Integrand size = 25, antiderivative size = 97 \[ \int \frac {\sqrt {x+\sqrt {1+x}}}{1+\sqrt {1+x}} \, dx=-\frac {3}{2} \sqrt {x+\sqrt {1+x}}+\sqrt {1+x} \sqrt {x+\sqrt {1+x}}-4 \arctan \left (1+\sqrt {1+x}-\sqrt {x+\sqrt {1+x}}\right )+\frac {1}{4} \log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}\right ) \]
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Time = 0.10 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.94, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {828, 857, 635, 212, 738, 210} \[ \int \frac {\sqrt {x+\sqrt {1+x}}}{1+\sqrt {1+x}} \, dx=-2 \arctan \left (\frac {\sqrt {x+1}+3}{2 \sqrt {x+\sqrt {x+1}}}\right )-\frac {1}{4} \text {arctanh}\left (\frac {2 \sqrt {x+1}+1}{2 \sqrt {x+\sqrt {x+1}}}\right )-\frac {1}{2} \sqrt {x+\sqrt {x+1}} \left (3-2 \sqrt {x+1}\right ) \]
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Rule 210
Rule 212
Rule 635
Rule 738
Rule 828
Rule 857
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x \sqrt {-1+x+x^2}}{1+x} \, dx,x,\sqrt {1+x}\right ) \\ & = -\frac {1}{2} \left (3-2 \sqrt {1+x}\right ) \sqrt {x+\sqrt {1+x}}-\frac {1}{2} \text {Subst}\left (\int \frac {-\frac {7}{2}+\frac {x}{2}}{(1+x) \sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right ) \\ & = -\frac {1}{2} \left (3-2 \sqrt {1+x}\right ) \sqrt {x+\sqrt {1+x}}-\frac {1}{4} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )+2 \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right ) \\ & = -\frac {1}{2} \left (3-2 \sqrt {1+x}\right ) \sqrt {x+\sqrt {1+x}}-\frac {1}{2} \text {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {1+2 \sqrt {1+x}}{\sqrt {x+\sqrt {1+x}}}\right )-4 \text {Subst}\left (\int \frac {1}{-4-x^2} \, dx,x,\frac {-3-\sqrt {1+x}}{\sqrt {x+\sqrt {1+x}}}\right ) \\ & = -\frac {1}{2} \left (3-2 \sqrt {1+x}\right ) \sqrt {x+\sqrt {1+x}}-2 \arctan \left (\frac {3+\sqrt {1+x}}{2 \sqrt {x+\sqrt {1+x}}}\right )-\frac {1}{4} \text {arctanh}\left (\frac {1+2 \sqrt {1+x}}{2 \sqrt {x+\sqrt {1+x}}}\right ) \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {x+\sqrt {1+x}}}{1+\sqrt {1+x}} \, dx=\frac {1}{2} \sqrt {x+\sqrt {1+x}} \left (-3+2 \sqrt {1+x}\right )-4 \arctan \left (1+\sqrt {1+x}-\sqrt {x+\sqrt {1+x}}\right )+\frac {1}{4} \log \left (-1-2 \sqrt {1+x}+2 \sqrt {x+\sqrt {1+x}}\right ) \]
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Time = 0.17 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.29
method | result | size |
derivativedivides | \(\frac {\left (2 \sqrt {1+x}+1\right ) \sqrt {x +\sqrt {1+x}}}{2}-\frac {5 \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )}{4}-2 \sqrt {\left (\sqrt {1+x}+1\right )^{2}-\sqrt {1+x}-2}+\ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {\left (\sqrt {1+x}+1\right )^{2}-\sqrt {1+x}-2}\right )+2 \arctan \left (\frac {-3-\sqrt {1+x}}{2 \sqrt {\left (\sqrt {1+x}+1\right )^{2}-\sqrt {1+x}-2}}\right )\) | \(125\) |
default | \(\frac {\left (2 \sqrt {1+x}+1\right ) \sqrt {x +\sqrt {1+x}}}{2}-\frac {5 \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )}{4}-2 \sqrt {\left (\sqrt {1+x}+1\right )^{2}-\sqrt {1+x}-2}+\ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {\left (\sqrt {1+x}+1\right )^{2}-\sqrt {1+x}-2}\right )+2 \arctan \left (\frac {-3-\sqrt {1+x}}{2 \sqrt {\left (\sqrt {1+x}+1\right )^{2}-\sqrt {1+x}-2}}\right )\) | \(125\) |
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Time = 1.36 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {x+\sqrt {1+x}}}{1+\sqrt {1+x}} \, dx=\frac {1}{2} \, \sqrt {x + \sqrt {x + 1}} {\left (2 \, \sqrt {x + 1} - 3\right )} + 2 \, \arctan \left (\frac {2 \, \sqrt {x + \sqrt {x + 1}} {\left (\sqrt {x + 1} - 3\right )}}{x - 8}\right ) + \frac {1}{8} \, \log \left (4 \, \sqrt {x + \sqrt {x + 1}} {\left (2 \, \sqrt {x + 1} + 1\right )} - 8 \, x - 8 \, \sqrt {x + 1} - 5\right ) \]
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\[ \int \frac {\sqrt {x+\sqrt {1+x}}}{1+\sqrt {1+x}} \, dx=\int \frac {\sqrt {x + \sqrt {x + 1}}}{\sqrt {x + 1} + 1}\, dx \]
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\[ \int \frac {\sqrt {x+\sqrt {1+x}}}{1+\sqrt {1+x}} \, dx=\int { \frac {\sqrt {x + \sqrt {x + 1}}}{\sqrt {x + 1} + 1} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt {x+\sqrt {1+x}}}{1+\sqrt {1+x}} \, dx=\frac {1}{2} \, \sqrt {x + \sqrt {x + 1}} {\left (2 \, \sqrt {x + 1} - 3\right )} + 4 \, \arctan \left (\sqrt {x + \sqrt {x + 1}} - \sqrt {x + 1} - 1\right ) + \frac {1}{4} \, \log \left (-2 \, \sqrt {x + \sqrt {x + 1}} + 2 \, \sqrt {x + 1} + 1\right ) \]
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Timed out. \[ \int \frac {\sqrt {x+\sqrt {1+x}}}{1+\sqrt {1+x}} \, dx=\int \frac {\sqrt {x+\sqrt {x+1}}}{\sqrt {x+1}+1} \,d x \]
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