Integrand size = 23, antiderivative size = 63 \[ \int \frac {1}{\left (1+\sqrt {x}\right ) \sqrt {-\sqrt {x}+x}} \, dx=4 \text {arctanh}\left (\frac {\sqrt {-\sqrt {x}+x}}{-1+\sqrt {x}}\right )-2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {-\sqrt {x}+x}}{-1+\sqrt {x}}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {1411, 857, 634, 212, 738} \[ \int \frac {1}{\left (1+\sqrt {x}\right ) \sqrt {-\sqrt {x}+x}} \, dx=\sqrt {2} \text {arctanh}\left (\frac {1-3 \sqrt {x}}{2 \sqrt {2} \sqrt {x-\sqrt {x}}}\right )+4 \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt {x-\sqrt {x}}}\right ) \]
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Rule 212
Rule 634
Rule 738
Rule 857
Rule 1411
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x}{(1+x) \sqrt {-x+x^2}} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \frac {1}{\sqrt {-x+x^2}} \, dx,x,\sqrt {x}\right )-2 \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {-x+x^2}} \, dx,x,\sqrt {x}\right ) \\ & = 4 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {-\sqrt {x}+x}}\right )+4 \text {Subst}\left (\int \frac {1}{8-x^2} \, dx,x,\frac {1-3 \sqrt {x}}{\sqrt {-\sqrt {x}+x}}\right ) \\ & = \sqrt {2} \text {arctanh}\left (\frac {1-3 \sqrt {x}}{2 \sqrt {2} \sqrt {-\sqrt {x}+x}}\right )+4 \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt {-\sqrt {x}+x}}\right ) \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (1+\sqrt {x}\right ) \sqrt {-\sqrt {x}+x}} \, dx=4 \text {arctanh}\left (\frac {\sqrt {-\sqrt {x}+x}}{-1+\sqrt {x}}\right )-2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {-\sqrt {x}+x}}{-1+\sqrt {x}}\right ) \]
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Time = 1.00 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.83
method | result | size |
derivativedivides | \(2 \ln \left (-\frac {1}{2}+\sqrt {x}+\sqrt {-\sqrt {x}+x}\right )+\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (1-3 \sqrt {x}\right ) \sqrt {2}}{4 \sqrt {\left (\sqrt {x}+1\right )^{2}-3 \sqrt {x}-1}}\right )\) | \(52\) |
default | \(\frac {\sqrt {-\sqrt {x}+x}\, \left (2 \ln \left (-\frac {1}{2}+\sqrt {x}+\sqrt {-\sqrt {x}+x}\right )-\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (-1+3 \sqrt {x}\right ) \sqrt {2}}{4 \sqrt {-\sqrt {x}+x}}\right )\right )}{\sqrt {\sqrt {x}\, \left (\sqrt {x}-1\right )}}\) | \(67\) |
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Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (47) = 94\).
Time = 0.94 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.62 \[ \int \frac {1}{\left (1+\sqrt {x}\right ) \sqrt {-\sqrt {x}+x}} \, dx=\frac {1}{2} \, \sqrt {2} \log \left (-\frac {17 \, x^{2} - 4 \, {\left (\sqrt {2} {\left (3 \, x + 5\right )} \sqrt {x} - \sqrt {2} {\left (7 \, x + 1\right )}\right )} \sqrt {x - \sqrt {x}} - 16 \, {\left (3 \, x + 1\right )} \sqrt {x} + 46 \, x + 1}{x^{2} - 2 \, x + 1}\right ) + \log \left (-4 \, \sqrt {x - \sqrt {x}} {\left (2 \, \sqrt {x} - 1\right )} - 8 \, x + 8 \, \sqrt {x} - 1\right ) \]
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\[ \int \frac {1}{\left (1+\sqrt {x}\right ) \sqrt {-\sqrt {x}+x}} \, dx=\int \frac {1}{\sqrt {- \sqrt {x} + x} \left (\sqrt {x} + 1\right )}\, dx \]
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\[ \int \frac {1}{\left (1+\sqrt {x}\right ) \sqrt {-\sqrt {x}+x}} \, dx=\int { \frac {1}{\sqrt {x - \sqrt {x}} {\left (\sqrt {x} + 1\right )}} \,d x } \]
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none
Time = 0.57 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.21 \[ \int \frac {1}{\left (1+\sqrt {x}\right ) \sqrt {-\sqrt {x}+x}} \, dx=-\sqrt {2} \log \left (\frac {2 \, {\left (\sqrt {2} - \sqrt {x - \sqrt {x}} + \sqrt {x} + 1\right )}}{{\left | 2 \, \sqrt {2} + 2 \, \sqrt {x - \sqrt {x}} - 2 \, \sqrt {x} - 2 \right |}}\right ) - 2 \, \log \left ({\left | 2 \, \sqrt {x - \sqrt {x}} - 2 \, \sqrt {x} + 1 \right |}\right ) \]
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Timed out. \[ \int \frac {1}{\left (1+\sqrt {x}\right ) \sqrt {-\sqrt {x}+x}} \, dx=\int \frac {1}{\sqrt {x-\sqrt {x}}\,\left (\sqrt {x}+1\right )} \,d x \]
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