Integrand size = 31, antiderivative size = 49 \[ \int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1+x^3-x^6\right )} \, dx=-\frac {1}{3} \text {RootSum}\left [-1+3 \text {$\#$1}^4+\text {$\#$1}^{12}\&,\frac {-\log (x)+\log \left (\sqrt [4]{-x^3+x^5}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]
[Out]
Result contains complex when optimal does not.
Time = 154.53 (sec) , antiderivative size = 18624, normalized size of antiderivative = 380.08, number of steps used = 1044, number of rules used = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.097, Rules used = {2081, 6847, 6860, 252, 251, 6857, 2184, 1452, 441, 440, 525, 524, 1576, 2185, 1505, 1254, 385, 218, 214, 211, 1483, 760, 408, 504, 1231, 226, 1721, 455, 65, 304, 209, 212, 210, 213} \[ \int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1+x^3-x^6\right )} \, dx=\frac {i \sqrt {\frac {2}{-1+\sqrt {5}}} \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {7}{8},\frac {1}{4},1,\frac {15}{8},x^2,\frac {2 i x^2}{\left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right ) x^{5/2}}{21 \sqrt [4]{x^5-x^3}}+\frac {i \sqrt {\frac {2}{-1+\sqrt {5}}} \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {7}{8},\frac {1}{4},1,\frac {15}{8},x^2,-\frac {2 \sqrt [6]{-1} x^2}{\left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right ) x^{5/2}}{21 \sqrt [4]{x^5-x^3}}+\frac {i \sqrt {\frac {2}{-1+\sqrt {5}}} \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {7}{8},\frac {1}{4},1,\frac {15}{8},x^2,-\frac {2 (-1)^{5/6} x^2}{\left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right ) x^{5/2}}{21 \sqrt [4]{x^5-x^3}}+\frac {i \sqrt {\frac {2}{-1+\sqrt {5}}} \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {7}{8},\frac {1}{4},1,\frac {15}{8},x^2,\frac {2 i x^2}{\left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right ) x^{5/2}}{21 \sqrt [4]{x^5-x^3}}+\frac {i \sqrt {\frac {2}{-1+\sqrt {5}}} \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {7}{8},\frac {1}{4},1,\frac {15}{8},x^2,-\frac {2 \sqrt [6]{-1} x^2}{\left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right ) x^{5/2}}{21 \sqrt [4]{x^5-x^3}}+\frac {i \sqrt {\frac {2}{-1+\sqrt {5}}} \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {7}{8},\frac {1}{4},1,\frac {15}{8},x^2,-\frac {2 (-1)^{5/6} x^2}{\left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right ) x^{5/2}}{21 \sqrt [4]{x^5-x^3}}-\frac {2 i \sqrt {\frac {2}{-1+\sqrt {5}}} \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {7}{8},\frac {1}{4},1,\frac {15}{8},x^2,\left (-\frac {2}{-1+\sqrt {5}}\right )^{2/3} x^2\right ) x^{5/2}}{21 \sqrt [4]{x^5-x^3}}-\frac {2 i \sqrt {\frac {2}{-1+\sqrt {5}}} \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {7}{8},\frac {1}{4},1,\frac {15}{8},x^2,\left (\frac {2}{-1+\sqrt {5}}\right )^{2/3} x^2\right ) x^{5/2}}{21 \sqrt [4]{x^5-x^3}}-\frac {2 i \sqrt {\frac {2}{-1+\sqrt {5}}} \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {7}{8},\frac {1}{4},1,\frac {15}{8},x^2,-\sqrt [3]{-1} \left (\frac {2}{-1+\sqrt {5}}\right )^{2/3} x^2\right ) x^{5/2}}{21 \sqrt [4]{x^5-x^3}}+\frac {i \sqrt [3]{1-i} \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{4},1,\frac {7}{4},x^2,\frac {2 i x^2}{\left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right ) x^{9/4}}{9\ 2^{3/4} \left (-1+\sqrt {5}\right )^{5/12} \sqrt [4]{x^5-x^3}}-\frac {\sqrt [6]{-1} \sqrt [3]{1-i} \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{4},1,\frac {7}{4},x^2,-\frac {2 \sqrt [6]{-1} x^2}{\left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right ) x^{9/4}}{9\ 2^{3/4} \left (-1+\sqrt {5}\right )^{5/12} \sqrt [4]{x^5-x^3}}-\frac {i \sqrt [3]{-1+i} \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{4},1,\frac {7}{4},x^2,-\frac {2 (-1)^{5/6} x^2}{\left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right ) x^{9/4}}{9\ 2^{3/4} \left (-1+\sqrt {5}\right )^{5/12} \sqrt [4]{x^5-x^3}}+\frac {i \sqrt [3]{-1+i} \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{4},1,\frac {7}{4},x^2,\frac {2 i x^2}{\left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right ) x^{9/4}}{9\ 2^{3/4} \left (-1+\sqrt {5}\right )^{5/12} \sqrt [4]{x^5-x^3}}-\frac {\sqrt [6]{-1} \sqrt [3]{-1+i} \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{4},1,\frac {7}{4},x^2,-\frac {2 \sqrt [6]{-1} x^2}{\left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right ) x^{9/4}}{9\ 2^{3/4} \left (-1+\sqrt {5}\right )^{5/12} \sqrt [4]{x^5-x^3}}-\frac {(-1)^{5/6} \sqrt [3]{-1+i} \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{4},1,\frac {7}{4},x^2,-\frac {2 (-1)^{5/6} x^2}{\left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right ) x^{9/4}}{9\ 2^{3/4} \left (-1+\sqrt {5}\right )^{5/12} \sqrt [4]{x^5-x^3}}-\frac {\sqrt [6]{-1} \left (1-\sqrt {5}\right )^{2/3} \sqrt [4]{1+\sqrt {5}} \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{4},1,\frac {7}{4},x^2,-\sqrt [3]{-1} \left (\frac {2}{1+\sqrt {5}}\right )^{2/3} x^2\right ) x^{9/4}}{18\ 2^{11/12} \sqrt [4]{x^5-x^3}}+\frac {(-1)^{5/6} \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{4},1,\frac {7}{4},x^2,-\sqrt [3]{-1} \left (\frac {2}{1+\sqrt {5}}\right )^{2/3} x^2\right ) x^{9/4}}{9\ 2^{7/12} \left (1+\sqrt {5}\right )^{5/12} \sqrt [4]{x^5-x^3}}+\frac {i (1-i)^{2/3} \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {5}{8},\frac {1}{4},1,\frac {13}{8},x^2,\frac {2 i x^2}{\left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right ) x^2}{15 \sqrt [3]{-1+\sqrt {5}} \sqrt [4]{x^5-x^3}}-\frac {(-1)^{5/6} (1-i)^{2/3} \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {5}{8},\frac {1}{4},1,\frac {13}{8},x^2,-\frac {2 \sqrt [6]{-1} x^2}{\left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right ) x^2}{15 \sqrt [3]{-1+\sqrt {5}} \sqrt [4]{x^5-x^3}}+\frac {i (-1+i)^{2/3} \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {5}{8},\frac {1}{4},1,\frac {13}{8},x^2,-\frac {2 (-1)^{5/6} x^2}{\left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right ) x^2}{15 \sqrt [3]{-1+\sqrt {5}} \sqrt [4]{x^5-x^3}}+\frac {i (-1+i)^{2/3} \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {5}{8},\frac {1}{4},1,\frac {13}{8},x^2,\frac {2 i x^2}{\left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right ) x^2}{15 \sqrt [3]{-1+\sqrt {5}} \sqrt [4]{x^5-x^3}}-\frac {(-1)^{5/6} (-1+i)^{2/3} \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {5}{8},\frac {1}{4},1,\frac {13}{8},x^2,-\frac {2 \sqrt [6]{-1} x^2}{\left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right ) x^2}{15 \sqrt [3]{-1+\sqrt {5}} \sqrt [4]{x^5-x^3}}-\frac {\sqrt [6]{-1} (-1+i)^{2/3} \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {5}{8},\frac {1}{4},1,\frac {13}{8},x^2,-\frac {2 (-1)^{5/6} x^2}{\left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right ) x^2}{15 \sqrt [3]{-1+\sqrt {5}} \sqrt [4]{x^5-x^3}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{\frac {2}{-1+\sqrt {5}}} \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {5}{8},\frac {1}{4},1,\frac {13}{8},x^2,\left (-\frac {2}{-1+\sqrt {5}}\right )^{2/3} x^2\right ) x^2}{15 \sqrt [4]{x^5-x^3}}-\frac {2 \sqrt [3]{\frac {2}{-1+\sqrt {5}}} \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {5}{8},\frac {1}{4},1,\frac {13}{8},x^2,\left (\frac {2}{-1+\sqrt {5}}\right )^{2/3} x^2\right ) x^2}{15 \sqrt [4]{x^5-x^3}}-\frac {2 (-1)^{2/3} \sqrt [3]{\frac {2}{-1+\sqrt {5}}} \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {5}{8},\frac {1}{4},1,\frac {13}{8},x^2,-\sqrt [3]{-1} \left (\frac {2}{-1+\sqrt {5}}\right )^{2/3} x^2\right ) x^2}{15 \sqrt [4]{x^5-x^3}}-\frac {4 \sqrt [3]{-\frac {2}{1+\sqrt {5}}} \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {5}{8},\frac {1}{4},1,\frac {13}{8},x^2,\left (-\frac {2}{1+\sqrt {5}}\right )^{2/3} x^2\right ) x^2}{15 \sqrt [4]{x^5-x^3}}+\frac {4 \sqrt [3]{\frac {2}{1+\sqrt {5}}} \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {5}{8},\frac {1}{4},1,\frac {13}{8},x^2,\left (\frac {2}{1+\sqrt {5}}\right )^{2/3} x^2\right ) x^2}{15 \sqrt [4]{x^5-x^3}}+\frac {4 (-1)^{2/3} \sqrt [3]{\frac {2}{1+\sqrt {5}}} \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {5}{8},\frac {1}{4},1,\frac {13}{8},x^2,-\sqrt [3]{-1} \left (\frac {2}{1+\sqrt {5}}\right )^{2/3} x^2\right ) x^2}{15 \sqrt [4]{x^5-x^3}}+\frac {\sqrt {2} \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {3}{8},\frac {1}{4},1,\frac {11}{8},x^2,\frac {2 i x^2}{\left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right ) x^{3/2}}{9 (1-i)^{2/3} \sqrt [6]{-1+\sqrt {5}} \sqrt [4]{x^5-x^3}}+\frac {(-1)^{2/3} \sqrt {2} \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {3}{8},\frac {1}{4},1,\frac {11}{8},x^2,-\frac {2 \sqrt [6]{-1} x^2}{\left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right ) x^{3/2}}{9 (1-i)^{2/3} \sqrt [6]{-1+\sqrt {5}} \sqrt [4]{x^5-x^3}}-\frac {\sqrt [3]{-1} \sqrt {2} \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {3}{8},\frac {1}{4},1,\frac {11}{8},x^2,-\frac {2 (-1)^{5/6} x^2}{\left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right ) x^{3/2}}{9 (1-i)^{2/3} \sqrt [6]{-1+\sqrt {5}} \sqrt [4]{x^5-x^3}}+\frac {\sqrt {2} \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {3}{8},\frac {1}{4},1,\frac {11}{8},x^2,\frac {2 i x^2}{\left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right ) x^{3/2}}{9 (-1+i)^{2/3} \sqrt [6]{-1+\sqrt {5}} \sqrt [4]{x^5-x^3}}-\frac {(-1+i)^{4/3} \sqrt [6]{-\frac {1}{-1+\sqrt {5}}} \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {3}{8},\frac {1}{4},1,\frac {11}{8},x^2,-\frac {2 \sqrt [6]{-1} x^2}{\left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right ) x^{3/2}}{9 \sqrt {2} \sqrt [4]{x^5-x^3}}-\frac {\sqrt [3]{-1} \sqrt {2} \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {3}{8},\frac {1}{4},1,\frac {11}{8},x^2,-\frac {2 (-1)^{5/6} x^2}{\left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right ) x^{3/2}}{9 (-1+i)^{2/3} \sqrt [6]{-1+\sqrt {5}} \sqrt [4]{x^5-x^3}}-\frac {\left (\frac {1}{9}-\frac {i}{9}\right ) (-1)^{5/12} 2^{2/3} \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {3}{8},\frac {1}{4},1,\frac {11}{8},x^2,\left (-\frac {2}{-1+\sqrt {5}}\right )^{2/3} x^2\right ) x^{3/2}}{\sqrt [6]{-1+\sqrt {5}} \sqrt [4]{x^5-x^3}}+\frac {2 i \sqrt [6]{\frac {2}{-1+\sqrt {5}}} \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {3}{8},\frac {1}{4},1,\frac {11}{8},x^2,\left (\frac {2}{-1+\sqrt {5}}\right )^{2/3} x^2\right ) x^{3/2}}{9 \sqrt [4]{x^5-x^3}}-\frac {2 (-1)^{5/6} \sqrt [6]{\frac {2}{-1+\sqrt {5}}} \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {3}{8},\frac {1}{4},1,\frac {11}{8},x^2,-\sqrt [3]{-1} \left (\frac {2}{-1+\sqrt {5}}\right )^{2/3} x^2\right ) x^{3/2}}{9 \sqrt [4]{x^5-x^3}}+\frac {\sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {1}{8},\frac {1}{4},1,\frac {9}{8},x^2,\frac {2 i x^2}{\left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right ) x}{3 \sqrt [4]{x^5-x^3}}+\frac {\sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {1}{8},\frac {1}{4},1,\frac {9}{8},x^2,-\frac {2 \sqrt [6]{-1} x^2}{\left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right ) x}{3 \sqrt [4]{x^5-x^3}}+\frac {\sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {1}{8},\frac {1}{4},1,\frac {9}{8},x^2,-\frac {2 (-1)^{5/6} x^2}{\left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right ) x}{3 \sqrt [4]{x^5-x^3}}+\frac {\sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {1}{8},\frac {1}{4},1,\frac {9}{8},x^2,\frac {2 i x^2}{\left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right ) x}{3 \sqrt [4]{x^5-x^3}}+\frac {\sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {1}{8},\frac {1}{4},1,\frac {9}{8},x^2,-\frac {2 \sqrt [6]{-1} x^2}{\left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right ) x}{3 \sqrt [4]{x^5-x^3}}+\frac {\sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {1}{8},\frac {1}{4},1,\frac {9}{8},x^2,-\frac {2 (-1)^{5/6} x^2}{\left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right ) x}{3 \sqrt [4]{x^5-x^3}}+\frac {2 \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {1}{8},\frac {1}{4},1,\frac {9}{8},x^2,\left (-\frac {2}{-1+\sqrt {5}}\right )^{2/3} x^2\right ) x}{3 \sqrt [4]{x^5-x^3}}+\frac {2 \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {1}{8},\frac {1}{4},1,\frac {9}{8},x^2,\left (\frac {2}{-1+\sqrt {5}}\right )^{2/3} x^2\right ) x}{3 \sqrt [4]{x^5-x^3}}+\frac {2 \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {1}{8},\frac {1}{4},1,\frac {9}{8},x^2,-\sqrt [3]{-1} \left (\frac {2}{-1+\sqrt {5}}\right )^{2/3} x^2\right ) x}{3 \sqrt [4]{x^5-x^3}}+\frac {4 \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {1}{8},\frac {1}{4},1,\frac {9}{8},x^2,\left (-\frac {2}{1+\sqrt {5}}\right )^{2/3} x^2\right ) x}{3 \sqrt [4]{x^5-x^3}}+\frac {4 \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {1}{8},\frac {1}{4},1,\frac {9}{8},x^2,\left (\frac {2}{1+\sqrt {5}}\right )^{2/3} x^2\right ) x}{3 \sqrt [4]{x^5-x^3}}+\frac {4 \sqrt [4]{1-x^2} \operatorname {AppellF1}\left (\frac {1}{8},\frac {1}{4},1,\frac {9}{8},x^2,-\sqrt [3]{-1} \left (\frac {2}{1+\sqrt {5}}\right )^{2/3} x^2\right ) x}{3 \sqrt [4]{x^5-x^3}}-\frac {4 \sqrt [4]{1-x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{8},\frac {1}{4},\frac {9}{8},x^2\right ) x}{\sqrt [4]{x^5-x^3}}+\frac {\left (\frac {1}{12}-\frac {i}{12}\right ) \sqrt [3]{-1} \sqrt [12]{1+\sqrt {5}} \sqrt [4]{x^2-1} \arctan \left (\frac {\sqrt [12]{-1} \sqrt [4]{2^{2/3}-\left (-1-\sqrt {5}\right )^{2/3}} \sqrt {x}}{\sqrt [6]{1+\sqrt {5}} \sqrt [4]{x^2-1}}\right ) x^{3/4}}{2^{5/12} \sqrt [4]{2^{2/3}-\left (-1-\sqrt {5}\right )^{2/3}} \sqrt [4]{x^5-x^3}}+\frac {\sqrt [12]{-1+\sqrt {5}} \sqrt [4]{x^2-1} \arctan \left (\frac {\sqrt [4]{-2 i+\left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt {x}}{\sqrt [6]{(-1+i) \left (1-\sqrt {5}\right )} \sqrt [4]{x^2-1}}\right ) x^{3/4}}{6 \sqrt [6]{1-i} 2^{3/4} \sqrt [4]{-2 i+\left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt [4]{x^5-x^3}}+\frac {(-1)^{2/3} \sqrt [12]{-1+\sqrt {5}} \sqrt [4]{x^2-1} \arctan \left (\frac {\sqrt [4]{2 (-1)^{5/6}+\left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt {x}}{\sqrt [6]{(-1+i) \left (1-\sqrt {5}\right )} \sqrt [4]{x^2-1}}\right ) x^{3/4}}{6 \sqrt [6]{1-i} 2^{3/4} \sqrt [4]{2 (-1)^{5/6}+\left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt [4]{x^5-x^3}}-\frac {\sqrt [3]{-1} \sqrt [12]{-1+\sqrt {5}} \sqrt [4]{x^2-1} \arctan \left (\frac {\sqrt [4]{i+\sqrt {3}+\left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt {x}}{\sqrt [6]{(-1+i) \left (1-\sqrt {5}\right )} \sqrt [4]{x^2-1}}\right ) x^{3/4}}{6 \sqrt [6]{1-i} 2^{3/4} \sqrt [4]{i+\sqrt {3}+\left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt [4]{x^5-x^3}}+\frac {\sqrt [12]{-1+\sqrt {5}} \sqrt [4]{x^2-1} \arctan \left (\frac {\sqrt [4]{-2 i+\left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt {x}}{\sqrt [6]{(1-i) \left (1-\sqrt {5}\right )} \sqrt [4]{x^2-1}}\right ) x^{3/4}}{6 \sqrt [6]{-1+i} 2^{3/4} \sqrt [4]{-2 i+\left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt [4]{x^5-x^3}}+\frac {i \sqrt [6]{1+i} \sqrt [12]{-1+\sqrt {5}} \sqrt [4]{x^2-1} \arctan \left (\frac {\sqrt [4]{2 (-1)^{5/6}+\left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt {x}}{\sqrt [6]{(1-i) \left (1-\sqrt {5}\right )} \sqrt [4]{x^2-1}}\right ) x^{3/4}}{6\ 2^{11/12} \sqrt [4]{2 (-1)^{5/6}+\left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt [4]{x^5-x^3}}-\frac {\sqrt [3]{-1} \sqrt [12]{-1+\sqrt {5}} \sqrt [4]{x^2-1} \arctan \left (\frac {\sqrt [4]{i+\sqrt {3}+\left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt {x}}{\sqrt [6]{(1-i) \left (1-\sqrt {5}\right )} \sqrt [4]{x^2-1}}\right ) x^{3/4}}{6 \sqrt [6]{-1+i} 2^{3/4} \sqrt [4]{i+\sqrt {3}+\left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt [4]{x^5-x^3}}-\frac {\sqrt [6]{-1} \sqrt [12]{1+\sqrt {5}} \sqrt [4]{x^2-1} \arctan \left (\frac {\sqrt [4]{\sqrt [3]{-1} 2^{2/3}+\left (1+\sqrt {5}\right )^{2/3}} \sqrt {x}}{\sqrt [6]{1+\sqrt {5}} \sqrt [4]{x^2-1}}\right ) x^{3/4}}{6\ 2^{11/12} \sqrt [4]{\sqrt [3]{-1} 2^{2/3}+\left (1+\sqrt {5}\right )^{2/3}} \sqrt [4]{x^5-x^3}}-\frac {\left (\frac {1}{12}-\frac {i}{12}\right ) \sqrt [6]{-1} \sqrt [12]{1+\sqrt {5}} \sqrt [4]{x^2-1} \arctan \left (\frac {(-1)^{11/12} \sqrt [4]{2^{2/3}+\sqrt [3]{-1} \left (1+\sqrt {5}\right )^{2/3}} \sqrt {x}}{\sqrt [6]{1+\sqrt {5}} \sqrt [4]{x^2-1}}\right ) x^{3/4}}{2^{5/12} \sqrt [4]{2^{2/3}+\sqrt [3]{-1} \left (1+\sqrt {5}\right )^{2/3}} \sqrt [4]{x^5-x^3}}-\frac {(-1)^{11/12} \sqrt [12]{1+\sqrt {5}} \sqrt [4]{x^2-1} \arctan \left (\frac {(-1)^{11/12} \sqrt [4]{2^{2/3}+\sqrt [3]{-1} \left (1+\sqrt {5}\right )^{2/3}} \sqrt {x}}{\sqrt [6]{1+\sqrt {5}} \sqrt [4]{x^2-1}}\right ) x^{3/4}}{6\ 2^{11/12} \sqrt [4]{2^{2/3}+\sqrt [3]{-1} \left (1+\sqrt {5}\right )^{2/3}} \sqrt [4]{x^5-x^3}}-\frac {\sqrt [3]{1-i} \sqrt [12]{-1+\sqrt {5}} \sqrt [4]{x^2-1} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x^2-1}}{\sqrt [4]{-2-i \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}}}\right ) x^{3/4}}{12 \sqrt [4]{-2-i \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt [4]{x^5-x^3}}+\frac {\sqrt [3]{-1+i} \sqrt [12]{-1+\sqrt {5}} \sqrt [4]{x^2-1} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x^2-1}}{\sqrt [4]{-2+\sqrt [6]{-1} \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}}}\right ) x^{3/4}}{12 \sqrt [4]{-2+\sqrt [6]{-1} \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt [4]{x^5-x^3}}-\frac {(-1)^{2/3} \sqrt [3]{1-i} \sqrt [12]{-1+\sqrt {5}} \sqrt [4]{x^2-1} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x^2-1}}{\sqrt [4]{-2+(-1)^{5/6} \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}}}\right ) x^{3/4}}{12 \sqrt [4]{-2+(-1)^{5/6} \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt [4]{x^5-x^3}}+\frac {i \sqrt [3]{-1+i} \sqrt [12]{-1+\sqrt {5}} \sqrt [4]{-\frac {i}{2 i-\left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}}} \sqrt [4]{x^2-1} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x^2-1}}{\sqrt [4]{-2-i \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}}}\right ) x^{3/4}}{12 \sqrt [4]{x^5-x^3}}+\frac {\sqrt [3]{-1} \sqrt [3]{-1+i} \sqrt [12]{-1+\sqrt {5}} \sqrt [4]{x^2-1} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x^2-1}}{\sqrt [4]{-2+\sqrt [6]{-1} \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}}}\right ) x^{3/4}}{12 \sqrt [4]{-2+\sqrt [6]{-1} \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt [4]{x^5-x^3}}-\frac {(-1)^{2/3} \sqrt [3]{-1+i} \sqrt [12]{-1+\sqrt {5}} \sqrt [4]{x^2-1} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x^2-1}}{\sqrt [4]{-2+(-1)^{5/6} \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}}}\right ) x^{3/4}}{12 \sqrt [4]{-2+(-1)^{5/6} \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt [4]{x^5-x^3}}+\frac {\left (\frac {1}{12}-\frac {i}{12}\right ) \sqrt [3]{-1} \sqrt [12]{1+\sqrt {5}} \sqrt [4]{x^2-1} \text {arctanh}\left (\frac {\sqrt [12]{-1} \sqrt [4]{2^{2/3}-\left (-1-\sqrt {5}\right )^{2/3}} \sqrt {x}}{\sqrt [6]{1+\sqrt {5}} \sqrt [4]{x^2-1}}\right ) x^{3/4}}{2^{5/12} \sqrt [4]{2^{2/3}-\left (-1-\sqrt {5}\right )^{2/3}} \sqrt [4]{x^5-x^3}}+\frac {\sqrt [12]{-1+\sqrt {5}} \sqrt [4]{x^2-1} \text {arctanh}\left (\frac {\sqrt [4]{-2 i+\left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt {x}}{\sqrt [6]{(-1+i) \left (1-\sqrt {5}\right )} \sqrt [4]{x^2-1}}\right ) x^{3/4}}{6 \sqrt [6]{1-i} 2^{3/4} \sqrt [4]{-2 i+\left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt [4]{x^5-x^3}}+\frac {(-1)^{2/3} \sqrt [12]{-1+\sqrt {5}} \sqrt [4]{x^2-1} \text {arctanh}\left (\frac {\sqrt [4]{2 (-1)^{5/6}+\left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt {x}}{\sqrt [6]{(-1+i) \left (1-\sqrt {5}\right )} \sqrt [4]{x^2-1}}\right ) x^{3/4}}{6 \sqrt [6]{1-i} 2^{3/4} \sqrt [4]{2 (-1)^{5/6}+\left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt [4]{x^5-x^3}}-\frac {\sqrt [3]{-1} \sqrt [12]{-1+\sqrt {5}} \sqrt [4]{x^2-1} \text {arctanh}\left (\frac {\sqrt [4]{i+\sqrt {3}+\left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt {x}}{\sqrt [6]{(-1+i) \left (1-\sqrt {5}\right )} \sqrt [4]{x^2-1}}\right ) x^{3/4}}{6 \sqrt [6]{1-i} 2^{3/4} \sqrt [4]{i+\sqrt {3}+\left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt [4]{x^5-x^3}}+\frac {\sqrt [12]{-1+\sqrt {5}} \sqrt [4]{x^2-1} \text {arctanh}\left (\frac {\sqrt [4]{-2 i+\left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt {x}}{\sqrt [6]{(1-i) \left (1-\sqrt {5}\right )} \sqrt [4]{x^2-1}}\right ) x^{3/4}}{6 \sqrt [6]{-1+i} 2^{3/4} \sqrt [4]{-2 i+\left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt [4]{x^5-x^3}}+\frac {i \sqrt [6]{1+i} \sqrt [12]{-1+\sqrt {5}} \sqrt [4]{x^2-1} \text {arctanh}\left (\frac {\sqrt [4]{2 (-1)^{5/6}+\left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt {x}}{\sqrt [6]{(1-i) \left (1-\sqrt {5}\right )} \sqrt [4]{x^2-1}}\right ) x^{3/4}}{6\ 2^{11/12} \sqrt [4]{2 (-1)^{5/6}+\left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt [4]{x^5-x^3}}-\frac {\sqrt [3]{-1} \sqrt [12]{-1+\sqrt {5}} \sqrt [4]{x^2-1} \text {arctanh}\left (\frac {\sqrt [4]{i+\sqrt {3}+\left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt {x}}{\sqrt [6]{(1-i) \left (1-\sqrt {5}\right )} \sqrt [4]{x^2-1}}\right ) x^{3/4}}{6 \sqrt [6]{-1+i} 2^{3/4} \sqrt [4]{i+\sqrt {3}+\left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt [4]{x^5-x^3}}-\frac {\sqrt [6]{-1} \sqrt [12]{1+\sqrt {5}} \sqrt [4]{x^2-1} \text {arctanh}\left (\frac {\sqrt [4]{\sqrt [3]{-1} 2^{2/3}+\left (1+\sqrt {5}\right )^{2/3}} \sqrt {x}}{\sqrt [6]{1+\sqrt {5}} \sqrt [4]{x^2-1}}\right ) x^{3/4}}{6\ 2^{11/12} \sqrt [4]{\sqrt [3]{-1} 2^{2/3}+\left (1+\sqrt {5}\right )^{2/3}} \sqrt [4]{x^5-x^3}}-\frac {\left (\frac {1}{12}-\frac {i}{12}\right ) \sqrt [6]{-1} \sqrt [12]{1+\sqrt {5}} \sqrt [4]{x^2-1} \text {arctanh}\left (\frac {(-1)^{11/12} \sqrt [4]{2^{2/3}+\sqrt [3]{-1} \left (1+\sqrt {5}\right )^{2/3}} \sqrt {x}}{\sqrt [6]{1+\sqrt {5}} \sqrt [4]{x^2-1}}\right ) x^{3/4}}{2^{5/12} \sqrt [4]{2^{2/3}+\sqrt [3]{-1} \left (1+\sqrt {5}\right )^{2/3}} \sqrt [4]{x^5-x^3}}-\frac {(-1)^{11/12} \sqrt [12]{1+\sqrt {5}} \sqrt [4]{x^2-1} \text {arctanh}\left (\frac {(-1)^{11/12} \sqrt [4]{2^{2/3}+\sqrt [3]{-1} \left (1+\sqrt {5}\right )^{2/3}} \sqrt {x}}{\sqrt [6]{1+\sqrt {5}} \sqrt [4]{x^2-1}}\right ) x^{3/4}}{6\ 2^{11/12} \sqrt [4]{2^{2/3}+\sqrt [3]{-1} \left (1+\sqrt {5}\right )^{2/3}} \sqrt [4]{x^5-x^3}}+\frac {\sqrt [3]{1-i} \sqrt [12]{-1+\sqrt {5}} \sqrt [4]{x^2-1} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x^2-1}}{\sqrt [4]{-2-i \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}}}\right ) x^{3/4}}{12 \sqrt [4]{-2-i \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt [4]{x^5-x^3}}-\frac {\sqrt [3]{-1+i} \sqrt [12]{-1+\sqrt {5}} \sqrt [4]{x^2-1} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x^2-1}}{\sqrt [4]{-2+\sqrt [6]{-1} \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}}}\right ) x^{3/4}}{12 \sqrt [4]{-2+\sqrt [6]{-1} \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt [4]{x^5-x^3}}+\frac {(-1)^{2/3} \sqrt [3]{1-i} \sqrt [12]{-1+\sqrt {5}} \sqrt [4]{x^2-1} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x^2-1}}{\sqrt [4]{-2+(-1)^{5/6} \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}}}\right ) x^{3/4}}{12 \sqrt [4]{-2+(-1)^{5/6} \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt [4]{x^5-x^3}}-\frac {i \sqrt [3]{-1+i} \sqrt [12]{-1+\sqrt {5}} \sqrt [4]{-\frac {i}{2 i-\left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}}} \sqrt [4]{x^2-1} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x^2-1}}{\sqrt [4]{-2-i \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}}}\right ) x^{3/4}}{12 \sqrt [4]{x^5-x^3}}-\frac {\sqrt [3]{-1} \sqrt [3]{-1+i} \sqrt [12]{-1+\sqrt {5}} \sqrt [4]{x^2-1} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x^2-1}}{\sqrt [4]{-2+\sqrt [6]{-1} \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}}}\right ) x^{3/4}}{12 \sqrt [4]{-2+\sqrt [6]{-1} \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt [4]{x^5-x^3}}+\frac {(-1)^{2/3} \sqrt [3]{-1+i} \sqrt [12]{-1+\sqrt {5}} \sqrt [4]{x^2-1} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x^2-1}}{\sqrt [4]{-2+(-1)^{5/6} \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}}}\right ) x^{3/4}}{12 \sqrt [4]{-2+(-1)^{5/6} \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt [4]{x^5-x^3}}-\frac {i \sqrt [3]{1-i} \sqrt [12]{-1+\sqrt {5}} \sqrt {x^2} \sqrt [4]{x^2-1} \arctan \left (\frac {(1+i) \sqrt [3]{(-1+i) \left (1-\sqrt {5}\right )} \sqrt [4]{x^2-1}}{2^{3/4} \sqrt [4]{-2-i \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt {x^2}}\right )}{24 \sqrt [4]{-2-i \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt [4]{x^5-x^3} \sqrt [4]{x}}+\frac {\sqrt [12]{-1+\sqrt {5}} \left (2 i-2 \sqrt {3}+\left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}\right ) \sqrt {x^2} \sqrt [4]{x^2-1} \arctan \left (\frac {\sqrt [12]{-1} \sqrt [3]{(-1+i) \left (1-\sqrt {5}\right )} \sqrt [4]{x^2-1}}{\sqrt [4]{2} \sqrt [4]{-2+\sqrt [6]{-1} \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt {x^2}}\right )}{12 (1-i)^{2/3} \sqrt {2} \sqrt [4]{2-\sqrt [6]{-1} \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}} \left (4-\sqrt [6]{-1} \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}\right ) \sqrt [4]{x^5-x^3} \sqrt [4]{x}}-\frac {(-1)^{7/12} \sqrt [12]{-1+\sqrt {5}} \sqrt {x^2} \sqrt [4]{x^2-1} \arctan \left (\frac {(-1)^{7/12} \sqrt [3]{(-1+i) \left (1-\sqrt {5}\right )} \sqrt [4]{x^2-1}}{\sqrt [4]{2} \sqrt [4]{-2+\sqrt [6]{-1} \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt {x^2}}\right )}{12 (1-i)^{2/3} \sqrt {2} \sqrt [4]{-2+\sqrt [6]{-1} \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt [4]{x^5-x^3} \sqrt [4]{x}}+\frac {\sqrt [3]{1-i} \sqrt [12]{-1+\sqrt {5}} \left (2-i \left (2 \sqrt {3}+\left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}\right )\right ) \sqrt {x^2} \sqrt [4]{x^2-1} \arctan \left (\frac {(-1)^{5/12} \sqrt [3]{(-1+i) \left (1-\sqrt {5}\right )} \sqrt [4]{x^2-1}}{\sqrt [4]{2} \sqrt [4]{-2+(-1)^{5/6} \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt {x^2}}\right )}{24 \left (4-(-1)^{5/6} \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}\right ) \sqrt [4]{-2+(-1)^{5/6} \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt [4]{x^5-x^3} \sqrt [4]{x}}-\frac {(-1)^{11/12} \sqrt [12]{-1+\sqrt {5}} \sqrt {x^2} \sqrt [4]{x^2-1} \arctan \left (\frac {(-1)^{11/12} \sqrt [3]{(-1+i) \left (1-\sqrt {5}\right )} \sqrt [4]{x^2-1}}{\sqrt [4]{2} \sqrt [4]{-2+(-1)^{5/6} \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt {x^2}}\right )}{12 (1-i)^{2/3} \sqrt {2} \sqrt [4]{-2+(-1)^{5/6} \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt [4]{x^5-x^3} \sqrt [4]{x}}+\frac {\sqrt [3]{-1+i} \sqrt [12]{-1+\sqrt {5}} \sqrt [4]{-\frac {i}{2 i-\left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}}} \sqrt {x^2} \sqrt [4]{x^2-1} \arctan \left (\frac {(1+i) \sqrt [3]{(1-i) \left (1-\sqrt {5}\right )} \sqrt [4]{x^2-1}}{2^{3/4} \sqrt [4]{-2-i \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt {x^2}}\right )}{24 \sqrt [4]{x^5-x^3} \sqrt [4]{x}}+\frac {\sqrt [12]{-1+\sqrt {5}} \left (2 i-2 \sqrt {3}+\left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}\right ) \sqrt {x^2} \sqrt [4]{x^2-1} \arctan \left (\frac {\sqrt [12]{-1} \sqrt [3]{(1-i) \left (1-\sqrt {5}\right )} \sqrt [4]{x^2-1}}{\sqrt [4]{2} \sqrt [4]{-2+\sqrt [6]{-1} \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt {x^2}}\right )}{12 (-1+i)^{2/3} \sqrt {2} \sqrt [4]{2-\sqrt [6]{-1} \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}} \left (4-\sqrt [6]{-1} \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}\right ) \sqrt [4]{x^5-x^3} \sqrt [4]{x}}-\frac {(-1)^{7/12} \sqrt [12]{-1+\sqrt {5}} \sqrt {x^2} \sqrt [4]{x^2-1} \arctan \left (\frac {(-1)^{7/12} \sqrt [3]{(1-i) \left (1-\sqrt {5}\right )} \sqrt [4]{x^2-1}}{\sqrt [4]{2} \sqrt [4]{-2+\sqrt [6]{-1} \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt {x^2}}\right )}{12 (-1+i)^{2/3} \sqrt {2} \sqrt [4]{-2+\sqrt [6]{-1} \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt [4]{x^5-x^3} \sqrt [4]{x}}-\frac {\sqrt [3]{-1+i} \sqrt [12]{-1+\sqrt {5}} \left (2-i \left (2 \sqrt {3}+\left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}\right )\right ) \sqrt {x^2} \sqrt [4]{x^2-1} \arctan \left (\frac {(-1)^{5/12} \sqrt [3]{(1-i) \left (1-\sqrt {5}\right )} \sqrt [4]{x^2-1}}{\sqrt [4]{2} \sqrt [4]{-2+(-1)^{5/6} \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt {x^2}}\right )}{24 \left (4-(-1)^{5/6} \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}\right ) \sqrt [4]{-2+(-1)^{5/6} \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt [4]{x^5-x^3} \sqrt [4]{x}}-\frac {(-1)^{11/12} \sqrt [12]{-1+\sqrt {5}} \sqrt {x^2} \sqrt [4]{x^2-1} \arctan \left (\frac {(-1)^{11/12} \sqrt [3]{(1-i) \left (1-\sqrt {5}\right )} \sqrt [4]{x^2-1}}{\sqrt [4]{2} \sqrt [4]{-2+(-1)^{5/6} \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt {x^2}}\right )}{12 (-1+i)^{2/3} \sqrt {2} \sqrt [4]{-2+(-1)^{5/6} \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt [4]{x^5-x^3} \sqrt [4]{x}}+\frac {i \sqrt [3]{-1+i} \sqrt [12]{-1+\sqrt {5}} \sqrt [4]{-\frac {i}{2 i-\left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}}} \sqrt {x^2} \sqrt [4]{x^2-1} \arctan \left (\frac {(-1)^{3/4} \sqrt [3]{(1-i) \left (1-\sqrt {5}\right )} \sqrt [4]{x^2-1}}{\sqrt [4]{-4+(-1+i)^{8/3} \left (-1+\sqrt {5}\right )^{2/3}} \sqrt {x^2}}\right )}{24 \sqrt [4]{x^5-x^3} \sqrt [4]{x}}+\frac {\sqrt [3]{1-i} \sqrt [12]{-1+\sqrt {5}} \sqrt {x^2} \sqrt [4]{x^2-1} \arctan \left (\frac {(-1)^{3/4} \sqrt [3]{(-1+i) \left (1-\sqrt {5}\right )} \sqrt [4]{x^2-1}}{\sqrt [4]{-4+(1-i)^{8/3} \left (-1+\sqrt {5}\right )^{2/3}} \sqrt {x^2}}\right )}{24 \sqrt [4]{-2-i \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt [4]{x^5-x^3} \sqrt [4]{x}}-\frac {\sqrt [4]{-1} \left (-1+\sqrt {5}\right )^{5/12} \left (2+\sqrt {-4+(1-i)^{8/3} \left (-1+\sqrt {5}\right )^{2/3}}\right ) \sqrt [4]{x^2-1} \sqrt {\frac {x^2}{\left (\sqrt {x^2-1}+1\right )^2}} \left (\sqrt {x^2-1}+1\right ) \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{x^2-1}\right ),\frac {1}{2}\right )}{24\ 2^{3/4} \left (2 (1-i)^{4/3}+\left (-1+\sqrt {5}\right )^{2/3}\right ) \sqrt [4]{x^5-x^3} \sqrt [4]{x}}-\frac {\sqrt [4]{-1} \left (-1+\sqrt {5}\right )^{5/12} \left (2-\sqrt {-4+(1-i)^{8/3} \left (-1+\sqrt {5}\right )^{2/3}}\right ) \sqrt [4]{x^2-1} \sqrt {\frac {x^2}{\left (\sqrt {x^2-1}+1\right )^2}} \left (\sqrt {x^2-1}+1\right ) \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{x^2-1}\right ),\frac {1}{2}\right )}{24\ 2^{3/4} \left (2 (1-i)^{4/3}+\left (-1+\sqrt {5}\right )^{2/3}\right ) \sqrt [4]{x^5-x^3} \sqrt [4]{x}}+\frac {\sqrt [4]{-1} \left (-1+\sqrt {5}\right )^{5/12} \left (2+\sqrt {-4+(-1+i)^{8/3} \left (-1+\sqrt {5}\right )^{2/3}}\right ) \sqrt [4]{x^2-1} \sqrt {\frac {x^2}{\left (\sqrt {x^2-1}+1\right )^2}} \left (\sqrt {x^2-1}+1\right ) \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{x^2-1}\right ),\frac {1}{2}\right )}{24\ 2^{3/4} \left (2 (-1+i)^{4/3}+\left (-1+\sqrt {5}\right )^{2/3}\right ) \sqrt [4]{x^5-x^3} \sqrt [4]{x}}+\frac {\sqrt [4]{-1} \left (-1+\sqrt {5}\right )^{5/12} \left (2-\sqrt {-4+(-1+i)^{8/3} \left (-1+\sqrt {5}\right )^{2/3}}\right ) \sqrt [4]{x^2-1} \sqrt {\frac {x^2}{\left (\sqrt {x^2-1}+1\right )^2}} \left (\sqrt {x^2-1}+1\right ) \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{x^2-1}\right ),\frac {1}{2}\right )}{24\ 2^{3/4} \left (2 (-1+i)^{4/3}+\left (-1+\sqrt {5}\right )^{2/3}\right ) \sqrt [4]{x^5-x^3} \sqrt [4]{x}}-\frac {(1-i)^{5/3} \left (-1+\sqrt {5}\right )^{5/12} \left (2+\sqrt {2 \left (-2+(-1)^{5/6} \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}\right )}\right ) \sqrt [4]{x^2-1} \sqrt {\frac {x^2}{\left (\sqrt {x^2-1}+1\right )^2}} \left (\sqrt {x^2-1}+1\right ) \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{x^2-1}\right ),\frac {1}{2}\right )}{48 \sqrt [4]{2} \left (4-(-1)^{5/6} \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}\right ) \sqrt [4]{x^5-x^3} \sqrt [4]{x}}-\frac {(1-i)^{5/3} \left (-1+\sqrt {5}\right )^{5/12} \left (2-\sqrt {2 \left (-2+(-1)^{5/6} \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}\right )}\right ) \sqrt [4]{x^2-1} \sqrt {\frac {x^2}{\left (\sqrt {x^2-1}+1\right )^2}} \left (\sqrt {x^2-1}+1\right ) \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{x^2-1}\right ),\frac {1}{2}\right )}{48 \sqrt [4]{2} \left (4-(-1)^{5/6} \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}\right ) \sqrt [4]{x^5-x^3} \sqrt [4]{x}}+\frac {(-1)^{2/3} (-1+i)^{5/3} \left (-1+\sqrt {5}\right )^{5/12} \left (2+\sqrt {2 \left (-2+\sqrt [6]{-1} \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}\right )}\right ) \sqrt [4]{x^2-1} \sqrt {\frac {x^2}{\left (\sqrt {x^2-1}+1\right )^2}} \left (\sqrt {x^2-1}+1\right ) \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{x^2-1}\right ),\frac {1}{2}\right )}{48 \sqrt [4]{2} \left (4-\sqrt [6]{-1} \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}\right ) \sqrt [4]{x^5-x^3} \sqrt [4]{x}}+\frac {(-1)^{2/3} (-1+i)^{5/3} \left (-1+\sqrt {5}\right )^{5/12} \left (2-\sqrt {2 \left (-2+\sqrt [6]{-1} \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}\right )}\right ) \sqrt [4]{x^2-1} \sqrt {\frac {x^2}{\left (\sqrt {x^2-1}+1\right )^2}} \left (\sqrt {x^2-1}+1\right ) \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{x^2-1}\right ),\frac {1}{2}\right )}{48 \sqrt [4]{2} \left (4-\sqrt [6]{-1} \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}\right ) \sqrt [4]{x^5-x^3} \sqrt [4]{x}}+\frac {(-1)^{5/6} \left (-1+\sqrt {5}\right )^{5/12} \left (2+\sqrt {2 \left (-2+(-1)^{5/6} \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}\right )}\right ) \sqrt [4]{x^2-1} \sqrt {\frac {x^2}{\left (\sqrt {x^2-1}+1\right )^2}} \left (\sqrt {x^2-1}+1\right ) \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{x^2-1}\right ),\frac {1}{2}\right )}{24 \sqrt [3]{1-i} \sqrt [4]{2} \left (4-(-1)^{5/6} \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}\right ) \sqrt [4]{x^5-x^3} \sqrt [4]{x}}+\frac {(-1)^{5/6} \left (-1+\sqrt {5}\right )^{5/12} \left (2-\sqrt {2 \left (-2+(-1)^{5/6} \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}\right )}\right ) \sqrt [4]{x^2-1} \sqrt {\frac {x^2}{\left (\sqrt {x^2-1}+1\right )^2}} \left (\sqrt {x^2-1}+1\right ) \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{x^2-1}\right ),\frac {1}{2}\right )}{24 \sqrt [3]{1-i} \sqrt [4]{2} \left (4-(-1)^{5/6} \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}\right ) \sqrt [4]{x^5-x^3} \sqrt [4]{x}}+\frac {(-1)^{2/3} (1-i)^{5/3} \left (-1+\sqrt {5}\right )^{5/12} \left (2+\sqrt {2 \left (-2+\sqrt [6]{-1} \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}\right )}\right ) \sqrt [4]{x^2-1} \sqrt {\frac {x^2}{\left (\sqrt {x^2-1}+1\right )^2}} \left (\sqrt {x^2-1}+1\right ) \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{x^2-1}\right ),\frac {1}{2}\right )}{48 \sqrt [4]{2} \left (4-\sqrt [6]{-1} \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}\right ) \sqrt [4]{x^5-x^3} \sqrt [4]{x}}+\frac {(-1)^{2/3} (1-i)^{5/3} \left (-1+\sqrt {5}\right )^{5/12} \left (2-\sqrt {2 \left (-2+\sqrt [6]{-1} \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}\right )}\right ) \sqrt [4]{x^2-1} \sqrt {\frac {x^2}{\left (\sqrt {x^2-1}+1\right )^2}} \left (\sqrt {x^2-1}+1\right ) \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{x^2-1}\right ),\frac {1}{2}\right )}{48 \sqrt [4]{2} \left (4-\sqrt [6]{-1} \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}\right ) \sqrt [4]{x^5-x^3} \sqrt [4]{x}}+\frac {\sqrt [6]{-1-\sqrt {5}} \sqrt [4]{1+\sqrt {5}} \left (\sqrt {2}+\sqrt {-2+\sqrt [3]{2} \left (-1-\sqrt {5}\right )^{2/3}}\right ) \sqrt [4]{x^2-1} \sqrt {\frac {x^2}{\left (\sqrt {x^2-1}+1\right )^2}} \left (\sqrt {x^2-1}+1\right ) \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{x^2-1}\right ),\frac {1}{2}\right )}{12\ 2^{11/12} \left (4-\sqrt [3]{2} \left (-1-\sqrt {5}\right )^{2/3}\right ) \sqrt [4]{x^5-x^3} \sqrt [4]{x}}+\frac {(-1)^{5/6} \sqrt [3]{-1-\sqrt {5}} \sqrt [12]{1+\sqrt {5}} \left (\sqrt {2}+\sqrt {-2+\sqrt [3]{2} \left (-1-\sqrt {5}\right )^{2/3}}\right ) \sqrt [4]{x^2-1} \sqrt {\frac {x^2}{\left (\sqrt {x^2-1}+1\right )^2}} \left (\sqrt {x^2-1}+1\right ) \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{x^2-1}\right ),\frac {1}{2}\right )}{12\ 2^{11/12} \left (4-\sqrt [3]{2} \left (-1-\sqrt {5}\right )^{2/3}\right ) \sqrt [4]{x^5-x^3} \sqrt [4]{x}}+\frac {\sqrt [6]{-1-\sqrt {5}} \sqrt [4]{1+\sqrt {5}} \left (\sqrt {2}-\sqrt {-2+\sqrt [3]{2} \left (-1-\sqrt {5}\right )^{2/3}}\right ) \sqrt [4]{x^2-1} \sqrt {\frac {x^2}{\left (\sqrt {x^2-1}+1\right )^2}} \left (\sqrt {x^2-1}+1\right ) \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{x^2-1}\right ),\frac {1}{2}\right )}{12\ 2^{11/12} \left (4-\sqrt [3]{2} \left (-1-\sqrt {5}\right )^{2/3}\right ) \sqrt [4]{x^5-x^3} \sqrt [4]{x}}+\frac {(-1)^{5/6} \sqrt [3]{-1-\sqrt {5}} \sqrt [12]{1+\sqrt {5}} \left (\sqrt {2}-\sqrt {-2+\sqrt [3]{2} \left (-1-\sqrt {5}\right )^{2/3}}\right ) \sqrt [4]{x^2-1} \sqrt {\frac {x^2}{\left (\sqrt {x^2-1}+1\right )^2}} \left (\sqrt {x^2-1}+1\right ) \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{x^2-1}\right ),\frac {1}{2}\right )}{12\ 2^{11/12} \left (4-\sqrt [3]{2} \left (-1-\sqrt {5}\right )^{2/3}\right ) \sqrt [4]{x^5-x^3} \sqrt [4]{x}}-\frac {\sqrt [6]{-1-\sqrt {5}} \sqrt [4]{1+\sqrt {5}} \left (\sqrt {2}+\sqrt {-2+\sqrt [3]{2} \left (-1-\sqrt {5}\right )^{2/3}}\right )^2 \sqrt [4]{x^2-1} \sqrt {\frac {x^2}{\left (\sqrt {x^2-1}+1\right )^2}} \left (\sqrt {x^2-1}+1\right ) \operatorname {EllipticPi}\left (\frac {1}{8} \left (4-\frac {2^{5/6} \left (-1-\sqrt {5}\right )^{2/3}}{\sqrt {-2+\sqrt [3]{2} \left (-1-\sqrt {5}\right )^{2/3}}}\right ),2 \arctan \left (\sqrt [4]{x^2-1}\right ),\frac {1}{2}\right )}{24\ 2^{11/12} \left (4-\sqrt [3]{2} \left (-1-\sqrt {5}\right )^{2/3}\right ) \sqrt {-2+\sqrt [3]{2} \left (-1-\sqrt {5}\right )^{2/3}} \sqrt [4]{x^5-x^3} \sqrt [4]{x}}-\frac {(-1)^{5/6} \sqrt [3]{-1-\sqrt {5}} \sqrt [12]{1+\sqrt {5}} \left (\sqrt {2}+\sqrt {-2+\sqrt [3]{2} \left (-1-\sqrt {5}\right )^{2/3}}\right )^2 \sqrt [4]{x^2-1} \sqrt {\frac {x^2}{\left (\sqrt {x^2-1}+1\right )^2}} \left (\sqrt {x^2-1}+1\right ) \operatorname {EllipticPi}\left (\frac {1}{8} \left (4-\frac {2^{5/6} \left (-1-\sqrt {5}\right )^{2/3}}{\sqrt {-2+\sqrt [3]{2} \left (-1-\sqrt {5}\right )^{2/3}}}\right ),2 \arctan \left (\sqrt [4]{x^2-1}\right ),\frac {1}{2}\right )}{24\ 2^{11/12} \left (4-\sqrt [3]{2} \left (-1-\sqrt {5}\right )^{2/3}\right ) \sqrt {-2+\sqrt [3]{2} \left (-1-\sqrt {5}\right )^{2/3}} \sqrt [4]{x^5-x^3} \sqrt [4]{x}}+\frac {\sqrt [6]{-1-\sqrt {5}} \sqrt [4]{1+\sqrt {5}} \left (\sqrt {2}-\sqrt {-2+\sqrt [3]{2} \left (-1-\sqrt {5}\right )^{2/3}}\right )^2 \sqrt [4]{x^2-1} \sqrt {\frac {x^2}{\left (\sqrt {x^2-1}+1\right )^2}} \left (\sqrt {x^2-1}+1\right ) \operatorname {EllipticPi}\left (\frac {1}{8} \left (4+\frac {2^{5/6} \left (-1-\sqrt {5}\right )^{2/3}}{\sqrt {-2+\sqrt [3]{2} \left (-1-\sqrt {5}\right )^{2/3}}}\right ),2 \arctan \left (\sqrt [4]{x^2-1}\right ),\frac {1}{2}\right )}{24\ 2^{11/12} \left (4-\sqrt [3]{2} \left (-1-\sqrt {5}\right )^{2/3}\right ) \sqrt {-2+\sqrt [3]{2} \left (-1-\sqrt {5}\right )^{2/3}} \sqrt [4]{x^5-x^3} \sqrt [4]{x}}+\frac {(-1)^{5/6} \sqrt [3]{-1-\sqrt {5}} \sqrt [12]{1+\sqrt {5}} \left (\sqrt {2}-\sqrt {-2+\sqrt [3]{2} \left (-1-\sqrt {5}\right )^{2/3}}\right )^2 \sqrt [4]{x^2-1} \sqrt {\frac {x^2}{\left (\sqrt {x^2-1}+1\right )^2}} \left (\sqrt {x^2-1}+1\right ) \operatorname {EllipticPi}\left (\frac {1}{8} \left (4+\frac {2^{5/6} \left (-1-\sqrt {5}\right )^{2/3}}{\sqrt {-2+\sqrt [3]{2} \left (-1-\sqrt {5}\right )^{2/3}}}\right ),2 \arctan \left (\sqrt [4]{x^2-1}\right ),\frac {1}{2}\right )}{24\ 2^{11/12} \left (4-\sqrt [3]{2} \left (-1-\sqrt {5}\right )^{2/3}\right ) \sqrt {-2+\sqrt [3]{2} \left (-1-\sqrt {5}\right )^{2/3}} \sqrt [4]{x^5-x^3} \sqrt [4]{x}}-\frac {\left (i+\sqrt {3}\right ) \left (-1+\sqrt {5}\right )^{5/12} \left (2+\sqrt {-4+\left (i+\sqrt {3}\right ) \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right )^2 \sqrt [4]{x^2-1} \sqrt {\frac {x^2}{\left (\sqrt {x^2-1}+1\right )^2}} \left (\sqrt {x^2-1}+1\right ) \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {2}-\sqrt {-2+\sqrt [6]{-1} \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right )^2}{4 \sqrt {2 \left (-2+\sqrt [6]{-1} \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}\right )}},2 \arctan \left (\sqrt [4]{x^2-1}\right ),\frac {1}{2}\right )}{96 \sqrt [3]{1-i} \sqrt [4]{2} \left (4-\sqrt [6]{-1} \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}\right ) \sqrt {-4+\left (i+\sqrt {3}\right ) \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt [4]{x^5-x^3} \sqrt [4]{x}}+\frac {\left (i+\sqrt {3}\right ) \left (-1+\sqrt {5}\right )^{5/12} \left (2-\sqrt {-4+\left (i+\sqrt {3}\right ) \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right )^2 \sqrt [4]{x^2-1} \sqrt {\frac {x^2}{\left (\sqrt {x^2-1}+1\right )^2}} \left (\sqrt {x^2-1}+1\right ) \operatorname {EllipticPi}\left (\frac {\left (\sqrt {2}+\sqrt {-2+\sqrt [6]{-1} \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right )^2}{4 \sqrt {2 \left (-2+\sqrt [6]{-1} \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}\right )}},2 \arctan \left (\sqrt [4]{x^2-1}\right ),\frac {1}{2}\right )}{96 \sqrt [3]{1-i} \sqrt [4]{2} \left (4-\sqrt [6]{-1} \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}\right ) \sqrt {-4+\left (i+\sqrt {3}\right ) \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt [4]{x^5-x^3} \sqrt [4]{x}}-\frac {(-1)^{5/6} \left (-1+\sqrt {5}\right )^{5/12} \left (2+\sqrt {-4+\left (i-\sqrt {3}\right ) \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right )^2 \sqrt [4]{x^2-1} \sqrt {\frac {x^2}{\left (\sqrt {x^2-1}+1\right )^2}} \left (\sqrt {x^2-1}+1\right ) \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {2}-\sqrt {-2+(-1)^{5/6} \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right )^2}{4 \sqrt {2 \left (-2+(-1)^{5/6} \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}\right )}},2 \arctan \left (\sqrt [4]{x^2-1}\right ),\frac {1}{2}\right )}{48 \sqrt [3]{1-i} \sqrt [4]{2} \left (4-(-1)^{5/6} \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}\right ) \sqrt {-4+\left (i-\sqrt {3}\right ) \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt [4]{x^5-x^3} \sqrt [4]{x}}+\frac {\left (i-\sqrt {3}\right ) \left (-1+\sqrt {5}\right )^{5/12} \left (2-\sqrt {-4+\left (i-\sqrt {3}\right ) \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right )^2 \sqrt [4]{x^2-1} \sqrt {\frac {x^2}{\left (\sqrt {x^2-1}+1\right )^2}} \left (\sqrt {x^2-1}+1\right ) \operatorname {EllipticPi}\left (\frac {\left (\sqrt {2}+\sqrt {-2+(-1)^{5/6} \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right )^2}{4 \sqrt {2 \left (-2+(-1)^{5/6} \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}\right )}},2 \arctan \left (\sqrt [4]{x^2-1}\right ),\frac {1}{2}\right )}{96 \sqrt [3]{1-i} \sqrt [4]{2} \left (4-(-1)^{5/6} \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}\right ) \sqrt {-4+\left (i-\sqrt {3}\right ) \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt [4]{x^5-x^3} \sqrt [4]{x}}+\frac {\sqrt [3]{-1} \sqrt [3]{-1+i} (1+i)^{2/3} \left (i+\sqrt {3}\right ) \left (-1+\sqrt {5}\right )^{5/12} \left (2+\sqrt {-4+\left (i+\sqrt {3}\right ) \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right )^2 \sqrt [4]{x^2-1} \sqrt {\frac {x^2}{\left (\sqrt {x^2-1}+1\right )^2}} \left (\sqrt {x^2-1}+1\right ) \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {2}-\sqrt {-2+\sqrt [6]{-1} \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right )^2}{4 \sqrt {2 \left (-2+\sqrt [6]{-1} \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}\right )}},2 \arctan \left (\sqrt [4]{x^2-1}\right ),\frac {1}{2}\right )}{96\ 2^{11/12} \left (4-\sqrt [6]{-1} \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}\right ) \sqrt {-4+\left (i+\sqrt {3}\right ) \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt [4]{x^5-x^3} \sqrt [4]{x}}-\frac {\sqrt [3]{-1} \sqrt [3]{-1+i} (1+i)^{2/3} \left (i+\sqrt {3}\right ) \left (-1+\sqrt {5}\right )^{5/12} \left (2-\sqrt {-4+\left (i+\sqrt {3}\right ) \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right )^2 \sqrt [4]{x^2-1} \sqrt {\frac {x^2}{\left (\sqrt {x^2-1}+1\right )^2}} \left (\sqrt {x^2-1}+1\right ) \operatorname {EllipticPi}\left (\frac {\left (\sqrt {2}+\sqrt {-2+\sqrt [6]{-1} \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right )^2}{4 \sqrt {2 \left (-2+\sqrt [6]{-1} \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}\right )}},2 \arctan \left (\sqrt [4]{x^2-1}\right ),\frac {1}{2}\right )}{96\ 2^{11/12} \left (4-\sqrt [6]{-1} \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}\right ) \sqrt {-4+\left (i+\sqrt {3}\right ) \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt [4]{x^5-x^3} \sqrt [4]{x}}+\frac {(1-i)^{5/3} \left (-1+\sqrt {5}\right )^{5/12} \left (2+\sqrt {-4+\left (i-\sqrt {3}\right ) \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right )^2 \sqrt [4]{x^2-1} \sqrt {\frac {x^2}{\left (\sqrt {x^2-1}+1\right )^2}} \left (\sqrt {x^2-1}+1\right ) \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {2}-\sqrt {-2+(-1)^{5/6} \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right )^2}{4 \sqrt {2 \left (-2+(-1)^{5/6} \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}\right )}},2 \arctan \left (\sqrt [4]{x^2-1}\right ),\frac {1}{2}\right )}{96 \sqrt [4]{2} \left (4-(-1)^{5/6} \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}\right ) \sqrt {-4+\left (i-\sqrt {3}\right ) \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt [4]{x^5-x^3} \sqrt [4]{x}}+\frac {\left (i-\sqrt {3}\right ) \left (-1+\sqrt {5}\right )^{5/12} \left (2-\sqrt {-4+\left (i-\sqrt {3}\right ) \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right )^2 \sqrt [4]{x^2-1} \sqrt {\frac {x^2}{\left (\sqrt {x^2-1}+1\right )^2}} \left (\sqrt {x^2-1}+1\right ) \operatorname {EllipticPi}\left (\frac {\left (\sqrt {2}+\sqrt {-2+(-1)^{5/6} \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}}\right )^2}{4 \sqrt {2 \left (-2+(-1)^{5/6} \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}\right )}},2 \arctan \left (\sqrt [4]{x^2-1}\right ),\frac {1}{2}\right )}{96 \sqrt [3]{-1+i} \sqrt [4]{2} \left (4-(-1)^{5/6} \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}\right ) \sqrt {-4+\left (i-\sqrt {3}\right ) \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}} \sqrt [4]{x^5-x^3} \sqrt [4]{x}}-\frac {\left (-1+\sqrt {5}\right )^{5/12} \left (4+\frac {i \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}}{\sqrt {-1-\frac {1}{2} i \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}}}\right ) \sqrt [4]{x^2-1} \sqrt {\frac {x^2}{\left (\sqrt {x^2-1}+1\right )^2}} \left (\sqrt {x^2-1}+1\right ) \operatorname {EllipticPi}\left (\frac {1}{4} \left (2-\frac {i \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}}{\sqrt {-4+(-1+i)^{8/3} \left (-1+\sqrt {5}\right )^{2/3}}}\right ),2 \arctan \left (\sqrt [4]{x^2-1}\right ),\frac {1}{2}\right )}{48 \sqrt [3]{-1+i} \sqrt [4]{2} \left (4 i-\left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}\right ) \sqrt [4]{x^5-x^3} \sqrt [4]{x}}-\frac {\left (-1+\sqrt {5}\right )^{5/12} \left (4-\frac {i \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}}{\sqrt {-1-\frac {1}{2} i \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}}}\right ) \sqrt [4]{x^2-1} \sqrt {\frac {x^2}{\left (\sqrt {x^2-1}+1\right )^2}} \left (\sqrt {x^2-1}+1\right ) \operatorname {EllipticPi}\left (\frac {1}{4} \left (2+\frac {i \left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}}{\sqrt {-4+(-1+i)^{8/3} \left (-1+\sqrt {5}\right )^{2/3}}}\right ),2 \arctan \left (\sqrt [4]{x^2-1}\right ),\frac {1}{2}\right )}{48 \sqrt [3]{-1+i} \sqrt [4]{2} \left (4 i-\left ((1-i) \left (1-\sqrt {5}\right )\right )^{2/3}\right ) \sqrt [4]{x^5-x^3} \sqrt [4]{x}}-\frac {\left (-1+\sqrt {5}\right )^{5/12} \left (4+\frac {i \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}}{\sqrt {-1-\frac {1}{2} i \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}}}\right ) \sqrt [4]{x^2-1} \sqrt {\frac {x^2}{\left (\sqrt {x^2-1}+1\right )^2}} \left (\sqrt {x^2-1}+1\right ) \operatorname {EllipticPi}\left (\frac {1}{4} \left (2-\frac {i \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}}{\sqrt {-4+(1-i)^{8/3} \left (-1+\sqrt {5}\right )^{2/3}}}\right ),2 \arctan \left (\sqrt [4]{x^2-1}\right ),\frac {1}{2}\right )}{48 \sqrt [3]{1-i} \sqrt [4]{2} \left (4 i-\left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}\right ) \sqrt [4]{x^5-x^3} \sqrt [4]{x}}-\frac {\left (-1+\sqrt {5}\right )^{5/12} \left (4-\frac {i \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}}{\sqrt {-1-\frac {1}{2} i \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}}}\right ) \sqrt [4]{x^2-1} \sqrt {\frac {x^2}{\left (\sqrt {x^2-1}+1\right )^2}} \left (\sqrt {x^2-1}+1\right ) \operatorname {EllipticPi}\left (\frac {1}{4} \left (2+\frac {i \left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}}{\sqrt {-4+(1-i)^{8/3} \left (-1+\sqrt {5}\right )^{2/3}}}\right ),2 \arctan \left (\sqrt [4]{x^2-1}\right ),\frac {1}{2}\right )}{48 \sqrt [3]{1-i} \sqrt [4]{2} \left (4 i-\left ((-1+i) \left (1-\sqrt {5}\right )\right )^{2/3}\right ) \sqrt [4]{x^5-x^3} \sqrt [4]{x}}-\frac {i \left (1+\sqrt {5}\right )^{5/12} \left (1+\sqrt {\frac {2}{-2+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}}}\right ) \left (\sqrt {2}+\sqrt {-2+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}}\right ) \sqrt [4]{x^2-1} \sqrt {\frac {x^2}{\left (\sqrt {x^2-1}+1\right )^2}} \left (\sqrt {x^2-1}+1\right ) \operatorname {EllipticPi}\left (\frac {1}{8} \left (4-\frac {2^{5/6} \left (1+\sqrt {5}\right )^{2/3}}{\sqrt {-2+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}}}\right ),2 \arctan \left (\sqrt [4]{x^2-1}\right ),\frac {1}{2}\right )}{24\ 2^{11/12} \left (4-\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right ) \sqrt [4]{x^5-x^3} \sqrt [4]{x}}+\frac {\sqrt [6]{-1} \sqrt [3]{-1-\sqrt {5}} \sqrt [12]{1+\sqrt {5}} \left (1+\sqrt {\frac {2}{-2+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}}}\right ) \left (\sqrt {2}+\sqrt {-2+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}}\right ) \sqrt [4]{x^2-1} \sqrt {\frac {x^2}{\left (\sqrt {x^2-1}+1\right )^2}} \left (\sqrt {x^2-1}+1\right ) \operatorname {EllipticPi}\left (\frac {1}{8} \left (4-\frac {2^{5/6} \left (1+\sqrt {5}\right )^{2/3}}{\sqrt {-2+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}}}\right ),2 \arctan \left (\sqrt [4]{x^2-1}\right ),\frac {1}{2}\right )}{24\ 2^{11/12} \left (4-\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right ) \sqrt [4]{x^5-x^3} \sqrt [4]{x}}-\frac {i \left (1+\sqrt {5}\right )^{5/12} \left (1-\sqrt {\frac {2}{-2+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}}}\right ) \left (\sqrt {2}-\sqrt {-2+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}}\right ) \sqrt [4]{x^2-1} \sqrt {\frac {x^2}{\left (\sqrt {x^2-1}+1\right )^2}} \left (\sqrt {x^2-1}+1\right ) \operatorname {EllipticPi}\left (\frac {1}{8} \left (4+\frac {2^{5/6} \left (1+\sqrt {5}\right )^{2/3}}{\sqrt {-2+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}}}\right ),2 \arctan \left (\sqrt [4]{x^2-1}\right ),\frac {1}{2}\right )}{24\ 2^{11/12} \left (4-\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right ) \sqrt [4]{x^5-x^3} \sqrt [4]{x}}+\frac {\sqrt [6]{-1} \sqrt [3]{-1-\sqrt {5}} \sqrt [12]{1+\sqrt {5}} \left (1-\sqrt {\frac {2}{-2+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}}}\right ) \left (\sqrt {2}-\sqrt {-2+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}}\right ) \sqrt [4]{x^2-1} \sqrt {\frac {x^2}{\left (\sqrt {x^2-1}+1\right )^2}} \left (\sqrt {x^2-1}+1\right ) \operatorname {EllipticPi}\left (\frac {1}{8} \left (4+\frac {2^{5/6} \left (1+\sqrt {5}\right )^{2/3}}{\sqrt {-2+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}}}\right ),2 \arctan \left (\sqrt [4]{x^2-1}\right ),\frac {1}{2}\right )}{24\ 2^{11/12} \left (4-\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right ) \sqrt [4]{x^5-x^3} \sqrt [4]{x}} \]
[In]
[Out]
Rule 65
Rule 209
Rule 210
Rule 211
Rule 212
Rule 213
Rule 214
Rule 218
Rule 226
Rule 251
Rule 252
Rule 304
Rule 385
Rule 408
Rule 440
Rule 441
Rule 455
Rule 504
Rule 524
Rule 525
Rule 760
Rule 1231
Rule 1254
Rule 1452
Rule 1483
Rule 1505
Rule 1576
Rule 1721
Rule 2081
Rule 2184
Rule 2185
Rule 6847
Rule 6857
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{3/4} \sqrt [4]{-1+x^2}\right ) \int \frac {1+x^6}{x^{3/4} \sqrt [4]{-1+x^2} \left (1+x^3-x^6\right )} \, dx}{\sqrt [4]{-x^3+x^5}} \\ & = \frac {\left (4 x^{3/4} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1+x^{24}}{\sqrt [4]{-1+x^8} \left (1+x^{12}-x^{24}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x^3+x^5}} \\ & = \frac {\left (4 x^{3/4} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \left (-\frac {1}{\sqrt [4]{-1+x^8}}+\frac {2+x^{12}}{\sqrt [4]{-1+x^8} \left (1+x^{12}-x^{24}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x^3+x^5}} \\ & = -\frac {\left (4 x^{3/4} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-1+x^8}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x^3+x^5}}+\frac {\left (4 x^{3/4} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {2+x^{12}}{\sqrt [4]{-1+x^8} \left (1+x^{12}-x^{24}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x^3+x^5}} \\ & = -\frac {\left (4 x^{3/4} \sqrt [4]{1-x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1-x^8}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x^3+x^5}}+\frac {\left (4 x^{3/4} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \left (\frac {1-\sqrt {5}}{\sqrt [4]{-1+x^8} \left (1-\sqrt {5}-2 x^{12}\right )}+\frac {1+\sqrt {5}}{\sqrt [4]{-1+x^8} \left (1+\sqrt {5}-2 x^{12}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x^3+x^5}} \\ & = -\frac {4 x \sqrt [4]{1-x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{8},\frac {1}{4},\frac {9}{8},x^2\right )}{\sqrt [4]{-x^3+x^5}}+\frac {\left (4 \left (1-\sqrt {5}\right ) x^{3/4} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-1+x^8} \left (1-\sqrt {5}-2 x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x^3+x^5}}+\frac {\left (4 \left (1+\sqrt {5}\right ) x^{3/4} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-1+x^8} \left (1+\sqrt {5}-2 x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x^3+x^5}} \\ & = -\frac {4 x \sqrt [4]{1-x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{8},\frac {1}{4},\frac {9}{8},x^2\right )}{\sqrt [4]{-x^3+x^5}}+\frac {\left (4 \left (1-\sqrt {5}\right ) x^{3/4} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \left (\frac {i \sqrt {-1+\sqrt {5}}}{2 \left (1-\sqrt {5}\right ) \left (i \sqrt {-1+\sqrt {5}}-\sqrt {2} x^6\right ) \sqrt [4]{-1+x^8}}+\frac {i \sqrt {-1+\sqrt {5}}}{2 \left (1-\sqrt {5}\right ) \left (i \sqrt {-1+\sqrt {5}}+\sqrt {2} x^6\right ) \sqrt [4]{-1+x^8}}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x^3+x^5}}+\frac {\left (4 \left (1+\sqrt {5}\right ) x^{3/4} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {1+\sqrt {5}} \left (\sqrt {1+\sqrt {5}}-\sqrt {2} x^6\right ) \sqrt [4]{-1+x^8}}+\frac {1}{2 \sqrt {1+\sqrt {5}} \left (\sqrt {1+\sqrt {5}}+\sqrt {2} x^6\right ) \sqrt [4]{-1+x^8}}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x^3+x^5}} \\ & = -\frac {4 x \sqrt [4]{1-x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{8},\frac {1}{4},\frac {9}{8},x^2\right )}{\sqrt [4]{-x^3+x^5}}-\frac {\left (2 i \left (1-\sqrt {5}\right ) x^{3/4} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (i \sqrt {-1+\sqrt {5}}-\sqrt {2} x^6\right ) \sqrt [4]{-1+x^8}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt {-1+\sqrt {5}} \sqrt [4]{-x^3+x^5}}-\frac {\left (2 i \left (1-\sqrt {5}\right ) x^{3/4} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (i \sqrt {-1+\sqrt {5}}+\sqrt {2} x^6\right ) \sqrt [4]{-1+x^8}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt {-1+\sqrt {5}} \sqrt [4]{-x^3+x^5}}+\frac {\left (2 \sqrt {1+\sqrt {5}} x^{3/4} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {1+\sqrt {5}}-\sqrt {2} x^6\right ) \sqrt [4]{-1+x^8}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x^3+x^5}}+\frac {\left (2 \sqrt {1+\sqrt {5}} x^{3/4} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {1+\sqrt {5}}+\sqrt {2} x^6\right ) \sqrt [4]{-1+x^8}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x^3+x^5}} \\ & = -\frac {4 x \sqrt [4]{1-x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{8},\frac {1}{4},\frac {9}{8},x^2\right )}{\sqrt [4]{-x^3+x^5}}-\frac {\left (2 i \left (1-\sqrt {5}\right ) x^{3/4} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \left (-\frac {i \sqrt [4]{1-\sqrt {5}}}{2 \sqrt {-1+\sqrt {5}} \left (\sqrt [4]{1-\sqrt {5}}-\sqrt [4]{2} x^3\right ) \sqrt [4]{-1+x^8}}-\frac {i \sqrt [4]{1-\sqrt {5}}}{2 \sqrt {-1+\sqrt {5}} \left (\sqrt [4]{1-\sqrt {5}}+\sqrt [4]{2} x^3\right ) \sqrt [4]{-1+x^8}}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt {-1+\sqrt {5}} \sqrt [4]{-x^3+x^5}}-\frac {\left (2 i \left (1-\sqrt {5}\right ) x^{3/4} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \left (-\frac {\sqrt [4]{-\frac {1}{-1+\sqrt {5}}}}{2 \left (-(-1)^{3/4} \sqrt [4]{-1+\sqrt {5}}-\sqrt [4]{2} x^3\right ) \sqrt [4]{-1+x^8}}-\frac {\sqrt [4]{-\frac {1}{-1+\sqrt {5}}}}{2 \left (-(-1)^{3/4} \sqrt [4]{-1+\sqrt {5}}+\sqrt [4]{2} x^3\right ) \sqrt [4]{-1+x^8}}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt {-1+\sqrt {5}} \sqrt [4]{-x^3+x^5}}+\frac {\left (2 \sqrt {1+\sqrt {5}} x^{3/4} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \left (\frac {i}{2 \sqrt [4]{1+\sqrt {5}} \left (i \sqrt [4]{1+\sqrt {5}}-\sqrt [4]{2} x^3\right ) \sqrt [4]{-1+x^8}}+\frac {i}{2 \sqrt [4]{1+\sqrt {5}} \left (i \sqrt [4]{1+\sqrt {5}}+\sqrt [4]{2} x^3\right ) \sqrt [4]{-1+x^8}}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x^3+x^5}}+\frac {\left (2 \sqrt {1+\sqrt {5}} x^{3/4} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{2 \sqrt [4]{1+\sqrt {5}} \left (\sqrt [4]{1+\sqrt {5}}-\sqrt [4]{2} x^3\right ) \sqrt [4]{-1+x^8}}+\frac {1}{2 \sqrt [4]{1+\sqrt {5}} \left (\sqrt [4]{1+\sqrt {5}}+\sqrt [4]{2} x^3\right ) \sqrt [4]{-1+x^8}}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x^3+x^5}} \\ & = -\frac {4 x \sqrt [4]{1-x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{8},\frac {1}{4},\frac {9}{8},x^2\right )}{\sqrt [4]{-x^3+x^5}}+\frac {\left ((-1)^{3/4} \left (1-\sqrt {5}\right ) x^{3/4} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-(-1)^{3/4} \sqrt [4]{-1+\sqrt {5}}-\sqrt [4]{2} x^3\right ) \sqrt [4]{-1+x^8}} \, dx,x,\sqrt [4]{x}\right )}{\left (-1+\sqrt {5}\right )^{3/4} \sqrt [4]{-x^3+x^5}}+\frac {\left ((-1)^{3/4} \left (1-\sqrt {5}\right ) x^{3/4} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-(-1)^{3/4} \sqrt [4]{-1+\sqrt {5}}+\sqrt [4]{2} x^3\right ) \sqrt [4]{-1+x^8}} \, dx,x,\sqrt [4]{x}\right )}{\left (-1+\sqrt {5}\right )^{3/4} \sqrt [4]{-x^3+x^5}}-\frac {\left ((1+i) \left (1-\sqrt {5}\right ) x^{3/4} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [4]{1-\sqrt {5}}-\sqrt [4]{2} x^3\right ) \sqrt [4]{-1+x^8}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt {2} \left (-1+\sqrt {5}\right )^{3/4} \sqrt [4]{-x^3+x^5}}-\frac {\left ((1+i) \left (1-\sqrt {5}\right ) x^{3/4} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [4]{1-\sqrt {5}}+\sqrt [4]{2} x^3\right ) \sqrt [4]{-1+x^8}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt {2} \left (-1+\sqrt {5}\right )^{3/4} \sqrt [4]{-x^3+x^5}}+\frac {\left (i \sqrt [4]{1+\sqrt {5}} x^{3/4} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (i \sqrt [4]{1+\sqrt {5}}-\sqrt [4]{2} x^3\right ) \sqrt [4]{-1+x^8}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x^3+x^5}}+\frac {\left (i \sqrt [4]{1+\sqrt {5}} x^{3/4} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (i \sqrt [4]{1+\sqrt {5}}+\sqrt [4]{2} x^3\right ) \sqrt [4]{-1+x^8}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x^3+x^5}}+\frac {\left (\sqrt [4]{1+\sqrt {5}} x^{3/4} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [4]{1+\sqrt {5}}-\sqrt [4]{2} x^3\right ) \sqrt [4]{-1+x^8}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x^3+x^5}}+\frac {\left (\sqrt [4]{1+\sqrt {5}} x^{3/4} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [4]{1+\sqrt {5}}+\sqrt [4]{2} x^3\right ) \sqrt [4]{-1+x^8}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x^3+x^5}} \\ & = \text {Too large to display} \\ \end{align*}
\[ \int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1+x^3-x^6\right )} \, dx=\int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1+x^3-x^6\right )} \, dx \]
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Timed out.
\[\int \frac {x^{6}+1}{\left (x^{5}-x^{3}\right )^{\frac {1}{4}} \left (-x^{6}+x^{3}+1\right )}d x\]
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Timed out. \[ \int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1+x^3-x^6\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1+x^3-x^6\right )} \, dx=\text {Timed out} \]
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Not integrable
Time = 0.24 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.67 \[ \int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1+x^3-x^6\right )} \, dx=\int { -\frac {x^{6} + 1}{{\left (x^{6} - x^{3} - 1\right )} {\left (x^{5} - x^{3}\right )}^{\frac {1}{4}}} \,d x } \]
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Not integrable
Time = 0.33 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.65 \[ \int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1+x^3-x^6\right )} \, dx=\int { -\frac {x^{6} + 1}{{\left (x^{6} - x^{3} - 1\right )} {\left (x^{5} - x^{3}\right )}^{\frac {1}{4}}} \,d x } \]
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Not integrable
Time = 6.16 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.63 \[ \int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1+x^3-x^6\right )} \, dx=\int \frac {x^6+1}{{\left (x^5-x^3\right )}^{1/4}\,\left (-x^6+x^3+1\right )} \,d x \]
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