Integrand size = 33, antiderivative size = 147 \[ \int \frac {\sqrt {1+x} \left (1+x^2\right )}{\left (-1+x^2\right ) \sqrt {x+\sqrt {1+x}}} \, dx=-\frac {3}{2} \sqrt {x+\sqrt {1+x}}+\sqrt {1+x} \sqrt {x+\sqrt {1+x}}-\frac {7}{4} \log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}\right )-16 \text {RootSum}\left [25+20 \text {$\#$1}-18 \text {$\#$1}^2+4 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {\log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}+\text {$\#$1}\right ) \text {$\#$1}}{5-9 \text {$\#$1}+3 \text {$\#$1}^2+\text {$\#$1}^3}\&\right ] \]
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Time = 0.52 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.48, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1600, 6857, 756, 654, 635, 212, 998, 738, 210} \[ \int \frac {\sqrt {1+x} \left (1+x^2\right )}{\left (-1+x^2\right ) \sqrt {x+\sqrt {1+x}}} \, dx=\sqrt {2 \left (1+\sqrt {2}\right )} \arctan \left (\frac {\left (4-\sqrt {2}\right ) \sqrt {x+1}+2 \left (1+\sqrt {2}\right )}{2 \sqrt {2 \left (\sqrt {2}-1\right )} \sqrt {x+\sqrt {x+1}}}\right )+\frac {7}{4} \text {arctanh}\left (\frac {2 \sqrt {x+1}+1}{2 \sqrt {x+\sqrt {x+1}}}\right )-\sqrt {2 \left (\sqrt {2}-1\right )} \text {arctanh}\left (\frac {\left (4+\sqrt {2}\right ) \sqrt {x+1}+2 \left (1-\sqrt {2}\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {x+\sqrt {x+1}}}\right )+\sqrt {x+1} \sqrt {x+\sqrt {x+1}}-\frac {3}{2} \sqrt {x+\sqrt {x+1}} \]
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Rule 210
Rule 212
Rule 635
Rule 654
Rule 738
Rule 756
Rule 998
Rule 1600
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \int \frac {1+x^2}{(-1+x) \sqrt {1+x} \sqrt {x+\sqrt {1+x}}} \, dx \\ & = 2 \text {Subst}\left (\int \frac {1+\left (-1+x^2\right )^2}{\left (-2+x^2\right ) \sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right ) \\ & = 2 \text {Subst}\left (\int \left (\frac {x^2}{\sqrt {-1+x+x^2}}+\frac {2}{\left (-2+x^2\right ) \sqrt {-1+x+x^2}}\right ) \, dx,x,\sqrt {1+x}\right ) \\ & = 2 \text {Subst}\left (\int \frac {x^2}{\sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )+4 \text {Subst}\left (\int \frac {1}{\left (-2+x^2\right ) \sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right ) \\ & = \sqrt {1+x} \sqrt {x+\sqrt {1+x}}+2 \text {Subst}\left (\int \frac {1}{\left (-2-\sqrt {2} x\right ) \sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )+2 \text {Subst}\left (\int \frac {1}{\left (-2+\sqrt {2} x\right ) \sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )+\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 )-4 \text {Subst}\left (\int \frac {1}{8-8 \sqrt {2}-x^2} \, dx,x,\frac {2+2 \sqrt {2}-\left (-4+\sqrt {2}\right ) \sqrt {1+x}}{\sqrt {x+\sqrt {1+x}}}\right )-4 \text {Subst}\left (\int \frac {1}{8+8 \sqrt {2}-x^2} \, dx,x,\frac {2-2 \sqrt {2}+\left (4+\sqrt {2}\right ) \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}}+\sqrt {2 \left (1+\sqrt {2}\right )} \arctan \left (\frac {2 \left (1+\sqrt {2}\right )+\left (4-\sqrt {2}\right ) \sqrt {1+x}}{2 \sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {x+\sqrt {1+x}}}\right )-\sqrt {2 \left (-1+\sqrt {2}\right )} \text {arctanh}\left (\frac {2 \left (1-\sqrt {2}\right )+\left (4+\sqrt {2}\right ) \sqrt {1+x}}{2 \sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {x+\sqrt {1+x}}}\right )+\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}}+\sqrt {2 \left (1+\sqrt {2}\right )} \arctan \left (\frac {2 \left (1+\sqrt {2}\right )+\left (4-\sqrt {2}\right ) \sqrt {1+x}}{2 \sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {x+\sqrt {1+x}}}\right )+\frac {7}{4} \text {arctanh}\left (\frac {1+2 \sqrt {1+x}}{2 \sqrt {x+\sqrt {1+x}}}\right )-\sqrt {2 \left (-1+\sqrt {2}\right )} \text {arctanh}\left (\frac {2 \left (1-\sqrt {2}\right )+\left (4+\sqrt {2}\right ) \sqrt {1+x}}{2 \sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {x+\sqrt {1+x}}}\right ) \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {1+x} \left (1+x^2\right )}{\left (-1+x^2\right ) \sqrt {x+\sqrt {1+x}}} \, dx=\frac {1}{2} \sqrt {x+\sqrt {1+x}} \left (-3+2 \sqrt {1+x}\right )-\frac {7}{4} \log \left (-1-2 \sqrt {1+x}+2 \sqrt {x+\sqrt {1+x}}\right )-2 \text {RootSum}\left [-1+8 \text {$\#$1}-6 \text {$\#$1}^2+\text {$\#$1}^4\&,\frac {-\log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right )+2 \log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}}{2-3 \text {$\#$1}+\text {$\#$1}^3}\&\right ] \]
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Time = 0.23 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.48
method | result | size |
derivativedivides | \(\sqrt {1+x}\, \sqrt {x +\sqrt {1+x}}-\frac {3 \sqrt {x +\sqrt {1+x}}}{2}+\frac {7 \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )}{4}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {2+2 \sqrt {2}+\left (1+2 \sqrt {2}\right ) \left (\sqrt {1+x}-\sqrt {2}\right )}{2 \sqrt {1+\sqrt {2}}\, \sqrt {\left (\sqrt {1+x}-\sqrt {2}\right )^{2}+\left (1+2 \sqrt {2}\right ) \left (\sqrt {1+x}-\sqrt {2}\right )+1+\sqrt {2}}}\right )}{\sqrt {1+\sqrt {2}}}-\frac {\sqrt {2}\, \arctan \left (\frac {2-2 \sqrt {2}+\left (1-2 \sqrt {2}\right ) \left (\sqrt {1+x}+\sqrt {2}\right )}{2 \sqrt {\sqrt {2}-1}\, \sqrt {\left (\sqrt {1+x}+\sqrt {2}\right )^{2}+\left (1-2 \sqrt {2}\right ) \left (\sqrt {1+x}+\sqrt {2}\right )+1-\sqrt {2}}}\right )}{\sqrt {\sqrt {2}-1}}\) | \(217\) |
default | \(\sqrt {1+x}\, \sqrt {x +\sqrt {1+x}}-\frac {3 \sqrt {x +\sqrt {1+x}}}{2}+\frac {7 \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )}{4}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {2+2 \sqrt {2}+\left (1+2 \sqrt {2}\right ) \left (\sqrt {1+x}-\sqrt {2}\right )}{2 \sqrt {1+\sqrt {2}}\, \sqrt {\left (\sqrt {1+x}-\sqrt {2}\right )^{2}+\left (1+2 \sqrt {2}\right ) \left (\sqrt {1+x}-\sqrt {2}\right )+1+\sqrt {2}}}\right )}{\sqrt {1+\sqrt {2}}}-\frac {\sqrt {2}\, \arctan \left (\frac {2-2 \sqrt {2}+\left (1-2 \sqrt {2}\right ) \left (\sqrt {1+x}+\sqrt {2}\right )}{2 \sqrt {\sqrt {2}-1}\, \sqrt {\left (\sqrt {1+x}+\sqrt {2}\right )^{2}+\left (1-2 \sqrt {2}\right ) \left (\sqrt {1+x}+\sqrt {2}\right )+1-\sqrt {2}}}\right )}{\sqrt {\sqrt {2}-1}}\) | \(217\) |
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 4.26 (sec) , antiderivative size = 417, normalized size of antiderivative = 2.84 \[ \int \frac {\sqrt {1+x} \left (1+x^2\right )}{\left (-1+x^2\right ) \sqrt {x+\sqrt {1+x}}} \, dx=-\frac {1}{2} \, \sqrt {2} \sqrt {\sqrt {2} - 1} \log \left (-\frac {2 \, {\left (\sqrt {2} {\left (\sqrt {2} {\left (5 \, x + 3\right )} + 2 \, \sqrt {x + 1} {\left (3 \, \sqrt {2} + 4\right )} + 6 \, x + 6\right )} \sqrt {\sqrt {2} - 1} + 4 \, {\left (\sqrt {x + 1} {\left (\sqrt {2} + 1\right )} + \sqrt {2} + 2\right )} \sqrt {x + \sqrt {x + 1}}\right )}}{x - 1}\right ) + \frac {1}{2} \, \sqrt {2} \sqrt {\sqrt {2} - 1} \log \left (\frac {2 \, {\left (\sqrt {2} {\left (\sqrt {2} {\left (5 \, x + 3\right )} + 2 \, \sqrt {x + 1} {\left (3 \, \sqrt {2} + 4\right )} + 6 \, x + 6\right )} \sqrt {\sqrt {2} - 1} - 4 \, {\left (\sqrt {x + 1} {\left (\sqrt {2} + 1\right )} + \sqrt {2} + 2\right )} \sqrt {x + \sqrt {x + 1}}\right )}}{x - 1}\right ) + \frac {1}{2} \, \sqrt {x + \sqrt {x + 1}} {\left (2 \, \sqrt {x + 1} - 3\right )} - \frac {1}{4} \, \sqrt {-8 \, \sqrt {2} - 8} \log \left (\frac {2 \, \sqrt {x + 1} {\left (3 \, \sqrt {2} - 4\right )} \sqrt {-8 \, \sqrt {2} - 8} + 8 \, {\left (\sqrt {x + 1} {\left (\sqrt {2} - 1\right )} + \sqrt {2} - 2\right )} \sqrt {x + \sqrt {x + 1}} + {\left (\sqrt {2} {\left (5 \, x + 3\right )} - 6 \, x - 6\right )} \sqrt {-8 \, \sqrt {2} - 8}}{x - 1}\right ) + \frac {1}{4} \, \sqrt {-8 \, \sqrt {2} - 8} \log \left (-\frac {2 \, \sqrt {x + 1} {\left (3 \, \sqrt {2} - 4\right )} \sqrt {-8 \, \sqrt {2} - 8} - 8 \, {\left (\sqrt {x + 1} {\left (\sqrt {2} - 1\right )} + \sqrt {2} - 2\right )} \sqrt {x + \sqrt {x + 1}} + {\left (\sqrt {2} {\left (5 \, x + 3\right )} - 6 \, x - 6\right )} \sqrt {-8 \, \sqrt {2} - 8}}{x - 1}\right ) + \frac {7}{8} \, \log \left (4 \, \sqrt {x + \sqrt {x + 1}} {\left (2 \, \sqrt {x + 1} + 1\right )} + 8 \, x + 8 \, \sqrt {x + 1} + 5\right ) \]
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Not integrable
Time = 15.80 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.18 \[ \int \frac {\sqrt {1+x} \left (1+x^2\right )}{\left (-1+x^2\right ) \sqrt {x+\sqrt {1+x}}} \, dx=\int \frac {x^{2} + 1}{\left (x - 1\right ) \sqrt {x + 1} \sqrt {x + \sqrt {x + 1}}}\, dx \]
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Not integrable
Time = 0.23 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.20 \[ \int \frac {\sqrt {1+x} \left (1+x^2\right )}{\left (-1+x^2\right ) \sqrt {x+\sqrt {1+x}}} \, dx=\int { \frac {{\left (x^{2} + 1\right )} \sqrt {x + 1}}{{\left (x^{2} - 1\right )} \sqrt {x + \sqrt {x + 1}}} \,d x } \]
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.76 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.41 \[ \int \frac {\sqrt {1+x} \left (1+x^2\right )}{\left (-1+x^2\right ) \sqrt {x+\sqrt {1+x}}} \, dx=\frac {1}{2} \, \sqrt {x + \sqrt {x + 1}} {\left (2 \, \sqrt {x + 1} - 3\right )} + 4 \, \sqrt {\frac {1}{2} \, \sqrt {2} + \frac {1}{2}} \arctan \left (\frac {\sqrt {2} - \sqrt {x + \sqrt {x + 1}} + \sqrt {x + 1}}{\sqrt {\sqrt {2} - 1}}\right ) - 2 \, \sqrt {\frac {1}{2} \, \sqrt {2} - \frac {1}{2}} \log \left ({\left | 10 \, \sqrt {2} \sqrt {2 \, \sqrt {2} - 2} + 20 \, \sqrt {2} + 20 \, \sqrt {x + \sqrt {x + 1}} - 20 \, \sqrt {x + 1} + 20 \, \sqrt {2 \, \sqrt {2} - 2} \right |}\right ) + 2 \, \sqrt {\frac {1}{2} \, \sqrt {2} - \frac {1}{2}} \log \left ({\left | -2 \, \sqrt {2} \sqrt {2 \, \sqrt {2} - 2} + 4 \, \sqrt {2} + 4 \, \sqrt {x + \sqrt {x + 1}} - 4 \, \sqrt {x + 1} - 4 \, \sqrt {2 \, \sqrt {2} - 2} \right |}\right ) - \frac {7}{4} \, \log \left (-2 \, \sqrt {x + \sqrt {x + 1}} + 2 \, \sqrt {x + 1} + 1\right ) \]
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Not integrable
Time = 6.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.20 \[ \int \frac {\sqrt {1+x} \left (1+x^2\right )}{\left (-1+x^2\right ) \sqrt {x+\sqrt {1+x}}} \, dx=\int \frac {\left (x^2+1\right )\,\sqrt {x+1}}{\sqrt {x+\sqrt {x+1}}\,\left (x^2-1\right )} \,d x \]
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