Integrand size = 11, antiderivative size = 22 \[ \int \sqrt {\frac {x}{1+x}} \, dx=\sqrt {x} \sqrt {1+x}-\text {arcsinh}\left (\sqrt {x}\right ) \]
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Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {1978, 52, 56, 221} \[ \int \sqrt {\frac {x}{1+x}} \, dx=\sqrt {x} \sqrt {x+1}-\text {arcsinh}\left (\sqrt {x}\right ) \]
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Rule 52
Rule 56
Rule 221
Rule 1978
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {x}}{\sqrt {1+x}} \, dx \\ & = \sqrt {x} \sqrt {1+x}-\frac {1}{2} \int \frac {1}{\sqrt {x} \sqrt {1+x}} \, dx \\ & = \sqrt {x} \sqrt {1+x}-\text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\sqrt {x}\right ) \\ & = \sqrt {x} \sqrt {1+x}-\sinh ^{-1}\left (\sqrt {x}\right ) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(51\) vs. \(2(22)=44\).
Time = 0.00 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.32 \[ \int \sqrt {\frac {x}{1+x}} \, dx=\frac {\sqrt {\frac {x}{1+x}} \left (\sqrt {x} (1+x)+\sqrt {1+x} \log \left (-\sqrt {x}+\sqrt {1+x}\right )\right )}{\sqrt {x}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(44\) vs. \(2(16)=32\).
Time = 0.11 (sec) , antiderivative size = 45, normalized size of antiderivative = 2.05
method | result | size |
default | \(\frac {\sqrt {\frac {x}{x +1}}\, \left (x +1\right ) \left (2 \sqrt {x^{2}+x}-\ln \left (x +\frac {1}{2}+\sqrt {x^{2}+x}\right )\right )}{2 \sqrt {\left (x +1\right ) x}}\) | \(45\) |
risch | \(\left (x +1\right ) \sqrt {\frac {x}{x +1}}-\frac {\ln \left (x +\frac {1}{2}+\sqrt {x^{2}+x}\right ) \sqrt {\frac {x}{x +1}}\, \sqrt {\left (x +1\right ) x}}{2 x}\) | \(47\) |
trager | \(2 \left (\frac {x}{2}+\frac {1}{2}\right ) \sqrt {\frac {x}{x +1}}-\frac {\ln \left (2 \sqrt {\frac {x}{x +1}}\, x +2 \sqrt {\frac {x}{x +1}}+2 x +1\right )}{2}\) | \(49\) |
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Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (16) = 32\).
Time = 0.31 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.91 \[ \int \sqrt {\frac {x}{1+x}} \, dx={\left (x + 1\right )} \sqrt {\frac {x}{x + 1}} - \frac {1}{2} \, \log \left (\sqrt {\frac {x}{x + 1}} + 1\right ) + \frac {1}{2} \, \log \left (\sqrt {\frac {x}{x + 1}} - 1\right ) \]
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\[ \int \sqrt {\frac {x}{1+x}} \, dx=\int \sqrt {\frac {x}{x + 1}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (16) = 32\).
Time = 0.19 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.32 \[ \int \sqrt {\frac {x}{1+x}} \, dx=-\frac {\sqrt {\frac {x}{x + 1}}}{\frac {x}{x + 1} - 1} - \frac {1}{2} \, \log \left (\sqrt {\frac {x}{x + 1}} + 1\right ) + \frac {1}{2} \, \log \left (\sqrt {\frac {x}{x + 1}} - 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (16) = 32\).
Time = 0.34 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.59 \[ \int \sqrt {\frac {x}{1+x}} \, dx=\frac {1}{2} \, \log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} + x} - 1 \right |}\right ) \mathrm {sgn}\left (x + 1\right ) + \sqrt {x^{2} + x} \mathrm {sgn}\left (x + 1\right ) \]
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Time = 18.84 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.59 \[ \int \sqrt {\frac {x}{1+x}} \, dx=-\mathrm {atanh}\left (\sqrt {\frac {x}{x+1}}\right )-\frac {\sqrt {\frac {x}{x+1}}}{\frac {x}{x+1}-1} \]
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