Integrand size = 21, antiderivative size = 137 \[ \int \frac {1}{\sqrt [4]{-x^2+x^4} \left (-1+x^8\right )} \, dx=-\frac {\left (-x^2+x^4\right )^{3/4}}{2 x \left (-1+x^2\right )}-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^2+x^4}}\right )}{4 \sqrt [4]{2}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^2+x^4}}\right )}{4 \sqrt [4]{2}}+\frac {1}{8} \text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{-x^2+x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]
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Result contains complex when optimal does not.
Time = 0.38 (sec) , antiderivative size = 471, normalized size of antiderivative = 3.44, number of steps used = 21, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {2081, 6847, 2098, 1166, 388, 385, 218, 212, 209, 1443} \[ \int \frac {1}{\sqrt [4]{-x^2+x^4} \left (-1+x^8\right )} \, dx=-\frac {\sqrt {x} \sqrt [4]{x^2-1} \arctan \left (\frac {\sqrt [4]{1-i} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{4 \sqrt [4]{1-i} \sqrt [4]{x^4-x^2}}-\frac {\sqrt {x} \sqrt [4]{x^2-1} \arctan \left (\frac {\sqrt [4]{1+i} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{4 \sqrt [4]{1+i} \sqrt [4]{x^4-x^2}}-\frac {\sqrt {x} \sqrt [4]{x^2-1} \arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{4 \sqrt [4]{2} \sqrt [4]{x^4-x^2}}-\frac {\sqrt {x} \sqrt [4]{x^2-1} \text {arctanh}\left (\frac {\sqrt [4]{1-i} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{4 \sqrt [4]{1-i} \sqrt [4]{x^4-x^2}}-\frac {\sqrt {x} \sqrt [4]{x^2-1} \text {arctanh}\left (\frac {\sqrt [4]{1+i} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{4 \sqrt [4]{1+i} \sqrt [4]{x^4-x^2}}-\frac {\sqrt {x} \sqrt [4]{x^2-1} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{4 \sqrt [4]{2} \sqrt [4]{x^4-x^2}}-\frac {x (x+1) \left (\frac {1-x}{x+1}\right )^{5/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {3}{2},\frac {2 x}{x+1}\right )}{4 (1-x) \sqrt [4]{x^4-x^2}}-\frac {x \sqrt [4]{\frac {1-x}{x+1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {3}{2},\frac {2 x}{x+1}\right )}{4 \sqrt [4]{x^4-x^2}} \]
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Rule 209
Rule 212
Rule 218
Rule 385
Rule 388
Rule 1166
Rule 1443
Rule 2081
Rule 2098
Rule 6847
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt [4]{-1+x^2} \left (-1+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 {1}{\sqrt [4]{-1+x^4} \left (-1+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 {1}{8 \left (-1+x^2\right ) \sqrt [4]{-1+x^4}}-\frac {1}{8 \left (1+x^2\right ) \sqrt [4]{-1+x^4}}-\frac {1}{4 \sqrt [4]{-1+x^4} \left (1+x^4\right )}-\frac {1}{2 \sqrt [4]{-1+x^4} \left (1+x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-x^2+x^4}} \\ & = \frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+x^2\right ) \sqrt [4]{-1+x^4}} \, dx,x,\sqrt {x}\right )}{4 \sqrt [4]{-x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt [4]{-1+x^4}} \, dx,x,\sqrt {x}\right )}{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]{-1+x^4} \left (1+x^4\right )} \, dx,x,\sqrt {x}\right )}{2 \sqrt [4]{-x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-1+x^4} \left (1+x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-x^2+x^4}} \\ & = -\frac {\left (\sqrt [4]{-1+x} \sqrt {x} \sqrt [4]{1+x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-1+x^2} \left (1+x^2\right )^{5/4}} \, dx,x,\sqrt {x}\right )}{4 \sqrt [4]{-x^2+x^4}}+\frac {\left (\sqrt [4]{-1+x} \sqrt {x} \sqrt [4]{1+x}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+x^2\right )^{5/4} \sqrt [4]{1+x^2}} \, dx,x,\sqrt {x}\right )}{4 \sqrt [4]{-x^2+x^4}}-\frac {\left (i \sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (i-x^4\right ) \sqrt [4]{-1+x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt [4]{-x^2+x^4}}-\frac {\left (i \sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-1+x^4} \left (i+x^4\right )} \, dx,x,\sqrt {x}\right )}{2 \sqrt [4]{-x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{1-2 x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{2 \sqrt [4]{-x^2+x^4}} \\ & = -\frac {x \sqrt [4]{\frac {1-x}{1+x}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {3}{2},\frac {2 x}{1+x}\right )}{4 \sqrt [4]{-x^2+x^4}}-\frac {x \left (\frac {1-x}{1+x}\right )^{5/4} (1+x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {3}{2},\frac {2 x}{1+x}\right )}{4 (1-x) \sqrt [4]{-x^2+x^4}}-\frac {\left (i \sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{i-(1+i) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{2 \sqrt [4]{-x^2+x^4}}-\frac {\left (i \sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{i+(1-i) x^4} \, 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 )}{4 \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 )}{4 \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 )}{4 \sqrt [4]{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 )}{4 \sqrt [4]{2} \sqrt [4]{-x^2+x^4}}-\frac {x \sqrt [4]{\frac {1-x}{1+x}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {3}{2},\frac {2 x}{1+x}\right )}{4 \sqrt [4]{-x^2+x^4}}-\frac {x \left (\frac {1-x}{1+x}\right )^{5/4} (1+x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {3}{2},\frac {2 x}{1+x}\right )}{4 (1-x) \sqrt [4]{-x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {1-i} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{4 \sqrt [4]{-x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {1-i} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{4 \sqrt [4]{-x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {1+i} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{4 \sqrt [4]{-x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {1+i} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{4 \sqrt [4]{-x^2+x^4}} \\ & = -\frac {\sqrt {x} \sqrt [4]{-1+x^2} \arctan \left (\frac {\sqrt [4]{1-i} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{4 \sqrt [4]{1-i} \sqrt [4]{-x^2+x^4}}-\frac {\sqrt {x} \sqrt [4]{-1+x^2} \arctan \left (\frac {\sqrt [4]{1+i} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{4 \sqrt [4]{1+i} \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 )}{4 \sqrt [4]{2} \sqrt [4]{-x^2+x^4}}-\frac {\sqrt {x} \sqrt [4]{-1+x^2} \text {arctanh}\left (\frac {\sqrt [4]{1-i} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{4 \sqrt [4]{1-i} \sqrt [4]{-x^2+x^4}}-\frac {\sqrt {x} \sqrt [4]{-1+x^2} \text {arctanh}\left (\frac {\sqrt [4]{1+i} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{4 \sqrt [4]{1+i} \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 )}{4 \sqrt [4]{2} \sqrt [4]{-x^2+x^4}}-\frac {x \sqrt [4]{\frac {1-x}{1+x}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {3}{2},\frac {2 x}{1+x}\right )}{4 \sqrt [4]{-x^2+x^4}}-\frac {x \left (\frac {1-x}{1+x}\right )^{5/4} (1+x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {3}{2},\frac {2 x}{1+x}\right )}{4 (1-x) \sqrt [4]{-x^2+x^4}} \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.18 \[ \int \frac {1}{\sqrt [4]{-x^2+x^4} \left (-1+x^8\right )} \, dx=-\frac {\sqrt {x} \left (4 \sqrt {x}+2^{3/4} \sqrt [4]{-1+x^2} \arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )+2^{3/4} \sqrt [4]{-1+x^2} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )-\sqrt [4]{-1+x^2} \text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right )+\log \left (\sqrt [4]{-1+x^2}-\sqrt {x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right )}{8 \sqrt [4]{x^2 \left (-1+x^2\right )}} \]
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Timed out.
\[\int \frac {1}{\left (x^{4}-x^{2}\right )^{\frac {1}{4}} \left (x^{8}-1\right )}d x\]
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 3.83 (sec) , antiderivative size = 1469, normalized size of antiderivative = 10.72 \[ \int \frac {1}{\sqrt [4]{-x^2+x^4} \left (-1+x^8\right )} \, dx=\text {Too large to display} \]
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Not integrable
Time = 0.87 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.23 \[ \int \frac {1}{\sqrt [4]{-x^2+x^4} \left (-1+x^8\right )} \, dx=\int \frac {1}{\sqrt [4]{x^{2} \left (x - 1\right ) \left (x + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right )}\, dx \]
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Not integrable
Time = 0.30 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.15 \[ \int \frac {1}{\sqrt [4]{-x^2+x^4} \left (-1+x^8\right )} \, dx=\int { \frac {1}{{\left (x^{8} - 1\right )} {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}}} \,d x } \]
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.33 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.78 \[ \int \frac {1}{\sqrt [4]{-x^2+x^4} \left (-1+x^8\right )} \, dx=\frac {1}{8} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{16} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {1}{4}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{16} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {1}{4}} - {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + 8 i \, \left (-\frac {1}{33554432} i + \frac {1}{33554432}\right )^{\frac {1}{4}} \log \left (i \, \left (-37778931862957161709568 i + 37778931862957161709568\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} - 524288\right ) - 8 i \, \left (-\frac {1}{33554432} i + \frac {1}{33554432}\right )^{\frac {1}{4}} \log \left (-i \, \left (-37778931862957161709568 i + 37778931862957161709568\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} - 524288\right ) + i \, \left (\frac {1}{8192} i + \frac {1}{8192}\right )^{\frac {1}{4}} \log \left (\left (549755813888 i + 549755813888\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 1024 i\right ) - \left (\frac {1}{8192} i + \frac {1}{8192}\right )^{\frac {1}{4}} \log \left (i \, \left (549755813888 i + 549755813888\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 1024 i\right ) + \left (\frac {1}{8192} i + \frac {1}{8192}\right )^{\frac {1}{4}} \log \left (-i \, \left (549755813888 i + 549755813888\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 1024 i\right ) - i \, \left (\frac {1}{8192} i + \frac {1}{8192}\right )^{\frac {1}{4}} \log \left (-\left (549755813888 i + 549755813888\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 1024 i\right ) + \left (-\frac {1}{8192} i + \frac {1}{8192}\right )^{\frac {1}{4}} \log \left (\left (-549755813888 i + 549755813888\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} - 1024\right ) - \left (-\frac {1}{8192} i + \frac {1}{8192}\right )^{\frac {1}{4}} \log \left (-\left (-549755813888 i + 549755813888\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} - 1024\right ) - \frac {1}{2 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}} \]
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Not integrable
Time = 5.80 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.15 \[ \int \frac {1}{\sqrt [4]{-x^2+x^4} \left (-1+x^8\right )} \, dx=\int \frac {1}{\left (x^8-1\right )\,{\left (x^4-x^2\right )}^{1/4}} \,d x \]
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