Optimal. Leaf size=20 \[ -\frac {1}{3} \tanh ^{-1}\left (\frac {1}{3} \sqrt {4 x^2+9}\right ) \]
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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 = {266, 63, 207} \[ -\frac {1}{3} \tanh ^{-1}\left (\frac {1}{3} \sqrt {4 x^2+9}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 207
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {9+4 x^2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {9+4 x}} \, dx,x,x^2\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{-\frac {9}{4}+\frac {x^2}{4}} \, dx,x,\sqrt {9+4 x^2}\right )\\ &=-\frac {1}{3} \tanh ^{-1}\left (\frac {1}{3} \sqrt {9+4 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 20, normalized size = 1.00 \[ -\frac {1}{3} \tanh ^{-1}\left (\frac {1}{3} \sqrt {4 x^2+9}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.93, size = 35, normalized size = 1.75 \[ -\frac {1}{3} \, \log \left (-2 \, x + \sqrt {4 \, x^{2} + 9} + 3\right ) + \frac {1}{3} \, \log \left (-2 \, x + \sqrt {4 \, x^{2} + 9} - 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.06, size = 29, normalized size = 1.45 \[ -\frac {1}{6} \, \log \left (\sqrt {4 \, x^{2} + 9} + 3\right ) + \frac {1}{6} \, \log \left (\sqrt {4 \, x^{2} + 9} - 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 15, normalized size = 0.75 \[ -\frac {\arctanh \left (\frac {3}{\sqrt {4 x^{2}+9}}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.96, size = 9, normalized size = 0.45 \[ -\frac {1}{3} \, \operatorname {arsinh}\left (\frac {3}{2 \, {\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 12, normalized size = 0.60 \[ -\frac {\mathrm {atanh}\left (\frac {2\,\sqrt {x^2+\frac {9}{4}}}{3}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.02, size = 8, normalized size = 0.40 \[ - \frac {\operatorname {asinh}{\left (\frac {3}{2 x} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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