Integrand size = 22, antiderivative size = 75 \[ \int \frac {\sqrt {-1-x+x^2}}{1-x^2} \, dx=-\frac {1}{2} \arctan \left (\frac {3-x}{2 \sqrt {-1-x+x^2}}\right )+\text {arctanh}\left (\frac {1-2 x}{2 \sqrt {-1-x+x^2}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {1+3 x}{2 \sqrt {-1-x+x^2}}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1004, 635, 212, 1047, 738, 210} \[ \int \frac {\sqrt {-1-x+x^2}}{1-x^2} \, dx=-\frac {1}{2} \arctan \left (\frac {3-x}{2 \sqrt {x^2-x-1}}\right )+\text {arctanh}\left (\frac {1-2 x}{2 \sqrt {x^2-x-1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {3 x+1}{2 \sqrt {x^2-x-1}}\right ) \]
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Rule 210
Rule 212
Rule 635
Rule 738
Rule 1004
Rule 1047
Rubi steps \begin{align*} \text {integral}& = -\int \frac {1}{\sqrt {-1-x+x^2}} \, dx-\int \frac {x}{\left (1-x^2\right ) \sqrt {-1-x+x^2}} \, dx \\ & = -\left (\frac {1}{2} \int \frac {1}{(-1-x) \sqrt {-1-x+x^2}} \, dx\right )-\frac {1}{2} \int \frac {1}{(1-x) \sqrt {-1-x+x^2}} \, dx-2 \text {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {-1+2 x}{\sqrt {-1-x+x^2}}\right ) \\ & = \tanh ^{-1}\left (\frac {1-2 x}{2 \sqrt {-1-x+x^2}}\right )+\text {Subst}\left (\int \frac {1}{-4-x^2} \, dx,x,\frac {3-x}{\sqrt {-1-x+x^2}}\right )+\text {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {1+3 x}{\sqrt {-1-x+x^2}}\right ) \\ & = -\frac {1}{2} \tan ^{-1}\left (\frac {3-x}{2 \sqrt {-1-x+x^2}}\right )+\tanh ^{-1}\left (\frac {1-2 x}{2 \sqrt {-1-x+x^2}}\right )+\frac {1}{2} \tanh ^{-1}\left (\frac {1+3 x}{2 \sqrt {-1-x+x^2}}\right ) \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {-1-x+x^2}}{1-x^2} \, dx=\arctan \left (1-x+\sqrt {-1-x+x^2}\right )+\text {arctanh}\left (1+x-\sqrt {-1-x+x^2}\right )+\log \left (1-2 x+2 \sqrt {-1-x+x^2}\right ) \]
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Time = 0.69 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.36
method | result | size |
default | \(-\frac {\sqrt {\left (-1+x \right )^{2}-2+x}}{2}-\frac {\ln \left (-\frac {1}{2}+x +\sqrt {\left (-1+x \right )^{2}-2+x}\right )}{4}+\frac {\arctan \left (\frac {-3+x}{2 \sqrt {\left (-1+x \right )^{2}-2+x}}\right )}{2}+\frac {\sqrt {\left (1+x \right )^{2}-2-3 x}}{2}-\frac {3 \ln \left (-\frac {1}{2}+x +\sqrt {\left (1+x \right )^{2}-2-3 x}\right )}{4}-\frac {\operatorname {arctanh}\left (\frac {-1-3 x}{2 \sqrt {\left (1+x \right )^{2}-2-3 x}}\right )}{2}\) | \(102\) |
trager | \(-\frac {\ln \left (\frac {8 \sqrt {x^{2}-x -1}\, x^{2}+8 x^{3}+12 \sqrt {x^{2}-x -1}\, x +8 x^{2}+2 \sqrt {x^{2}-x -1}-9 x -11}{1+x}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +2 \sqrt {x^{2}-x -1}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{-1+x}\right )}{2}\) | \(112\) |
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Time = 0.31 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {-1-x+x^2}}{1-x^2} \, dx=\arctan \left (-x + \sqrt {x^{2} - x - 1} + 1\right ) - \frac {1}{2} \, \log \left (-x + \sqrt {x^{2} - x - 1}\right ) + \frac {1}{2} \, \log \left (-x + \sqrt {x^{2} - x - 1} - 2\right ) + \log \left (-2 \, x + 2 \, \sqrt {x^{2} - x - 1} + 1\right ) \]
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\[ \int \frac {\sqrt {-1-x+x^2}}{1-x^2} \, dx=- \int \frac {\sqrt {x^{2} - x - 1}}{x^{2} - 1}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.11 \[ \int \frac {\sqrt {-1-x+x^2}}{1-x^2} \, dx=\frac {1}{2} \, \arcsin \left (\frac {2 \, \sqrt {5} x}{5 \, {\left | 2 \, x - 2 \right |}} - \frac {6 \, \sqrt {5}}{5 \, {\left | 2 \, x - 2 \right |}}\right ) - \log \left (x + \sqrt {x^{2} - x - 1} - \frac {1}{2}\right ) - \frac {1}{2} \, \log \left (\frac {2 \, \sqrt {x^{2} - x - 1}}{{\left | 2 \, x + 2 \right |}} + \frac {2}{{\left | 2 \, x + 2 \right |}} - \frac {3}{2}\right ) \]
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Time = 0.28 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {-1-x+x^2}}{1-x^2} \, dx=\arctan \left (-x + \sqrt {x^{2} - x - 1} + 1\right ) - \frac {1}{2} \, \log \left ({\left | -x + \sqrt {x^{2} - x - 1} \right |}\right ) + \frac {1}{2} \, \log \left ({\left | -x + \sqrt {x^{2} - x - 1} - 2 \right |}\right ) + \log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} - x - 1} + 1 \right |}\right ) \]
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Timed out. \[ \int \frac {\sqrt {-1-x+x^2}}{1-x^2} \, dx=-\int \frac {\sqrt {x^2-x-1}}{x^2-1} \,d x \]
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