Optimal. Leaf size=39 \[ \frac{2}{3} \tanh ^{-1}\left (\frac{1}{3} \sqrt{9-4 x^2}\right )-\frac{\sqrt{9-4 x^2}}{2 x^2} \]
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Rubi [A] time = 0.016247, antiderivative size = 39, normalized size of antiderivative = 1., 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, 47, 63, 206} \[ \frac{2}{3} \tanh ^{-1}\left (\frac{1}{3} \sqrt{9-4 x^2}\right )-\frac{\sqrt{9-4 x^2}}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 47
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{9-4 x^2}}{x^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt{9-4 x}}{x^2} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{9-4 x^2}}{2 x^2}-\operatorname{Subst}\left (\int \frac{1}{\sqrt{9-4 x} x} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{9-4 x^2}}{2 x^2}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\frac{9}{4}-\frac{x^2}{4}} \, dx,x,\sqrt{9-4 x^2}\right )\\ &=-\frac{\sqrt{9-4 x^2}}{2 x^2}+\frac{2}{3} \tanh ^{-1}\left (\frac{1}{3} \sqrt{9-4 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0251954, size = 37, normalized size = 0.95 \[ \frac{2}{3} \tanh ^{-1}\left (\sqrt{1-\frac{4 x^2}{9}}\right )-\frac{\sqrt{9-4 x^2}}{2 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 41, normalized size = 1.1 \begin{align*} -{\frac{1}{18\,{x}^{2}} \left ( -4\,{x}^{2}+9 \right ) ^{{\frac{3}{2}}}}-{\frac{2}{9}\sqrt{-4\,{x}^{2}+9}}+{\frac{2}{3}{\it Artanh} \left ( 3\,{\frac{1}{\sqrt{-4\,{x}^{2}+9}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 3.90331, size = 69, normalized size = 1.77 \begin{align*} -\frac{2}{9} \, \sqrt{-4 \, x^{2} + 9} - \frac{{\left (-4 \, x^{2} + 9\right )}^{\frac{3}{2}}}{18 \, x^{2}} + \frac{2}{3} \, \log \left (\frac{6 \, \sqrt{-4 \, x^{2} + 9}}{{\left | x \right |}} + \frac{18}{{\left | x \right |}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50812, size = 93, normalized size = 2.38 \begin{align*} -\frac{4 \, x^{2} \log \left (\frac{\sqrt{-4 \, x^{2} + 9} - 3}{x}\right ) + 3 \, \sqrt{-4 \, x^{2} + 9}}{6 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.80749, size = 97, normalized size = 2.49 \begin{align*} \begin{cases} \frac{2 \operatorname{acosh}{\left (\frac{3}{2 x} \right )}}{3} + \frac{1}{x \sqrt{-1 + \frac{9}{4 x^{2}}}} - \frac{9}{4 x^{3} \sqrt{-1 + \frac{9}{4 x^{2}}}} & \text{for}\: \frac{9}{4 \left |{x^{2}}\right |} > 1 \\- \frac{2 i \operatorname{asin}{\left (\frac{3}{2 x} \right )}}{3} - \frac{i}{x \sqrt{1 - \frac{9}{4 x^{2}}}} + \frac{9 i}{4 x^{3} \sqrt{1 - \frac{9}{4 x^{2}}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.5649, size = 61, normalized size = 1.56 \begin{align*} -\frac{\sqrt{-4 \, x^{2} + 9}}{2 \, x^{2}} + \frac{1}{3} \, \log \left (\sqrt{-4 \, x^{2} + 9} + 3\right ) - \frac{1}{3} \, \log \left (-\sqrt{-4 \, x^{2} + 9} + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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