Optimal. Leaf size=13 \[ \tan ^{-1}\left (\sqrt{x^2-4 x+3}\right ) \]
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Rubi [A] time = 0.008268, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {688, 203} \[ \tan ^{-1}\left (\sqrt{x^2-4 x+3}\right ) \]
Antiderivative was successfully verified.
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Rule 688
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{(-2+x) \sqrt{3-4 x+x^2}} \, dx &=4 \operatorname{Subst}\left (\int \frac{1}{4+4 x^2} \, dx,x,\sqrt{3-4 x+x^2}\right )\\ &=\tan ^{-1}\left (\sqrt{3-4 x+x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0037221, size = 12, normalized size = 0.92 \[ \tan ^{-1}\left (\sqrt{(x-2)^2-1}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 13, normalized size = 1. \begin{align*} -\arctan \left ({\frac{1}{\sqrt{ \left ( -2+x \right ) ^{2}-1}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.5334, size = 12, normalized size = 0.92 \begin{align*} -\arcsin \left (\frac{1}{{\left | x - 2 \right |}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.05789, size = 54, normalized size = 4.15 \begin{align*} 2 \, \arctan \left (-x + \sqrt{x^{2} - 4 \, x + 3} + 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\left (x - 3\right ) \left (x - 1\right )} \left (x - 2\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.48421, size = 24, normalized size = 1.85 \begin{align*} 2 \, \arctan \left (-x + \sqrt{x^{2} - 4 \, x + 3} + 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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