Optimal. Leaf size=30 \[ \sqrt{9-x^2}-3 \tanh ^{-1}\left (\frac{\sqrt{9-x^2}}{3}\right ) \]
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Rubi [A] time = 0.0149736, antiderivative size = 30, 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, 50, 63, 206} \[ \sqrt{9-x^2}-3 \tanh ^{-1}\left (\frac{\sqrt{9-x^2}}{3}\right ) \]
Antiderivative was successfully verified.
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Rule 266
Rule 50
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{9-x^2}}{x} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt{9-x}}{x} \, dx,x,x^2\right )\\ &=\sqrt{9-x^2}+\frac{9}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{9-x} x} \, dx,x,x^2\right )\\ &=\sqrt{9-x^2}-9 \operatorname{Subst}\left (\int \frac{1}{9-x^2} \, dx,x,\sqrt{9-x^2}\right )\\ &=\sqrt{9-x^2}-3 \tanh ^{-1}\left (\frac{\sqrt{9-x^2}}{3}\right )\\ \end{align*}
Mathematica [A] time = 0.0051874, size = 30, normalized size = 1. \[ \sqrt{9-x^2}-3 \tanh ^{-1}\left (\frac{\sqrt{9-x^2}}{3}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 25, normalized size = 0.8 \begin{align*} \sqrt{-{x}^{2}+9}-3\,{\it Artanh} \left ( 3\,{\frac{1}{\sqrt{-{x}^{2}+9}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.42481, size = 47, normalized size = 1.57 \begin{align*} \sqrt{-x^{2} + 9} - 3 \, \log \left (\frac{6 \, \sqrt{-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 = 1.87756, size = 65, normalized size = 2.17 \begin{align*} \sqrt{-x^{2} + 9} + 3 \, \log \left (\frac{\sqrt{-x^{2} + 9} - 3}{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.30047, size = 73, normalized size = 2.43 \begin{align*} \begin{cases} - \frac{x}{\sqrt{-1 + \frac{9}{x^{2}}}} - 3 \operatorname{acosh}{\left (\frac{3}{x} \right )} + \frac{9}{x \sqrt{-1 + \frac{9}{x^{2}}}} & \text{for}\: \frac{9}{\left |{x^{2}}\right |} > 1 \\\frac{i x}{\sqrt{1 - \frac{9}{x^{2}}}} + 3 i \operatorname{asin}{\left (\frac{3}{x} \right )} - \frac{9 i}{x \sqrt{1 - \frac{9}{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 = 1.06007, size = 54, normalized size = 1.8 \begin{align*} \sqrt{-x^{2} + 9} - \frac{3}{2} \, \log \left (\sqrt{-x^{2} + 9} + 3\right ) + \frac{3}{2} \, \log \left (-\sqrt{-x^{2} + 9} + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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