Optimal. Leaf size=39 \[ -\frac{\sqrt{-4 x^2-9}}{2 x^2}-\frac{2}{3} \tan ^{-1}\left (\frac{1}{3} \sqrt{-4 x^2-9}\right ) \]
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Rubi [A] time = 0.0161465, 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, 204} \[ -\frac{\sqrt{-4 x^2-9}}{2 x^2}-\frac{2}{3} \tan ^{-1}\left (\frac{1}{3} \sqrt{-4 x^2-9}\right ) \]
Antiderivative was successfully verified.
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Rule 266
Rule 47
Rule 63
Rule 204
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} \tan ^{-1}\left (\frac{1}{3} \sqrt{-9-4 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0136417, size = 55, normalized size = 1.41 \[ \frac{12 x^2+4 \sqrt{4 x^2+9} x^2 \tanh ^{-1}\left (\sqrt{\frac{4 x^2}{9}+1}\right )+27}{6 x^2 \sqrt{-4 x^2-9}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, 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}\arctan \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 [C] time = 3.57064, 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} i \, \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 [C] time = 1.25728, size = 170, normalized size = 4.36 \begin{align*} \frac{-2 i \, x^{2} \log \left (-\frac{4 \,{\left (i \, \sqrt{-4 \, x^{2} - 9} - 3\right )}}{3 \, x}\right ) + 2 i \, x^{2} \log \left (-\frac{4 \,{\left (-i \, \sqrt{-4 \, x^{2} - 9} - 3\right )}}{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 [C] time = 1.74531, size = 27, normalized size = 0.69 \begin{align*} - \frac{2 i \operatorname{asinh}{\left (\frac{3}{2 x} \right )}}{3} - \frac{i \sqrt{1 + \frac{9}{4 x^{2}}}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 2.00082, size = 39, normalized size = 1. \begin{align*} -\frac{i \, \sqrt{4 \, x^{2} + 9}}{2 \, x^{2}} - \frac{2}{3} \, \arctan \left (\frac{1}{3} i \, \sqrt{4 \, x^{2} + 9}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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