Optimal. Leaf size=40 \[ \frac {b e^{-b^2 x^2}}{\sqrt {\pi } x}+b^2 \text {Erf}(b x)-\frac {\text {Erfc}(b x)}{2 x^2} \]
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Rubi [A]
time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6497, 2245,
2236} \begin {gather*} b^2 \text {Erf}(b x)+\frac {b e^{-b^2 x^2}}{\sqrt {\pi } x}-\frac {\text {Erfc}(b x)}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2236
Rule 2245
Rule 6497
Rubi steps
\begin {align*} \int \frac {\text {erfc}(b x)}{x^3} \, dx &=-\frac {\text {erfc}(b x)}{2 x^2}-\frac {b \int \frac {e^{-b^2 x^2}}{x^2} \, dx}{\sqrt {\pi }}\\ &=\frac {b e^{-b^2 x^2}}{\sqrt {\pi } x}-\frac {\text {erfc}(b x)}{2 x^2}+\frac {\left (2 b^3\right ) \int e^{-b^2 x^2} \, dx}{\sqrt {\pi }}\\ &=\frac {b e^{-b^2 x^2}}{\sqrt {\pi } x}+b^2 \text {erf}(b x)-\frac {\text {erfc}(b x)}{2 x^2}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 40, normalized size = 1.00 \begin {gather*} \frac {b e^{-b^2 x^2}}{\sqrt {\pi } x}+b^2 \text {Erf}(b x)-\frac {\text {Erfc}(b x)}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 51, normalized size = 1.28
method | result | size |
derivativedivides | \(b^{2} \left (-\frac {\mathrm {erfc}\left (b x \right )}{2 b^{2} x^{2}}-\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}}}{x b}-\erf \left (b x \right ) \sqrt {\pi }}{\sqrt {\pi }}\right )\) | \(51\) |
default | \(b^{2} \left (-\frac {\mathrm {erfc}\left (b x \right )}{2 b^{2} x^{2}}-\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}}}{x b}-\erf \left (b x \right ) \sqrt {\pi }}{\sqrt {\pi }}\right )\) | \(51\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 35, normalized size = 0.88 \begin {gather*} \frac {b^{2} \sqrt {x^{2}} \Gamma \left (-\frac {1}{2}, b^{2} x^{2}\right )}{2 \, \sqrt {\pi } x} - \frac {\operatorname {erfc}\left (b x\right )}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 43, normalized size = 1.08 \begin {gather*} -\frac {\pi - 2 \, \sqrt {\pi } b x e^{\left (-b^{2} x^{2}\right )} - {\left (\pi + 2 \, \pi b^{2} x^{2}\right )} \operatorname {erf}\left (b x\right )}{2 \, \pi x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.22, size = 34, normalized size = 0.85 \begin {gather*} - b^{2} \operatorname {erfc}{\left (b x \right )} + \frac {b e^{- b^{2} x^{2}}}{\sqrt {\pi } x} - \frac {\operatorname {erfc}{\left (b x \right )}}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.13, size = 38, normalized size = 0.95 \begin {gather*} -b^2\,\mathrm {erfc}\left (b\,x\right )-\frac {\frac {\mathrm {erfc}\left (b\,x\right )}{2}-\frac {b\,x\,{\mathrm {e}}^{-b^2\,x^2}}{\sqrt {\pi }}}{x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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