Integrand size = 30, antiderivative size = 242 \[ \int \frac {\left (-1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2} \, dx=\frac {1}{2} x \sqrt {x^2+\sqrt {1+x^4}}+\frac {\arctan \left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )}{\sqrt {2}}-2 \sqrt {-1+\sqrt {2}} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )+2 \sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right ) \]
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\[ \int \frac {\left (-1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2} \, dx=\int \frac {\left (-1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\sqrt {x^2+\sqrt {1+x^4}}-\frac {2 \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2}\right ) \, dx \\ & = -\left (2 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+x^2} \, dx\right )+\int \sqrt {x^2+\sqrt {1+x^4}} \, dx \\ & = -\left (2 \int \left (\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{2 (i-x)}+\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{2 (i+x)}\right ) \, dx\right )+\int \sqrt {x^2+\sqrt {1+x^4}} \, dx \\ & = -\left (i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{i-x} \, dx\right )-i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{i+x} \, dx+\int \sqrt {x^2+\sqrt {1+x^4}} \, dx \\ \end{align*}
Time = 0.79 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2} \, dx=\frac {1}{2} \left (x \sqrt {x^2+\sqrt {1+x^4}}+\sqrt {2} \arctan \left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}\right )-4 \sqrt {-1+\sqrt {2}} \arctan \left (\frac {\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} \left (-1+x^2+\sqrt {1+x^4}\right )}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )-4 \sqrt {2} \text {arctanh}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}\right )+4 \sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}\right )\right ) \]
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\[\int \frac {\left (x^{2}-1\right ) \sqrt {x^{2}+\sqrt {x^{4}+1}}}{x^{2}+1}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 503 vs. \(2 (189) = 378\).
Time = 2.42 (sec) , antiderivative size = 503, normalized size of antiderivative = 2.08 \[ \int \frac {\left (-1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2} \, dx=\frac {1}{2} \, \sqrt {x^{2} + \sqrt {x^{4} + 1}} x - \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {{\left (\sqrt {2} x^{2} - \sqrt {2} \sqrt {x^{4} + 1}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{2 \, x}\right ) + \frac {1}{2} \, \sqrt {2} \log \left (4 \, x^{4} + 4 \, \sqrt {x^{4} + 1} x^{2} - 2 \, {\left (\sqrt {2} x^{3} + \sqrt {2} \sqrt {x^{4} + 1} x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + 1\right ) + \frac {1}{4} \, \sqrt {-4 \, \sqrt {2} + 4} \log \left (-\frac {2 \, \sqrt {2} x^{2} - 4 \, x^{2} + \sqrt {x^{2} + \sqrt {x^{4} + 1}} {\left (\sqrt {x^{4} + 1} x \sqrt {-4 \, \sqrt {2} + 4} - {\left (x^{3} - \sqrt {2} x + x\right )} \sqrt {-4 \, \sqrt {2} + 4}\right )} + 2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} - 1\right )}}{x^{2} + 1}\right ) - \frac {1}{4} \, \sqrt {-4 \, \sqrt {2} + 4} \log \left (-\frac {2 \, \sqrt {2} x^{2} - 4 \, x^{2} - \sqrt {x^{2} + \sqrt {x^{4} + 1}} {\left (\sqrt {x^{4} + 1} x \sqrt {-4 \, \sqrt {2} + 4} - {\left (x^{3} - \sqrt {2} x + x\right )} \sqrt {-4 \, \sqrt {2} + 4}\right )} + 2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} - 1\right )}}{x^{2} + 1}\right ) + \frac {1}{2} \, \sqrt {\sqrt {2} + 1} \log \left (\frac {2 \, {\left (\sqrt {2} x^{2} + 2 \, x^{2} + {\left (x^{3} + \sqrt {2} x - \sqrt {x^{4} + 1} x + x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\sqrt {2} + 1} + \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}\right )}}{x^{2} + 1}\right ) - \frac {1}{2} \, \sqrt {\sqrt {2} + 1} \log \left (\frac {2 \, {\left (\sqrt {2} x^{2} + 2 \, x^{2} - {\left (x^{3} + \sqrt {2} x - \sqrt {x^{4} + 1} x + x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\sqrt {2} + 1} + \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}\right )}}{x^{2} + 1}\right ) \]
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\[ \int \frac {\left (-1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{x^{2} + 1}\, dx \]
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\[ \int \frac {\left (-1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2} \, dx=\int { \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}} {\left (x^{2} - 1\right )}}{x^{2} + 1} \,d x } \]
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\[ \int \frac {\left (-1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2} \, dx=\int { \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}} {\left (x^{2} - 1\right )}}{x^{2} + 1} \,d x } \]
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Timed out. \[ \int \frac {\left (-1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2} \, dx=\int \frac {\left (x^2-1\right )\,\sqrt {\sqrt {x^4+1}+x^2}}{x^2+1} \,d x \]
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