Integrand size = 13, antiderivative size = 16 \[ \int \frac {1}{x^2 \sqrt {4+x^2}} \, dx=-\frac {\sqrt {4+x^2}}{4 x} \]
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Time = 0.00 (sec) , antiderivative size = 16, 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 = {270} \[ \int \frac {1}{x^2 \sqrt {4+x^2}} \, dx=-\frac {\sqrt {x^2+4}}{4 x} \]
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Rule 270
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {4+x^2}}{4 x} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \sqrt {4+x^2}} \, dx=-\frac {\sqrt {4+x^2}}{4 x} \]
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Time = 0.14 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81
method | result | size |
gosper | \(-\frac {\sqrt {x^{2}+4}}{4 x}\) | \(13\) |
default | \(-\frac {\sqrt {x^{2}+4}}{4 x}\) | \(13\) |
trager | \(-\frac {\sqrt {x^{2}+4}}{4 x}\) | \(13\) |
risch | \(-\frac {\sqrt {x^{2}+4}}{4 x}\) | \(13\) |
pseudoelliptic | \(-\frac {\sqrt {x^{2}+4}}{4 x}\) | \(13\) |
meijerg | \(-\frac {\sqrt {1+\frac {x^{2}}{4}}}{2 x}\) | \(15\) |
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none
Time = 0.23 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^2 \sqrt {4+x^2}} \, dx=-\frac {x + \sqrt {x^{2} + 4}}{4 \, x} \]
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Time = 0.39 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^2 \sqrt {4+x^2}} \, dx=- \frac {\sqrt {1 + \frac {4}{x^{2}}}}{4} \]
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none
Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^2 \sqrt {4+x^2}} \, dx=-\frac {\sqrt {x^{2} + 4}}{4 \, x} \]
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none
Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.19 \[ \int \frac {1}{x^2 \sqrt {4+x^2}} \, dx=\frac {2}{{\left (x - \sqrt {x^{2} + 4}\right )}^{2} - 4} \]
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Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^2 \sqrt {4+x^2}} \, dx=-\frac {\sqrt {x^2+4}}{4\,x} \]
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