Integrand size = 31, antiderivative size = 127 \[ \int \frac {2+x^4}{\sqrt [4]{x^2+x^4} \left (-1-x^4+2 x^8\right )} \, dx=-\frac {\left (x^2+x^4\right )^{3/4}}{x \left (1+x^2\right )}-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^4}}\right )}{2 \sqrt [4]{2}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^4}}\right )}{2 \sqrt [4]{2}}+\frac {1}{4} \text {RootSum}\left [3-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{x^2+x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]
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Result contains complex when optimal does not.
Time = 0.58 (sec) , antiderivative size = 441, normalized size of antiderivative = 3.47, number of steps used = 19, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.323, Rules used = {2081, 6847, 6860, 1418, 390, 385, 218, 212, 209, 1443} \[ \int \frac {2+x^4}{\sqrt [4]{x^2+x^4} \left (-1-x^4+2 x^8\right )} \, dx=-\frac {\sqrt [4]{x^2+1} \sqrt {x} \arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2+1}}\right )}{2 \sqrt [4]{2} \sqrt [4]{x^4+x^2}}-\frac {\sqrt [4]{x^2+1} \sqrt {x} \arctan \left (\frac {\sqrt [4]{\sqrt {2}-2 i} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{x^2+1}}\right )}{2^{7/8} \sqrt [4]{\sqrt {2}-2 i} \sqrt [4]{x^4+x^2}}-\frac {\sqrt [4]{x^2+1} \sqrt {x} \arctan \left (\frac {\sqrt [4]{\sqrt {2}+2 i} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{x^2+1}}\right )}{2^{7/8} \sqrt [4]{\sqrt {2}+2 i} \sqrt [4]{x^4+x^2}}-\frac {\sqrt [4]{x^2+1} \sqrt {x} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2+1}}\right )}{2 \sqrt [4]{2} \sqrt [4]{x^4+x^2}}-\frac {\sqrt [4]{x^2+1} \sqrt {x} \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {2}-2 i} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{x^2+1}}\right )}{2^{7/8} \sqrt [4]{\sqrt {2}-2 i} \sqrt [4]{x^4+x^2}}-\frac {\sqrt [4]{x^2+1} \sqrt {x} \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {2}+2 i} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{x^2+1}}\right )}{2^{7/8} \sqrt [4]{\sqrt {2}+2 i} \sqrt [4]{x^4+x^2}}-\frac {x}{\sqrt [4]{x^4+x^2}} \]
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Rule 209
Rule 212
Rule 218
Rule 385
Rule 390
Rule 1418
Rule 1443
Rule 2081
Rule 6847
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt [4]{1+x^2}\right ) \int \frac {2+x^4}{\sqrt {x} \sqrt [4]{1+x^2} \left (-1-x^4+2 x^8\right )} \, dx}{\sqrt [4]{x^2+x^4}} \\ & = \frac {\left (2 \sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {2+x^8}{\sqrt [4]{1+x^4} \left (-1-x^8+2 x^{16}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^4}} \\ & = \frac {\left (2 \sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \left (\frac {4}{\sqrt [4]{1+x^4} \left (-4+4 x^8\right )}-\frac {2}{\sqrt [4]{1+x^4} \left (2+4 x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^4}} \\ & = -\frac {\left (4 \sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4} \left (2+4 x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^4}}+\frac {\left (8 \sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4} \left (-4+4 x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^4}} \\ & = \frac {\left (8 \sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1+x^4\right )^{5/4} \left (-4+4 x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^4}}-\frac {\left (2 i \sqrt {2} \sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (2 i \sqrt {2}-4 x^4\right ) \sqrt [4]{1+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^4}}-\frac {\left (2 i \sqrt {2} \sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4} \left (2 i \sqrt {2}+4 x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^4}} \\ & = -\frac {x}{\sqrt [4]{x^2+x^4}}+\frac {\left (4 \sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4} \left (-4+4 x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^4}}-\frac {\left (2 i \sqrt {2} \sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{2 i \sqrt {2}-\left (-4+2 i \sqrt {2}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt [4]{x^2+x^4}}-\frac {\left (2 i \sqrt {2} \sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{2 i \sqrt {2}-\left (4+2 i \sqrt {2}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt [4]{x^2+x^4}} \\ & = -\frac {x}{\sqrt [4]{x^2+x^4}}+\frac {\left (4 \sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{-4+8 x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt [4]{x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2}-\sqrt {-2 i+\sqrt {2}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{2^{3/4} \sqrt [4]{x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2}+\sqrt {-2 i+\sqrt {2}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{2^{3/4} \sqrt [4]{x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2}-\sqrt {2 i+\sqrt {2}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{2^{3/4} \sqrt [4]{x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2}+\sqrt {2 i+\sqrt {2}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{2^{3/4} \sqrt [4]{x^2+x^4}} \\ & = -\frac {x}{\sqrt [4]{x^2+x^4}}-\frac {\sqrt {x} \sqrt [4]{1+x^2} \arctan \left (\frac {\sqrt [4]{-2 i+\sqrt {2}} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{1+x^2}}\right )}{2^{7/8} \sqrt [4]{-2 i+\sqrt {2}} \sqrt [4]{x^2+x^4}}-\frac {\sqrt {x} \sqrt [4]{1+x^2} \arctan \left (\frac {\sqrt [4]{2 i+\sqrt {2}} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{1+x^2}}\right )}{2^{7/8} \sqrt [4]{2 i+\sqrt {2}} \sqrt [4]{x^2+x^4}}-\frac {\sqrt {x} \sqrt [4]{1+x^2} \text {arctanh}\left (\frac {\sqrt [4]{-2 i+\sqrt {2}} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{1+x^2}}\right )}{2^{7/8} \sqrt [4]{-2 i+\sqrt {2}} \sqrt [4]{x^2+x^4}}-\frac {\sqrt {x} \sqrt [4]{1+x^2} \text {arctanh}\left (\frac {\sqrt [4]{2 i+\sqrt {2}} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{1+x^2}}\right )}{2^{7/8} \sqrt [4]{2 i+\sqrt {2}} \sqrt [4]{x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{2 \sqrt [4]{x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{2 \sqrt [4]{x^2+x^4}} \\ & = -\frac {x}{\sqrt [4]{x^2+x^4}}-\frac {\sqrt {x} \sqrt [4]{1+x^2} \arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{2 \sqrt [4]{2} \sqrt [4]{x^2+x^4}}-\frac {\sqrt {x} \sqrt [4]{1+x^2} \arctan \left (\frac {\sqrt [4]{-2 i+\sqrt {2}} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{1+x^2}}\right )}{2^{7/8} \sqrt [4]{-2 i+\sqrt {2}} \sqrt [4]{x^2+x^4}}-\frac {\sqrt {x} \sqrt [4]{1+x^2} \arctan \left (\frac {\sqrt [4]{2 i+\sqrt {2}} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{1+x^2}}\right )}{2^{7/8} \sqrt [4]{2 i+\sqrt {2}} \sqrt [4]{x^2+x^4}}-\frac {\sqrt {x} \sqrt [4]{1+x^2} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{2 \sqrt [4]{2} \sqrt [4]{x^2+x^4}}-\frac {\sqrt {x} \sqrt [4]{1+x^2} \text {arctanh}\left (\frac {\sqrt [4]{-2 i+\sqrt {2}} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{1+x^2}}\right )}{2^{7/8} \sqrt [4]{-2 i+\sqrt {2}} \sqrt [4]{x^2+x^4}}-\frac {\sqrt {x} \sqrt [4]{1+x^2} \text {arctanh}\left (\frac {\sqrt [4]{2 i+\sqrt {2}} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{1+x^2}}\right )}{2^{7/8} \sqrt [4]{2 i+\sqrt {2}} \sqrt [4]{x^2+x^4}} \\ \end{align*}
Time = 0.46 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.26 \[ \int \frac {2+x^4}{\sqrt [4]{x^2+x^4} \left (-1-x^4+2 x^8\right )} \, dx=-\frac {\sqrt {x} \left (4 \sqrt {x}+2^{3/4} \sqrt [4]{1+x^2} \arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^2}}\right )+2^{3/4} \sqrt [4]{1+x^2} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^2}}\right )-\sqrt [4]{1+x^2} \text {RootSum}\left [3-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right )+\log \left (\sqrt [4]{1+x^2}-\sqrt {x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right )}{4 \sqrt [4]{x^2+x^4}} \]
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Timed out.
\[\int \frac {x^{4}+2}{\left (x^{4}+x^{2}\right )^{\frac {1}{4}} \left (2 x^{8}-x^{4}-1\right )}d x\]
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 5.33 (sec) , antiderivative size = 2054, normalized size of antiderivative = 16.17 \[ \int \frac {2+x^4}{\sqrt [4]{x^2+x^4} \left (-1-x^4+2 x^8\right )} \, dx=\text {Too large to display} \]
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Not integrable
Time = 4.76 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.28 \[ \int \frac {2+x^4}{\sqrt [4]{x^2+x^4} \left (-1-x^4+2 x^8\right )} \, dx=\int \frac {x^{4} + 2}{\sqrt [4]{x^{2} \left (x^{2} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (2 x^{4} + 1\right )}\, dx \]
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Not integrable
Time = 0.34 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.21 \[ \int \frac {2+x^4}{\sqrt [4]{x^2+x^4} \left (-1-x^4+2 x^8\right )} \, dx=\int { \frac {x^{4} + 2}{{\left (2 \, x^{8} - x^{4} - 1\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}}} \,d x } \]
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Exception generated. \[ \int \frac {2+x^4}{\sqrt [4]{x^2+x^4} \left (-1-x^4+2 x^8\right )} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 5.75 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.24 \[ \int \frac {2+x^4}{\sqrt [4]{x^2+x^4} \left (-1-x^4+2 x^8\right )} \, dx=\int -\frac {x^4+2}{{\left (x^4+x^2\right )}^{1/4}\,\left (-2\,x^8+x^4+1\right )} \,d x \]
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