Optimal. Leaf size=34 \[ \sqrt{x^2-4 x+3}-\tanh ^{-1}\left (\frac{2-x}{\sqrt{x^2-4 x+3}}\right ) \]
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Rubi [A] time = 0.009237, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {640, 621, 206} \[ \sqrt{x^2-4 x+3}-\tanh ^{-1}\left (\frac{2-x}{\sqrt{x^2-4 x+3}}\right ) \]
Antiderivative was successfully verified.
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Rule 640
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{-1+x}{\sqrt{3-4 x+x^2}} \, dx &=\sqrt{3-4 x+x^2}+\int \frac{1}{\sqrt{3-4 x+x^2}} \, dx\\ &=\sqrt{3-4 x+x^2}+2 \operatorname{Subst}\left (\int \frac{1}{4-x^2} \, dx,x,\frac{-4+2 x}{\sqrt{3-4 x+x^2}}\right )\\ &=\sqrt{3-4 x+x^2}-\tanh ^{-1}\left (\frac{2-x}{\sqrt{3-4 x+x^2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0191737, size = 30, normalized size = 0.88 \[ \sqrt{x^2-4 x+3}+\tanh ^{-1}\left (\frac{x-2}{\sqrt{x^2-4 x+3}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 26, normalized size = 0.8 \begin{align*} \ln \left ( x-2+\sqrt{{x}^{2}-4\,x+3} \right ) +\sqrt{{x}^{2}-4\,x+3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.994683, size = 39, normalized size = 1.15 \begin{align*} \sqrt{x^{2} - 4 \, x + 3} + \log \left (2 \, x + 2 \, \sqrt{x^{2} - 4 \, x + 3} - 4\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.29784, size = 77, normalized size = 2.26 \begin{align*} \sqrt{x^{2} - 4 \, x + 3} - \log \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{x - 1}{\sqrt{\left (x - 3\right ) \left (x - 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.38208, size = 41, normalized size = 1.21 \begin{align*} \sqrt{x^{2} - 4 \, x + 3} - \log \left ({\left | -x + \sqrt{x^{2} - 4 \, x + 3} + 2 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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