Integrand size = 28, antiderivative size = 112 \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x^2}}}{1+x^4} \, dx=\frac {4 x}{3 \sqrt {1+\sqrt {1+x^2}}}+\frac {2 x \sqrt {1+x^2}}{3 \sqrt {1+\sqrt {1+x^2}}}-\frac {1}{2} \text {RootSum}\left [1+4 \text {$\#$1}^4+4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\frac {x}{\sqrt {1+\sqrt {1+x^2}}}-\text {$\#$1}\right )}{2 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ] \]
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\[ \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x^2}}}{1+x^4} \, dx=\int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x^2}}}{1+x^4} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\sqrt {1+\sqrt {1+x^2}}-\frac {2 \sqrt {1+\sqrt {1+x^2}}}{1+x^4}\right ) \, dx \\ & = -\left (2 \int \frac {\sqrt {1+\sqrt {1+x^2}}}{1+x^4} \, dx\right )+\int \sqrt {1+\sqrt {1+x^2}} \, dx \\ & = \frac {2 x^3}{3 \left (1+\sqrt {1+x^2}\right )^{3/2}}+\frac {2 x}{\sqrt {1+\sqrt {1+x^2}}}-2 \int \left (\frac {i \sqrt {1+\sqrt {1+x^2}}}{2 \left (i-x^2\right )}+\frac {i \sqrt {1+\sqrt {1+x^2}}}{2 \left (i+x^2\right )}\right ) \, dx \\ & = \frac {2 x^3}{3 \left (1+\sqrt {1+x^2}\right )^{3/2}}+\frac {2 x}{\sqrt {1+\sqrt {1+x^2}}}-i \int \frac {\sqrt {1+\sqrt {1+x^2}}}{i-x^2} \, dx-i \int \frac {\sqrt {1+\sqrt {1+x^2}}}{i+x^2} \, dx \\ & = \frac {2 x^3}{3 \left (1+\sqrt {1+x^2}\right )^{3/2}}+\frac {2 x}{\sqrt {1+\sqrt {1+x^2}}}-i \int \left (-\frac {(-1)^{3/4} \sqrt {1+\sqrt {1+x^2}}}{2 \left (\sqrt [4]{-1}-x\right )}-\frac {(-1)^{3/4} \sqrt {1+\sqrt {1+x^2}}}{2 \left (\sqrt [4]{-1}+x\right )}\right ) \, dx-i \int \left (-\frac {\sqrt [4]{-1} \sqrt {1+\sqrt {1+x^2}}}{2 \left (-(-1)^{3/4}-x\right )}-\frac {\sqrt [4]{-1} \sqrt {1+\sqrt {1+x^2}}}{2 \left (-(-1)^{3/4}+x\right )}\right ) \, dx \\ & = \frac {2 x^3}{3 \left (1+\sqrt {1+x^2}\right )^{3/2}}+\frac {2 x}{\sqrt {1+\sqrt {1+x^2}}}-\frac {1}{2} \sqrt [4]{-1} \int \frac {\sqrt {1+\sqrt {1+x^2}}}{\sqrt [4]{-1}-x} \, dx-\frac {1}{2} \sqrt [4]{-1} \int \frac {\sqrt {1+\sqrt {1+x^2}}}{\sqrt [4]{-1}+x} \, dx+\frac {1}{2} (-1)^{3/4} \int \frac {\sqrt {1+\sqrt {1+x^2}}}{-(-1)^{3/4}-x} \, dx+\frac {1}{2} (-1)^{3/4} \int \frac {\sqrt {1+\sqrt {1+x^2}}}{-(-1)^{3/4}+x} \, dx \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.84 \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x^2}}}{1+x^4} \, dx=\frac {2 x \left (2+\sqrt {1+x^2}\right )}{3 \sqrt {1+\sqrt {1+x^2}}}-\frac {1}{2} \text {RootSum}\left [1+4 \text {$\#$1}^4+4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\frac {x}{\sqrt {1+\sqrt {1+x^2}}}-\text {$\#$1}\right )}{2 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ] \]
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Not integrable
Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.21
\[\int \frac {\left (x^{4}-1\right ) \sqrt {1+\sqrt {x^{2}+1}}}{x^{4}+1}d x\]
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 9.68 (sec) , antiderivative size = 6296, normalized size of antiderivative = 56.21 \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x^2}}}{1+x^4} \, dx=\text {Too large to display} \]
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Not integrable
Time = 10.70 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.28 \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x^2}}}{1+x^4} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \sqrt {\sqrt {x^{2} + 1} + 1}}{x^{4} + 1}\, dx \]
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Not integrable
Time = 0.32 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.23 \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x^2}}}{1+x^4} \, dx=\int { \frac {{\left (x^{4} - 1\right )} \sqrt {\sqrt {x^{2} + 1} + 1}}{x^{4} + 1} \,d x } \]
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Not integrable
Time = 0.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.23 \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x^2}}}{1+x^4} \, dx=\int { \frac {{\left (x^{4} - 1\right )} \sqrt {\sqrt {x^{2} + 1} + 1}}{x^{4} + 1} \,d x } \]
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Not integrable
Time = 5.66 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.23 \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x^2}}}{1+x^4} \, dx=\int \frac {\left (x^4-1\right )\,\sqrt {\sqrt {x^2+1}+1}}{x^4+1} \,d x \]
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