Optimal. Leaf size=62 \[ -\frac {e^{-b^2 x^2} \text {erf}(b x)}{2 x^2}-\sqrt {2} b^2 \text {erf}\left (\sqrt {2} b x\right )-\frac {b e^{-2 b^2 x^2}}{\sqrt {\pi } x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {6391, 2214, 2205} \[ -\frac {e^{-b^2 x^2} \text {Erf}(b x)}{2 x^2}-\sqrt {2} b^2 \text {Erf}\left (\sqrt {2} b x\right )-\frac {b e^{-2 b^2 x^2}}{\sqrt {\pi } x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2205
Rule 2214
Rule 6391
Rubi steps
\begin {align*} \int \left (\frac {e^{-b^2 x^2} \text {erf}(b x)}{x^3}+\frac {b^2 e^{-b^2 x^2} \text {erf}(b x)}{x}\right ) \, dx &=b^2 \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x} \, dx+\int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^3} \, dx\\ &=-\frac {e^{-b^2 x^2} \text {erf}(b x)}{2 x^2}+\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 {erf}(b x)}{2 x^2}-\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}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{2 x^2}-\sqrt {2} b^2 \text {erf}\left (\sqrt {2} b x\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.13, size = 62, normalized size = 1.00 \[ -\frac {e^{-b^2 x^2} \text {erf}(b x)}{2 x^2}-\sqrt {2} b^2 \text {erf}\left (\sqrt {2} b x\right )-\frac {b e^{-2 b^2 x^2}}{\sqrt {\pi } x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.49, size = 66, normalized size = 1.06 \[ -\frac {2 \, \sqrt {2} \pi \sqrt {b^{2}} b x^{2} \operatorname {erf}\left (\sqrt {2} \sqrt {b^{2}} x\right ) + 2 \, \sqrt {\pi } b x e^{\left (-2 \, b^{2} x^{2}\right )} + \pi \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{2 \, \pi x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b^{2} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x} + \frac {\operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.24, size = 67, normalized size = 1.08 \[ \frac {-\frac {\erf \left (b x \right ) b \,{\mathrm e}^{-b^{2} x^{2}}}{2 x^{2}}+\frac {b^{3} \left (-\frac {{\mathrm e}^{-2 b^{2} x^{2}}}{b x}-\sqrt {2}\, \sqrt {\pi }\, \erf \left (b x \sqrt {2}\right )\right )}{\sqrt {\pi }}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {-\frac {\sqrt {2} b^{2} \sqrt {x^{2}} \Gamma \left (-\frac {1}{2}, 2 \, b^{2} x^{2}\right )}{2 \, x}}{\sqrt {\pi }} - \frac {\operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.22, size = 52, normalized size = 0.84 \[ -\frac {\frac {{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erf}\left (b\,x\right )}{2}+\frac {b\,x\,{\mathrm {e}}^{-2\,b^2\,x^2}}{\sqrt {\pi }}}{x^2}-\sqrt {2}\,b^2\,\mathrm {erf}\left (\sqrt {2}\,b\,x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (b^{2} x^{2} + 1\right ) e^{- b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________