Integrand size = 31, antiderivative size = 99 \[ \int \frac {\sqrt {1+x^2}}{x^2+\sqrt {x+\sqrt {1+x^2}}} \, dx=\log \left (x+\sqrt {1+x^2}\right )-2 \text {RootSum}\left [1-2 \text {$\#$1}^4+4 \text {$\#$1}^5+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right )+\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}}{-2+5 \text {$\#$1}+2 \text {$\#$1}^4}\&\right ] \]
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\[ \int \frac {\sqrt {1+x^2}}{x^2+\sqrt {x+\sqrt {1+x^2}}} \, dx=\int \frac {\sqrt {1+x^2}}{x^2+\sqrt {x+\sqrt {1+x^2}}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {x^2 \left (1+x^2\right )}{-1-2 x^5+x^8}+\frac {x^3 \sqrt {1+x^2} \left (-1+x^3\right )}{-1-2 x^5+x^8}+\frac {x \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8}-\frac {x^4 \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8}-\frac {\left (1+x^2\right ) \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8}\right ) \, dx \\ & = \int \frac {x^2 \left (1+x^2\right )}{-1-2 x^5+x^8} \, dx+\int \frac {x^3 \sqrt {1+x^2} \left (-1+x^3\right )}{-1-2 x^5+x^8} \, dx+\int \frac {x \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8} \, dx-\int \frac {x^4 \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8} \, dx-\int \frac {\left (1+x^2\right ) \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8} \, dx \\ & = \int \frac {x \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8} \, dx-\int \frac {x^4 \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8} \, dx+\int \left (\frac {x^2}{-1-2 x^5+x^8}+\frac {x^4}{-1-2 x^5+x^8}\right ) \, dx+\int \left (-\frac {x^3 \sqrt {1+x^2}}{-1-2 x^5+x^8}+\frac {x^6 \sqrt {1+x^2}}{-1-2 x^5+x^8}\right ) \, dx-\int \left (\frac {\sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8}+\frac {x^2 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8}\right ) \, dx \\ & = \int \frac {x^2}{-1-2 x^5+x^8} \, dx+\int \frac {x^4}{-1-2 x^5+x^8} \, dx-\int \frac {x^3 \sqrt {1+x^2}}{-1-2 x^5+x^8} \, dx+\int \frac {x^6 \sqrt {1+x^2}}{-1-2 x^5+x^8} \, dx-\int \frac {\sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8} \, dx-\int \frac {x^2 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8} \, dx+\int \frac {x \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8} \, dx-\int \frac {x^4 \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8} \, dx \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1+x^2}}{x^2+\sqrt {x+\sqrt {1+x^2}}} \, dx=\log \left (x+\sqrt {1+x^2}\right )-2 \text {RootSum}\left [1-2 \text {$\#$1}^4+4 \text {$\#$1}^5+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right )+\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}}{-2+5 \text {$\#$1}+2 \text {$\#$1}^4}\&\right ] \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.25
\[\int \frac {\sqrt {x^{2}+1}}{x^{2}+\sqrt {x +\sqrt {x^{2}+1}}}d x\]
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Timed out. \[ \int \frac {\sqrt {1+x^2}}{x^2+\sqrt {x+\sqrt {1+x^2}}} \, dx=\text {Timed out} \]
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Not integrable
Time = 1.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.26 \[ \int \frac {\sqrt {1+x^2}}{x^2+\sqrt {x+\sqrt {1+x^2}}} \, dx=\int \frac {\sqrt {x^{2} + 1}}{x^{2} + \sqrt {x + \sqrt {x^{2} + 1}}}\, dx \]
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Not integrable
Time = 0.31 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.27 \[ \int \frac {\sqrt {1+x^2}}{x^2+\sqrt {x+\sqrt {1+x^2}}} \, dx=\int { \frac {\sqrt {x^{2} + 1}}{x^{2} + \sqrt {x + \sqrt {x^{2} + 1}}} \,d x } \]
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Not integrable
Time = 0.37 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.27 \[ \int \frac {\sqrt {1+x^2}}{x^2+\sqrt {x+\sqrt {1+x^2}}} \, dx=\int { \frac {\sqrt {x^{2} + 1}}{x^{2} + \sqrt {x + \sqrt {x^{2} + 1}}} \,d x } \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.27 \[ \int \frac {\sqrt {1+x^2}}{x^2+\sqrt {x+\sqrt {1+x^2}}} \, dx=\int \frac {\sqrt {x^2+1}}{\sqrt {x+\sqrt {x^2+1}}+x^2} \,d x \]
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