Integrand size = 21, antiderivative size = 102 \[ \int \frac {x}{x+\sqrt {x+\sqrt {1+x^2}}} \, dx=x-2 \sqrt {x+\sqrt {1+x^2}}+2 \text {RootSum}\left [-1+2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right )+\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}^3}{3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ] \]
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\[ \int \frac {x}{x+\sqrt {x+\sqrt {1+x^2}}} \, dx=\int \frac {x}{x+\sqrt {x+\sqrt {1+x^2}}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (1+\frac {x^2 \sqrt {1+x^2}}{-1-2 x^3+x^4}+\frac {1+x^3}{-1-2 x^3+x^4}+\frac {x^2 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4}-\frac {x^3 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4}-\frac {x \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4}\right ) \, dx \\ & = x+\int \frac {x^2 \sqrt {1+x^2}}{-1-2 x^3+x^4} \, dx+\int \frac {1+x^3}{-1-2 x^3+x^4} \, dx+\int \frac {x^2 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx-\int \frac {x^3 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx-\int \frac {x \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx \\ & = x+\frac {1}{4} \log \left (-1-2 x^3+x^4\right )+\frac {1}{4} \int \frac {4+6 x^2}{-1-2 x^3+x^4} \, dx+\int \frac {x^2 \sqrt {1+x^2}}{-1-2 x^3+x^4} \, dx+\int \frac {x^2 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx-\int \frac {x^3 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx-\int \frac {x \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx \\ & = x+\frac {1}{4} \log \left (-1-2 x^3+x^4\right )+\frac {1}{4} \int \left (\frac {4}{-1-2 x^3+x^4}+\frac {6 x^2}{-1-2 x^3+x^4}\right ) \, dx+\int \frac {x^2 \sqrt {1+x^2}}{-1-2 x^3+x^4} \, dx+\int \frac {x^2 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx-\int \frac {x^3 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx-\int \frac {x \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx \\ & = x+\frac {1}{4} \log \left (-1-2 x^3+x^4\right )+\frac {3}{2} \int \frac {x^2}{-1-2 x^3+x^4} \, dx+\int \frac {1}{-1-2 x^3+x^4} \, dx+\int \frac {x^2 \sqrt {1+x^2}}{-1-2 x^3+x^4} \, dx+\int \frac {x^2 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx-\int \frac {x^3 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx-\int \frac {x \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx \\ \end{align*}
Time = 0.59 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00 \[ \int \frac {x}{x+\sqrt {x+\sqrt {1+x^2}}} \, dx=x-2 \sqrt {x+\sqrt {1+x^2}}+2 \text {RootSum}\left [-1+2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right )+\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}^3}{3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ] \]
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Not integrable
Time = 0.05 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.17
\[\int \frac {x}{x +\sqrt {x +\sqrt {x^{2}+1}}}d x\]
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.98 (sec) , antiderivative size = 7672, normalized size of antiderivative = 75.22 \[ \int \frac {x}{x+\sqrt {x+\sqrt {1+x^2}}} \, dx=\text {Too large to display} \]
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Not integrable
Time = 0.84 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.17 \[ \int \frac {x}{x+\sqrt {x+\sqrt {1+x^2}}} \, dx=\int \frac {x}{x + \sqrt {x + \sqrt {x^{2} + 1}}}\, dx \]
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Not integrable
Time = 0.34 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.19 \[ \int \frac {x}{x+\sqrt {x+\sqrt {1+x^2}}} \, dx=\int { \frac {x}{x + \sqrt {x + \sqrt {x^{2} + 1}}} \,d x } \]
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Not integrable
Time = 0.41 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.19 \[ \int \frac {x}{x+\sqrt {x+\sqrt {1+x^2}}} \, dx=\int { \frac {x}{x + \sqrt {x + \sqrt {x^{2} + 1}}} \,d x } \]
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Not integrable
Time = 6.19 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.19 \[ \int \frac {x}{x+\sqrt {x+\sqrt {1+x^2}}} \, dx=\int \frac {x}{x+\sqrt {x+\sqrt {x^2+1}}} \,d x \]
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