Integrand size = 19, antiderivative size = 49 \[ \int \frac {1}{\left (-1+x^2\right ) \sqrt {2 x+x^2}} \, dx=-\frac {1}{2} \arctan \left (\sqrt {2 x+x^2}\right )-\frac {\text {arctanh}\left (\frac {1+2 x}{\sqrt {3} \sqrt {2 x+x^2}}\right )}{2 \sqrt {3}} \]
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Time = 0.03 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {998, 702, 210, 738, 212} \[ \int \frac {1}{\left (-1+x^2\right ) \sqrt {2 x+x^2}} \, dx=-\frac {1}{2} \arctan \left (\sqrt {x^2+2 x}\right )-\frac {\text {arctanh}\left (\frac {2 x+1}{\sqrt {3} \sqrt {x^2+2 x}}\right )}{2 \sqrt {3}} \]
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Rule 210
Rule 212
Rule 702
Rule 738
Rule 998
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \int \frac {1}{(-1-x) \sqrt {2 x+x^2}} \, dx+\frac {1}{2} \int \frac {1}{(-1+x) \sqrt {2 x+x^2}} \, dx \\ & = 2 \text {Subst}\left (\int \frac {1}{-4-4 x^2} \, dx,x,\sqrt {2 x+x^2}\right )-\text {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {2+4 x}{\sqrt {2 x+x^2}}\right ) \\ & = -\frac {1}{2} \arctan \left (\sqrt {2 x+x^2}\right )-\frac {\text {arctanh}\left (\frac {2+4 x}{2 \sqrt {3} \sqrt {2 x+x^2}}\right )}{2 \sqrt {3}} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.59 \[ \int \frac {1}{\left (-1+x^2\right ) \sqrt {2 x+x^2}} \, dx=\frac {\sqrt {x} \sqrt {2+x} \left (3 \arctan \left (1+x-\sqrt {x} \sqrt {2+x}\right )-\sqrt {3} \text {arctanh}\left (\frac {1-x+\sqrt {x} \sqrt {2+x}}{\sqrt {3}}\right )\right )}{3 \sqrt {x (2+x)}} \]
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Time = 0.47 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.71
method | result | size |
pseudoelliptic | \(-\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {3}\, \sqrt {x \left (2+x \right )}}{3 x}\right )}{3}+\arctan \left (\frac {\sqrt {x \left (2+x \right )}}{x}\right )\) | \(35\) |
default | \(-\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (2+4 x \right ) \sqrt {3}}{6 \sqrt {\left (-1+x \right )^{2}-1+4 x}}\right )}{6}+\frac {\arctan \left (\frac {1}{\sqrt {\left (1+x \right )^{2}-1}}\right )}{2}\) | \(42\) |
trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{2}+2 x}}{1+x}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (\frac {-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +3 \sqrt {x^{2}+2 x}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )}{-1+x}\right )}{6}\) | \(77\) |
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Time = 0.25 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.27 \[ \int \frac {1}{\left (-1+x^2\right ) \sqrt {2 x+x^2}} \, dx=\frac {1}{6} \, \sqrt {3} \log \left (-\frac {\sqrt {3} {\left (2 \, x + 1\right )} + \sqrt {x^{2} + 2 \, x} {\left (2 \, \sqrt {3} - 3\right )} - 4 \, x - 2}{x - 1}\right ) - \arctan \left (-x + \sqrt {x^{2} + 2 \, x} - 1\right ) \]
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\[ \int \frac {1}{\left (-1+x^2\right ) \sqrt {2 x+x^2}} \, dx=\int \frac {1}{\sqrt {x \left (x + 2\right )} \left (x - 1\right ) \left (x + 1\right )}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\left (-1+x^2\right ) \sqrt {2 x+x^2}} \, dx=-\frac {1}{6} \, \sqrt {3} \log \left (\frac {2 \, \sqrt {3} \sqrt {x^{2} + 2 \, x}}{{\left | 2 \, x - 2 \right |}} + \frac {6}{{\left | 2 \, x - 2 \right |}} + 2\right ) + \frac {1}{2} \, \arcsin \left (\frac {2}{{\left | 2 \, x + 2 \right |}}\right ) \]
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Time = 0.32 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.45 \[ \int \frac {1}{\left (-1+x^2\right ) \sqrt {2 x+x^2}} \, dx=\frac {1}{6} \, \sqrt {3} \log \left (\frac {{\left | -2 \, x - 2 \, \sqrt {3} + 2 \, \sqrt {x^{2} + 2 \, x} + 2 \right |}}{{\left | -2 \, x + 2 \, \sqrt {3} + 2 \, \sqrt {x^{2} + 2 \, x} + 2 \right |}}\right ) - \arctan \left (-x + \sqrt {x^{2} + 2 \, x} - 1\right ) \]
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Timed out. \[ \int \frac {1}{\left (-1+x^2\right ) \sqrt {2 x+x^2}} \, dx=\int \frac {1}{\sqrt {x^2+2\,x}\,\left (x^2-1\right )} \,d x \]
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