\left (2 b^2\right ) \int e^{-b^2 x^2} \text {erf}(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 {erf}(b x)}{x}+\frac {b \text {Ei}\left (-2 b^2 x^2\right )}{\sqrt {\pi }}-\left (b \sqrt {\pi }\right ) \operatorname {Subst}(\int x \, dx,x,\text {erf}(b x))\\ &=-\frac {e^{-b^2 x^2} \text {erf}(b x)}{x}-\frac {1}{2} b \sqrt {\pi } \text {erf}(b x)^2+\frac {b \text {Ei}\left (-2 b^2 x^2\right )}{\sqrt {\pi }}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 52, normalized size = 1.00 \[ -\frac {e^{-b^2 x^2} \text {erf}(b x)}{x}+\frac {b \text {Ei}\left (-2 b^2 x^2\right )}{\sqrt {\pi }}-\frac {1}{2} \sqrt {\pi } b \text {erf}(b x)^2 \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 53, normalized size = 1.02 \[ -\frac {2 \, \pi \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} + \sqrt {\pi } {\left (\pi b x \operatorname {erf}\left (b x\right )^{2} - 2 \, b x {\rm Ei}\left (-2 \, b^{2} x^{2}\right )\right )}}{2 \, \pi x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.20, size = 0, normalized size = 0.00 \[ \int \frac {\erf \left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erf}\left (b\,x\right )}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{- b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________