Optimal. Leaf size=39 \[ \frac{\sqrt{-4 x^2-9}}{18 x^2}-\frac{2}{27} \tan ^{-1}\left (\frac{1}{3} \sqrt{-4 x^2-9}\right ) \]
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Rubi [A] time = 0.0160702, 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, 51, 63, 204} \[ \frac{\sqrt{-4 x^2-9}}{18 x^2}-\frac{2}{27} \tan ^{-1}\left (\frac{1}{3} \sqrt{-4 x^2-9}\right ) \]
Antiderivative was successfully verified.
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Rule 266
Rule 51
Rule 63
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{x^3 \sqrt{-9-4 x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-9-4 x} x^2} \, dx,x,x^2\right )\\ &=\frac{\sqrt{-9-4 x^2}}{18 x^2}-\frac{1}{9} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-9-4 x} x} \, dx,x,x^2\right )\\ &=\frac{\sqrt{-9-4 x^2}}{18 x^2}+\frac{1}{18} \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}}{18 x^2}-\frac{2}{27} \tan ^{-1}\left (\frac{1}{3} \sqrt{-9-4 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0215214, size = 54, normalized size = 1.38 \[ -\frac{4}{81} \sqrt{-4 x^2-9} \left (\frac{\tanh ^{-1}\left (\sqrt{\frac{4 x^2}{9}+1}\right )}{2 \sqrt{\frac{4 x^2}{9}+1}}-\frac{9}{8 x^2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 30, normalized size = 0.8 \begin{align*}{\frac{1}{18\,{x}^{2}}\sqrt{-4\,{x}^{2}-9}}+{\frac{2}{27}\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 = 2.95667, size = 54, normalized size = 1.38 \begin{align*} \frac{\sqrt{-4 \, x^{2} - 9}}{18 \, x^{2}} + \frac{2}{27} 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 = 1.19176, size = 174, normalized size = 4.46 \begin{align*} \frac{-2 i \, x^{2} \log \left (-\frac{4 \,{\left (i \, \sqrt{-4 \, x^{2} - 9} - 3\right )}}{27 \, x}\right ) + 2 i \, x^{2} \log \left (-\frac{4 \,{\left (-i \, \sqrt{-4 \, x^{2} - 9} - 3\right )}}{27 \, x}\right ) + 3 \, \sqrt{-4 \, x^{2} - 9}}{54 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.20793, size = 46, normalized size = 1.18 \begin{align*} - \frac{2 i \operatorname{asinh}{\left (\frac{3}{2 x} \right )}}{27} + \frac{i}{9 x \sqrt{1 + \frac{9}{4 x^{2}}}} + \frac{i}{4 x^{3} \sqrt{1 + \frac{9}{4 x^{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 2.67715, size = 39, normalized size = 1. \begin{align*} \frac{i \, \sqrt{4 \, x^{2} + 9}}{18 \, x^{2}} - \frac{2}{27} \, \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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