Integrand size = 30, antiderivative size = 80 \[ \int \frac {\sqrt [4]{2+x^4} \left (-4+x^8\right )}{x^6 \left (-4-2 x^4+x^8\right )} \, dx=\frac {\sqrt [4]{2+x^4} \left (-1+2 x^4\right )}{5 x^5}+\frac {1}{8} \text {RootSum}\left [-1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{2+x^4}-x \text {$\#$1}\right )}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \]
[Out]
Leaf count is larger than twice the leaf count of optimal. \(877\) vs. \(2(80)=160\).
Time = 2.19 (sec) , antiderivative size = 877, normalized size of antiderivative = 10.96, number of steps used = 53, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.567, Rules used = {6860, 270, 283, 338, 304, 209, 212, 1542, 524, 1532, 508, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {\sqrt [4]{2+x^4} \left (-4+x^8\right )}{x^6 \left (-4-2 x^4+x^8\right )} \, dx=-\frac {\operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},-\frac {x^4}{2},\frac {x^4}{1-\sqrt {5}}\right ) x^3}{3\ 2^{3/4} \sqrt {5} \left (1-\sqrt {5}\right )}+\frac {\operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {x^4}{1+\sqrt {5}},-\frac {x^4}{2}\right ) x^3}{3\ 2^{3/4} \sqrt {5} \left (1+\sqrt {5}\right )}+\frac {\sqrt [4]{\frac {1}{2} \left (-29+13 \sqrt {5}\right )} \arctan \left (\frac {\sqrt [4]{\frac {2}{-1+\sqrt {5}}} x}{\sqrt [4]{x^4+2}}\right )}{4 \sqrt {5}}+\frac {\sqrt [4]{-2+\sqrt {5}} \arctan \left (\frac {\sqrt [4]{\frac {2}{-1+\sqrt {5}}} x}{\sqrt [4]{x^4+2}}\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{29+13 \sqrt {5}} \arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{1+\sqrt {5}} \sqrt [4]{x^4+2}}\right )}{4\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{2+\sqrt {5}} \arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{1+\sqrt {5}} \sqrt [4]{x^4+2}}\right )}{2 \sqrt {10}}-\frac {\sqrt [4]{29+13 \sqrt {5}} \arctan \left (\frac {2^{3/4} x}{\sqrt [4]{1+\sqrt {5}} \sqrt [4]{x^4+2}}+1\right )}{4\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{2+\sqrt {5}} \arctan \left (\frac {2^{3/4} x}{\sqrt [4]{1+\sqrt {5}} \sqrt [4]{x^4+2}}+1\right )}{2 \sqrt {10}}-\frac {\sqrt [4]{\frac {1}{2} \left (-29+13 \sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt [4]{\frac {2}{-1+\sqrt {5}}} x}{\sqrt [4]{x^4+2}}\right )}{4 \sqrt {5}}-\frac {\sqrt [4]{-2+\sqrt {5}} \text {arctanh}\left (\frac {\sqrt [4]{\frac {2}{-1+\sqrt {5}}} x}{\sqrt [4]{x^4+2}}\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{29+13 \sqrt {5}} \log \left (\frac {2 x^2}{\sqrt {x^4+2}}-\frac {2 \sqrt [4]{2 \left (1+\sqrt {5}\right )} x}{\sqrt [4]{x^4+2}}+\sqrt {2 \left (1+\sqrt {5}\right )}\right )}{8\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{2+\sqrt {5}} \log \left (\frac {2 x^2}{\sqrt {x^4+2}}-\frac {2 \sqrt [4]{2 \left (1+\sqrt {5}\right )} x}{\sqrt [4]{x^4+2}}+\sqrt {2 \left (1+\sqrt {5}\right )}\right )}{4 \sqrt {10}}+\frac {\sqrt [4]{29+13 \sqrt {5}} \log \left (\frac {2 x^2}{\sqrt {x^4+2}}+\frac {2 \sqrt [4]{2 \left (1+\sqrt {5}\right )} x}{\sqrt [4]{x^4+2}}+\sqrt {2 \left (1+\sqrt {5}\right )}\right )}{8\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{2+\sqrt {5}} \log \left (\frac {2 x^2}{\sqrt {x^4+2}}+\frac {2 \sqrt [4]{2 \left (1+\sqrt {5}\right )} x}{\sqrt [4]{x^4+2}}+\sqrt {2 \left (1+\sqrt {5}\right )}\right )}{4 \sqrt {10}}+\frac {\sqrt [4]{x^4+2}}{2 x}-\frac {\left (x^4+2\right )^{5/4}}{10 x^5} \]
[In]
[Out]
Rule 209
Rule 210
Rule 212
Rule 270
Rule 283
Rule 303
Rule 304
Rule 338
Rule 508
Rule 524
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1532
Rule 1542
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\sqrt [4]{2+x^4}}{x^6}-\frac {\sqrt [4]{2+x^4}}{2 x^2}+\frac {x^2 \left (-2+x^4\right ) \sqrt [4]{2+x^4}}{2 \left (-4-2 x^4+x^8\right )}\right ) \, dx \\ & = -\left (\frac {1}{2} \int \frac {\sqrt [4]{2+x^4}}{x^2} \, dx\right )+\frac {1}{2} \int \frac {x^2 \left (-2+x^4\right ) \sqrt [4]{2+x^4}}{-4-2 x^4+x^8} \, dx+\int \frac {\sqrt [4]{2+x^4}}{x^6} \, dx \\ & = \frac {\sqrt [4]{2+x^4}}{2 x}-\frac {\left (2+x^4\right )^{5/4}}{10 x^5}-\frac {1}{2} \int \frac {x^2}{\left (2+x^4\right )^{3/4}} \, dx+\frac {1}{2} \int \left (-\frac {2 x^2 \sqrt [4]{2+x^4}}{-4-2 x^4+x^8}+\frac {x^6 \sqrt [4]{2+x^4}}{-4-2 x^4+x^8}\right ) \, dx \\ & = \frac {\sqrt [4]{2+x^4}}{2 x}-\frac {\left (2+x^4\right )^{5/4}}{10 x^5}+\frac {1}{2} \int \frac {x^6 \sqrt [4]{2+x^4}}{-4-2 x^4+x^8} \, dx-\frac {1}{2} \text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )-\int \frac {x^2 \sqrt [4]{2+x^4}}{-4-2 x^4+x^8} \, dx \\ & = \frac {\sqrt [4]{2+x^4}}{2 x}-\frac {\left (2+x^4\right )^{5/4}}{10 x^5}-\frac {1}{4} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )+\frac {1}{4} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )+\frac {1}{2} \int \frac {x^2}{\left (2+x^4\right )^{3/4}} \, dx-\frac {1}{2} \int \frac {x^2 \left (-4-4 x^4\right )}{\left (2+x^4\right )^{3/4} \left (-4-2 x^4+x^8\right )} \, dx-\int \left (-\frac {x^2 \sqrt [4]{2+x^4}}{\sqrt {5} \left (2+2 \sqrt {5}-2 x^4\right )}-\frac {x^2 \sqrt [4]{2+x^4}}{\sqrt {5} \left (-2+2 \sqrt {5}+2 x^4\right )}\right ) \, dx \\ & = \frac {\sqrt [4]{2+x^4}}{2 x}-\frac {\left (2+x^4\right )^{5/4}}{10 x^5}+\frac {1}{4} \arctan \left (\frac {x}{\sqrt [4]{2+x^4}}\right )-\frac {1}{4} \text {arctanh}\left (\frac {x}{\sqrt [4]{2+x^4}}\right )-\frac {1}{2} \int \left (-\frac {4 x^2}{\left (2+x^4\right )^{3/4} \left (-4-2 x^4+x^8\right )}-\frac {4 x^6}{\left (2+x^4\right )^{3/4} \left (-4-2 x^4+x^8\right )}\right ) \, dx+\frac {1}{2} \text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )+\frac {\int \frac {x^2 \sqrt [4]{2+x^4}}{2+2 \sqrt {5}-2 x^4} \, dx}{\sqrt {5}}+\frac {\int \frac {x^2 \sqrt [4]{2+x^4}}{-2+2 \sqrt {5}+2 x^4} \, dx}{\sqrt {5}} \\ & = \frac {\sqrt [4]{2+x^4}}{2 x}-\frac {\left (2+x^4\right )^{5/4}}{10 x^5}-\frac {x^3 \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},-\frac {x^4}{2},\frac {x^4}{1-\sqrt {5}}\right )}{3\ 2^{3/4} \sqrt {5} \left (1-\sqrt {5}\right )}+\frac {x^3 \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {x^4}{1+\sqrt {5}},-\frac {x^4}{2}\right )}{3\ 2^{3/4} \sqrt {5} \left (1+\sqrt {5}\right )}+\frac {1}{4} \arctan \left (\frac {x}{\sqrt [4]{2+x^4}}\right )-\frac {1}{4} \text {arctanh}\left (\frac {x}{\sqrt [4]{2+x^4}}\right )+\frac {1}{4} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )-\frac {1}{4} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )+2 \int \frac {x^2}{\left (2+x^4\right )^{3/4} \left (-4-2 x^4+x^8\right )} \, dx+2 \int \frac {x^6}{\left (2+x^4\right )^{3/4} \left (-4-2 x^4+x^8\right )} \, dx \\ & = \frac {\sqrt [4]{2+x^4}}{2 x}-\frac {\left (2+x^4\right )^{5/4}}{10 x^5}-\frac {x^3 \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},-\frac {x^4}{2},\frac {x^4}{1-\sqrt {5}}\right )}{3\ 2^{3/4} \sqrt {5} \left (1-\sqrt {5}\right )}+\frac {x^3 \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {x^4}{1+\sqrt {5}},-\frac {x^4}{2}\right )}{3\ 2^{3/4} \sqrt {5} \left (1+\sqrt {5}\right )}+2 \int \left (-\frac {x^2}{\sqrt {5} \left (2+2 \sqrt {5}-2 x^4\right ) \left (2+x^4\right )^{3/4}}-\frac {x^2}{\sqrt {5} \left (2+x^4\right )^{3/4} \left (-2+2 \sqrt {5}+2 x^4\right )}\right ) \, dx+2 \int \left (-\frac {\left (2+2 \sqrt {5}\right ) x^2}{2 \sqrt {5} \left (2+2 \sqrt {5}-2 x^4\right ) \left (2+x^4\right )^{3/4}}+\frac {\left (-2+2 \sqrt {5}\right ) x^2}{2 \sqrt {5} \left (2+x^4\right )^{3/4} \left (-2+2 \sqrt {5}+2 x^4\right )}\right ) \, dx \\ & = \frac {\sqrt [4]{2+x^4}}{2 x}-\frac {\left (2+x^4\right )^{5/4}}{10 x^5}-\frac {x^3 \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},-\frac {x^4}{2},\frac {x^4}{1-\sqrt {5}}\right )}{3\ 2^{3/4} \sqrt {5} \left (1-\sqrt {5}\right )}+\frac {x^3 \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {x^4}{1+\sqrt {5}},-\frac {x^4}{2}\right )}{3\ 2^{3/4} \sqrt {5} \left (1+\sqrt {5}\right )}-\frac {2 \int \frac {x^2}{\left (2+2 \sqrt {5}-2 x^4\right ) \left (2+x^4\right )^{3/4}} \, dx}{\sqrt {5}}-\frac {2 \int \frac {x^2}{\left (2+x^4\right )^{3/4} \left (-2+2 \sqrt {5}+2 x^4\right )} \, dx}{\sqrt {5}}+\frac {1}{5} \left (2 \left (5-\sqrt {5}\right )\right ) \int \frac {x^2}{\left (2+x^4\right )^{3/4} \left (-2+2 \sqrt {5}+2 x^4\right )} \, dx-\frac {1}{5} \left (2 \left (5+\sqrt {5}\right )\right ) \int \frac {x^2}{\left (2+2 \sqrt {5}-2 x^4\right ) \left (2+x^4\right )^{3/4}} \, dx \\ & = \frac {\sqrt [4]{2+x^4}}{2 x}-\frac {\left (2+x^4\right )^{5/4}}{10 x^5}-\frac {x^3 \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},-\frac {x^4}{2},\frac {x^4}{1-\sqrt {5}}\right )}{3\ 2^{3/4} \sqrt {5} \left (1-\sqrt {5}\right )}+\frac {x^3 \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {x^4}{1+\sqrt {5}},-\frac {x^4}{2}\right )}{3\ 2^{3/4} \sqrt {5} \left (1+\sqrt {5}\right )}-\frac {2 \text {Subst}\left (\int \frac {x^2}{-2+2 \sqrt {5}-\left (-6+2 \sqrt {5}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )}{\sqrt {5}}-\frac {2 \text {Subst}\left (\int \frac {x^2}{2+2 \sqrt {5}-\left (6+2 \sqrt {5}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )}{\sqrt {5}}+\frac {1}{5} \left (2 \left (5-\sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {x^2}{-2+2 \sqrt {5}-\left (-6+2 \sqrt {5}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )-\frac {1}{5} \left (2 \left (5+\sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {x^2}{2+2 \sqrt {5}-\left (6+2 \sqrt {5}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right ) \\ & = \frac {\sqrt [4]{2+x^4}}{2 x}-\frac {\left (2+x^4\right )^{5/4}}{10 x^5}-\frac {x^3 \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},-\frac {x^4}{2},\frac {x^4}{1-\sqrt {5}}\right )}{3\ 2^{3/4} \sqrt {5} \left (1-\sqrt {5}\right )}+\frac {x^3 \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {x^4}{1+\sqrt {5}},-\frac {x^4}{2}\right )}{3\ 2^{3/4} \sqrt {5} \left (1+\sqrt {5}\right )}+\frac {\text {Subst}\left (\int \frac {\sqrt {1+\sqrt {5}}-\sqrt {2} x^2}{-2+2 \sqrt {5}+\left (6-2 \sqrt {5}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )}{\sqrt {10}}-\frac {\text {Subst}\left (\int \frac {\sqrt {1+\sqrt {5}}+\sqrt {2} x^2}{-2+2 \sqrt {5}+\left (6-2 \sqrt {5}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )}{\sqrt {10}}+\frac {\left (1-\sqrt {5}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\sqrt {5}}-\sqrt {2} x^2}{-2+2 \sqrt {5}+\left (6-2 \sqrt {5}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )}{\sqrt {10}}-\frac {\left (1-\sqrt {5}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\sqrt {5}}+\sqrt {2} x^2}{-2+2 \sqrt {5}+\left (6-2 \sqrt {5}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )}{\sqrt {10}}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {-1+\sqrt {5}}-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )}{\sqrt {10} \left (3+\sqrt {5}\right )}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {-1+\sqrt {5}}+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )}{\sqrt {10} \left (3+\sqrt {5}\right )}-\frac {\left (1+\sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+\sqrt {5}}-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )}{\sqrt {10} \left (3+\sqrt {5}\right )}+\frac {\left (1+\sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+\sqrt {5}}+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )}{\sqrt {10} \left (3+\sqrt {5}\right )} \\ & = \frac {\sqrt [4]{2+x^4}}{2 x}-\frac {\left (2+x^4\right )^{5/4}}{10 x^5}-\frac {x^3 \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},-\frac {x^4}{2},\frac {x^4}{1-\sqrt {5}}\right )}{3\ 2^{3/4} \sqrt {5} \left (1-\sqrt {5}\right )}+\frac {x^3 \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {x^4}{1+\sqrt {5}},-\frac {x^4}{2}\right )}{3\ 2^{3/4} \sqrt {5} \left (1+\sqrt {5}\right )}+\frac {\sqrt [4]{-2+\sqrt {5}} \arctan \left (\frac {\sqrt [4]{\frac {2}{-1+\sqrt {5}}} x}{\sqrt [4]{2+x^4}}\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (-29+13 \sqrt {5}\right )} \arctan \left (\frac {\sqrt [4]{\frac {2}{-1+\sqrt {5}}} x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {5}}-\frac {\sqrt [4]{-2+\sqrt {5}} \text {arctanh}\left (\frac {\sqrt [4]{\frac {2}{-1+\sqrt {5}}} x}{\sqrt [4]{2+x^4}}\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (-29+13 \sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt [4]{\frac {2}{-1+\sqrt {5}}} x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {5}}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )}-\sqrt [4]{2 \left (1+\sqrt {5}\right )} x+x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {5} \left (3-\sqrt {5}\right )}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )}+\sqrt [4]{2 \left (1+\sqrt {5}\right )} x+x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {5} \left (3-\sqrt {5}\right )}-\frac {\left (1-\sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )}-\sqrt [4]{2 \left (1+\sqrt {5}\right )} x+x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {5} \left (3-\sqrt {5}\right )}-\frac {\left (1-\sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )}+\sqrt [4]{2 \left (1+\sqrt {5}\right )} x+x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {5} \left (3-\sqrt {5}\right )}+\frac {\sqrt [4]{2+\sqrt {5}} \text {Subst}\left (\int \frac {\sqrt [4]{2 \left (1+\sqrt {5}\right )}+2 x}{-\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )}-\sqrt [4]{2 \left (1+\sqrt {5}\right )} x-x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {10}}+\frac {\sqrt [4]{2+\sqrt {5}} \text {Subst}\left (\int \frac {\sqrt [4]{2 \left (1+\sqrt {5}\right )}-2 x}{-\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )}+\sqrt [4]{2 \left (1+\sqrt {5}\right )} x-x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {10}}-\frac {\sqrt [4]{29+13 \sqrt {5}} \text {Subst}\left (\int \frac {\sqrt [4]{2 \left (1+\sqrt {5}\right )}+2 x}{-\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )}-\sqrt [4]{2 \left (1+\sqrt {5}\right )} x-x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )}{8\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{29+13 \sqrt {5}} \text {Subst}\left (\int \frac {\sqrt [4]{2 \left (1+\sqrt {5}\right )}-2 x}{-\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )}+\sqrt [4]{2 \left (1+\sqrt {5}\right )} x-x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )}{8\ 2^{3/4} \sqrt {5}} \\ & = \frac {\sqrt [4]{2+x^4}}{2 x}-\frac {\left (2+x^4\right )^{5/4}}{10 x^5}-\frac {x^3 \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},-\frac {x^4}{2},\frac {x^4}{1-\sqrt {5}}\right )}{3\ 2^{3/4} \sqrt {5} \left (1-\sqrt {5}\right )}+\frac {x^3 \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {x^4}{1+\sqrt {5}},-\frac {x^4}{2}\right )}{3\ 2^{3/4} \sqrt {5} \left (1+\sqrt {5}\right )}+\frac {\sqrt [4]{-2+\sqrt {5}} \arctan \left (\frac {\sqrt [4]{\frac {2}{-1+\sqrt {5}}} x}{\sqrt [4]{2+x^4}}\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (-29+13 \sqrt {5}\right )} \arctan \left (\frac {\sqrt [4]{\frac {2}{-1+\sqrt {5}}} x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {5}}-\frac {\sqrt [4]{-2+\sqrt {5}} \text {arctanh}\left (\frac {\sqrt [4]{\frac {2}{-1+\sqrt {5}}} x}{\sqrt [4]{2+x^4}}\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (-29+13 \sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt [4]{\frac {2}{-1+\sqrt {5}}} x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {5}}+\frac {\sqrt [4]{2+\sqrt {5}} \log \left (\sqrt {2 \left (1+\sqrt {5}\right )}+\frac {2 x^2}{\sqrt {2+x^4}}-\frac {2 \sqrt [4]{2 \left (1+\sqrt {5}\right )} x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {10}}-\frac {\sqrt [4]{29+13 \sqrt {5}} \log \left (\sqrt {2 \left (1+\sqrt {5}\right )}+\frac {2 x^2}{\sqrt {2+x^4}}-\frac {2 \sqrt [4]{2 \left (1+\sqrt {5}\right )} x}{\sqrt [4]{2+x^4}}\right )}{8\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{2+\sqrt {5}} \log \left (\sqrt {2 \left (1+\sqrt {5}\right )}+\frac {2 x^2}{\sqrt {2+x^4}}+\frac {2 \sqrt [4]{2 \left (1+\sqrt {5}\right )} x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {10}}+\frac {\sqrt [4]{29+13 \sqrt {5}} \log \left (\sqrt {2 \left (1+\sqrt {5}\right )}+\frac {2 x^2}{\sqrt {2+x^4}}+\frac {2 \sqrt [4]{2 \left (1+\sqrt {5}\right )} x}{\sqrt [4]{2+x^4}}\right )}{8\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{2+\sqrt {5}} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {2^{3/4} x}{\sqrt [4]{1+\sqrt {5}} \sqrt [4]{2+x^4}}\right )}{2 \sqrt {10}}-\frac {\sqrt [4]{2+\sqrt {5}} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {2^{3/4} x}{\sqrt [4]{1+\sqrt {5}} \sqrt [4]{2+x^4}}\right )}{2 \sqrt {10}}-\frac {\sqrt [4]{29+13 \sqrt {5}} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {2^{3/4} x}{\sqrt [4]{1+\sqrt {5}} \sqrt [4]{2+x^4}}\right )}{4\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{29+13 \sqrt {5}} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {2^{3/4} x}{\sqrt [4]{1+\sqrt {5}} \sqrt [4]{2+x^4}}\right )}{4\ 2^{3/4} \sqrt {5}} \\ & = \frac {\sqrt [4]{2+x^4}}{2 x}-\frac {\left (2+x^4\right )^{5/4}}{10 x^5}-\frac {x^3 \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},-\frac {x^4}{2},\frac {x^4}{1-\sqrt {5}}\right )}{3\ 2^{3/4} \sqrt {5} \left (1-\sqrt {5}\right )}+\frac {x^3 \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {x^4}{1+\sqrt {5}},-\frac {x^4}{2}\right )}{3\ 2^{3/4} \sqrt {5} \left (1+\sqrt {5}\right )}+\frac {\sqrt [4]{-2+\sqrt {5}} \arctan \left (\frac {\sqrt [4]{\frac {2}{-1+\sqrt {5}}} x}{\sqrt [4]{2+x^4}}\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (-29+13 \sqrt {5}\right )} \arctan \left (\frac {\sqrt [4]{\frac {2}{-1+\sqrt {5}}} x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {5}}-\frac {\sqrt [4]{2+\sqrt {5}} \arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{1+\sqrt {5}} \sqrt [4]{2+x^4}}\right )}{2 \sqrt {10}}+\frac {\sqrt [4]{29+13 \sqrt {5}} \arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{1+\sqrt {5}} \sqrt [4]{2+x^4}}\right )}{4\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{2+\sqrt {5}} \arctan \left (1+\frac {2^{3/4} x}{\sqrt [4]{1+\sqrt {5}} \sqrt [4]{2+x^4}}\right )}{2 \sqrt {10}}-\frac {\sqrt [4]{29+13 \sqrt {5}} \arctan \left (1+\frac {2^{3/4} x}{\sqrt [4]{1+\sqrt {5}} \sqrt [4]{2+x^4}}\right )}{4\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{-2+\sqrt {5}} \text {arctanh}\left (\frac {\sqrt [4]{\frac {2}{-1+\sqrt {5}}} x}{\sqrt [4]{2+x^4}}\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (-29+13 \sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt [4]{\frac {2}{-1+\sqrt {5}}} x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {5}}+\frac {\sqrt [4]{2+\sqrt {5}} \log \left (\sqrt {2 \left (1+\sqrt {5}\right )}+\frac {2 x^2}{\sqrt {2+x^4}}-\frac {2 \sqrt [4]{2 \left (1+\sqrt {5}\right )} x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {10}}-\frac {\sqrt [4]{29+13 \sqrt {5}} \log \left (\sqrt {2 \left (1+\sqrt {5}\right )}+\frac {2 x^2}{\sqrt {2+x^4}}-\frac {2 \sqrt [4]{2 \left (1+\sqrt {5}\right )} x}{\sqrt [4]{2+x^4}}\right )}{8\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{2+\sqrt {5}} \log \left (\sqrt {2 \left (1+\sqrt {5}\right )}+\frac {2 x^2}{\sqrt {2+x^4}}+\frac {2 \sqrt [4]{2 \left (1+\sqrt {5}\right )} x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {10}}+\frac {\sqrt [4]{29+13 \sqrt {5}} \log \left (\sqrt {2 \left (1+\sqrt {5}\right )}+\frac {2 x^2}{\sqrt {2+x^4}}+\frac {2 \sqrt [4]{2 \left (1+\sqrt {5}\right )} x}{\sqrt [4]{2+x^4}}\right )}{8\ 2^{3/4} \sqrt {5}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [4]{2+x^4} \left (-4+x^8\right )}{x^6 \left (-4-2 x^4+x^8\right )} \, dx=\frac {\sqrt [4]{2+x^4} \left (-1+2 x^4\right )}{5 x^5}+\frac {1}{8} \text {RootSum}\left [-1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{2+x^4}-x \text {$\#$1}\right )}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \]
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Result contains higher order function than in optimal. Order 9 vs. order 1.
Time = 0.03 (sec) , antiderivative size = 6505, normalized size of antiderivative = 81.31
\[\text {output too large to display}\]
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 4.40 (sec) , antiderivative size = 1398, normalized size of antiderivative = 17.48 \[ \int \frac {\sqrt [4]{2+x^4} \left (-4+x^8\right )}{x^6 \left (-4-2 x^4+x^8\right )} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\sqrt [4]{2+x^4} \left (-4+x^8\right )}{x^6 \left (-4-2 x^4+x^8\right )} \, dx=\text {Timed out} \]
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Not integrable
Time = 0.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.38 \[ \int \frac {\sqrt [4]{2+x^4} \left (-4+x^8\right )}{x^6 \left (-4-2 x^4+x^8\right )} \, dx=\int { \frac {{\left (x^{8} - 4\right )} {\left (x^{4} + 2\right )}^{\frac {1}{4}}}{{\left (x^{8} - 2 \, x^{4} - 4\right )} x^{6}} \,d x } \]
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Not integrable
Time = 0.35 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.38 \[ \int \frac {\sqrt [4]{2+x^4} \left (-4+x^8\right )}{x^6 \left (-4-2 x^4+x^8\right )} \, dx=\int { \frac {{\left (x^{8} - 4\right )} {\left (x^{4} + 2\right )}^{\frac {1}{4}}}{{\left (x^{8} - 2 \, x^{4} - 4\right )} x^{6}} \,d x } \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.42 \[ \int \frac {\sqrt [4]{2+x^4} \left (-4+x^8\right )}{x^6 \left (-4-2 x^4+x^8\right )} \, dx=-\int \frac {{\left (x^4+2\right )}^{1/4}\,\left (x^8-4\right )}{x^6\,\left (-x^8+2\,x^4+4\right )} \,d x \]
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