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 = 0.01, 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 = {272, 52, 65,
212} \begin {gather*} \sqrt {3-x^2}-\sqrt {3} \tanh ^{-1}\left (\frac {\sqrt {3-x^2}}{\sqrt {3}}\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 {3-x^2}}{x} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {3-x}}{x} \, dx,x,x^2\right )\\ &=\sqrt {3-x^2}+\frac {3}{2} \text {Subst}\left (\int \frac {1}{\sqrt {3-x} x} \, dx,x,x^2\right )\\ &=\sqrt {3-x^2}-3 \text {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 = 0.03, size = 33, normalized size = 0.89 \begin {gather*} \sqrt {3-x^2}-\sqrt {3} \tanh ^{-1}\left (\sqrt {1-\frac {x^2}{3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 2.61, size = 84, normalized size = 2.27 \begin {gather*} \text {Piecewise}\left [\left \{\left \{-\sqrt {3} \text {Log}\left [x\right ]+I \sqrt {3} \text {ArcSin}\left [\frac {\sqrt {3}}{x}\right ]+I \sqrt {-3+x^2}+\frac {\sqrt {3} \text {Log}\left [x^2\right ]}{2},\text {Abs}\left [x^2\right ]>3\right \}\right \},\sqrt {3-x^2}-\sqrt {3} \text {Log}\left [1+\sqrt {1-\frac {x^2}{3}}\right ]+\frac {\sqrt {3} \text {Log}\left [x^2\right ]}{2}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.09, size = 30, normalized size = 0.81
method | result | size |
default | \(\sqrt {-x^{2}+3}-\sqrt {3}\, \arctanh \left (\frac {\sqrt {3}}{\sqrt {-x^{2}+3}}\right )\) | \(30\) |
trager | \(\sqrt {-x^{2}+3}-\RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (\frac {\sqrt {-x^{2}+3}+\RootOf \left (\textit {\_Z}^{2}-3\right )}{x}\right )\) | \(40\) |
meijerg | \(-\frac {\sqrt {3}\, \left (-2 \left (2-2 \ln \left (2\right )+2 \ln \left (x \right )-\ln \left (3\right )+i \pi \right ) \sqrt {\pi }+4 \sqrt {\pi }-4 \sqrt {\pi }\, \sqrt {-\frac {x^{2}}{3}+1}+4 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-\frac {x^{2}}{3}+1}}{2}\right )\right )}{4 \sqrt {\pi }}\) | \(71\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 41, normalized size = 1.11 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 40, normalized size = 1.08 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.70, size = 87, normalized size = 2.35 \begin {gather*} \begin {cases} i \sqrt {x^{2} - 3} - \sqrt {3} \log {\left (x \right )} + \frac {\sqrt {3} \log {\left (x^{2} \right )}}{2} + \sqrt {3} i \operatorname {asin}{\left (\frac {\sqrt {3}}{x} \right )} & \text {for}\: \left |{x^{2}}\right | > 3 \\\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 65, normalized size = 1.76 \begin {gather*} \sqrt {-x^{2}+3}+\frac {3 \ln \left (\frac {-2 \sqrt {-x^{2}+3}+2 \sqrt {3}}{2 \sqrt {-x^{2}+3}+2 \sqrt {3}}\right )}{2 \sqrt {3}} \end {gather*}
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 \begin {gather*} \sqrt {3}\,\ln \left (\sqrt {\frac {3}{x^2}-1}-\sqrt {3}\,\sqrt {\frac {1}{x^2}}\right )+\sqrt {3-x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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