Optimal. Leaf size=30 \[ \sqrt {4-3 x^2}-2 \tanh ^{-1}\left (\frac {1}{2} \sqrt {4-3 x^2}\right ) \]
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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} \[ \sqrt {4-3 x^2}-2 \tanh ^{-1}\left (\frac {1}{2} \sqrt {4-3 x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 206
Rule 266
Rubi steps
\begin {align*} \int \frac {\sqrt {4-3 x^2}}{x} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {4-3 x}}{x} \, dx,x,x^2\right )\\ &=\sqrt {4-3 x^2}+2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {4-3 x} x} \, dx,x,x^2\right )\\ &=\sqrt {4-3 x^2}-\frac {4}{3} \operatorname {Subst}\left (\int \frac {1}{\frac {4}{3}-\frac {x^2}{3}} \, dx,x,\sqrt {4-3 x^2}\right )\\ &=\sqrt {4-3 x^2}-2 \tanh ^{-1}\left (\frac {1}{2} \sqrt {4-3 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 1.00 \[ \sqrt {4-3 x^2}-2 \tanh ^{-1}\left (\frac {1}{2} \sqrt {4-3 x^2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 28, normalized size = 0.93 \[ \sqrt {-3 \, x^{2} + 4} + 2 \, \log \left (\frac {\sqrt {-3 \, x^{2} + 4} - 2}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.95, size = 38, normalized size = 1.27 \[ \sqrt {-3 \, x^{2} + 4} - \log \left (\sqrt {-3 \, x^{2} + 4} + 2\right ) + \log \left (-\sqrt {-3 \, x^{2} + 4} + 2\right ) \]
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 \[ -2 \arctanh \left (\frac {2}{\sqrt {-3 x^{2}+4}}\right )+\sqrt {-3 x^{2}+4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.19, size = 35, normalized size = 1.17 \[ \sqrt {-3 \, x^{2} + 4} - 2 \, \log \left (\frac {4 \, \sqrt {-3 \, x^{2} + 4}}{{\left | x \right |}} + \frac {8}{{\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 37, normalized size = 1.23 \[ 2\,\ln \left (\sqrt {\frac {4}{3\,x^2}-1}-\frac {2\,\sqrt {3}\,\sqrt {\frac {1}{x^2}}}{3}\right )+\sqrt {3}\,\sqrt {\frac {4}{3}-x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.56, size = 75, normalized size = 2.50 \[ \begin {cases} i \sqrt {3 x^{2} - 4} - 2 \log {\relax (x )} + \log {\left (x^{2} \right )} + 2 i \operatorname {asin}{\left (\frac {2 \sqrt {3}}{3 x} \right )} & \text {for}\: \frac {3 \left |{x^{2}}\right |}{4} > 1 \\\sqrt {4 - 3 x^{2}} + \log {\left (x^{2} \right )} - 2 \log {\left (\sqrt {1 - \frac {3 x^{2}}{4}} + 1 \right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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