Integrand size = 24, antiderivative size = 87 \[ \int \frac {\sqrt {x}}{(-1+x) \sqrt {-\sqrt {x}+x}} \, dx=-\frac {2 \sqrt {-\sqrt {x}+x}}{-1+\sqrt {x}}+4 \text {arctanh}\left (\frac {\sqrt {-\sqrt {x}+x}}{-1+\sqrt {x}}\right )-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {-\sqrt {x}+x}}{-1+\sqrt {x}}\right ) \]
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Time = 0.16 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.41, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2081, 1563, 862, 100, 163, 56, 221, 95, 212} \[ \int \frac {\sqrt {x}}{(-1+x) \sqrt {-\sqrt {x}+x}} \, dx=\frac {4 \sqrt {\sqrt {x}-1} \sqrt [4]{x} \text {arcsinh}\left (\sqrt {\sqrt {x}-1}\right )}{\sqrt {x-\sqrt {x}}}-\frac {\sqrt {2} \sqrt {\sqrt {x}-1} \sqrt [4]{x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{x}}{\sqrt {\sqrt {x}-1}}\right )}{\sqrt {x-\sqrt {x}}}-\frac {2 \sqrt {x}}{\sqrt {x-\sqrt {x}}} \]
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Rule 56
Rule 95
Rule 100
Rule 163
Rule 212
Rule 221
Rule 862
Rule 1563
Rule 2081
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {-1+\sqrt {x}} \sqrt [4]{x}\right ) \int \frac {\sqrt [4]{x}}{\sqrt {-1+\sqrt {x}} (-1+x)} \, dx}{\sqrt {-\sqrt {x}+x}} \\ & = \frac {\left (2 \sqrt {-1+\sqrt {x}} \sqrt [4]{x}\right ) \text {Subst}\left (\int \frac {x^{3/2}}{\sqrt {-1+x} \left (-1+x^2\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-\sqrt {x}+x}} \\ & = \frac {\left (2 \sqrt {-1+\sqrt {x}} \sqrt [4]{x}\right ) \text {Subst}\left (\int \frac {x^{3/2}}{(-1+x)^{3/2} (1+x)} \, dx,x,\sqrt {x}\right )}{\sqrt {-\sqrt {x}+x}} \\ & = -\frac {2 \sqrt {x}}{\sqrt {-\sqrt {x}+x}}-\frac {\left (2 \sqrt {-1+\sqrt {x}} \sqrt [4]{x}\right ) \text {Subst}\left (\int \frac {-\frac {1}{2}-x}{\sqrt {-1+x} \sqrt {x} (1+x)} \, dx,x,\sqrt {x}\right )}{\sqrt {-\sqrt {x}+x}} \\ & = -\frac {2 \sqrt {x}}{\sqrt {-\sqrt {x}+x}}-\frac {\left (\sqrt {-1+\sqrt {x}} \sqrt [4]{x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {x} (1+x)} \, dx,x,\sqrt {x}\right )}{\sqrt {-\sqrt {x}+x}}+\frac {\left (2 \sqrt {-1+\sqrt {x}} \sqrt [4]{x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {x}} \, dx,x,\sqrt {x}\right )}{\sqrt {-\sqrt {x}+x}} \\ & = -\frac {2 \sqrt {x}}{\sqrt {-\sqrt {x}+x}}-\frac {\left (2 \sqrt {-1+\sqrt {x}} \sqrt [4]{x}\right ) \text {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt {-1+\sqrt {x}}}\right )}{\sqrt {-\sqrt {x}+x}}+\frac {\left (4 \sqrt {-1+\sqrt {x}} \sqrt [4]{x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\sqrt {-1+\sqrt {x}}\right )}{\sqrt {-\sqrt {x}+x}} \\ & = -\frac {2 \sqrt {x}}{\sqrt {-\sqrt {x}+x}}+\frac {4 \sqrt {-1+\sqrt {x}} \sqrt [4]{x} \text {arcsinh}\left (\sqrt {-1+\sqrt {x}}\right )}{\sqrt {-\sqrt {x}+x}}-\frac {\sqrt {2} \sqrt {-1+\sqrt {x}} \sqrt [4]{x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{x}}{\sqrt {-1+\sqrt {x}}}\right )}{\sqrt {-\sqrt {x}+x}} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {x}}{(-1+x) \sqrt {-\sqrt {x}+x}} \, dx=-\frac {2 \sqrt {-\sqrt {x}+x}}{-1+\sqrt {x}}+4 \text {arctanh}\left (\frac {\sqrt {-\sqrt {x}+x}}{-1+\sqrt {x}}\right )-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {-\sqrt {x}+x}}{-1+\sqrt {x}}\right ) \]
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Time = 1.13 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.87
method | result | size |
derivativedivides | \(2 \ln \left (-\frac {1}{2}+\sqrt {x}+\sqrt {-\sqrt {x}+x}\right )-\frac {2 \sqrt {\left (\sqrt {x}-1\right )^{2}+\sqrt {x}-1}}{\sqrt {x}-1}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (1-3 \sqrt {x}\right ) \sqrt {2}}{4 \sqrt {\left (\sqrt {x}+1\right )^{2}-3 \sqrt {x}-1}}\right )}{2}\) | \(76\) |
default | \(-\frac {\sqrt {-\sqrt {x}+x}\, \left (-2 \sqrt {x}\, \operatorname {arctanh}\left (\frac {\left (-1+3 \sqrt {x}\right ) \sqrt {2}}{4 \sqrt {-\sqrt {x}+x}}\right ) \sqrt {2}+4 \left (-\sqrt {x}+x \right )^{\frac {3}{2}}+x \,\operatorname {arctanh}\left (\frac {\left (-1+3 \sqrt {x}\right ) \sqrt {2}}{4 \sqrt {-\sqrt {x}+x}}\right ) \sqrt {2}+8 \sqrt {x}\, \sqrt {-\sqrt {x}+x}+8 \sqrt {x}\, \ln \left (-\frac {1}{2}+\sqrt {x}+\sqrt {-\sqrt {x}+x}\right )-4 x \sqrt {-\sqrt {x}+x}-4 x \ln \left (-\frac {1}{2}+\sqrt {x}+\sqrt {-\sqrt {x}+x}\right )+\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (-1+3 \sqrt {x}\right ) \sqrt {2}}{4 \sqrt {-\sqrt {x}+x}}\right )-4 \sqrt {-\sqrt {x}+x}-4 \ln \left (-\frac {1}{2}+\sqrt {x}+\sqrt {-\sqrt {x}+x}\right )\right )}{2 \sqrt {\sqrt {x}\, \left (\sqrt {x}-1\right )}\, \left (\sqrt {x}-1\right )^{2}}\) | \(217\) |
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Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (65) = 130\).
Time = 1.19 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.52 \[ \int \frac {\sqrt {x}}{(-1+x) \sqrt {-\sqrt {x}+x}} \, dx=\frac {\sqrt {2} {\left (x - 1\right )} \log \left (-\frac {17 \, x^{2} - 4 \, {\left (\sqrt {2} {\left (3 \, x + 5\right )} \sqrt {x} - \sqrt {2} {\left (7 \, x + 1\right )}\right )} \sqrt {x - \sqrt {x}} - 16 \, {\left (3 \, x + 1\right )} \sqrt {x} + 46 \, x + 1}{x^{2} - 2 \, x + 1}\right ) + 4 \, {\left (x - 1\right )} \log \left (-4 \, \sqrt {x - \sqrt {x}} {\left (2 \, \sqrt {x} - 1\right )} - 8 \, x + 8 \, \sqrt {x} - 1\right ) - 8 \, \sqrt {x - \sqrt {x}} {\left (\sqrt {x} + 1\right )}}{4 \, {\left (x - 1\right )}} \]
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\[ \int \frac {\sqrt {x}}{(-1+x) \sqrt {-\sqrt {x}+x}} \, dx=\int \frac {\sqrt {x}}{\sqrt {- \sqrt {x} + x} \left (x - 1\right )}\, dx \]
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\[ \int \frac {\sqrt {x}}{(-1+x) \sqrt {-\sqrt {x}+x}} \, dx=\int { \frac {\sqrt {x}}{\sqrt {x - \sqrt {x}} {\left (x - 1\right )}} \,d x } \]
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none
Time = 0.58 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt {x}}{(-1+x) \sqrt {-\sqrt {x}+x}} \, dx=-\frac {1}{2} \, \sqrt {2} \log \left (\frac {2 \, {\left (\sqrt {2} - \sqrt {x - \sqrt {x}} + \sqrt {x} + 1\right )}}{{\left | 2 \, \sqrt {2} + 2 \, \sqrt {x - \sqrt {x}} - 2 \, \sqrt {x} - 2 \right |}}\right ) - \frac {2}{\sqrt {x - \sqrt {x}} - \sqrt {x} + 1} - 2 \, \log \left ({\left | 2 \, \sqrt {x - \sqrt {x}} - 2 \, \sqrt {x} + 1 \right |}\right ) \]
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Timed out. \[ \int \frac {\sqrt {x}}{(-1+x) \sqrt {-\sqrt {x}+x}} \, dx=\int \frac {\sqrt {x}}{\sqrt {x-\sqrt {x}}\,\left (x-1\right )} \,d x \]
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