Integrand size = 40, antiderivative size = 392 \[ \int \frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx=\frac {\sqrt {1+x^2} \left (-11 x+8 x^3\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (-5-7 x^2+8 x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 x \left (-1+x^2\right ) \sqrt {1+x^2}+\left (-1+x^2\right ) \left (1+2 x^2\right )}-4 \text {arctanh}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )+\frac {1}{8} \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-6 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2+5 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{2 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]-\frac {1}{8} \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-6 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}+5 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^3}{-2+4 \text {$\#$1}^2-3 \text {$\#$1}^4+\text {$\#$1}^6}\&\right ] \]
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\[ \int \frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx=\int \frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{4 (1-x)^2}+\frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{4 (1+x)^2}+\frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 \left (1-x^2\right )}\right ) \, dx \\ & = \frac {1}{4} \int \frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1-x)^2} \, dx+\frac {1}{4} \int \frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1+x)^2} \, dx+\frac {1}{2} \int \frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^2} \, dx \\ & = \frac {1}{4} \int \frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1-x)^2} \, dx+\frac {1}{4} \int \frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1+x)^2} \, dx+\frac {1}{2} \int \left (\frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 (1-x)}+\frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 (1+x)}\right ) \, dx \\ & = \frac {1}{4} \int \frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1-x)^2} \, dx+\frac {1}{4} \int \frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x} \, dx+\frac {1}{4} \int \frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1+x)^2} \, dx+\frac {1}{4} \int \frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1+x} \, dx \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 689, normalized size of antiderivative = 1.76 \[ \int \frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx=\frac {\left (-5-7 x^2+8 x^4-11 x \sqrt {1+x^2}+8 x^3 \sqrt {1+x^2}\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (-1+x^2\right ) \left (1+2 x^2+2 x \sqrt {1+x^2}\right )}-4 \text {arctanh}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )+\text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{2 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]-\frac {1}{8} \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {14 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2+3 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{2 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]-\text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{-2 \text {$\#$1}+4 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]+\frac {1}{8} \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-16 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2+3 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{-2 \text {$\#$1}+4 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ] \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.08
\[\int \frac {\left (x^{2}+1\right )^{\frac {3}{2}} \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{\left (-x^{2}+1\right )^{2}}d x\]
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 1.08 (sec) , antiderivative size = 6658, normalized size of antiderivative = 16.98 \[ \int \frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx=\text {Too large to display} \]
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Not integrable
Time = 47.39 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.09 \[ \int \frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx=\int \frac {\left (x^{2} + 1\right )^{\frac {3}{2}} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{\left (x - 1\right )^{2} \left (x + 1\right )^{2}}\, dx \]
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Not integrable
Time = 0.57 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.08 \[ \int \frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx=\int { \frac {{\left (x^{2} + 1\right )}^{\frac {3}{2}} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{{\left (x^{2} - 1\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx=\text {Timed out} \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.08 \[ \int \frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx=\int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}\,{\left (x^2+1\right )}^{3/2}}{{\left (x^2-1\right )}^2} \,d x \]
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