Optimal. Leaf size=37 \[ \sqrt {3-x^2}-\sqrt {3} \tanh ^{-1}\left (\frac {\sqrt {3-x^2}}{\sqrt {3}}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {266, 50, 63, 206} \[ \sqrt {3-x^2}-\sqrt {3} \tanh ^{-1}\left (\frac {\sqrt {3-x^2}}{\sqrt {3}}\right ) \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 206
Rule 266
Rubi steps
\begin {align*} \int \frac {\sqrt {3-x^2}}{x} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {3-x}}{x} \, dx,x,x^2\right )\\ &=\sqrt {3-x^2}+\frac {3}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {3-x} x} \, dx,x,x^2\right )\\ &=\sqrt {3-x^2}-3 \operatorname {Subst}\left (\int \frac {1}{3-x^2} \, dx,x,\sqrt {3-x^2}\right )\\ &=\sqrt {3-x^2}-\sqrt {3} \tanh ^{-1}\left (\frac {\sqrt {3-x^2}}{\sqrt {3}}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 33, normalized size = 0.89 \[ \sqrt {3-x^2}-\sqrt {3} \tanh ^{-1}\left (\sqrt {1-\frac {x^2}{3}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 40, normalized size = 1.08 \[ \frac {1}{2} \, \sqrt {3} \log \left (-\frac {x^{2} + 2 \, \sqrt {3} \sqrt {-x^{2} + 3} - 6}{x^{2}}\right ) + \sqrt {-x^{2} + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.01, size = 47, normalized size = 1.27 \[ \frac {1}{2} \, \sqrt {3} \log \left (\frac {\sqrt {3} - \sqrt {-x^{2} + 3}}{\sqrt {3} + \sqrt {-x^{2} + 3}}\right ) + \sqrt {-x^{2} + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 30, normalized size = 0.81 \[ -\sqrt {3}\, \arctanh \left (\frac {\sqrt {3}}{\sqrt {-x^{2}+3}}\right )+\sqrt {-x^{2}+3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.20, size = 41, normalized size = 1.11 \[ -\sqrt {3} \log \left (\frac {2 \, \sqrt {3} \sqrt {-x^{2} + 3}}{{\left | x \right |}} + \frac {6}{{\left | x \right |}}\right ) + \sqrt {-x^{2} + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 35, normalized size = 0.95 \[ \sqrt {3}\,\ln \left (\sqrt {\frac {3}{x^2}-1}-\sqrt {3}\,\sqrt {\frac {1}{x^2}}\right )+\sqrt {3-x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.38, size = 88, normalized size = 2.38 \[ \begin {cases} i \sqrt {x^{2} - 3} - \sqrt {3} \log {\relax (x )} + \frac {\sqrt {3} \log {\left (x^{2} \right )}}{2} + \sqrt {3} i \operatorname {asin}{\left (\frac {\sqrt {3}}{x} \right )} & \text {for}\: \frac {\left |{x^{2}}\right |}{3} > 1 \\\sqrt {3 - x^{2}} + \frac {\sqrt {3} \log {\left (x^{2} \right )}}{2} - \sqrt {3} \log {\left (\sqrt {1 - \frac {x^{2}}{3}} + 1 \right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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