Integrand size = 15, antiderivative size = 11 \[ \int \frac {\sqrt {1+\frac {1}{x}}}{(1+x)^2} \, dx=\frac {2}{\sqrt {1+\frac {1}{x}}} \]
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Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {25, 267} \[ \int \frac {\sqrt {1+\frac {1}{x}}}{(1+x)^2} \, dx=\frac {2}{\sqrt {\frac {1}{x}+1}} \]
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Rule 25
Rule 267
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\left (1+\frac {1}{x}\right )^{3/2} x^2} \, dx \\ & = \frac {2}{\sqrt {1+\frac {1}{x}}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.73 \[ \int \frac {\sqrt {1+\frac {1}{x}}}{(1+x)^2} \, dx=\frac {2 x \sqrt {\frac {1+x}{x}}}{1+x} \]
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Time = 1.08 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.64
method | result | size |
gosper | \(\frac {2 x \sqrt {\frac {x +1}{x}}}{x +1}\) | \(18\) |
risch | \(\frac {2 x \sqrt {\frac {x +1}{x}}}{x +1}\) | \(18\) |
trager | \(\frac {2 x \sqrt {-\frac {-x -1}{x}}}{x +1}\) | \(21\) |
default | \(\frac {2 \sqrt {x^{2}+x}\, x \sqrt {\frac {x +1}{x}}}{\left (x +1\right ) \sqrt {\left (x +1\right ) x}}\) | \(32\) |
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none
Time = 0.27 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.55 \[ \int \frac {\sqrt {1+\frac {1}{x}}}{(1+x)^2} \, dx=\frac {2 \, x \sqrt {\frac {x + 1}{x}}}{x + 1} \]
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\[ \int \frac {\sqrt {1+\frac {1}{x}}}{(1+x)^2} \, dx=\int \frac {\sqrt {1 + \frac {1}{x}}}{\left (x + 1\right )^{2}}\, dx \]
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\[ \int \frac {\sqrt {1+\frac {1}{x}}}{(1+x)^2} \, dx=\int { \frac {\sqrt {\frac {1}{x} + 1}}{{\left (x + 1\right )}^{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 23 vs. \(2 (9) = 18\).
Time = 0.33 (sec) , antiderivative size = 23, normalized size of antiderivative = 2.09 \[ \int \frac {\sqrt {1+\frac {1}{x}}}{(1+x)^2} \, dx=\frac {2 \, \mathrm {sgn}\left (x\right )}{x - \sqrt {x^{2} + x} + 1} - 2 \, \mathrm {sgn}\left (x\right ) \]
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Time = 18.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.36 \[ \int \frac {\sqrt {1+\frac {1}{x}}}{(1+x)^2} \, dx=\frac {2\,x\,\sqrt {\frac {1}{x}+1}}{x+1} \]
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