Optimal. Leaf size=30 \[ \sqrt {9 x^2-4}-2 \tan ^{-1}\left (\frac {1}{2} \sqrt {9 x^2-4}\right ) \]
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Rubi [A] time = 0.02, 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 = {266, 50, 63, 203} \[ \sqrt {9 x^2-4}-2 \tan ^{-1}\left (\frac {1}{2} \sqrt {9 x^2-4}\right ) \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 203
Rule 266
Rubi steps
\begin {align*} \int \frac {\sqrt {-4+9 x^2}}{x} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {-4+9 x}}{x} \, dx,x,x^2\right )\\ &=\sqrt {-4+9 x^2}-2 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {-4+9 x}} \, dx,x,x^2\right )\\ &=\sqrt {-4+9 x^2}-\frac {4}{9} \operatorname {Subst}\left (\int \frac {1}{\frac {4}{9}+\frac {x^2}{9}} \, dx,x,\sqrt {-4+9 x^2}\right )\\ &=\sqrt {-4+9 x^2}-2 \tan ^{-1}\left (\frac {1}{2} \sqrt {-4+9 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 1.00 \[ \sqrt {9 x^2-4}-2 \tan ^{-1}\left (\frac {1}{2} \sqrt {9 x^2-4}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 28, normalized size = 0.93 \[ \sqrt {9 \, x^{2} - 4} - 4 \, \arctan \left (-\frac {3}{2} \, x + \frac {1}{2} \, \sqrt {9 \, x^{2} - 4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.93, size = 24, normalized size = 0.80 \[ \sqrt {9 \, x^{2} - 4} - 2 \, \arctan \left (\frac {1}{2} \, \sqrt {9 \, x^{2} - 4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 25, normalized size = 0.83 \[ 2 \arctan \left (\frac {2}{\sqrt {9 x^{2}-4}}\right )+\sqrt {9 x^{2}-4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 19, normalized size = 0.63 \[ \sqrt {9 \, x^{2} - 4} + 2 \, \arcsin \left (\frac {2}{3 \, {\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.31, size = 24, normalized size = 0.80 \[ \sqrt {9\,x^2-4}-2\,\mathrm {atan}\left (\frac {\sqrt {9\,x^2-4}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.40, size = 92, normalized size = 3.07 \[ \begin {cases} - \frac {3 i x}{\sqrt {-1 + \frac {4}{9 x^{2}}}} - 2 i \operatorname {acosh}{\left (\frac {2}{3 x} \right )} + \frac {4 i}{3 x \sqrt {-1 + \frac {4}{9 x^{2}}}} & \text {for}\: \frac {4}{9 \left |{x^{2}}\right |} > 1 \\\frac {3 x}{\sqrt {1 - \frac {4}{9 x^{2}}}} + 2 \operatorname {asin}{\left (\frac {2}{3 x} \right )} - \frac {4}{3 x \sqrt {1 - \frac {4}{9 x^{2}}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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