Integrand size = 32, antiderivative size = 171 \[ \int \left (\frac {1}{\sqrt {1-\sqrt {x}}}-\sqrt {1-\sqrt {x}-x}\right ) \, dx=-\frac {8}{3} \sqrt {1-\sqrt {x}}+\left (\frac {2}{3}+\frac {1}{4} \sqrt {\left (1+2 \sqrt {x}\right )^2}\right ) \sqrt {1-\sqrt {x}-x}-\frac {4}{3} \sqrt {1-\sqrt {x}} \sqrt {x}-\frac {2}{3} \sqrt {1-\sqrt {x}-x} \sqrt {x}-\frac {2}{3} \sqrt {1-\sqrt {x}-x} x+\frac {5}{8} i \log \left (i \sqrt {\left (1+2 \sqrt {x}\right )^2}-2 \sqrt {1-\sqrt {x}-x}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.60, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {196, 45, 1355, 654, 626, 633, 222} \[ \int \left (\frac {1}{\sqrt {1-\sqrt {x}}}-\sqrt {1-\sqrt {x}-x}\right ) \, dx=\frac {5}{8} \arcsin \left (\frac {2 \sqrt {x}+1}{\sqrt {5}}\right )+\frac {4}{3} \left (1-\sqrt {x}\right )^{3/2}-4 \sqrt {1-\sqrt {x}}+\frac {2}{3} \left (-x-\sqrt {x}+1\right )^{3/2}+\frac {1}{4} \left (2 \sqrt {x}+1\right ) \sqrt {-x-\sqrt {x}+1} \]
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Rule 45
Rule 196
Rule 222
Rule 626
Rule 633
Rule 654
Rule 1355
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\sqrt {1-\sqrt {x}}} \, dx-\int \sqrt {1-\sqrt {x}-x} \, dx \\ & = 2 \text {Subst}\left (\int \frac {x}{\sqrt {1-x}} \, dx,x,\sqrt {x}\right )-2 \text {Subst}\left (\int x \sqrt {1-x-x^2} \, dx,x,\sqrt {x}\right ) \\ & = \frac {2}{3} \left (1-\sqrt {x}-x\right )^{3/2}+2 \text {Subst}\left (\int \left (\frac {1}{\sqrt {1-x}}-\sqrt {1-x}\right ) \, dx,x,\sqrt {x}\right )+\text {Subst}\left (\int \sqrt {1-x-x^2} \, dx,x,\sqrt {x}\right ) \\ & = -4 \sqrt {1-\sqrt {x}}+\frac {4}{3} \left (1-\sqrt {x}\right )^{3/2}+\frac {1}{4} \left (1+2 \sqrt {x}\right ) \sqrt {1-\sqrt {x}-x}+\frac {2}{3} \left (1-\sqrt {x}-x\right )^{3/2}+\frac {5}{8} \text {Subst}\left (\int \frac {1}{\sqrt {1-x-x^2}} \, dx,x,\sqrt {x}\right ) \\ & = -4 \sqrt {1-\sqrt {x}}+\frac {4}{3} \left (1-\sqrt {x}\right )^{3/2}+\frac {1}{4} \left (1+2 \sqrt {x}\right ) \sqrt {1-\sqrt {x}-x}+\frac {2}{3} \left (1-\sqrt {x}-x\right )^{3/2}-\frac {1}{8} \sqrt {5} \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{5}}} \, dx,x,-1-2 \sqrt {x}\right ) \\ & = -4 \sqrt {1-\sqrt {x}}+\frac {4}{3} \left (1-\sqrt {x}\right )^{3/2}+\frac {1}{4} \left (1+2 \sqrt {x}\right ) \sqrt {1-\sqrt {x}-x}+\frac {2}{3} \left (1-\sqrt {x}-x\right )^{3/2}+\frac {5}{8} \arcsin \left (\frac {1+2 \sqrt {x}}{\sqrt {5}}\right ) \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.51 \[ \int \left (\frac {1}{\sqrt {1-\sqrt {x}}}-\sqrt {1-\sqrt {x}-x}\right ) \, dx=-\frac {4}{3} \sqrt {1-\sqrt {x}} \left (2+\sqrt {x}\right )-\frac {1}{12} \sqrt {1-\sqrt {x}-x} \left (-11+2 \sqrt {x}+8 x\right )+\frac {5}{4} \arctan \left (\frac {\sqrt {x}}{-1+\sqrt {1-\sqrt {x}-x}}\right ) \]
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Time = 0.10 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.42
method | result | size |
derivativedivides | \(\frac {2 \left (1-\sqrt {x}-x \right )^{\frac {3}{2}}}{3}-\frac {\left (-2 \sqrt {x}-1\right ) \sqrt {1-\sqrt {x}-x}}{4}+\frac {5 \arcsin \left (\frac {2 \sqrt {5}\, \left (\sqrt {x}+\frac {1}{2}\right )}{5}\right )}{8}+\frac {4 \left (1-\sqrt {x}\right )^{\frac {3}{2}}}{3}-4 \sqrt {1-\sqrt {x}}\) | \(72\) |
default | \(\frac {2 \left (1-\sqrt {x}-x \right )^{\frac {3}{2}}}{3}-\frac {\left (-2 \sqrt {x}-1\right ) \sqrt {1-\sqrt {x}-x}}{4}+\frac {5 \arcsin \left (\frac {2 \sqrt {5}\, \left (\sqrt {x}+\frac {1}{2}\right )}{5}\right )}{8}+\frac {4 \left (1-\sqrt {x}\right )^{\frac {3}{2}}}{3}-4 \sqrt {1-\sqrt {x}}\) | \(72\) |
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Time = 0.77 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.58 \[ \int \left (\frac {1}{\sqrt {1-\sqrt {x}}}-\sqrt {1-\sqrt {x}-x}\right ) \, dx=-\frac {1}{12} \, {\left (8 \, x + 2 \, \sqrt {x} - 11\right )} \sqrt {-x - \sqrt {x} + 1} - \frac {4}{3} \, {\left (\sqrt {x} + 2\right )} \sqrt {-\sqrt {x} + 1} - \frac {5}{16} \, \arctan \left (-\frac {{\left (8 \, x^{2} - {\left (16 \, x^{2} - 38 \, x + 11\right )} \sqrt {x} - 9 \, x + 3\right )} \sqrt {-x - \sqrt {x} + 1}}{4 \, {\left (4 \, x^{3} - 13 \, x^{2} + 7 \, x - 1\right )}}\right ) \]
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Time = 0.99 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.73 \[ \int \left (\frac {1}{\sqrt {1-\sqrt {x}}}-\sqrt {1-\sqrt {x}-x}\right ) \, dx=- 2 \sqrt {- \sqrt {x} - x + 1} \left (\frac {\sqrt {x}}{12} + \frac {x}{3} - \frac {11}{24}\right ) + \begin {cases} \frac {4 i x^{\frac {5}{2}} \sqrt {\sqrt {x} - 1}}{- 3 x^{\frac {5}{2}} + 3 x^{2}} - \frac {8 x^{\frac {5}{2}}}{- 3 x^{\frac {5}{2}} + 3 x^{2}} + \frac {4 i x^{3} \sqrt {\sqrt {x} - 1}}{- 3 x^{\frac {5}{2}} + 3 x^{2}} - \frac {8 i x^{2} \sqrt {\sqrt {x} - 1}}{- 3 x^{\frac {5}{2}} + 3 x^{2}} + \frac {8 x^{2}}{- 3 x^{\frac {5}{2}} + 3 x^{2}} & \text {for}\: \left |{\sqrt {x}}\right | > 1 \\\frac {4 x^{\frac {5}{2}} \sqrt {1 - \sqrt {x}}}{- 3 x^{\frac {5}{2}} + 3 x^{2}} - \frac {8 x^{\frac {5}{2}}}{- 3 x^{\frac {5}{2}} + 3 x^{2}} + \frac {4 x^{3} \sqrt {1 - \sqrt {x}}}{- 3 x^{\frac {5}{2}} + 3 x^{2}} - \frac {8 x^{2} \sqrt {1 - \sqrt {x}}}{- 3 x^{\frac {5}{2}} + 3 x^{2}} + \frac {8 x^{2}}{- 3 x^{\frac {5}{2}} + 3 x^{2}} & \text {otherwise} \end {cases} + \frac {5 \operatorname {asin}{\left (\frac {2 \sqrt {5} \left (\sqrt {x} + \frac {1}{2}\right )}{5} \right )}}{8} \]
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\[ \int \left (\frac {1}{\sqrt {1-\sqrt {x}}}-\sqrt {1-\sqrt {x}-x}\right ) \, dx=\int { -\sqrt {-x - \sqrt {x} + 1} + \frac {1}{\sqrt {-\sqrt {x} + 1}} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.39 \[ \int \left (\frac {1}{\sqrt {1-\sqrt {x}}}-\sqrt {1-\sqrt {x}-x}\right ) \, dx=-\frac {1}{12} \, {\left (2 \, \sqrt {x} {\left (4 \, \sqrt {x} + 1\right )} - 11\right )} \sqrt {-x - \sqrt {x} + 1} + \frac {4}{3} \, {\left (-\sqrt {x} + 1\right )}^{\frac {3}{2}} - 4 \, \sqrt {-\sqrt {x} + 1} + \frac {5}{8} \, \arcsin \left (\frac {1}{5} \, \sqrt {5} {\left (2 \, \sqrt {x} + 1\right )}\right ) \]
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Timed out. \[ \int \left (\frac {1}{\sqrt {1-\sqrt {x}}}-\sqrt {1-\sqrt {x}-x}\right ) \, dx=-\int \sqrt {1-\sqrt {x}-x}-\frac {1}{\sqrt {1-\sqrt {x}}} \,d x \]
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