Integrand size = 15, antiderivative size = 9 \[ \int \frac {1-x^2}{\left (1+x^2\right )^2} \, dx=\frac {x}{1+x^2} \]
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Time = 0.00 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {391} \[ \int \frac {1-x^2}{\left (1+x^2\right )^2} \, dx=\frac {x}{x^2+1} \]
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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 = 9, normalized size of antiderivative = 1.00 \[ \int \frac {1-x^2}{\left (1+x^2\right )^2} \, dx=\frac {x}{1+x^2} \]
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Time = 2.52 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.11
method | result | size |
gosper | \(\frac {x}{x^{2}+1}\) | \(10\) |
default | \(\frac {x}{x^{2}+1}\) | \(10\) |
norman | \(\frac {x}{x^{2}+1}\) | \(10\) |
risch | \(\frac {x}{x^{2}+1}\) | \(10\) |
parallelrisch | \(\frac {x}{x^{2}+1}\) | \(10\) |
meijerg | \(\frac {2 x}{2 x^{2}+2}\) | \(23\) |
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none
Time = 0.24 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \frac {1-x^2}{\left (1+x^2\right )^2} \, dx=\frac {x}{x^{2} + 1} \]
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Time = 0.04 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.56 \[ \int \frac {1-x^2}{\left (1+x^2\right )^2} \, dx=\frac {x}{x^{2} + 1} \]
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none
Time = 0.20 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \frac {1-x^2}{\left (1+x^2\right )^2} \, dx=\frac {x}{x^{2} + 1} \]
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none
Time = 0.29 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \frac {1-x^2}{\left (1+x^2\right )^2} \, dx=\frac {1}{x + \frac {1}{x}} \]
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Time = 0.03 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \frac {1-x^2}{\left (1+x^2\right )^2} \, dx=\frac {x}{x^2+1} \]
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