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 = 0.01, 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 = {272, 52, 65,
212} \begin {gather*} \sqrt {4-3 x^2}-2 \tanh ^{-1}\left (\frac {1}{2} \sqrt {4-3 x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 212
Rule 272
Rubi steps
\begin {align*} \int \frac {\sqrt {4-3 x^2}}{x} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {4-3 x}}{x} \, dx,x,x^2\right )\\ &=\sqrt {4-3 x^2}+2 \text {Subst}\left (\int \frac {1}{\sqrt {4-3 x} x} \, dx,x,x^2\right )\\ &=\sqrt {4-3 x^2}-\frac {4}{3} \text {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 = 0.02, size = 30, normalized size = 1.00 \begin {gather*} \sqrt {4-3 x^2}-2 \tanh ^{-1}\left (\frac {1}{2} \sqrt {4-3 x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 25, normalized size = 0.83
method | result | size |
default | \(\sqrt {-3 x^{2}+4}-2 \arctanh \left (\frac {2}{\sqrt {-3 x^{2}+4}}\right )\) | \(25\) |
trager | \(\sqrt {-3 x^{2}+4}-2 \ln \left (\frac {\sqrt {-3 x^{2}+4}+2}{x}\right )\) | \(29\) |
meijerg | \(-\frac {-2 \left (2-4 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (3\right )+i \pi \right ) \sqrt {\pi }+4 \sqrt {\pi }-4 \sqrt {\pi }\, \sqrt {1-\frac {3 x^{2}}{4}}+4 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {1-\frac {3 x^{2}}{4}}}{2}\right )}{2 \sqrt {\pi }}\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 3.52, size = 35, normalized size = 1.17 \begin {gather*} \sqrt {-3 \, x^{2} + 4} - 2 \, \log \left (\frac {4 \, \sqrt {-3 \, x^{2} + 4}}{{\left | x \right |}} + \frac {8}{{\left | x \right |}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.70, size = 28, normalized size = 0.93 \begin {gather*} \sqrt {-3 \, x^{2} + 4} + 2 \, \log \left (\frac {\sqrt {-3 \, x^{2} + 4} - 2}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.69, size = 73, normalized size = 2.43 \begin {gather*} \begin {cases} i \sqrt {3 x^{2} - 4} - 2 \log {\left (x \right )} + \log {\left (x^{2} \right )} + 2 i \operatorname {asin}{\left (\frac {2 \sqrt {3}}{3 x} \right )} & \text {for}\: \left |{x^{2}}\right | > \frac {4}{3} \\\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.38, size = 38, normalized size = 1.27 \begin {gather*} \sqrt {-3 \, x^{2} + 4} - \log \left (\sqrt {-3 \, x^{2} + 4} + 2\right ) + \log \left (-\sqrt {-3 \, x^{2} + 4} + 2\right ) \end {gather*}
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 \begin {gather*} 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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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