Integrand size = 34, antiderivative size = 118 \[ \int \frac {\sqrt {1+x} \sqrt {1+\sqrt {1+x}}}{x-\sqrt {1+x}} \, dx=\frac {16}{3} \sqrt {1+\sqrt {1+x}}+\frac {4}{3} \sqrt {1+x} \sqrt {1+\sqrt {1+x}}-\frac {4}{5} \left (-5+2 \sqrt {5}\right ) \text {arctanh}\left (\frac {2 \sqrt {1+\sqrt {1+x}}}{-1+\sqrt {5}}\right )-\frac {4}{5} \left (5+2 \sqrt {5}\right ) \text {arctanh}\left (\frac {2 \sqrt {1+\sqrt {1+x}}}{1+\sqrt {5}}\right ) \]
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Time = 0.32 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {911, 1301, 1180, 213} \[ \int \frac {\sqrt {1+x} \sqrt {1+\sqrt {1+x}}}{x-\sqrt {1+x}} \, dx=-4 \sqrt {\frac {1}{5} \left (9+4 \sqrt {5}\right )} \text {arctanh}\left (\sqrt {\frac {2}{3+\sqrt {5}}} \sqrt {\sqrt {x+1}+1}\right )+4 \sqrt {\frac {1}{5} \left (9-4 \sqrt {5}\right )} \text {arctanh}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt {\sqrt {x+1}+1}\right )+\frac {4}{3} \left (\sqrt {x+1}+1\right )^{3/2}+4 \sqrt {\sqrt {x+1}+1} \]
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Rule 213
Rule 911
Rule 1180
Rule 1301
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x^2 \sqrt {1+x}}{-1-x+x^2} \, dx,x,\sqrt {1+x}\right ) \\ & = 4 \text {Subst}\left (\int \frac {x^2 \left (-1+x^2\right )^2}{1-3 x^2+x^4} \, dx,x,\sqrt {1+\sqrt {1+x}}\right ) \\ & = 4 \text {Subst}\left (\int \left (1+x^2-\frac {1-3 x^2}{1-3 x^2+x^4}\right ) \, dx,x,\sqrt {1+\sqrt {1+x}}\right ) \\ & = 4 \sqrt {1+\sqrt {1+x}}+\frac {4}{3} \left (1+\sqrt {1+x}\right )^{3/2}-4 \text {Subst}\left (\int \frac {1-3 x^2}{1-3 x^2+x^4} \, dx,x,\sqrt {1+\sqrt {1+x}}\right ) \\ & = 4 \sqrt {1+\sqrt {1+x}}+\frac {4}{3} \left (1+\sqrt {1+x}\right )^{3/2}+\frac {1}{5} \left (2 \left (15-7 \sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {3}{2}+\frac {\sqrt {5}}{2}+x^2} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )+\frac {1}{5} \left (2 \left (15+7 \sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {3}{2}-\frac {\sqrt {5}}{2}+x^2} \, dx,x,\sqrt {1+\sqrt {1+x}}\right ) \\ & = 4 \sqrt {1+\sqrt {1+x}}+\frac {4}{3} \left (1+\sqrt {1+x}\right )^{3/2}-4 \sqrt {\frac {1}{5} \left (9+4 \sqrt {5}\right )} \text {arctanh}\left (\sqrt {\frac {2}{3+\sqrt {5}}} \sqrt {1+\sqrt {1+x}}\right )+4 \sqrt {\frac {1}{5} \left (9-4 \sqrt {5}\right )} \text {arctanh}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt {1+\sqrt {1+x}}\right ) \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {1+x} \sqrt {1+\sqrt {1+x}}}{x-\sqrt {1+x}} \, dx=\frac {4}{15} \left (5 \sqrt {1+\sqrt {1+x}} \left (4+\sqrt {1+x}\right )-3 \left (5+2 \sqrt {5}\right ) \text {arctanh}\left (\frac {1}{2} \left (-1+\sqrt {5}\right ) \sqrt {1+\sqrt {1+x}}\right )+\left (15-6 \sqrt {5}\right ) \text {arctanh}\left (\frac {1}{2} \left (1+\sqrt {5}\right ) \sqrt {1+\sqrt {1+x}}\right )\right ) \]
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Time = 0.14 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.93
method | result | size |
derivativedivides | \(\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {3}{2}}}{3}+4 \sqrt {1+\sqrt {1+x}}-2 \ln \left (\sqrt {1+x}+\sqrt {1+\sqrt {1+x}}\right )-\frac {8 \sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (2 \sqrt {1+\sqrt {1+x}}+1\right ) \sqrt {5}}{5}\right )}{5}+2 \ln \left (\sqrt {1+x}-\sqrt {1+\sqrt {1+x}}\right )-\frac {8 \sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (2 \sqrt {1+\sqrt {1+x}}-1\right ) \sqrt {5}}{5}\right )}{5}\) | \(110\) |
default | \(\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {3}{2}}}{3}+4 \sqrt {1+\sqrt {1+x}}-2 \ln \left (\sqrt {1+x}+\sqrt {1+\sqrt {1+x}}\right )-\frac {8 \sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (2 \sqrt {1+\sqrt {1+x}}+1\right ) \sqrt {5}}{5}\right )}{5}+2 \ln \left (\sqrt {1+x}-\sqrt {1+\sqrt {1+x}}\right )-\frac {8 \sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (2 \sqrt {1+\sqrt {1+x}}-1\right ) \sqrt {5}}{5}\right )}{5}\) | \(110\) |
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Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (84) = 168\).
Time = 0.28 (sec) , antiderivative size = 243, normalized size of antiderivative = 2.06 \[ \int \frac {\sqrt {1+x} \sqrt {1+\sqrt {1+x}}}{x-\sqrt {1+x}} \, dx=\frac {4}{3} \, {\left (\sqrt {x + 1} + 4\right )} \sqrt {\sqrt {x + 1} + 1} + \frac {4}{5} \, \sqrt {5} \log \left (\frac {2 \, x^{2} + \sqrt {5} {\left (3 \, x + 1\right )} + {\left (\sqrt {5} {\left (x + 2\right )} + 5 \, x\right )} \sqrt {x + 1} - {\left (\sqrt {5} {\left (x + 2\right )} + {\left (\sqrt {5} {\left (2 \, x - 1\right )} + 5\right )} \sqrt {x + 1} + 5 \, x\right )} \sqrt {\sqrt {x + 1} + 1} + 3 \, x + 3}{x^{2} - x - 1}\right ) + \frac {4}{5} \, \sqrt {5} \log \left (\frac {2 \, x^{2} - \sqrt {5} {\left (3 \, x + 1\right )} - {\left (\sqrt {5} {\left (x + 2\right )} - 5 \, x\right )} \sqrt {x + 1} - {\left (\sqrt {5} {\left (x + 2\right )} + {\left (\sqrt {5} {\left (2 \, x - 1\right )} - 5\right )} \sqrt {x + 1} - 5 \, x\right )} \sqrt {\sqrt {x + 1} + 1} + 3 \, x + 3}{x^{2} - x - 1}\right ) - 2 \, \log \left (\sqrt {x + 1} + \sqrt {\sqrt {x + 1} + 1}\right ) + 2 \, \log \left (\sqrt {x + 1} - \sqrt {\sqrt {x + 1} + 1}\right ) \]
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Time = 4.33 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.56 \[ \int \frac {\sqrt {1+x} \sqrt {1+\sqrt {1+x}}}{x-\sqrt {1+x}} \, dx=\frac {4 \left (\sqrt {x + 1} + 1\right )^{\frac {3}{2}}}{3} + 4 \sqrt {\sqrt {x + 1} + 1} + \frac {4 \sqrt {5} \left (- \log {\left (\sqrt {\sqrt {x + 1} + 1} - \frac {1}{2} + \frac {\sqrt {5}}{2} \right )} + \log {\left (\sqrt {\sqrt {x + 1} + 1} - \frac {\sqrt {5}}{2} - \frac {1}{2} \right )}\right )}{5} + \frac {4 \sqrt {5} \left (- \log {\left (\sqrt {\sqrt {x + 1} + 1} + \frac {1}{2} + \frac {\sqrt {5}}{2} \right )} + \log {\left (\sqrt {\sqrt {x + 1} + 1} - \frac {\sqrt {5}}{2} + \frac {1}{2} \right )}\right )}{5} + 2 \log {\left (\sqrt {x + 1} - \sqrt {\sqrt {x + 1} + 1} \right )} - 2 \log {\left (\sqrt {x + 1} + \sqrt {\sqrt {x + 1} + 1} \right )} \]
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Time = 0.32 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.23 \[ \int \frac {\sqrt {1+x} \sqrt {1+\sqrt {1+x}}}{x-\sqrt {1+x}} \, dx=\frac {4}{3} \, {\left (\sqrt {x + 1} + 1\right )}^{\frac {3}{2}} + \frac {4}{5} \, \sqrt {5} \log \left (-\frac {\sqrt {5} - 2 \, \sqrt {\sqrt {x + 1} + 1} + 1}{\sqrt {5} + 2 \, \sqrt {\sqrt {x + 1} + 1} - 1}\right ) + \frac {4}{5} \, \sqrt {5} \log \left (-\frac {\sqrt {5} - 2 \, \sqrt {\sqrt {x + 1} + 1} - 1}{\sqrt {5} + 2 \, \sqrt {\sqrt {x + 1} + 1} + 1}\right ) + 4 \, \sqrt {\sqrt {x + 1} + 1} - 2 \, \log \left (\sqrt {x + 1} + \sqrt {\sqrt {x + 1} + 1}\right ) + 2 \, \log \left (\sqrt {x + 1} - \sqrt {\sqrt {x + 1} + 1}\right ) \]
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Time = 0.53 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.26 \[ \int \frac {\sqrt {1+x} \sqrt {1+\sqrt {1+x}}}{x-\sqrt {1+x}} \, dx=\frac {4}{3} \, {\left (\sqrt {x + 1} + 1\right )}^{\frac {3}{2}} + \frac {4}{5} \, \sqrt {5} \log \left (-\frac {\sqrt {5} - 2 \, \sqrt {\sqrt {x + 1} + 1} - 1}{\sqrt {5} + 2 \, \sqrt {\sqrt {x + 1} + 1} + 1}\right ) + \frac {4}{5} \, \sqrt {5} \log \left (\frac {{\left | -\sqrt {5} + 2 \, \sqrt {\sqrt {x + 1} + 1} - 1 \right |}}{{\left | \sqrt {5} + 2 \, \sqrt {\sqrt {x + 1} + 1} - 1 \right |}}\right ) + 4 \, \sqrt {\sqrt {x + 1} + 1} - 2 \, \log \left (\sqrt {x + 1} + \sqrt {\sqrt {x + 1} + 1}\right ) + 2 \, \log \left ({\left | \sqrt {x + 1} - \sqrt {\sqrt {x + 1} + 1} \right |}\right ) \]
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Timed out. \[ \int \frac {\sqrt {1+x} \sqrt {1+\sqrt {1+x}}}{x-\sqrt {1+x}} \, dx=\int \frac {\sqrt {\sqrt {x+1}+1}\,\sqrt {x+1}}{x-\sqrt {x+1}} \,d x \]
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