Optimal. Leaf size=17 \[ x-2 \sqrt {3} \tanh ^{-1}\left (\frac {x}{\sqrt {3}}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {388, 207} \[ x-2 \sqrt {3} \tanh ^{-1}\left (\frac {x}{\sqrt {3}}\right ) \]
Antiderivative was successfully verified.
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Rule 207
Rule 388
Rubi steps
\begin {align*} \int \frac {3+x^2}{-3+x^2} \, dx &=x+6 \int \frac {1}{-3+x^2} \, dx\\ &=x-2 \sqrt {3} \tanh ^{-1}\left (\frac {x}{\sqrt {3}}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 33, normalized size = 1.94 \[ x+\sqrt {3} \log \left (\sqrt {3}-x\right )-\sqrt {3} \log \left (x+\sqrt {3}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 26, normalized size = 1.53 \[ \sqrt {3} \log \left (\frac {x^{2} - 2 \, \sqrt {3} x + 3}{x^{2} - 3}\right ) + x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.41, size = 30, normalized size = 1.76 \[ \sqrt {3} \log \left (\frac {{\left | 2 \, x - 2 \, \sqrt {3} \right |}}{{\left | 2 \, x + 2 \, \sqrt {3} \right |}}\right ) + x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 15, normalized size = 0.88 \[ x -2 \sqrt {3}\, \arctanh \left (\frac {\sqrt {3}\, x}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.88, size = 22, normalized size = 1.29 \[ \sqrt {3} \log \left (\frac {x - \sqrt {3}}{x + \sqrt {3}}\right ) + x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.92, size = 14, normalized size = 0.82 \[ x-2\,\sqrt {3}\,\mathrm {atanh}\left (\frac {\sqrt {3}\,x}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.19, size = 27, normalized size = 1.59 \[ x + \sqrt {3} \log {\left (x - \sqrt {3} \right )} - \sqrt {3} \log {\left (x + \sqrt {3} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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