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, 203} \[ \frac {1}{3} \tan ^{-1}\left (\frac {1}{3} \sqrt {4 x^2-9}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 203
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}{x \sqrt {-9+4 x}} \, dx,x,x^2\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{\frac {9}{4}+\frac {x^2}{4}} \, dx,x,\sqrt {-9+4 x^2}\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 [A] time = 0.55, size = 18, normalized size = 0.90 \[ \frac {2}{3} \, \arctan \left (-\frac {2}{3} \, x + \frac {1}{3} \, \sqrt {4 \, x^{2} - 9}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.06, 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 [A] time = 2.91, size = 9, normalized size = 0.45 \[ -\frac {1}{3} \, \arcsin \left (\frac {3}{2 \, {\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 20, normalized size = 1.00 \[ \frac {\ln \left (\frac {\sqrt {4\,x^2-9}+3{}\mathrm {i}}{x}\right )\,1{}\mathrm {i}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.07, size = 26, normalized size = 1.30 \[ \begin {cases} \frac {i \operatorname {acosh}{\left (\frac {3}{2 x} \right )}}{3} & \text {for}\: \frac {9}{4 \left |{x^{2}}\right |} > 1 \\- \frac {\operatorname {asin}{\left (\frac {3}{2 x} \right )}}{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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