Optimal. Leaf size=18 \[ \frac{\sqrt{4 x^2-9}}{9 x} \]
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Rubi [A] time = 0.0029692, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {264} \[ \frac{\sqrt{4 x^2-9}}{9 x} \]
Antiderivative was successfully verified.
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Rule 264
Rubi steps
\begin{align*} \int \frac{1}{x^2 \sqrt{-9+4 x^2}} \, dx &=\frac{\sqrt{-9+4 x^2}}{9 x}\\ \end{align*}
Mathematica [A] time = 0.0023989, size = 18, normalized size = 1. \[ \frac{\sqrt{4 x^2-9}}{9 x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 25, normalized size = 1.4 \begin{align*}{\frac{ \left ( -3+2\,x \right ) \left ( 3+2\,x \right ) }{9\,x}{\frac{1}{\sqrt{4\,{x}^{2}-9}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 3.09762, size = 19, normalized size = 1.06 \begin{align*} \frac{\sqrt{4 \, x^{2} - 9}}{9 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28741, size = 42, normalized size = 2.33 \begin{align*} \frac{2 \, x + \sqrt{4 \, x^{2} - 9}}{9 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.745963, size = 37, normalized size = 2.06 \begin{align*} \begin{cases} \frac{2 i \sqrt{-1 + \frac{9}{4 x^{2}}}}{9} & \text{for}\: \frac{9}{4 \left |{x^{2}}\right |} > 1 \\\frac{2 \sqrt{1 - \frac{9}{4 x^{2}}}}{9} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.83537, size = 31, normalized size = 1.72 \begin{align*} \frac{4}{{\left (2 \, x - \sqrt{4 \, x^{2} - 9}\right )}^{2} + 9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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