Integrand size = 16, antiderivative size = 15 \[ \int \frac {\sqrt {-\frac {x}{1+x}}}{x} \, dx=2 \arctan \left (\sqrt {-\frac {x}{1+x}}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1980, 210} \[ \int \frac {\sqrt {-\frac {x}{1+x}}}{x} \, dx=2 \arctan \left (\sqrt {-\frac {x}{x+1}}\right ) \]
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Rule 210
Rule 1980
Rubi steps \begin{align*} \text {integral}& = -\left (2 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {-\frac {x}{1+x}}\right )\right ) \\ & = 2 \tan ^{-1}\left (\sqrt {-\frac {x}{1+x}}\right ) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(42\) vs. \(2(15)=30\).
Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.80 \[ \int \frac {\sqrt {-\frac {x}{1+x}}}{x} \, dx=-\frac {2 \sqrt {-\frac {x}{1+x}} \sqrt {1+x} \log \left (-\sqrt {x}+\sqrt {1+x}\right )}{\sqrt {x}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(32\) vs. \(2(13)=26\).
Time = 0.17 (sec) , antiderivative size = 33, normalized size of antiderivative = 2.20
method | result | size |
default | \(\frac {\sqrt {-\frac {x}{x +1}}\, \left (x +1\right ) \ln \left (x +\frac {1}{2}+\sqrt {x^{2}+x}\right )}{\sqrt {\left (x +1\right ) x}}\) | \(33\) |
trager | \(-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (2 \sqrt {-\frac {x}{x +1}}\, x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +2 \sqrt {-\frac {x}{x +1}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )\) | \(53\) |
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Time = 0.26 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {-\frac {x}{1+x}}}{x} \, dx=2 \, \arctan \left (\sqrt {-\frac {x}{x + 1}}\right ) \]
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\[ \int \frac {\sqrt {-\frac {x}{1+x}}}{x} \, dx=\int \frac {\sqrt {- \frac {x}{x + 1}}}{x}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {-\frac {x}{1+x}}}{x} \, dx=2 \, \arctan \left (\sqrt {-\frac {x}{x + 1}}\right ) \]
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Time = 0.33 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.33 \[ \int \frac {\sqrt {-\frac {x}{1+x}}}{x} \, dx=-\frac {1}{2} \, \pi \mathrm {sgn}\left (x + 1\right ) - \arcsin \left (2 \, x + 1\right ) \mathrm {sgn}\left (x + 1\right ) \]
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Time = 0.17 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {-\frac {x}{1+x}}}{x} \, dx=2\,\mathrm {atan}\left (\sqrt {-\frac {x}{x+1}}\right ) \]
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