Integrand size = 21, antiderivative size = 62 \[ \int \frac {x}{\left (-1+x^2\right ) \sqrt {4+2 x+x^2}} \, dx=-\frac {\text {arctanh}\left (\frac {5+2 x}{\sqrt {7} \sqrt {4+2 x+x^2}}\right )}{2 \sqrt {7}}-\frac {\text {arctanh}\left (\frac {\sqrt {4+2 x+x^2}}{\sqrt {3}}\right )}{2 \sqrt {3}} \]
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Time = 0.03 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {1047, 738, 212, 702, 213} \[ \int \frac {x}{\left (-1+x^2\right ) \sqrt {4+2 x+x^2}} \, dx=-\frac {\text {arctanh}\left (\frac {2 x+5}{\sqrt {7} \sqrt {x^2+2 x+4}}\right )}{2 \sqrt {7}}-\frac {\text {arctanh}\left (\frac {\sqrt {x^2+2 x+4}}{\sqrt {3}}\right )}{2 \sqrt {3}} \]
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Rule 212
Rule 213
Rule 702
Rule 738
Rule 1047
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \int \frac {1}{(-1+x) \sqrt {4+2 x+x^2}} \, dx+\frac {1}{2} \int \frac {1}{(1+x) \sqrt {4+2 x+x^2}} \, dx \\ & = 2 \text {Subst}\left (\int \frac {1}{-12+4 x^2} \, dx,x,\sqrt {4+2 x+x^2}\right )-\text {Subst}\left (\int \frac {1}{28-x^2} \, dx,x,\frac {10+4 x}{\sqrt {4+2 x+x^2}}\right ) \\ & = -\frac {\text {arctanh}\left (\frac {10+4 x}{2 \sqrt {7} \sqrt {4+2 x+x^2}}\right )}{2 \sqrt {7}}-\frac {\text {arctanh}\left (\frac {\sqrt {4+2 x+x^2}}{\sqrt {3}}\right )}{2 \sqrt {3}} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00 \[ \int \frac {x}{\left (-1+x^2\right ) \sqrt {4+2 x+x^2}} \, dx=\frac {\text {arctanh}\left (\frac {1+x-\sqrt {4+2 x+x^2}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\text {arctanh}\left (\frac {1-x+\sqrt {4+2 x+x^2}}{\sqrt {7}}\right )}{\sqrt {7}} \]
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Time = 0.56 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.79
method | result | size |
default | \(-\frac {\sqrt {7}\, \operatorname {arctanh}\left (\frac {\left (10+4 x \right ) \sqrt {7}}{14 \sqrt {\left (-1+x \right )^{2}+3+4 x}}\right )}{14}-\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {3}}{\sqrt {\left (1+x \right )^{2}+3}}\right )}{6}\) | \(49\) |
trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-7\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-7\right ) x +7 \sqrt {x^{2}+2 x +4}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-7\right )}{-1+x}\right )}{14}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (\frac {\sqrt {x^{2}+2 x +4}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )}{1+x}\right )}{6}\) | \(82\) |
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Time = 0.26 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.19 \[ \int \frac {x}{\left (-1+x^2\right ) \sqrt {4+2 x+x^2}} \, dx=\frac {1}{14} \, \sqrt {7} \log \left (\frac {\sqrt {7} {\left (2 \, x + 5\right )} + \sqrt {x^{2} + 2 \, x + 4} {\left (2 \, \sqrt {7} - 7\right )} - 4 \, x - 10}{x - 1}\right ) + \frac {1}{6} \, \sqrt {3} \log \left (-\frac {\sqrt {3} - \sqrt {x^{2} + 2 \, x + 4}}{x + 1}\right ) \]
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\[ \int \frac {x}{\left (-1+x^2\right ) \sqrt {4+2 x+x^2}} \, dx=\int \frac {x}{\left (x - 1\right ) \left (x + 1\right ) \sqrt {x^{2} + 2 x + 4}}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.87 \[ \int \frac {x}{\left (-1+x^2\right ) \sqrt {4+2 x+x^2}} \, dx=-\frac {1}{14} \, \sqrt {7} \operatorname {arsinh}\left (\frac {4 \, \sqrt {3} x}{3 \, {\left | 2 \, x - 2 \right |}} + \frac {10 \, \sqrt {3}}{3 \, {\left | 2 \, x - 2 \right |}}\right ) - \frac {1}{6} \, \sqrt {3} \operatorname {arsinh}\left (\frac {2 \, \sqrt {3}}{{\left | 2 \, x + 2 \right |}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (48) = 96\).
Time = 0.32 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.76 \[ \int \frac {x}{\left (-1+x^2\right ) \sqrt {4+2 x+x^2}} \, dx=\frac {1}{14} \, \sqrt {7} \log \left (\frac {{\left | -2 \, x - 2 \, \sqrt {7} + 2 \, \sqrt {x^{2} + 2 \, x + 4} + 2 \right |}}{{\left | -2 \, x + 2 \, \sqrt {7} + 2 \, \sqrt {x^{2} + 2 \, x + 4} + 2 \right |}}\right ) + \frac {1}{6} \, \sqrt {3} \log \left (-\frac {{\left | -2 \, x - 2 \, \sqrt {3} + 2 \, \sqrt {x^{2} + 2 \, x + 4} - 2 \right |}}{2 \, {\left (x - \sqrt {3} - \sqrt {x^{2} + 2 \, x + 4} + 1\right )}}\right ) \]
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Timed out. \[ \int \frac {x}{\left (-1+x^2\right ) \sqrt {4+2 x+x^2}} \, dx=\int \frac {x}{\left (x^2-1\right )\,\sqrt {x^2+2\,x+4}} \,d x \]
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