Optimal. Leaf size=30 \[ \sqrt {9-4 x^2}-3 \tanh ^{-1}\left (\frac {1}{3} \sqrt {9-4 x^2}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 30, 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} \[ \sqrt {9-4 x^2}-3 \tanh ^{-1}\left (\frac {1}{3} \sqrt {9-4 x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 206
Rule 266
Rubi steps
\begin {align*} \int \frac {\sqrt {9-4 x^2}}{x} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {9-4 x}}{x} \, dx,x,x^2\right )\\ &=\sqrt {9-4 x^2}+\frac {9}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {9-4 x} x} \, dx,x,x^2\right )\\ &=\sqrt {9-4 x^2}-\frac {9}{4} \operatorname {Subst}\left (\int \frac {1}{\frac {9}{4}-\frac {x^2}{4}} \, dx,x,\sqrt {9-4 x^2}\right )\\ &=\sqrt {9-4 x^2}-3 \tanh ^{-1}\left (\frac {1}{3} \sqrt {9-4 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 30, normalized size = 1.00 \[ \sqrt {9-4 x^2}-3 \tanh ^{-1}\left (\frac {1}{3} \sqrt {9-4 x^2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 28, normalized size = 0.93 \[ \sqrt {-4 \, x^{2} + 9} + 3 \, \log \left (\frac {\sqrt {-4 \, x^{2} + 9} - 3}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.18, size = 40, normalized size = 1.33 \[ \sqrt {-4 \, x^{2} + 9} - \frac {3}{2} \, \log \left (\sqrt {-4 \, x^{2} + 9} + 3\right ) + \frac {3}{2} \, \log \left (-\sqrt {-4 \, x^{2} + 9} + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 25, normalized size = 0.83 \[ -3 \arctanh \left (\frac {3}{\sqrt {-4 x^{2}+9}}\right )+\sqrt {-4 x^{2}+9} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.86, size = 35, normalized size = 1.17 \[ \sqrt {-4 \, x^{2} + 9} - 3 \, \log \left (\frac {6 \, \sqrt {-4 \, x^{2} + 9}}{{\left | x \right |}} + \frac {18}{{\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.55, size = 32, normalized size = 1.07 \[ 3\,\ln \left (\sqrt {\frac {9}{4\,x^2}-1}-\frac {3\,\sqrt {\frac {1}{x^2}}}{2}\right )+2\,\sqrt {\frac {9}{4}-x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.32, size = 76, normalized size = 2.53 \[ \begin {cases} i \sqrt {4 x^{2} - 9} - 3 \log {\relax (x )} + \frac {3 \log {\left (x^{2} \right )}}{2} + 3 i \operatorname {asin}{\left (\frac {3}{2 x} \right )} & \text {for}\: \frac {4 \left |{x^{2}}\right |}{9} > 1 \\\sqrt {9 - 4 x^{2}} + \frac {3 \log {\left (x^{2} \right )}}{2} - 3 \log {\left (\sqrt {1 - \frac {4 x^{2}}{9}} + 1 \right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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