Optimal. Leaf size=19 \[ \frac {1}{2} \tan ^{-1}\left (\frac {2 x}{\sqrt {-4 x^2-9}}\right ) \]
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Rubi [A] time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {217, 203} \[ \frac {1}{2} \tan ^{-1}\left (\frac {2 x}{\sqrt {-4 x^2-9}}\right ) \]
Antiderivative was successfully verified.
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Rule 203
Rule 217
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-9-4 x^2}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{1+4 x^2} \, dx,x,\frac {x}{\sqrt {-9-4 x^2}}\right )\\ &=\frac {1}{2} \tan ^{-1}\left (\frac {2 x}{\sqrt {-9-4 x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 19, normalized size = 1.00 \[ \frac {1}{2} \tan ^{-1}\left (\frac {2 x}{\sqrt {-4 x^2-9}}\right ) \]
Antiderivative was successfully verified.
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fricas [C] time = 0.79, size = 47, normalized size = 2.47 \[ \frac {1}{4} i \, \log \left (-\frac {8 \, x + 4 i \, \sqrt {-4 \, x^{2} - 9}}{x}\right ) - \frac {1}{4} i \, \log \left (-\frac {8 \, x - 4 i \, \sqrt {-4 \, x^{2} - 9}}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-4 \, x^{2} - 9}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 16, normalized size = 0.84 \[ \frac {\arctan \left (\frac {2 x}{\sqrt {-4 x^{2}-9}}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 2.98, size = 6, normalized size = 0.32 \[ -\frac {1}{2} i \, \operatorname {arsinh}\left (\frac {2}{3} \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 15, normalized size = 0.79 \[ \frac {\mathrm {atan}\left (\frac {2\,x}{\sqrt {-4\,x^2-9}}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.32, size = 17, normalized size = 0.89 \[ \frac {\operatorname {atan}{\left (\frac {2 x}{\sqrt {- 4 x^{2} - 9}} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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