Optimal. Leaf size=34 \[ \sqrt {3-4 x+x^2}-\tanh ^{-1}\left (\frac {2-x}{\sqrt {3-4 x+x^2}}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, 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 = {654, 635, 212}
\begin {gather*} \sqrt {x^2-4 x+3}-\tanh ^{-1}\left (\frac {2-x}{\sqrt {x^2-4 x+3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rule 654
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 \text {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*}
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Mathematica [A]
time = 0.08, size = 34, normalized size = 1.00 \begin {gather*} \sqrt {3-4 x+x^2}+2 \tanh ^{-1}\left (\frac {\sqrt {3-4 x+x^2}}{-3+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.73, size = 26, normalized size = 0.76
method | result | size |
default | \(\sqrt {x^{2}-4 x +3}+\ln \left (x -2+\sqrt {x^{2}-4 x +3}\right )\) | \(26\) |
risch | \(\sqrt {x^{2}-4 x +3}+\ln \left (x -2+\sqrt {x^{2}-4 x +3}\right )\) | \(26\) |
trager | \(\sqrt {x^{2}-4 x +3}-\ln \left (\sqrt {x^{2}-4 x +3}+2-x \right )\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 29, normalized size = 0.85 \begin {gather*} \sqrt {x^{2} - 4 \, x + 3} + \log \left (2 \, x + 2 \, \sqrt {x^{2} - 4 \, x + 3} - 4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.58, size = 29, normalized size = 0.85 \begin {gather*} \sqrt {x^{2} - 4 \, x + 3} - \log \left (-x + \sqrt {x^{2} - 4 \, x + 3} + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x - 1}{\sqrt {\left (x - 3\right ) \left (x - 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.80, size = 30, normalized size = 0.88 \begin {gather*} \sqrt {x^{2} - 4 \, x + 3} - \log \left ({\left | -x + \sqrt {x^{2} - 4 \, x + 3} + 2 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.52, size = 25, normalized size = 0.74 \begin {gather*} \ln \left (x+\sqrt {x^2-4\,x+3}-2\right )+\sqrt {x^2-4\,x+3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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