Integrand size = 13, antiderivative size = 11 \[ \int \frac {1+x^2}{\left (-1+x^2\right )^2} \, dx=\frac {x}{1-x^2} \]
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Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {391} \[ \int \frac {1+x^2}{\left (-1+x^2\right )^2} \, dx=\frac {x}{1-x^2} \]
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Rule 391
Rubi steps \begin{align*} \text {integral}& = \frac {x}{1-x^2} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.91 \[ \int \frac {1+x^2}{\left (-1+x^2\right )^2} \, dx=-\frac {x}{-1+x^2} \]
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Time = 2.51 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00
method | result | size |
gosper | \(-\frac {x}{x^{2}-1}\) | \(11\) |
norman | \(-\frac {x}{x^{2}-1}\) | \(11\) |
risch | \(-\frac {x}{x^{2}-1}\) | \(11\) |
parallelrisch | \(-\frac {x}{x^{2}-1}\) | \(11\) |
default | \(-\frac {1}{2 \left (1+x \right )}-\frac {1}{2 \left (-1+x \right )}\) | \(16\) |
meijerg | \(\frac {i \left (-\frac {i x}{-x^{2}+1}+i \operatorname {arctanh}\left (x \right )\right )}{2}-\frac {i \left (\frac {2 i x}{-2 x^{2}+2}+i \operatorname {arctanh}\left (x \right )\right )}{2}\) | \(46\) |
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none
Time = 0.28 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.91 \[ \int \frac {1+x^2}{\left (-1+x^2\right )^2} \, dx=-\frac {x}{x^{2} - 1} \]
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Time = 0.03 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.64 \[ \int \frac {1+x^2}{\left (-1+x^2\right )^2} \, dx=- \frac {x}{x^{2} - 1} \]
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none
Time = 0.20 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.91 \[ \int \frac {1+x^2}{\left (-1+x^2\right )^2} \, dx=-\frac {x}{x^{2} - 1} \]
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none
Time = 0.29 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {1+x^2}{\left (-1+x^2\right )^2} \, dx=-\frac {1}{x - \frac {1}{x}} \]
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Time = 0.03 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.91 \[ \int \frac {1+x^2}{\left (-1+x^2\right )^2} \, dx=-\frac {x}{x^2-1} \]
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