Integrand size = 31, antiderivative size = 124 \[ \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^{3/2}} \, dx=-\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1+x^2+x \sqrt {1+x^2}}+\frac {1}{4} \text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )}{-\text {$\#$1}+3 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ] \]
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\[ \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^{3/2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^{3/2}} \, dx \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^{3/2}} \, dx=-\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1+x^2+x \sqrt {1+x^2}}+\frac {1}{4} \text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )}{-\text {$\#$1}+3 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ] \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.19
\[\int \frac {\sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{\left (x^{2}+1\right )^{\frac {3}{2}}}d x\]
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.30 (sec) , antiderivative size = 682, normalized size of antiderivative = 5.50 \[ \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^{3/2}} \, dx=\frac {\sqrt {2} {\left (x^{2} + 1\right )} \sqrt {\sqrt {2 i \, \sqrt {2} + 1} - 1} \log \left ({\left ({\left (\sqrt {2} - i\right )} \sqrt {2 i \, \sqrt {2} + 1} + 3 \, \sqrt {2} - 3 i\right )} \sqrt {\sqrt {2 i \, \sqrt {2} + 1} - 1} + 6 \, \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}\right ) - \sqrt {2} {\left (x^{2} + 1\right )} \sqrt {\sqrt {2 i \, \sqrt {2} + 1} - 1} \log \left (-{\left ({\left (\sqrt {2} - i\right )} \sqrt {2 i \, \sqrt {2} + 1} + 3 \, \sqrt {2} - 3 i\right )} \sqrt {\sqrt {2 i \, \sqrt {2} + 1} - 1} + 6 \, \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}\right ) - \sqrt {2} {\left (x^{2} + 1\right )} \sqrt {-\sqrt {2 i \, \sqrt {2} + 1} - 1} \log \left ({\left ({\left (\sqrt {2} - i\right )} \sqrt {2 i \, \sqrt {2} + 1} - 3 \, \sqrt {2} + 3 i\right )} \sqrt {-\sqrt {2 i \, \sqrt {2} + 1} - 1} + 6 \, \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}\right ) + \sqrt {2} {\left (x^{2} + 1\right )} \sqrt {-\sqrt {2 i \, \sqrt {2} + 1} - 1} \log \left (-{\left ({\left (\sqrt {2} - i\right )} \sqrt {2 i \, \sqrt {2} + 1} - 3 \, \sqrt {2} + 3 i\right )} \sqrt {-\sqrt {2 i \, \sqrt {2} + 1} - 1} + 6 \, \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}\right ) + \sqrt {2} {\left (x^{2} + 1\right )} \sqrt {\sqrt {-2 i \, \sqrt {2} + 1} - 1} \log \left ({\left ({\left (\sqrt {2} + i\right )} \sqrt {-2 i \, \sqrt {2} + 1} + 3 \, \sqrt {2} + 3 i\right )} \sqrt {\sqrt {-2 i \, \sqrt {2} + 1} - 1} + 6 \, \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}\right ) - \sqrt {2} {\left (x^{2} + 1\right )} \sqrt {\sqrt {-2 i \, \sqrt {2} + 1} - 1} \log \left (-{\left ({\left (\sqrt {2} + i\right )} \sqrt {-2 i \, \sqrt {2} + 1} + 3 \, \sqrt {2} + 3 i\right )} \sqrt {\sqrt {-2 i \, \sqrt {2} + 1} - 1} + 6 \, \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}\right ) - \sqrt {2} {\left (x^{2} + 1\right )} \sqrt {-\sqrt {-2 i \, \sqrt {2} + 1} - 1} \log \left ({\left ({\left (\sqrt {2} + i\right )} \sqrt {-2 i \, \sqrt {2} + 1} - 3 \, \sqrt {2} - 3 i\right )} \sqrt {-\sqrt {-2 i \, \sqrt {2} + 1} - 1} + 6 \, \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}\right ) + \sqrt {2} {\left (x^{2} + 1\right )} \sqrt {-\sqrt {-2 i \, \sqrt {2} + 1} - 1} \log \left (-{\left ({\left (\sqrt {2} + i\right )} \sqrt {-2 i \, \sqrt {2} + 1} - 3 \, \sqrt {2} - 3 i\right )} \sqrt {-\sqrt {-2 i \, \sqrt {2} + 1} - 1} + 6 \, \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}\right ) - 8 \, {\left (x^{2} - \sqrt {x^{2} + 1} x + 1\right )} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{8 \, {\left (x^{2} + 1\right )}} \]
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Not integrable
Time = 1.88 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.22 \[ \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{\left (x^{2} + 1\right )^{\frac {3}{2}}}\, dx \]
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Not integrable
Time = 0.61 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.20 \[ \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{{\left (x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \]
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Not integrable
Time = 125.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.20 \[ \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{{\left (x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.20 \[ \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{{\left (x^2+1\right )}^{3/2}} \,d x \]
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