Integrand size = 32, antiderivative size = 119 \[ \int \frac {\left (-1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx=\frac {1}{2} x \sqrt {x^2+\sqrt {1+x^4}}-\frac {\arctan \left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )}{\sqrt {2}}-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right ) \]
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Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.95, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {6874, 2157, 212, 2158, 327, 222, 221} \[ \int \frac {\left (-1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx=-\frac {\arcsin \left (\sqrt [4]{-1} x\right )}{2 \sqrt {2}}+\frac {i \text {arcsinh}\left (\sqrt [4]{-1} x\right )}{2 \sqrt {2}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {\sqrt {x^4+1}+x^2}}\right )}{\sqrt {2}}+\left (\frac {1}{4}+\frac {i}{4}\right ) \sqrt {1-i x^2} x+\left (\frac {1}{4}-\frac {i}{4}\right ) \sqrt {1+i x^2} x \]
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Rule 212
Rule 221
Rule 222
Rule 327
Rule 2157
Rule 2158
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}}+\frac {x^2 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}}\right ) \, dx \\ & = -\int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx+\int \frac {x^2 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx \\ & = \left (\frac {1}{2}-\frac {i}{2}\right ) \int \frac {x^2}{\sqrt {1-i x^2}} \, dx+\left (\frac {1}{2}+\frac {i}{2}\right ) \int \frac {x^2}{\sqrt {1+i x^2}} \, dx-\text {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {x}{\sqrt {x^2+\sqrt {1+x^4}}}\right ) \\ & = \left (\frac {1}{4}+\frac {i}{4}\right ) x \sqrt {1-i x^2}+\left (\frac {1}{4}-\frac {i}{4}\right ) x \sqrt {1+i x^2}-\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}}+\left (-\frac {1}{4}-\frac {i}{4}\right ) \int \frac {1}{\sqrt {1-i x^2}} \, dx+\left (-\frac {1}{4}+\frac {i}{4}\right ) \int \frac {1}{\sqrt {1+i x^2}} \, dx \\ & = \left (\frac {1}{4}+\frac {i}{4}\right ) x \sqrt {1-i x^2}+\left (\frac {1}{4}-\frac {i}{4}\right ) x \sqrt {1+i x^2}-\frac {\arcsin \left (\sqrt [4]{-1} x\right )}{2 \sqrt {2}}+\frac {i \text {arcsinh}\left (\sqrt [4]{-1} x\right )}{2 \sqrt {2}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.01 \[ \int \frac {\left (-1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx=\frac {1}{2} \left (x \sqrt {x^2+\sqrt {1+x^4}}-\sqrt {2} \arctan \left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}\right )-2 \sqrt {2} \text {arctanh}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}\right )\right ) \]
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\[\int \frac {\left (x^{2}-1\right ) \sqrt {x^{2}+\sqrt {x^{4}+1}}}{\sqrt {x^{4}+1}}d x\]
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Timed out. \[ \int \frac {\left (-1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx=\text {Timed out} \]
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Time = 2.32 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.27 \[ \int \frac {\left (-1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx=- \frac {{G_{3, 3}^{2, 2}\left (\begin {matrix} 1, 1 & \frac {1}{2} \\\frac {1}{4}, \frac {3}{4} & 0 \end {matrix} \middle | {x^{4}} \right )}}{4 \sqrt {\pi }} + \frac {{G_{3, 3}^{2, 2}\left (\begin {matrix} \frac {3}{2}, 1 & 1 \\\frac {3}{4}, \frac {5}{4} & 0 \end {matrix} \middle | {x^{4}} \right )}}{4 \sqrt {\pi }} \]
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\[ \int \frac {\left (-1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx=\int { \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}} {\left (x^{2} - 1\right )}}{\sqrt {x^{4} + 1}} \,d x } \]
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\[ \int \frac {\left (-1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx=\int { \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}} {\left (x^{2} - 1\right )}}{\sqrt {x^{4} + 1}} \,d x } \]
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Timed out. \[ \int \frac {\left (-1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx=\int \frac {\left (x^2-1\right )\,\sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}} \,d x \]
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