Optimal. Leaf size=30 \[ \sqrt {-9-4 x^2}-3 \tan ^{-1}\left (\frac {1}{3} \sqrt {-9-4 x^2}\right ) \]
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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,
210} \begin {gather*} \sqrt {-4 x^2-9}-3 \text {ArcTan}\left (\frac {1}{3} \sqrt {-4 x^2-9}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 210
Rule 272
Rubi steps
\begin {align*} \int \frac {\sqrt {-9-4 x^2}}{x} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {-9-4 x}}{x} \, dx,x,x^2\right )\\ &=\sqrt {-9-4 x^2}-\frac {9}{2} \text {Subst}\left (\int \frac {1}{\sqrt {-9-4 x} x} \, dx,x,x^2\right )\\ &=\sqrt {-9-4 x^2}+\frac {9}{4} \text {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 \tan ^{-1}\left (\frac {1}{3} \sqrt {-9-4 x^2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 30, normalized size = 1.00 \begin {gather*} \sqrt {-9-4 x^2}-3 \tan ^{-1}\left (\frac {1}{3} \sqrt {-9-4 x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 25, normalized size = 0.83
method | result | size |
default | \(\sqrt {-4 x^{2}-9}+3 \arctan \left (\frac {3}{\sqrt {-4 x^{2}-9}}\right )\) | \(25\) |
trager | \(\sqrt {-4 x^{2}-9}+3 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\sqrt {-4 x^{2}-9}-3 \RootOf \left (\textit {\_Z}^{2}+1\right )}{x}\right )\) | \(42\) |
meijerg | \(-\frac {3 i \left (4 \sqrt {\pi }-4 \sqrt {\pi }\, \sqrt {1+\frac {4 x^{2}}{9}}+4 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {1+\frac {4 x^{2}}{9}}}{2}\right )-2 \left (2+2 \ln \left (x \right )-2 \ln \left (3\right )\right ) \sqrt {\pi }\right )}{4 \sqrt {\pi }}\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.49, size = 35, normalized size = 1.17 \begin {gather*} \sqrt {-4 \, x^{2} - 9} + 3 i \, \log \left (\frac {6 \, \sqrt {4 \, x^{2} + 9}}{{\left | x \right |}} + \frac {18}{{\left | x \right |}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains complex when optimal does not.
time = 0.65, size = 52, normalized size = 1.73 \begin {gather*} \sqrt {-4 \, x^{2} - 9} - \frac {3}{2} i \, \log \left (-\frac {6 \, {\left (i \, \sqrt {-4 \, x^{2} - 9} - 3\right )}}{x}\right ) + \frac {3}{2} i \, \log \left (-\frac {6 \, {\left (-i \, \sqrt {-4 \, x^{2} - 9} - 3\right )}}{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 = 0.61, size = 44, normalized size = 1.47 \begin {gather*} \frac {2 i x}{\sqrt {1 + \frac {9}{4 x^{2}}}} - 3 i \operatorname {asinh}{\left (\frac {3}{2 x} \right )} + \frac {9 i}{2 x \sqrt {1 + \frac {9}{4 x^{2}}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.53, size = 24, normalized size = 0.80 \begin {gather*} \sqrt {-4 \, x^{2} - 9} - 3 \, \arctan \left (\frac {1}{3} \, \sqrt {-4 \, x^{2} - 9}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.73, size = 24, normalized size = 0.80 \begin {gather*} \sqrt {-4\,x^2-9}-3\,\mathrm {atan}\left (\frac {\sqrt {-4\,x^2-9}}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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