Integrand size = 21, antiderivative size = 70 \[ \int \frac {\sqrt {1+x}}{\sqrt {x+\sqrt {1+x}}} \, dx=-\frac {3}{2} \sqrt {x+\sqrt {1+x}}+\sqrt {1+x} \sqrt {x+\sqrt {1+x}}-\frac {7}{4} \log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}\right ) \]
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Time = 0.12 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {756, 654, 635, 212} \[ \int \frac {\sqrt {1+x}}{\sqrt {x+\sqrt {1+x}}} \, dx=\frac {7}{4} \text {arctanh}\left (\frac {2 \sqrt {x+1}+1}{2 \sqrt {x+\sqrt {x+1}}}\right )+\sqrt {x+1} \sqrt {x+\sqrt {x+1}}-\frac {3}{2} \sqrt {x+\sqrt {x+1}} \]
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Rule 212
Rule 635
Rule 654
Rule 756
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x^2}{\sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right ) \\ & = \sqrt {1+x} \sqrt {x+\sqrt {1+x}}+\text {Subst}\left (\int \frac {1-\frac {3 x}{2}}{\sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right ) \\ & = -\frac {3}{2} \sqrt {x+\sqrt {1+x}}+\sqrt {1+x} \sqrt {x+\sqrt {1+x}}+\frac {7}{4} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right ) \\ & = -\frac {3}{2} \sqrt {x+\sqrt {1+x}}+\sqrt {1+x} \sqrt {x+\sqrt {1+x}}+\frac {7}{2} \text {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {1+2 \sqrt {1+x}}{\sqrt {x+\sqrt {1+x}}}\right ) \\ & = -\frac {3}{2} \sqrt {x+\sqrt {1+x}}+\sqrt {1+x} \sqrt {x+\sqrt {1+x}}+\frac {7}{4} \text {arctanh}\left (\frac {1+2 \sqrt {1+x}}{2 \sqrt {x+\sqrt {1+x}}}\right ) \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {1+x}}{\sqrt {x+\sqrt {1+x}}} \, dx=\frac {1}{2} \sqrt {x+\sqrt {1+x}} \left (-3+2 \sqrt {1+x}\right )-\frac {7}{4} \log \left (-1-2 \sqrt {1+x}+2 \sqrt {x+\sqrt {1+x}}\right ) \]
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Time = 0.89 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.67
method | result | size |
derivativedivides | \(\sqrt {1+x}\, \sqrt {x +\sqrt {1+x}}-\frac {3 \sqrt {x +\sqrt {1+x}}}{2}+\frac {7 \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )}{4}\) | \(47\) |
default | \(\sqrt {1+x}\, \sqrt {x +\sqrt {1+x}}-\frac {3 \sqrt {x +\sqrt {1+x}}}{2}+\frac {7 \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )}{4}\) | \(47\) |
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Time = 0.51 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.80 \[ \int \frac {\sqrt {1+x}}{\sqrt {x+\sqrt {1+x}}} \, dx=\frac {1}{2} \, \sqrt {x + \sqrt {x + 1}} {\left (2 \, \sqrt {x + 1} - 3\right )} + \frac {7}{8} \, \log \left (4 \, \sqrt {x + \sqrt {x + 1}} {\left (2 \, \sqrt {x + 1} + 1\right )} + 8 \, x + 8 \, \sqrt {x + 1} + 5\right ) \]
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Time = 0.41 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {1+x}}{\sqrt {x+\sqrt {1+x}}} \, dx=2 \sqrt {x + \sqrt {x + 1}} \left (\frac {\sqrt {x + 1}}{2} - \frac {3}{4}\right ) + \frac {7 \log {\left (2 \sqrt {x + 1} + 2 \sqrt {x + \sqrt {x + 1}} + 1 \right )}}{4} \]
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\[ \int \frac {\sqrt {1+x}}{\sqrt {x+\sqrt {1+x}}} \, dx=\int { \frac {\sqrt {x + 1}}{\sqrt {x + \sqrt {x + 1}}} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.63 \[ \int \frac {\sqrt {1+x}}{\sqrt {x+\sqrt {1+x}}} \, dx=\frac {1}{2} \, \sqrt {x + \sqrt {x + 1}} {\left (2 \, \sqrt {x + 1} - 3\right )} - \frac {7}{4} \, \log \left (-2 \, \sqrt {x + \sqrt {x + 1}} + 2 \, \sqrt {x + 1} + 1\right ) \]
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Timed out. \[ \int \frac {\sqrt {1+x}}{\sqrt {x+\sqrt {1+x}}} \, dx=\int \frac {\sqrt {x+1}}{\sqrt {x+\sqrt {x+1}}} \,d x \]
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