Integrand size = 28, antiderivative size = 70 \[ \int \frac {\left (-1+x^2\right ) \sqrt {1+\sqrt {1+x^2}}}{1+x^2} \, dx=\frac {4 x}{3 \sqrt {1+\sqrt {1+x^2}}}+\frac {2 x \sqrt {1+x^2}}{3 \sqrt {1+\sqrt {1+x^2}}}-4 \arctan \left (\frac {x}{\sqrt {1+\sqrt {1+x^2}}}\right ) \]
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\[ \int \frac {\left (-1+x^2\right ) \sqrt {1+\sqrt {1+x^2}}}{1+x^2} \, dx=\int \frac {\left (-1+x^2\right ) \sqrt {1+\sqrt {1+x^2}}}{1+x^2} \, dx \]
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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^2}\right ) \, dx \\ & = -\left (2 \int \frac {\sqrt {1+\sqrt {1+x^2}}}{1+x^2} \, 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 (i-x)}+\frac {i \sqrt {1+\sqrt {1+x^2}}}{2 (i+x)}\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} \, dx-i \int \frac {\sqrt {1+\sqrt {1+x^2}}}{i+x} \, dx \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.74 \[ \int \frac {\left (-1+x^2\right ) \sqrt {1+\sqrt {1+x^2}}}{1+x^2} \, dx=\frac {2 x \left (2+\sqrt {1+x^2}\right )}{3 \sqrt {1+\sqrt {1+x^2}}}-4 \arctan \left (\frac {x}{\sqrt {1+\sqrt {1+x^2}}}\right ) \]
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\[\int \frac {\left (x^{2}-1\right ) \sqrt {1+\sqrt {x^{2}+1}}}{x^{2}+1}d x\]
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none
Time = 0.89 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.24 \[ \int \frac {\left (-1+x^2\right ) \sqrt {1+\sqrt {1+x^2}}}{1+x^2} \, dx=\frac {3 \, x \arctan \left (\frac {4 \, {\left (x^{4} - 12 \, x^{2} + {\left (5 \, x^{2} - 3\right )} \sqrt {x^{2} + 1} + 3\right )} \sqrt {\sqrt {x^{2} + 1} + 1}}{x^{5} - 46 \, x^{3} + 17 \, x}\right ) + 2 \, {\left (x^{2} + \sqrt {x^{2} + 1} - 1\right )} \sqrt {\sqrt {x^{2} + 1} + 1}}{3 \, x} \]
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\[ \int \frac {\left (-1+x^2\right ) \sqrt {1+\sqrt {1+x^2}}}{1+x^2} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \sqrt {\sqrt {x^{2} + 1} + 1}}{x^{2} + 1}\, dx \]
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\[ \int \frac {\left (-1+x^2\right ) \sqrt {1+\sqrt {1+x^2}}}{1+x^2} \, dx=\int { \frac {{\left (x^{2} - 1\right )} \sqrt {\sqrt {x^{2} + 1} + 1}}{x^{2} + 1} \,d x } \]
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\[ \int \frac {\left (-1+x^2\right ) \sqrt {1+\sqrt {1+x^2}}}{1+x^2} \, dx=\int { \frac {{\left (x^{2} - 1\right )} \sqrt {\sqrt {x^{2} + 1} + 1}}{x^{2} + 1} \,d x } \]
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Timed out. \[ \int \frac {\left (-1+x^2\right ) \sqrt {1+\sqrt {1+x^2}}}{1+x^2} \, dx=\int \frac {\left (x^2-1\right )\,\sqrt {\sqrt {x^2+1}+1}}{x^2+1} \,d x \]
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