Integrand size = 21, antiderivative size = 70 \[ \int \frac {\sqrt {x+\sqrt {1+x}}}{\sqrt {1+x}} \, dx=\frac {1}{2} \sqrt {x+\sqrt {1+x}}+\sqrt {1+x} \sqrt {x+\sqrt {1+x}}+\frac {5}{4} \log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}\right ) \]
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Time = 0.09 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.89, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {626, 635, 212} \[ \int \frac {\sqrt {x+\sqrt {1+x}}}{\sqrt {1+x}} \, dx=\frac {1}{2} \sqrt {x+\sqrt {x+1}} \left (2 \sqrt {x+1}+1\right )-\frac {5}{4} \text {arctanh}\left (\frac {2 \sqrt {x+1}+1}{2 \sqrt {x+\sqrt {x+1}}}\right ) \]
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Rule 212
Rule 626
Rule 635
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \sqrt {-1+x+x^2} \, dx,x,\sqrt {1+x}\right ) \\ & = \frac {1}{2} \sqrt {x+\sqrt {1+x}} \left (1+2 \sqrt {1+x}\right )-\frac {5}{4} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right ) \\ & = \frac {1}{2} \sqrt {x+\sqrt {1+x}} \left (1+2 \sqrt {1+x}\right )-\frac {5}{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 {1}{2} \sqrt {x+\sqrt {1+x}} \left (1+2 \sqrt {1+x}\right )-\frac {5}{4} \text {arctanh}\left (\frac {1+2 \sqrt {1+x}}{2 \sqrt {x+\sqrt {1+x}}}\right ) \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {x+\sqrt {1+x}}}{\sqrt {1+x}} \, dx=\frac {1}{2} \sqrt {x+\sqrt {1+x}} \left (1+2 \sqrt {1+x}\right )+\frac {5}{4} \log \left (-1-2 \sqrt {1+x}+2 \sqrt {x+\sqrt {1+x}}\right ) \]
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Time = 0.88 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.59
method | result | size |
derivativedivides | \(\frac {\left (2 \sqrt {1+x}+1\right ) \sqrt {x +\sqrt {1+x}}}{2}-\frac {5 \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )}{4}\) | \(41\) |
default | \(\frac {\left (2 \sqrt {1+x}+1\right ) \sqrt {x +\sqrt {1+x}}}{2}-\frac {5 \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )}{4}\) | \(41\) |
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none
Time = 0.48 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.80 \[ \int \frac {\sqrt {x+\sqrt {1+x}}}{\sqrt {1+x}} \, dx=\frac {1}{2} \, \sqrt {x + \sqrt {x + 1}} {\left (2 \, \sqrt {x + 1} + 1\right )} + \frac {5}{8} \, \log \left (4 \, \sqrt {x + \sqrt {x + 1}} {\left (2 \, \sqrt {x + 1} + 1\right )} - 8 \, x - 8 \, \sqrt {x + 1} - 5\right ) \]
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Time = 0.36 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {x+\sqrt {1+x}}}{\sqrt {1+x}} \, dx=2 \sqrt {x + \sqrt {x + 1}} \left (\frac {\sqrt {x + 1}}{2} + \frac {1}{4}\right ) - \frac {5 \log {\left (2 \sqrt {x + 1} + 2 \sqrt {x + \sqrt {x + 1}} + 1 \right )}}{4} \]
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\[ \int \frac {\sqrt {x+\sqrt {1+x}}}{\sqrt {1+x}} \, dx=\int { \frac {\sqrt {x + \sqrt {x + 1}}}{\sqrt {x + 1}} \,d x } \]
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none
Time = 0.27 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.63 \[ \int \frac {\sqrt {x+\sqrt {1+x}}}{\sqrt {1+x}} \, dx=\frac {1}{2} \, \sqrt {x + \sqrt {x + 1}} {\left (2 \, \sqrt {x + 1} + 1\right )} + \frac {5}{4} \, \log \left (-2 \, \sqrt {x + \sqrt {x + 1}} + 2 \, \sqrt {x + 1} + 1\right ) \]
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Timed out. \[ \int \frac {\sqrt {x+\sqrt {1+x}}}{\sqrt {1+x}} \, dx=\int \frac {\sqrt {x+\sqrt {x+1}}}{\sqrt {x+1}} \,d x \]
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