Integrand size = 11, antiderivative size = 36 \[ \int \frac {x^6}{\left (-2+x^2\right )^2} \, dx=4 x+\frac {x^3}{3}-\frac {2 x}{-2+x^2}-5 \sqrt {2} \text {arctanh}\left (\frac {x}{\sqrt {2}}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.17, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {294, 308, 213} \[ \int \frac {x^6}{\left (-2+x^2\right )^2} \, dx=-5 \sqrt {2} \text {arctanh}\left (\frac {x}{\sqrt {2}}\right )+\frac {5 x^3}{6}+\frac {x^5}{2 \left (2-x^2\right )}+5 x \]
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Rule 213
Rule 294
Rule 308
Rubi steps \begin{align*} \text {integral}& = \frac {x^5}{2 \left (2-x^2\right )}+\frac {5}{2} \int \frac {x^4}{-2+x^2} \, dx \\ & = \frac {x^5}{2 \left (2-x^2\right )}+\frac {5}{2} \int \left (2+x^2+\frac {4}{-2+x^2}\right ) \, dx \\ & = 5 x+\frac {5 x^3}{6}+\frac {x^5}{2 \left (2-x^2\right )}+10 \int \frac {1}{-2+x^2} \, dx \\ & = 5 x+\frac {5 x^3}{6}+\frac {x^5}{2 \left (2-x^2\right )}-5 \sqrt {2} \text {arctanh}\left (\frac {x}{\sqrt {2}}\right ) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.47 \[ \int \frac {x^6}{\left (-2+x^2\right )^2} \, dx=4 x+\frac {x^3}{3}-\frac {2 x}{-2+x^2}+\frac {5 \log \left (\sqrt {2}-x\right )}{\sqrt {2}}-\frac {5 \log \left (\sqrt {2}+x\right )}{\sqrt {2}} \]
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Time = 0.23 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.89
method | result | size |
default | \(4 x +\frac {x^{3}}{3}-\frac {2 x}{x^{2}-2}-5 \,\operatorname {arctanh}\left (\frac {x \sqrt {2}}{2}\right ) \sqrt {2}\) | \(32\) |
risch | \(\frac {x^{3}}{3}+4 x -\frac {2 x}{x^{2}-2}+\frac {5 \sqrt {2}\, \ln \left (x -\sqrt {2}\right )}{2}-\frac {5 \sqrt {2}\, \ln \left (x +\sqrt {2}\right )}{2}\) | \(44\) |
meijerg | \(i \sqrt {2}\, \left (-\frac {i x \sqrt {2}\, \left (-\frac {7}{2} x^{4}-35 x^{2}+105\right )}{42 \left (-\frac {x^{2}}{2}+1\right )}+5 i \operatorname {arctanh}\left (\frac {x \sqrt {2}}{2}\right )\right )\) | \(46\) |
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Time = 0.24 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.47 \[ \int \frac {x^6}{\left (-2+x^2\right )^2} \, dx=\frac {2 \, x^{5} + 20 \, x^{3} + 15 \, \sqrt {2} {\left (x^{2} - 2\right )} \log \left (\frac {x^{2} - 2 \, \sqrt {2} x + 2}{x^{2} - 2}\right ) - 60 \, x}{6 \, {\left (x^{2} - 2\right )}} \]
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Time = 0.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.36 \[ \int \frac {x^6}{\left (-2+x^2\right )^2} \, dx=\frac {x^{3}}{3} + 4 x - \frac {2 x}{x^{2} - 2} + \frac {5 \sqrt {2} \log {\left (x - \sqrt {2} \right )}}{2} - \frac {5 \sqrt {2} \log {\left (x + \sqrt {2} \right )}}{2} \]
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Time = 0.27 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.11 \[ \int \frac {x^6}{\left (-2+x^2\right )^2} \, dx=\frac {1}{3} \, x^{3} + \frac {5}{2} \, \sqrt {2} \log \left (\frac {x - \sqrt {2}}{x + \sqrt {2}}\right ) + 4 \, x - \frac {2 \, x}{x^{2} - 2} \]
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Time = 0.30 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.33 \[ \int \frac {x^6}{\left (-2+x^2\right )^2} \, dx=\frac {1}{3} \, x^{3} + \frac {5}{2} \, \sqrt {2} \log \left (\frac {{\left | 2 \, x - 2 \, \sqrt {2} \right |}}{{\left | 2 \, x + 2 \, \sqrt {2} \right |}}\right ) + 4 \, x - \frac {2 \, x}{x^{2} - 2} \]
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Time = 0.27 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.92 \[ \int \frac {x^6}{\left (-2+x^2\right )^2} \, dx=4\,x-\frac {2\,x}{x^2-2}+\frac {x^3}{3}+\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x\,1{}\mathrm {i}}{2}\right )\,5{}\mathrm {i} \]
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