Integrand size = 27, antiderivative size = 136 \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1+x^4+x^8} \, dx=\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^2+x^6}}\right )-\frac {\arctan \left (\frac {\sqrt {2} x \sqrt [4]{x^2+x^6}}{-x^2+\sqrt {x^2+x^6}}\right )}{2 \sqrt {2}}-\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^2+x^6}}\right )+\frac {\text {arctanh}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {x^2+x^6}}{\sqrt {2}}}{x \sqrt [4]{x^2+x^6}}\right )}{2 \sqrt {2}} \]
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Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.28 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.20, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2081, 6860, 477, 524} \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1+x^4+x^8} \, dx=\frac {2 \left (-\sqrt {3}+i\right ) x \sqrt [4]{x^6+x^2} \operatorname {AppellF1}\left (\frac {3}{8},-\frac {1}{4},1,\frac {11}{8},-x^4,-\frac {2 x^4}{1-i \sqrt {3}}\right )}{3 \left (\sqrt {3}+i\right ) \sqrt [4]{x^4+1}}+\frac {2 \left (\sqrt {3}+i\right ) x \sqrt [4]{x^6+x^2} \operatorname {AppellF1}\left (\frac {3}{8},-\frac {1}{4},1,\frac {11}{8},-x^4,-\frac {2 x^4}{1+i \sqrt {3}}\right )}{3 \left (-\sqrt {3}+i\right ) \sqrt [4]{x^4+1}} \]
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Rule 477
Rule 524
Rule 2081
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [4]{x^2+x^6} \int \frac {\sqrt {x} \left (-1+x^4\right ) \sqrt [4]{1+x^4}}{1+x^4+x^8} \, dx}{\sqrt {x} \sqrt [4]{1+x^4}} \\ & = \frac {\sqrt [4]{x^2+x^6} \int \left (\frac {\left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^4}}{1-i \sqrt {3}+2 x^4}+\frac {\left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^4}}{1+i \sqrt {3}+2 x^4}\right ) \, dx}{\sqrt {x} \sqrt [4]{1+x^4}} \\ & = \frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [4]{x^2+x^6}\right ) \int \frac {\sqrt {x} \sqrt [4]{1+x^4}}{1+i \sqrt {3}+2 x^4} \, dx}{\sqrt {x} \sqrt [4]{1+x^4}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [4]{x^2+x^6}\right ) \int \frac {\sqrt {x} \sqrt [4]{1+x^4}}{1-i \sqrt {3}+2 x^4} \, dx}{\sqrt {x} \sqrt [4]{1+x^4}} \\ & = \frac {\left (2 \left (1-i \sqrt {3}\right ) \sqrt [4]{x^2+x^6}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{1+x^8}}{1+i \sqrt {3}+2 x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{1+x^4}}+\frac {\left (2 \left (1+i \sqrt {3}\right ) \sqrt [4]{x^2+x^6}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{1+x^8}}{1-i \sqrt {3}+2 x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{1+x^4}} \\ & = \frac {2 \left (i-\sqrt {3}\right ) x \sqrt [4]{x^2+x^6} \operatorname {AppellF1}\left (\frac {3}{8},-\frac {1}{4},1,\frac {11}{8},-x^4,-\frac {2 x^4}{1-i \sqrt {3}}\right )}{3 \left (i+\sqrt {3}\right ) \sqrt [4]{1+x^4}}+\frac {2 \left (i+\sqrt {3}\right ) x \sqrt [4]{x^2+x^6} \operatorname {AppellF1}\left (\frac {3}{8},-\frac {1}{4},1,\frac {11}{8},-x^4,-\frac {2 x^4}{1+i \sqrt {3}}\right )}{3 \left (i-\sqrt {3}\right ) \sqrt [4]{1+x^4}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.09 \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1+x^4+x^8} \, dx=\frac {\sqrt [4]{x^2+x^6} \left (2 \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{1+x^4}}\right )+\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {x} \sqrt [4]{1+x^4}}{x-\sqrt {1+x^4}}\right )-2 \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{1+x^4}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {x} \sqrt [4]{1+x^4}}{x+\sqrt {1+x^4}}\right )\right )}{4 \sqrt {x} \sqrt [4]{1+x^4}} \]
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Time = 4.55 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.40
method | result | size |
pseudoelliptic | \(\frac {\ln \left (\frac {\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}-x}{x}\right )}{4}-\frac {\ln \left (\frac {x +\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}}{x}\right )}{4}-\frac {\arctan \left (\frac {\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}}{x}\right )}{2}+\frac {\ln \left (\frac {\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{2} \left (x^{4}+1\right )}}{-\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{2} \left (x^{4}+1\right )}}\right ) \sqrt {2}}{8}+\frac {\arctan \left (\frac {\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}} \sqrt {2}+x}{x}\right ) \sqrt {2}}{4}+\frac {\arctan \left (\frac {\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}} \sqrt {2}-x}{x}\right ) \sqrt {2}}{4}\) | \(190\) |
trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {-2 \sqrt {x^{6}+x^{2}}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{5}-2 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{3}+2 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x}{\left (x^{2}+x +1\right ) \left (x^{2}-x +1\right ) x}\right )}{4}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{5}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{3}+2 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}-2 \sqrt {x^{6}+x^{2}}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x +2 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}}{\left (x^{2}+x +1\right ) \left (x^{2}-x +1\right ) x}\right )}{4}+\frac {\ln \left (-\frac {-x^{5}+2 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}-2 \sqrt {x^{6}+x^{2}}\, x +2 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} x^{2}-x^{3}-x}{\left (x^{4}-x^{2}+1\right ) x}\right )}{4}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{5}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \sqrt {x^{6}+x^{2}}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+2 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}-2 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x}{\left (x^{4}-x^{2}+1\right ) x}\right )}{4}\) | \(420\) |
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Result contains complex when optimal does not.
Time = 7.25 (sec) , antiderivative size = 407, normalized size of antiderivative = 2.99 \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1+x^4+x^8} \, dx=\left (\frac {1}{16} i - \frac {1}{16}\right ) \, \sqrt {2} \log \left (\frac {4 i \, {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} - \left (2 i - 2\right ) \, \sqrt {2} \sqrt {x^{6} + x^{2}} x + \sqrt {2} {\left (\left (i + 1\right ) \, x^{5} - \left (i + 1\right ) \, x^{3} + \left (i + 1\right ) \, x\right )} - 4 \, {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} + x^{3} + x}\right ) - \left (\frac {1}{16} i - \frac {1}{16}\right ) \, \sqrt {2} \log \left (\frac {4 i \, {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + \left (2 i - 2\right ) \, \sqrt {2} \sqrt {x^{6} + x^{2}} x + \sqrt {2} {\left (-\left (i + 1\right ) \, x^{5} + \left (i + 1\right ) \, x^{3} - \left (i + 1\right ) \, x\right )} - 4 \, {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} + x^{3} + x}\right ) - \left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2} \log \left (\frac {-4 i \, {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + \left (2 i + 2\right ) \, \sqrt {2} \sqrt {x^{6} + x^{2}} x + \sqrt {2} {\left (-\left (i - 1\right ) \, x^{5} + \left (i - 1\right ) \, x^{3} - \left (i - 1\right ) \, x\right )} - 4 \, {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} + x^{3} + x}\right ) + \left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2} \log \left (\frac {-4 i \, {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} - \left (2 i + 2\right ) \, \sqrt {2} \sqrt {x^{6} + x^{2}} x + \sqrt {2} {\left (\left (i - 1\right ) \, x^{5} - \left (i - 1\right ) \, x^{3} + \left (i - 1\right ) \, x\right )} - 4 \, {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} + x^{3} + x}\right ) + \frac {1}{4} \, \arctan \left (\frac {2 \, {\left ({\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}\right )}}{x^{5} - x^{3} + x}\right ) + \frac {1}{4} \, \log \left (-\frac {x^{5} + x^{3} - 2 \, {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 2 \, \sqrt {x^{6} + x^{2}} x + x - 2 \, {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} - x^{3} + x}\right ) \]
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\[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1+x^4+x^8} \, dx=\int \frac {\sqrt [4]{x^{2} \left (x^{4} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right ) \left (x^{4} - x^{2} + 1\right )}\, dx \]
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\[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1+x^4+x^8} \, dx=\int { \frac {{\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}}{x^{8} + x^{4} + 1} \,d x } \]
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\[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1+x^4+x^8} \, dx=\int { \frac {{\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}}{x^{8} + x^{4} + 1} \,d x } \]
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Timed out. \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1+x^4+x^8} \, dx=\int \frac {{\left (x^6+x^2\right )}^{1/4}\,\left (x^4-1\right )}{x^8+x^4+1} \,d x \]
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