Integrand size = 16, antiderivative size = 34 \[ \int \frac {-1+x}{\sqrt {3-4 x+x^2}} \, dx=\sqrt {3-4 x+x^2}-\text {arctanh}\left (\frac {2-x}{\sqrt {3-4 x+x^2}}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {654, 635, 212} \[ \int \frac {-1+x}{\sqrt {3-4 x+x^2}} \, dx=\sqrt {x^2-4 x+3}-\text {arctanh}\left (\frac {2-x}{\sqrt {x^2-4 x+3}}\right ) \]
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Rule 212
Rule 635
Rule 654
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.08 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x}{\sqrt {3-4 x+x^2}} \, dx=\sqrt {3-4 x+x^2}+2 \text {arctanh}\left (\frac {\sqrt {3-4 x+x^2}}{-3+x}\right ) \]
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Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.76
method | result | size |
default | \(\ln \left (-2+x +\sqrt {x^{2}-4 x +3}\right )+\sqrt {x^{2}-4 x +3}\) | \(26\) |
trager | \(\ln \left (-2+x +\sqrt {x^{2}-4 x +3}\right )+\sqrt {x^{2}-4 x +3}\) | \(26\) |
risch | \(\ln \left (-2+x +\sqrt {x^{2}-4 x +3}\right )+\sqrt {x^{2}-4 x +3}\) | \(26\) |
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Time = 0.34 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.85 \[ \int \frac {-1+x}{\sqrt {3-4 x+x^2}} \, dx=\sqrt {x^{2} - 4 \, x + 3} - \log \left (-x + \sqrt {x^{2} - 4 \, x + 3} + 2\right ) \]
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Time = 0.40 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.91 \[ \int \frac {-1+x}{\sqrt {3-4 x+x^2}} \, dx=\sqrt {x^{2} - 4 x + 3} + \log {\left (2 x + 2 \sqrt {x^{2} - 4 x + 3} - 4 \right )} \]
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Time = 0.19 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.85 \[ \int \frac {-1+x}{\sqrt {3-4 x+x^2}} \, dx=\sqrt {x^{2} - 4 \, x + 3} + \log \left (2 \, x + 2 \, \sqrt {x^{2} - 4 \, x + 3} - 4\right ) \]
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Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.88 \[ \int \frac {-1+x}{\sqrt {3-4 x+x^2}} \, dx=\sqrt {x^{2} - 4 \, x + 3} - \log \left ({\left | -x + \sqrt {x^{2} - 4 \, x + 3} + 2 \right |}\right ) \]
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Time = 10.30 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.74 \[ \int \frac {-1+x}{\sqrt {3-4 x+x^2}} \, dx=\ln \left (x+\sqrt {x^2-4\,x+3}-2\right )+\sqrt {x^2-4\,x+3} \]
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