Optimal. Leaf size=20 \[ \frac {1}{3} \tan ^{-1}\left (\frac {1}{3} \sqrt {-4 x^2-9}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {266, 63, 204} \[ \frac {1}{3} \tan ^{-1}\left (\frac {1}{3} \sqrt {-4 x^2-9}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 204
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {-9-4 x^2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-9-4 x} x} \, dx,x,x^2\right )\\ &=-\left (\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{-\frac {9}{4}-\frac {x^2}{4}} \, dx,x,\sqrt {-9-4 x^2}\right )\right )\\ &=\frac {1}{3} \tan ^{-1}\left (\frac {1}{3} \sqrt {-9-4 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 20, normalized size = 1.00 \[ \frac {1}{3} \tan ^{-1}\left (\frac {1}{3} \sqrt {-4 x^2-9}\right ) \]
Antiderivative was successfully verified.
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fricas [C] time = 0.76, size = 43, normalized size = 2.15 \[ -\frac {1}{6} i \, \log \left (-\frac {2 \, {\left (i \, \sqrt {-4 \, x^{2} - 9} + 3\right )}}{3 \, x}\right ) + \frac {1}{6} i \, \log \left (-\frac {2 \, {\left (-i \, \sqrt {-4 \, x^{2} - 9} + 3\right )}}{3 \, x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.13, size = 14, normalized size = 0.70 \[ \frac {1}{3} \, \arctan \left (\frac {1}{3} \, \sqrt {-4 \, x^{2} - 9}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 15, normalized size = 0.75 \[ -\frac {\arctan \left (\frac {3}{\sqrt {-4 x^{2}-9}}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 2.94, size = 25, normalized size = 1.25 \[ -\frac {1}{3} i \, \log \left (\frac {6 \, \sqrt {4 \, x^{2} + 9}}{{\left | x \right |}} + \frac {18}{{\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.27, size = 14, normalized size = 0.70 \[ \frac {\mathrm {atan}\left (\frac {\sqrt {-4\,x^2-9}}{3}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.02, size = 8, normalized size = 0.40 \[ \frac {i \operatorname {asinh}{\left (\frac {3}{2 x} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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