Optimal. Leaf size=60 \[ \frac {b e^{-2 b^2 x^2}}{\sqrt {\pi } x}+\sqrt {2} b^2 \text {Erf}\left (\sqrt {2} b x\right )-\frac {e^{-b^2 x^2} \text {Erfc}(b x)}{2 x^2} \]
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Rubi [A]
time = 0.10, antiderivative size = 60, 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 = {6527, 2245,
2236} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2236
Rule 2245
Rule 6527
Rubi steps
\begin {align*} \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 &=b^2 \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x} \, dx+\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}-\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}+\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}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 60, normalized size = 1.00 \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.81, size = 84, normalized size = 1.40
method | result | size |
default | \(\frac {-\frac {b \,{\mathrm e}^{-b^{2} x^{2}}}{2 x^{2}}+\frac {\erf \left (b x \right ) b \,{\mathrm e}^{-b^{2} x^{2}}}{2 x^{2}}-\frac {b^{3} \left (-\frac {{\mathrm e}^{-2 b^{2} x^{2}}}{b x}-\sqrt {2}\, \sqrt {\pi }\, \erf \left (b x \sqrt {2}\right )\right )}{\sqrt {\pi }}}{b}\) | \(84\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 71, normalized size = 1.18 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b^{2} x^{2} + 1\right ) e^{- b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}}{x^{3}}\, dx \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 [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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