Integrand size = 18, antiderivative size = 61 \[ \int \frac {\sqrt {-1-x+x^2}}{1+x} \, dx=\sqrt {-1-x+x^2}+\frac {3}{2} \text {arctanh}\left (\frac {1-2 x}{2 \sqrt {-1-x+x^2}}\right )+\text {arctanh}\left (\frac {1+3 x}{2 \sqrt {-1-x+x^2}}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {748, 857, 635, 212, 738} \[ \int \frac {\sqrt {-1-x+x^2}}{1+x} \, dx=\frac {3}{2} \text {arctanh}\left (\frac {1-2 x}{2 \sqrt {x^2-x-1}}\right )+\text {arctanh}\left (\frac {3 x+1}{2 \sqrt {x^2-x-1}}\right )+\sqrt {x^2-x-1} \]
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Rule 212
Rule 635
Rule 738
Rule 748
Rule 857
Rubi steps \begin{align*} \text {integral}& = \sqrt {-1-x+x^2}-\frac {1}{2} \int \frac {1+3 x}{(1+x) \sqrt {-1-x+x^2}} \, dx \\ & = \sqrt {-1-x+x^2}-\frac {3}{2} \int \frac {1}{\sqrt {-1-x+x^2}} \, dx+\int \frac {1}{(1+x) \sqrt {-1-x+x^2}} \, dx \\ & = \sqrt {-1-x+x^2}-2 \text {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {-1-3 x}{\sqrt {-1-x+x^2}}\right )-3 \text {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {-1+2 x}{\sqrt {-1-x+x^2}}\right ) \\ & = \sqrt {-1-x+x^2}+\frac {3}{2} \tanh ^{-1}\left (\frac {1-2 x}{2 \sqrt {-1-x+x^2}}\right )+\tanh ^{-1}\left (\frac {1+3 x}{2 \sqrt {-1-x+x^2}}\right ) \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {-1-x+x^2}}{1+x} \, dx=\sqrt {-1-x+x^2}+2 \text {arctanh}\left (1+x-\sqrt {-1-x+x^2}\right )+\frac {3}{2} \log \left (1-2 x+2 \sqrt {-1-x+x^2}\right ) \]
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Time = 0.27 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.82
method | result | size |
risch | \(\sqrt {x^{2}-x -1}-\frac {3 \ln \left (-\frac {1}{2}+x +\sqrt {x^{2}-x -1}\right )}{2}-\operatorname {arctanh}\left (\frac {-1-3 x}{2 \sqrt {\left (1+x \right )^{2}-2-3 x}}\right )\) | \(50\) |
default | \(\sqrt {\left (1+x \right )^{2}-2-3 x}-\frac {3 \ln \left (-\frac {1}{2}+x +\sqrt {\left (1+x \right )^{2}-2-3 x}\right )}{2}-\operatorname {arctanh}\left (\frac {-1-3 x}{2 \sqrt {\left (1+x \right )^{2}-2-3 x}}\right )\) | \(54\) |
trager | \(\sqrt {x^{2}-x -1}+\frac {\ln \left (\frac {32 \sqrt {x^{2}-x -1}\, x^{4}-32 x^{5}+96 \sqrt {x^{2}-x -1}\, x^{3}-80 x^{4}+78 \sqrt {x^{2}-x -1}\, x^{2}-10 x^{3}-16 \sqrt {x^{2}-x -1}\, x +125 x^{2}-38 \sqrt {x^{2}-x -1}+120 x +41}{\left (1+x \right )^{2}}\right )}{2}\) | \(116\) |
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Time = 0.30 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt {-1-x+x^2}}{1+x} \, dx=\sqrt {x^{2} - x - 1} - \log \left (-x + \sqrt {x^{2} - x - 1}\right ) + \log \left (-x + \sqrt {x^{2} - x - 1} - 2\right ) + \frac {3}{2} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} - x - 1} + 1\right ) \]
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\[ \int \frac {\sqrt {-1-x+x^2}}{1+x} \, dx=\int \frac {\sqrt {x^{2} - x - 1}}{x + 1}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt {-1-x+x^2}}{1+x} \, dx=\sqrt {x^{2} - x - 1} - \frac {3}{2} \, \log \left (2 \, x + 2 \, \sqrt {x^{2} - x - 1} - 1\right ) - \log \left (\frac {2 \, \sqrt {x^{2} - x - 1}}{{\left | x + 1 \right |}} + \frac {2}{{\left | x + 1 \right |}} - 3\right ) \]
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Time = 0.27 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt {-1-x+x^2}}{1+x} \, dx=\sqrt {x^{2} - x - 1} - \log \left ({\left | -x + \sqrt {x^{2} - x - 1} \right |}\right ) + \log \left ({\left | -x + \sqrt {x^{2} - x - 1} - 2 \right |}\right ) + \frac {3}{2} \, \log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} - x - 1} + 1 \right |}\right ) \]
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Timed out. \[ \int \frac {\sqrt {-1-x+x^2}}{1+x} \, dx=\int \frac {\sqrt {x^2-x-1}}{x+1} \,d x \]
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