Optimal. Leaf size=25 \[ -\frac{\sqrt{9-x^2}}{x}-\sin ^{-1}\left (\frac{x}{3}\right ) \]
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Rubi [A] time = 0.0042819, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {277, 216} \[ -\frac{\sqrt{9-x^2}}{x}-\sin ^{-1}\left (\frac{x}{3}\right ) \]
Antiderivative was successfully verified.
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Rule 277
Rule 216
Rubi steps
\begin{align*} \int \frac{\sqrt{9-x^2}}{x^2} \, dx &=-\frac{\sqrt{9-x^2}}{x}-\int \frac{1}{\sqrt{9-x^2}} \, dx\\ &=-\frac{\sqrt{9-x^2}}{x}-\sin ^{-1}\left (\frac{x}{3}\right )\\ \end{align*}
Mathematica [A] time = 0.0073502, size = 25, normalized size = 1. \[ -\frac{\sqrt{9-x^2}}{x}-\sin ^{-1}\left (\frac{x}{3}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 34, normalized size = 1.4 \begin{align*} -{\frac{1}{9\,x} \left ( -{x}^{2}+9 \right ) ^{{\frac{3}{2}}}}-{\frac{x}{9}\sqrt{-{x}^{2}+9}}-\arcsin \left ({\frac{x}{3}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.40731, size = 28, normalized size = 1.12 \begin{align*} -\frac{\sqrt{-x^{2} + 9}}{x} - \arcsin \left (\frac{1}{3} \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79534, size = 77, normalized size = 3.08 \begin{align*} \frac{2 \, x \arctan \left (\frac{\sqrt{-x^{2} + 9} - 3}{x}\right ) - \sqrt{-x^{2} + 9}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.226875, size = 15, normalized size = 0.6 \begin{align*} - \operatorname{asin}{\left (\frac{x}{3} \right )} - \frac{\sqrt{9 - x^{2}}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06067, size = 53, normalized size = 2.12 \begin{align*} \frac{x}{2 \,{\left (\sqrt{-x^{2} + 9} - 3\right )}} - \frac{\sqrt{-x^{2} + 9} - 3}{2 \, x} - \arcsin \left (\frac{1}{3} \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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