Optimal. Leaf size=25 \[ -\frac {\sqrt {9-x^2}}{x}-\sin ^{-1}\left (\frac {x}{3}\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 25, normalized size of antiderivative = 1.00, 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 = {283, 222}
\begin {gather*} -\frac {\sqrt {9-x^2}}{x}-\sin ^{-1}\left (\frac {x}{3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 283
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*}
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Mathematica [A]
time = 0.03, size = 37, normalized size = 1.48 \begin {gather*} -\frac {\sqrt {9-x^2}}{x}+2 \tan ^{-1}\left (\frac {\sqrt {9-x^2}}{3+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 2.04, size = 21, normalized size = 0.84 \begin {gather*} -\frac {\sqrt {9-x^2}}{x}-\text {ArcSin}\left [\frac {x}{3}\right ] \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 34, normalized size = 1.36
method | result | size |
risch | \(\frac {x^{2}-9}{x \sqrt {-x^{2}+9}}-\arcsin \left (\frac {x}{3}\right )\) | \(26\) |
default | \(-\frac {\left (-x^{2}+9\right )^{\frac {3}{2}}}{9 x}-\frac {x \sqrt {-x^{2}+9}}{9}-\arcsin \left (\frac {x}{3}\right )\) | \(34\) |
meijerg | \(-\frac {i \left (-\frac {12 i \sqrt {\pi }\, \sqrt {-\frac {x^{2}}{9}+1}}{x}-4 i \sqrt {\pi }\, \arcsin \left (\frac {x}{3}\right )\right )}{4 \sqrt {\pi }}\) | \(36\) |
trager | \(-\frac {\sqrt {-x^{2}+9}}{x}+\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}+1\right ) x +\sqrt {-x^{2}+9}\right )\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 21, normalized size = 0.84 \begin {gather*} -\frac {\sqrt {-x^{2} + 9}}{x} - \arcsin \left (\frac {1}{3} \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 35, normalized size = 1.40 \begin {gather*} \frac {2 \, x \arctan \left (\frac {\sqrt {-x^{2} + 9} - 3}{x}\right ) - \sqrt {-x^{2} + 9}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 15, normalized size = 0.60 \begin {gather*} - \operatorname {asin}{\left (\frac {x}{3} \right )} - \frac {\sqrt {9 - x^{2}}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 46, normalized size = 1.84 \begin {gather*} -\frac {x}{-2 \sqrt {-x^{2}+9}+6}+\frac {-2 \sqrt {-x^{2}+9}+6}{4 x}-\arcsin \left (\frac {x}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 21, normalized size = 0.84 \begin {gather*} -\mathrm {asin}\left (\frac {x}{3}\right )-\frac {\sqrt {9-x^2}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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