Optimal. Leaf size=20 \[ \frac {1}{3} \tan ^{-1}\left (\frac {1}{3} \sqrt {-9-4 x^2}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {272, 65, 210}
\begin {gather*} \frac {1}{3} \text {ArcTan}\left (\frac {1}{3} \sqrt {-4 x^2-9}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 210
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {-9-4 x^2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {-9-4 x} x} \, dx,x,x^2\right )\\ &=-\left (\frac {1}{4} \text {Subst}\left (\int \frac {1}{-\frac {9}{4}-\frac {x^2}{4}} \, dx,x,\sqrt {-9-4 x^2}\right )\right )\\ &=\frac {1}{3} \tan ^{-1}\left (\frac {1}{3} \sqrt {-9-4 x^2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 20, normalized size = 1.00 \begin {gather*} \frac {1}{3} \tan ^{-1}\left (\frac {1}{3} \sqrt {-9-4 x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 15, normalized size = 0.75
method | result | size |
default | \(-\frac {\arctan \left (\frac {3}{\sqrt {-4 x^{2}-9}}\right )}{3}\) | \(15\) |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\sqrt {-4 x^{2}-9}+3 \RootOf \left (\textit {\_Z}^{2}+1\right )}{x}\right )}{3}\) | \(32\) |
meijerg | \(-\frac {i \left (-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {1+\frac {4 x^{2}}{9}}}{2}\right )+\left (2 \ln \left (x \right )-2 \ln \left (3\right )\right ) \sqrt {\pi }\right )}{6 \sqrt {\pi }}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.51, size = 25, normalized size = 1.25 \begin {gather*} -\frac {1}{3} i \, \log \left (\frac {6 \, \sqrt {4 \, x^{2} + 9}}{{\left | x \right |}} + \frac {18}{{\left | x \right |}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains complex when optimal does not.
time = 0.96, size = 43, normalized size = 2.15 \begin {gather*} -\frac {1}{6} i \, \log \left (-\frac {2 \, {\left (i \, \sqrt {-4 \, x^{2} - 9} + 3\right )}}{3 \, x}\right ) + \frac {1}{6} i \, \log \left (-\frac {2 \, {\left (-i \, \sqrt {-4 \, x^{2} - 9} + 3\right )}}{3 \, x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.45, size = 8, normalized size = 0.40 \begin {gather*} \frac {i \operatorname {asinh}{\left (\frac {3}{2 x} \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.60, size = 14, normalized size = 0.70 \begin {gather*} \frac {1}{3} \, \arctan \left (\frac {1}{3} \, \sqrt {-4 \, x^{2} - 9}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.27, size = 14, normalized size = 0.70 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {-4\,x^2-9}}{3}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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