\(\int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} (1+x^3-x^6)} \, dx\) [629]

Optimal result

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]

Unintegrable

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 145.62 (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]

Int[(1 + x^6)/((-x^3 + x^5)^(1/4)*(1 + x^3 - x^6)),x]

[Out]

(x*(1 - x^2)^(1/4)*AppellF1[1/8, 1/4, 1, 9/8, x^2, ((2*I)*x^2)/((-1 + I)*(1 - Sqrt[5]))^(2/3)])/(3*(-x^3 + x^5
)^(1/4)) + (x*(1 - x^2)^(1/4)*AppellF1[1/8, 1/4, 1, 9/8, x^2, (-2*(-1)^(1/6)*x^2)/((-1 + I)*(1 - Sqrt[5]))^(2/
3)])/(3*(-x^3 + x^5)^(1/4)) + (x*(1 - x^2)^(1/4)*AppellF1[1/8, 1/4, 1, 9/8, x^2, (-2*(-1)^(5/6)*x^2)/((-1 + I)
*(1 - Sqrt[5]))^(2/3)])/(3*(-x^3 + x^5)^(1/4)) + (x*(1 - x^2)^(1/4)*AppellF1[1/8, 1/4, 1, 9/8, x^2, ((2*I)*x^2
)/((1 - I)*(1 - Sqrt[5]))^(2/3)])/(3*(-x^3 + x^5)^(1/4)) + (x*(1 - x^2)^(1/4)*AppellF1[1/8, 1/4, 1, 9/8, x^2,
(-2*(-1)^(1/6)*x^2)/((1 - I)*(1 - Sqrt[5]))^(2/3)])/(3*(-x^3 + x^5)^(1/4)) + (x*(1 - x^2)^(1/4)*AppellF1[1/8,
1/4, 1, 9/8, x^2, (-2*(-1)^(5/6)*x^2)/((1 - I)*(1 - Sqrt[5]))^(2/3)])/(3*(-x^3 + x^5)^(1/4)) + (2*x*(1 - x^2)^
(1/4)*AppellF1[1/8, 1/4, 1, 9/8, x^2, (-2/(-1 + Sqrt[5]))^(2/3)*x^2])/(3*(-x^3 + x^5)^(1/4)) + (2*x*(1 - x^2)^
(1/4)*AppellF1[1/8, 1/4, 1, 9/8, x^2, (2/(-1 + Sqrt[5]))^(2/3)*x^2])/(3*(-x^3 + x^5)^(1/4)) + (2*x*(1 - x^2)^(
1/4)*AppellF1[1/8, 1/4, 1, 9/8, x^2, -((-1)^(1/3)*(2/(-1 + Sqrt[5]))^(2/3)*x^2)])/(3*(-x^3 + x^5)^(1/4)) + (4*
x*(1 - x^2)^(1/4)*AppellF1[1/8, 1/4, 1, 9/8, x^2, (-2/(1 + Sqrt[5]))^(2/3)*x^2])/(3*(-x^3 + x^5)^(1/4)) + (4*x
*(1 - x^2)^(1/4)*AppellF1[1/8, 1/4, 1, 9/8, x^2, (2/(1 + Sqrt[5]))^(2/3)*x^2])/(3*(-x^3 + x^5)^(1/4)) + (4*x*(
1 - x^2)^(1/4)*AppellF1[1/8, 1/4, 1, 9/8, x^2, -((-1)^(1/3)*(2/(1 + Sqrt[5]))^(2/3)*x^2)])/(3*(-x^3 + x^5)^(1/
4)) + (Sqrt[2]*x^(3/2)*(1 - x^2)^(1/4)*AppellF1[3/8, 1/4, 1, 11/8, x^2, ((2*I)*x^2)/((-1 + I)*(1 - Sqrt[5]))^(
2/3)])/(9*(1 - I)^(2/3)*(-1 + Sqrt[5])^(1/6)*(-x^3 + x^5)^(1/4)) + ((-1)^(2/3)*Sqrt[2]*x^(3/2)*(1 - x^2)^(1/4)
*AppellF1[3/8, 1/4, 1, 11/8, x^2, (-2*(-1)^(1/6)*x^2)/((-1 + I)*(1 - Sqrt[5]))^(2/3)])/(9*(1 - I)^(2/3)*(-1 +
Sqrt[5])^(1/6)*(-x^3 + x^5)^(1/4)) - ((-1)^(1/3)*Sqrt[2]*x^(3/2)*(1 - x^2)^(1/4)*AppellF1[3/8, 1/4, 1, 11/8, x
^2, (-2*(-1)^(5/6)*x^2)/((-1 + I)*(1 - Sqrt[5]))^(2/3)])/(9*(1 - I)^(2/3)*(-1 + Sqrt[5])^(1/6)*(-x^3 + x^5)^(1
/4)) + (Sqrt[2]*x^(3/2)*(1 - x^2)^(1/4)*AppellF1[3/8, 1/4, 1, 11/8, x^2, ((2*I)*x^2)/((1 - I)*(1 - Sqrt[5]))^(
2/3)])/(9*(-1 + I)^(2/3)*(-1 + Sqrt[5])^(1/6)*(-x^3 + x^5)^(1/4)) - ((-1 + I)^(4/3)*(-(-1 + Sqrt[5])^(-1))^(1/
6)*x^(3/2)*(1 - x^2)^(1/4)*AppellF1[3/8, 1/4, 1, 11/8, x^2, (-2*(-1)^(1/6)*x^2)/((1 - I)*(1 - Sqrt[5]))^(2/3)]
)/(9*Sqrt[2]*(-x^3 + x^5)^(1/4)) - ((-1)^(1/3)*Sqrt[2]*x^(3/2)*(1 - x^2)^(1/4)*AppellF1[3/8, 1/4, 1, 11/8, x^2
, (-2*(-1)^(5/6)*x^2)/((1 - I)*(1 - Sqrt[5]))^(2/3)])/(9*(-1 + I)^(2/3)*(-1 + Sqrt[5])^(1/6)*(-x^3 + x^5)^(1/4
)) - ((1/9 - I/9)*(-1)^(5/12)*2^(2/3)*x^(3/2)*(1 - x^2)^(1/4)*AppellF1[3/8, 1/4, 1, 11/8, x^2, (-2/(-1 + Sqrt[
5]))^(2/3)*x^2])/((-1 + Sqrt[5])^(1/6)*(-x^3 + x^5)^(1/4)) + (((2*I)/9)*(2/(-1 + Sqrt[5]))^(1/6)*x^(3/2)*(1 -
x^2)^(1/4)*AppellF1[3/8, 1/4, 1, 11/8, x^2, (2/(-1 + Sqrt[5]))^(2/3)*x^2])/(-x^3 + x^5)^(1/4) - (2*(-1)^(5/6)*
(2/(-1 + Sqrt[5]))^(1/6)*x^(3/2)*(1 - x^2)^(1/4)*AppellF1[3/8, 1/4, 1, 11/8, x^2, -((-1)^(1/3)*(2/(-1 + Sqrt[5
]))^(2/3)*x^2)])/(9*(-x^3 + x^5)^(1/4)) + ((I/15)*(1 - I)^(2/3)*x^2*(1 - x^2)^(1/4)*AppellF1[5/8, 1/4, 1, 13/8
, x^2, ((2*I)*x^2)/((-1 + I)*(1 - Sqrt[5]))^(2/3)])/((-1 + Sqrt[5])^(1/3)*(-x^3 + x^5)^(1/4)) - ((-1)^(5/6)*(1
- I)^(2/3)*x^2*(1 - x^2)^(1/4)*AppellF1[5/8, 1/4, 1, 13/8, x^2, (-2*(-1)^(1/6)*x^2)/((-1 + I)*(1 - Sqrt[5]))^
(2/3)])/(15*(-1 + Sqrt[5])^(1/3)*(-x^3 + x^5)^(1/4)) + ((I/15)*(-1 + I)^(2/3)*x^2*(1 - x^2)^(1/4)*AppellF1[5/8
, 1/4, 1, 13/8, x^2, (-2*(-1)^(5/6)*x^2)/((-1 + I)*(1 - Sqrt[5]))^(2/3)])/((-1 + Sqrt[5])^(1/3)*(-x^3 + x^5)^(
1/4)) + ((I/15)*(-1 + I)^(2/3)*x^2*(1 - x^2)^(1/4)*AppellF1[5/8, 1/4, 1, 13/8, x^2, ((2*I)*x^2)/((1 - I)*(1 -
Sqrt[5]))^(2/3)])/((-1 + Sqrt[5])^(1/3)*(-x^3 + x^5)^(1/4)) - ((-1)^(5/6)*(-1 + I)^(2/3)*x^2*(1 - x^2)^(1/4)*A
ppellF1[5/8, 1/4, 1, 13/8, x^2, (-2*(-1)^(1/6)*x^2)/((1 - I)*(1 - Sqrt[5]))^(2/3)])/(15*(-1 + Sqrt[5])^(1/3)*(
-x^3 + x^5)^(1/4)) - ((-1)^(1/6)*(-1 + I)^(2/3)*x^2*(1 - x^2)^(1/4)*AppellF1[5/8, 1/4, 1, 13/8, x^2, (-2*(-1)^
(5/6)*x^2)/((1 - I)*(1 - Sqrt[5]))^(2/3)])/(15*(-1 + Sqrt[5])^(1/3)*(-x^3 + x^5)^(1/4)) + ((1 + I*Sqrt[3])*(2/
(-1 + Sqrt[5]))^(1/3)*x^2*(1 - x^2)^(1/4)*AppellF1[5/8, 1/4, 1, 13/8, x^2, (-2/(-1 + Sqrt[5]))^(2/3)*x^2])/(15
*(-x^3 + x^5)^(1/4)) - (2*(2/(-1 + Sqrt[5]))^(1/3)*x^2*(1 - x^2)^(1/4)*AppellF1[5/8, 1/4, 1, 13/8, x^2, (2/(-1
+ Sqrt[5]))^(2/3)*x^2])/(15*(-x^3 + x^5)^(1/4)) - (2*(-1)^(2/3)*(2/(-1 + Sqrt[5]))^(1/3)*x^2*(1 - x^2)^(1/4)*
AppellF1[5/8, 1/4, 1, 13/8, x^2, -((-1)^(1/3)*(2/(-1 + Sqrt[5]))^(2/3)*x^2)])/(15*(-x^3 + x^5)^(1/4)) - (4*(-2
/(1 + Sqrt[5]))^(1/3)*x^2*(1 - x^2)^(1/4)*AppellF1[5/8, 1/4, 1, 13/8, x^2, (-2/(1 + Sqrt[5]))^(2/3)*x^2])/(15*
(-x^3 + x^5)^(1/4)) + (4*(2/(1 + Sqrt[5]))^(1/3)*x^2*(1 - x^2)^(1/4)*AppellF1[5/8, 1/4, 1, 13/8, x^2, (2/(1 +
Sqrt[5]))^(2/3)*x^2])/(15*(-x^3 + x^5)^(1/4)) + (4*(-1)^(2/3)*(2/(1 + Sqrt[5]))^(1/3)*x^2*(1 - x^2)^(1/4)*Appe
llF1[5/8, 1/4, 1, 13/8, x^2, -((-1)^(1/3)*(2/(1 + Sqrt[5]))^(2/3)*x^2)])/(15*(-x^3 + x^5)^(1/4)) + ((I/9)*(1 -
I)^(1/3)*x^(9/4)*(1 - x^2)^(1/4)*AppellF1[3/4, 1/4, 1, 7/4, x^2, ((2*I)*x^2)/((-1 + I)*(1 - Sqrt[5]))^(2/3)])
/(2^(3/4)*(-1 + Sqrt[5])^(5/12)*(-x^3 + x^5)^(1/4)) - ((-1)^(1/6)*(1 - I)^(1/3)*x^(9/4)*(1 - x^2)^(1/4)*Appell
F1[3/4, 1/4, 1, 7/4, x^2, (-2*(-1)^(1/6)*x^2)/((-1 + I)*(1 - Sqrt[5]))^(2/3)])/(9*2^(3/4)*(-1 + Sqrt[5])^(5/12
)*(-x^3 + x^5)^(1/4)) - ((I/9)*(-1 + I)^(1/3)*x^(9/4)*(1 - x^2)^(1/4)*AppellF1[3/4, 1/4, 1, 7/4, x^2, (-2*(-1)
^(5/6)*x^2)/((-1 + I)*(1 - Sqrt[5]))^(2/3)])/(2^(3/4)*(-1 + Sqrt[5])^(5/12)*(-x^3 + x^5)^(1/4)) + ((I/9)*(-1 +
I)^(1/3)*x^(9/4)*(1 - x^2)^(1/4)*AppellF1[3/4, 1/4, 1, 7/4, x^2, ((2*I)*x^2)/((1 - I)*(1 - Sqrt[5]))^(2/3)])/
(2^(3/4)*(-1 + Sqrt[5])^(5/12)*(-x^3 + x^5)^(1/4)) - ((-1)^(1/6)*(-1 + I)^(1/3)*x^(9/4)*(1 - x^2)^(1/4)*Appell
F1[3/4, 1/4, 1, 7/4, x^2, (-2*(-1)^(1/6)*x^2)/((1 - I)*(1 - Sqrt[5]))^(2/3)])/(9*2^(3/4)*(-1 + Sqrt[5])^(5/12)
*(-x^3 + x^5)^(1/4)) - ((-1)^(5/6)*(-1 + I)^(1/3)*x^(9/4)*(1 - x^2)^(1/4)*AppellF1[3/4, 1/4, 1, 7/4, x^2, (-2*
(-1)^(5/6)*x^2)/((1 - I)*(1 - Sqrt[5]))^(2/3)])/(9*2^(3/4)*(-1 + Sqrt[5])^(5/12)*(-x^3 + x^5)^(1/4)) + ((-1)^(
5/6)*x^(9/4)*(1 - x^2)^(1/4)*AppellF1[3/4, 1/4, 1, 7/4, x^2, -((-1)^(1/3)*(2/(1 + Sqrt[5]))^(2/3)*x^2)])/(9*2^
(7/12)*(1 + Sqrt[5])^(5/12)*(-x^3 + x^5)^(1/4)) - ((-1)^(1/6)*(1 - Sqrt[5])^(2/3)*(1 + Sqrt[5])^(1/4)*x^(9/4)*
(1 - x^2)^(1/4)*AppellF1[3/4, 1/4, 1, 7/4, x^2, -((-1)^(1/3)*(2/(1 + Sqrt[5]))^(2/3)*x^2)])/(18*2^(11/12)*(-x^
3 + x^5)^(1/4)) + ((I/21)*Sqrt[2/(-1 + Sqrt[5])]*x^(5/2)*(1 - x^2)^(1/4)*AppellF1[7/8, 1/4, 1, 15/8, x^2, ((2*
I)*x^2)/((-1 + I)*(1 - Sqrt[5]))^(2/3)])/(-x^3 + x^5)^(1/4) + ((I/21)*Sqrt[2/(-1 + Sqrt[5])]*x^(5/2)*(1 - x^2)
^(1/4)*AppellF1[7/8, 1/4, 1, 15/8, x^2, (-2*(-1)^(1/6)*x^2)/((-1 + I)*(1 - Sqrt[5]))^(2/3)])/(-x^3 + x^5)^(1/4
) + ((I/21)*Sqrt[2/(-1 + Sqrt[5])]*x^(5/2)*(1 - x^2)^(1/4)*AppellF1[7/8, 1/4, 1, 15/8, x^2, (-2*(-1)^(5/6)*x^2
)/((-1 + I)*(1 - Sqrt[5]))^(2/3)])/(-x^3 + x^5)^(1/4) + ((I/21)*Sqrt[2/(-1 + Sqrt[5])]*x^(5/2)*(1 - x^2)^(1/4)
*AppellF1[7/8, 1/4, 1, 15/8, x^2, ((2*I)*x^2)/((1 - I)*(1 - Sqrt[5]))^(2/3)])/(-x^3 + x^5)^(1/4) + ((I/21)*Sqr
t[2/(-1 + Sqrt[5])]*x^(5/2)*(1 - x^2)^(1/4)*AppellF1[7/8, 1/4, 1, 15/8, x^2, (-2*(-1)^(1/6)*x^2)/((1 - I)*(1 -
Sqrt[5]))^(2/3)])/(-x^3 + x^5)^(1/4) + ((I/21)*Sqrt[2/(-1 + Sqrt[5])]*x^(5/2)*(1 - x^2)^(1/4)*AppellF1[7/8, 1
/4, 1, 15/8, x^2, (-2*(-1)^(5/6)*x^2)/((1 - I)*(1 - Sqrt[5]))^(2/3)])/(-x^3 + x^5)^(1/4) - (((2*I)/21)*Sqrt[2/
(-1 + Sqrt[5])]*x^(5/2)*(1 - x^2)^(1/4)*AppellF1[7/8, 1/4, 1, 15/8, x^2, (-2/(-1 + Sqrt[5]))^(2/3)*x^2])/(-x^3
+ x^5)^(1/4) - (((2*I)/21)*Sqrt[2/(-1 + Sqrt[5])]*x^(5/2)*(1 - x^2)^(1/4)*AppellF1[7/8, 1/4, 1, 15/8, x^2, (2
/(-1 + Sqrt[5]))^(2/3)*x^2])/(-x^3 + x^5)^(1/4) - (((2*I)/21)*Sqrt[2/(-1 + Sqrt[5])]*x^(5/2)*(1 - x^2)^(1/4)*A
ppellF1[7/8, 1/4, 1, 15/8, x^2, -((-1)^(1/3)*(2/(-1 + Sqrt[5]))^(2/3)*x^2)])/(-x^3 + x^5)^(1/4) + ((1/12 - I/1
2)*(-1)^(1/3)*(1 + Sqrt[5])^(1/12)*x^(3/4)*(-1 + x^2)^(1/4)*ArcTan[((-1)^(1/12)*(2^(2/3) - (-1 - Sqrt[5])^(2/3
))^(1/4)*Sqrt[x])/((1 + Sqrt[5])^(1/6)*(-1 + x^2)^(1/4))])/(2^(5/12)*(2^(2/3) - (-1 - Sqrt[5])^(2/3))^(1/4)*(-
x^3 + x^5)^(1/4)) + ((-1 + Sqrt[5])^(1/12)*x^(3/4)*(-1 + x^2)^(1/4)*ArcTan[((-2*I + ((-1 + I)*(1 - Sqrt[5]))^(
2/3))^(1/4)*Sqrt[x])/(((-1 + I)*(1 - Sqrt[5]))^(1/6)*(-1 + x^2)^(1/4))])/(6*(1 - I)^(1/6)*2^(3/4)*(-2*I + ((-1
+ I)*(1 - Sqrt[5]))^(2/3))^(1/4)*(-x^3 + x^5)^(1/4)) + ((-1)^(2/3)*(-1 + Sqrt[5])^(1/12)*x^(3/4)*(-1 + x^2)^(
1/4)*ArcTan[((2*(-1)^(5/6) + ((-1 + I)*(1 - Sqrt[5]))^(2/3))^(1/4)*Sqrt[x])/(((-1 + I)*(1 - Sqrt[5]))^(1/6)*(-
1 + x^2)^(1/4))])/(6*(1 - I)^(1/6)*2^(3/4)*(2*(-1)^(5/6) + ((-1 + I)*(1 - Sqrt[5]))^(2/3))^(1/4)*(-x^3 + x^5)^
(1/4)) - ((-1)^(1/3)*(-1 + Sqrt[5])^(1/12)*x^(3/4)*(-1 + x^2)^(1/4)*ArcTan[((I + Sqrt[3] + ((-1 + I)*(1 - Sqrt
[5]))^(2/3))^(1/4)*Sqrt[x])/(((-1 + I)*(1 - Sqrt[5]))^(1/6)*(-1 + x^2)^(1/4))])/(6*(1 - I)^(1/6)*2^(3/4)*(I +
Sqrt[3] + ((-1 + I)*(1 - Sqrt[5]))^(2/3))^(1/4)*(-x^3 + x^5)^(1/4)) + ((-1 + Sqrt[5])^(1/12)*x^(3/4)*(-1 + x^2
)^(1/4)*ArcTan[((-2*I + ((1 - I)*(1 - Sqrt[5]))^(2/3))^(1/4)*Sqrt[x])/(((1 - I)*(1 - Sqrt[5]))^(1/6)*(-1 + x^2
)^(1/4))])/(6*(-1 + I)^(1/6)*2^(3/4)*(-2*I + ((1 - I)*(1 - Sqrt[5]))^(2/3))^(1/4)*(-x^3 + x^5)^(1/4)) + ((I/6)
*(1 + I)^(1/6)*(-1 + Sqrt[5])^(1/12)*x^(3/4)*(-1 + x^2)^(1/4)*ArcTan[((2*(-1)^(5/6) + ((1 - I)*(1 - Sqrt[5]))^
(2/3))^(1/4)*Sqrt[x])/(((1 - I)*(1 - Sqrt[5]))^(1/6)*(-1 + x^2)^(1/4))])/(2^(11/12)*(2*(-1)^(5/6) + ((1 - I)*(
1 - Sqrt[5]))^(2/3))^(1/4)*(-x^3 + x^5)^(1/4)) - ((-1)^(1/3)*(-1 + Sqrt[5])^(1/12)*x^(3/4)*(-1 + x^2)^(1/4)*Ar
cTan[((I + Sqrt[3] + ((1 - I)*(1 - Sqrt[5]))^(2/3))^(1/4)*Sqrt[x])/(((1 - I)*(1 - Sqrt[5]))^(1/6)*(-1 + x^2)^(
1/4))])/(6*(-1 + I)^(1/6)*2^(3/4)*(I + Sqrt[3] + ((1 - I)*(1 - Sqrt[5]))^(2/3))^(1/4)*(-x^3 + x^5)^(1/4)) - ((
-1)^(1/6)*(1 + Sqrt[5])^(1/12)*x^(3/4)*(-1 + x^2)^(1/4)*ArcTan[(((-1)^(1/3)*2^(2/3) + (1 + Sqrt[5])^(2/3))^(1/
4)*Sqrt[x])/((1 + Sqrt[5])^(1/6)*(-1 + x^2)^(1/4))])/(6*2^(11/12)*((-1)^(1/3)*2^(2/3) + (1 + Sqrt[5])^(2/3))^(
1/4)*(-x^3 + x^5)^(1/4)) - ((-1)^(11/12)*(1 + Sqrt[5])^(1/12)*x^(3/4)*(-1 + x^2)^(1/4)*ArcTan[((-1)^(11/12)*(2
^(2/3) + (-1)^(1/3)*(1 + Sqrt[5])^(2/3))^(1/4)*Sqrt[x])/((1 + Sqrt[5])^(1/6)*(-1 + x^2)^(1/4))])/(6*2^(11/12)*
(2^(2/3) + (-1)^(1/3)*(1 + Sqrt[5])^(2/3))^(1/4)*(-x^3 + x^5)^(1/4)) - ((1/12 - I/12)*(-1)^(1/6)*(1 + Sqrt[5])
^(1/12)*x^(3/4)*(-1 + x^2)^(1/4)*ArcTan[((-1)^(11/12)*(2^(2/3) + (-1)^(1/3)*(1 + Sqrt[5])^(2/3))^(1/4)*Sqrt[x]
)/((1 + Sqrt[5])^(1/6)*(-1 + x^2)^(1/4))])/(2^(5/12)*(2^(2/3) + (-1)^(1/3)*(1 + Sqrt[5])^(2/3))^(1/4)*(-x^3 +
x^5)^(1/4)) - ((1 - I)^(1/3)*(-1 + Sqrt[5])^(1/12)*x^(3/4)*(-1 + x^2)^(1/4)*ArcTan[(2^(1/4)*(-1 + x^2)^(1/4))/
(-2 - I*((-1 + I)*(1 - Sqrt[5]))^(2/3))^(1/4)])/(12*(-2 - I*((-1 + I)*(1 - Sqrt[5]))^(2/3))^(1/4)*(-x^3 + x^5)
^(1/4)) + ((-1 + I)^(1/3)*(-1 + Sqrt[5])^(1/12)*x^(3/4)*(-1 + x^2)^(1/4)*ArcTan[(2^(1/4)*(-1 + x^2)^(1/4))/(-2
+ (-1)^(1/6)*((-1 + I)*(1 - Sqrt[5]))^(2/3))^(1/4)])/(12*(-2 + (-1)^(1/6)*((-1 + I)*(1 - Sqrt[5]))^(2/3))^(1/
4)*(-x^3 + x^5)^(1/4)) - ((-1)^(2/3)*(1 - I)^(1/3)*(-1 + Sqrt[5])^(1/12)*x^(3/4)*(-1 + x^2)^(1/4)*ArcTan[(2^(1
/4)*(-1 + x^2)^(1/4))/(-2 + (-1)^(5/6)*((-1 + I)*(1 - Sqrt[5]))^(2/3))^(1/4)])/(12*(-2 + (-1)^(5/6)*((-1 + I)*
(1 - Sqrt[5]))^(2/3))^(1/4)*(-x^3 + x^5)^(1/4)) + ((I/12)*(-1 + I)^(1/3)*(-1 + Sqrt[5])^(1/12)*((-I)/(2*I - ((
1 - I)*(1 - Sqrt[5]))^(2/3)))^(1/4)*x^(3/4)*(-1 + x^2)^(1/4)*ArcTan[(2^(1/4)*(-1 + x^2)^(1/4))/(-2 - I*((1 - I
)*(1 - Sqrt[5]))^(2/3))^(1/4)])/(-x^3 + x^5)^(1/4) + ((-1)^(1/3)*(-1 + I)^(1/3)*(-1 + Sqrt[5])^(1/12)*x^(3/4)*
(-1 + x^2)^(1/4)*ArcTan[(2^(1/4)*(-1 + x^2)^(1/4))/(-2 + (-1)^(1/6)*((1 - I)*(1 - Sqrt[5]))^(2/3))^(1/4)])/(12
*(-2 + (-1)^(1/6)*((1 - I)*(1 - Sqrt[5]))^(2/3))^(1/4)*(-x^3 + x^5)^(1/4)) - ((-1)^(2/3)*(-1 + I)^(1/3)*(-1 +
Sqrt[5])^(1/12)*x^(3/4)*(-1 + x^2)^(1/4)*ArcTan[(2^(1/4)*(-1 + x^2)^(1/4))/(-2 + (-1)^(5/6)*((1 - I)*(1 - Sqrt
[5]))^(2/3))^(1/4)])/(12*(-2 + (-1)^(5/6)*((1 - I)*(1 - Sqrt[5]))^(2/3))^(1/4)*(-x^3 + x^5)^(1/4)) - ((I/24)*(
1 - I)^(1/3)*(-1 + Sqrt[5])^(1/12)*Sqrt[x^2]*(-1 + x^2)^(1/4)*ArcTan[((1 + I)*((-1 + I)*(1 - Sqrt[5]))^(1/3)*(
-1 + x^2)^(1/4))/(2^(3/4)*(-2 - I*((-1 + I)*(1 - Sqrt[5]))^(2/3))^(1/4)*Sqrt[x^2])])/((-2 - I*((-1 + I)*(1 - S
qrt[5]))^(2/3))^(1/4)*x^(1/4)*(-x^3 + x^5)^(1/4)) + ((-1 + Sqrt[5])^(1/12)*(2*I - 2*Sqrt[3] + ((-1 + I)*(1 - S
qrt[5]))^(2/3))*Sqrt[x^2]*(-1 + x^2)^(1/4)*ArcTan[((-1)^(1/12)*((-1 + I)*(1 - Sqrt[5]))^(1/3)*(-1 + x^2)^(1/4)
)/(2^(1/4)*(-2 + (-1)^(1/6)*((-1 + I)*(1 - Sqrt[5]))^(2/3))^(1/4)*Sqrt[x^2])])/(12*(1 - I)^(2/3)*Sqrt[2]*(2 -
(-1)^(1/6)*((-1 + I)*(1 - Sqrt[5]))^(2/3))^(1/4)*(4 - (-1)^(1/6)*((-1 + I)*(1 - Sqrt[5]))^(2/3))*x^(1/4)*(-x^3
+ x^5)^(1/4)) - ((-1)^(7/12)*(-1 + Sqrt[5])^(1/12)*Sqrt[x^2]*(-1 + x^2)^(1/4)*ArcTan[((-1)^(7/12)*((-1 + I)*(
1 - Sqrt[5]))^(1/3)*(-1 + x^2)^(1/4))/(2^(1/4)*(-2 + (-1)^(1/6)*((-1 + I)*(1 - Sqrt[5]))^(2/3))^(1/4)*Sqrt[x^2
])])/(12*(1 - I)^(2/3)*Sqrt[2]*(-2 + (-1)^(1/6)*((-1 + I)*(1 - Sqrt[5]))^(2/3))^(1/4)*x^(1/4)*(-x^3 + x^5)^(1/
4)) + ((1 - I)^(1/3)*(-1 + Sqrt[5])^(1/12)*(2 - I*(2*Sqrt[3] + ((-1 + I)*(1 - Sqrt[5]))^(2/3)))*Sqrt[x^2]*(-1
+ x^2)^(1/4)*ArcTan[((-1)^(5/12)*((-1 + I)*(1 - Sqrt[5]))^(1/3)*(-1 + x^2)^(1/4))/(2^(1/4)*(-2 + (-1)^(5/6)*((
-1 + I)*(1 - Sqrt[5]))^(2/3))^(1/4)*Sqrt[x^2])])/(24*(4 - (-1)^(5/6)*((-1 + I)*(1 - Sqrt[5]))^(2/3))*(-2 + (-1
)^(5/6)*((-1 + I)*(1 - Sqrt[5]))^(2/3))^(1/4)*x^(1/4)*(-x^3 + x^5)^(1/4)) - ((-1)^(11/12)*(-1 + Sqrt[5])^(1/12
)*Sqrt[x^2]*(-1 + x^2)^(1/4)*ArcTan[((-1)^(11/12)*((-1 + I)*(1 - Sqrt[5]))^(1/3)*(-1 + x^2)^(1/4))/(2^(1/4)*(-
2 + (-1)^(5/6)*((-1 + I)*(1 - Sqrt[5]))^(2/3))^(1/4)*Sqrt[x^2])])/(12*(1 - I)^(2/3)*Sqrt[2]*(-2 + (-1)^(5/6)*(
(-1 + I)*(1 - Sqrt[5]))^(2/3))^(1/4)*x^(1/4)*(-x^3 + x^5)^(1/4)) + ((-1 + I)^(1/3)*(-1 + Sqrt[5])^(1/12)*((-I)
/(2*I - ((1 - I)*(1 - Sqrt[5]))^(2/3)))^(1/4)*Sqrt[x^2]*(-1 + x^2)^(1/4)*ArcTan[((1 + I)*((1 - I)*(1 - Sqrt[5]
))^(1/3)*(-1 + x^2)^(1/4))/(2^(3/4)*(-2 - I*((1 - I)*(1 - Sqrt[5]))^(2/3))^(1/4)*Sqrt[x^2])])/(24*x^(1/4)*(-x^
3 + x^5)^(1/4)) + ((-1 + Sqrt[5])^(1/12)*(2*I - 2*Sqrt[3] + ((1 - I)*(1 - Sqrt[5]))^(2/3))*Sqrt[x^2]*(-1 + x^2
)^(1/4)*ArcTan[((-1)^(1/12)*((1 - I)*(1 - Sqrt[5]))^(1/3)*(-1 + x^2)^(1/4))/(2^(1/4)*(-2 + (-1)^(1/6)*((1 - I)
*(1 - Sqrt[5]))^(2/3))^(1/4)*Sqrt[x^2])])/(12*(-1 + I)^(2/3)*Sqrt[2]*(2 - (-1)^(1/6)*((1 - I)*(1 - Sqrt[5]))^(
2/3))^(1/4)*(4 - (-1)^(1/6)*((1 - I)*(1 - Sqrt[5]))^(2/3))*x^(1/4)*(-x^3 + x^5)^(1/4)) - ((-1)^(7/12)*(-1 + Sq
rt[5])^(1/12)*Sqrt[x^2]*(-1 + x^2)^(1/4)*ArcTan[((-1)^(7/12)*((1 - I)*(1 - Sqrt[5]))^(1/3)*(-1 + x^2)^(1/4))/(
2^(1/4)*(-2 + (-1)^(1/6)*((1 - I)*(1 - Sqrt[5]))^(2/3))^(1/4)*Sqrt[x^2])])/(12*(-1 + I)^(2/3)*Sqrt[2]*(-2 + (-
1)^(1/6)*((1 - I)*(1 - Sqrt[5]))^(2/3))^(1/4)*x^(1/4)*(-x^3 + x^5)^(1/4)) - ((-1 + I)^(1/3)*(-1 + Sqrt[5])^(1/
12)*(2 - I*(2*Sqrt[3] + ((1 - I)*(1 - Sqrt[5]))^(2/3)))*Sqrt[x^2]*(-1 + x^2)^(1/4)*ArcTan[((-1)^(5/12)*((1 - I
)*(1 - Sqrt[5]))^(1/3)*(-1 + x^2)^(1/4))/(2^(1/4)*(-2 + (-1)^(5/6)*((1 - I)*(1 - Sqrt[5]))^(2/3))^(1/4)*Sqrt[x
^2])])/(24*(4 - (-1)^(5/6)*((1 - I)*(1 - Sqrt[5]))^(2/3))*(-2 + (-1)^(5/6)*((1 - I)*(1 - Sqrt[5]))^(2/3))^(1/4
)*x^(1/4)*(-x^3 + x^5)^(1/4)) - ((-1)^(11/12)*(-1 + Sqrt[5])^(1/12)*Sqrt[x^2]*(-1 + x^2)^(1/4)*ArcTan[((-1)^(1
1/12)*((1 - I)*(1 - Sqrt[5]))^(1/3)*(-1 + x^2)^(1/4))/(2^(1/4)*(-2 + (-1)^(5/6)*((1 - I)*(1 - Sqrt[5]))^(2/3))
^(1/4)*Sqrt[x^2])])/(12*(-1 + I)^(2/3)*Sqrt[2]*(-2 + (-1)^(5/6)*((1 - I)*(1 - Sqrt[5]))^(2/3))^(1/4)*x^(1/4)*(
-x^3 + x^5)^(1/4)) + ((I/24)*(-1 + I)^(1/3)*(-1 + Sqrt[5])^(1/12)*((-I)/(2*I - ((1 - I)*(1 - Sqrt[5]))^(2/3)))
^(1/4)*Sqrt[x^2]*(-1 + x^2)^(1/4)*ArcTan[((-1)^(3/4)*((1 - I)*(1 - Sqrt[5]))^(1/3)*(-1 + x^2)^(1/4))/((-4 + (-
1 + I)^(8/3)*(-1 + Sqrt[5])^(2/3))^(1/4)*Sqrt[x^2])])/(x^(1/4)*(-x^3 + x^5)^(1/4)) + ((1 - I)^(1/3)*(-1 + Sqrt
[5])^(1/12)*Sqrt[x^2]*(-1 + x^2)^(1/4)*ArcTan[((-1)^(3/4)*((-1 + I)*(1 - Sqrt[5]))^(1/3)*(-1 + x^2)^(1/4))/((-
4 + (1 - I)^(8/3)*(-1 + Sqrt[5])^(2/3))^(1/4)*Sqrt[x^2])])/(24*(-2 - I*((-1 + I)*(1 - Sqrt[5]))^(2/3))^(1/4)*x
^(1/4)*(-x^3 + x^5)^(1/4)) + ((1/12 - I/12)*(-1)^(1/3)*(1 + Sqrt[5])^(1/12)*x^(3/4)*(-1 + x^2)^(1/4)*ArcTanh[(
(-1)^(1/12)*(2^(2/3) - (-1 - Sqrt[5])^(2/3))^(1/4)*Sqrt[x])/((1 + Sqrt[5])^(1/6)*(-1 + x^2)^(1/4))])/(2^(5/12)
*(2^(2/3) - (-1 - Sqrt[5])^(2/3))^(1/4)*(-x^3 + x^5)^(1/4)) + ((-1 + Sqrt[5])^(1/12)*x^(3/4)*(-1 + x^2)^(1/4)*
ArcTanh[((-2*I + ((-1 + I)*(1 - Sqrt[5]))^(2/3))^(1/4)*Sqrt[x])/(((-1 + I)*(1 - Sqrt[5]))^(1/6)*(-1 + x^2)^(1/
4))])/(6*(1 - I)^(1/6)*2^(3/4)*(-2*I + ((-1 + I)*(1 - Sqrt[5]))^(2/3))^(1/4)*(-x^3 + x^5)^(1/4)) + ((-1)^(2/3)
*(-1 + Sqrt[5])^(1/12)*x^(3/4)*(-1 + x^2)^(1/4)*ArcTanh[((2*(-1)^(5/6) + ((-1 + I)*(1 - Sqrt[5]))^(2/3))^(1/4)
*Sqrt[x])/(((-1 + I)*(1 - Sqrt[5]))^(1/6)*(-1 + x^2)^(1/4))])/(6*(1 - I)^(1/6)*2^(3/4)*(2*(-1)^(5/6) + ((-1 +
I)*(1 - Sqrt[5]))^(2/3))^(1/4)*(-x^3 + x^5)^(1/4)) - ((-1)^(1/3)*(-1 + Sqrt[5])^(1/12)*x^(3/4)*(-1 + x^2)^(1/4
)*ArcTanh[((I + Sqrt[3] + ((-1 + I)*(1 - Sqrt[5]))^(2/3))^(1/4)*Sqrt[x])/(((-1 + I)*(1 - Sqrt[5]))^(1/6)*(-1 +
x^2)^(1/4))])/(6*(1 - I)^(1/6)*2^(3/4)*(I + Sqrt[3] + ((-1 + I)*(1 - Sqrt[5]))^(2/3))^(1/4)*(-x^3 + x^5)^(1/4
)) + ((-1 + Sqrt[5])^(1/12)*x^(3/4)*(-1 + x^2)^(1/4)*ArcTanh[((-2*I + ((1 - I)*(1 - Sqrt[5]))^(2/3))^(1/4)*Sqr
t[x])/(((1 - I)*(1 - Sqrt[5]))^(1/6)*(-1 + x^2)^(1/4))])/(6*(-1 + I)^(1/6)*2^(3/4)*(-2*I + ((1 - I)*(1 - Sqrt[
5]))^(2/3))^(1/4)*(-x^3 + x^5)^(1/4)) + ((I/6)*(1 + I)^(1/6)*(-1 + Sqrt[5])^(1/12)*x^(3/4)*(-1 + x^2)^(1/4)*Ar
cTanh[((2*(-1)^(5/6) + ((1 - I)*(1 - Sqrt[5]))^(2/3))^(1/4)*Sqrt[x])/(((1 - I)*(1 - Sqrt[5]))^(1/6)*(-1 + x^2)
^(1/4))])/(2^(11/12)*(2*(-1)^(5/6) + ((1 - I)*(1 - Sqrt[5]))^(2/3))^(1/4)*(-x^3 + x^5)^(1/4)) - ((-1)^(1/3)*(-
1 + Sqrt[5])^(1/12)*x^(3/4)*(-1 + x^2)^(1/4)*ArcTanh[((I + Sqrt[3] + ((1 - I)*(1 - Sqrt[5]))^(2/3))^(1/4)*Sqrt
[x])/(((1 - I)*(1 - Sqrt[5]))^(1/6)*(-1 + x^2)^(1/4))])/(6*(-1 + I)^(1/6)*2^(3/4)*(I + Sqrt[3] + ((1 - I)*(1 -
Sqrt[5]))^(2/3))^(1/4)*(-x^3 + x^5)^(1/4)) - ((-1)^(1/6)*(1 + Sqrt[5])^(1/12)*x^(3/4)*(-1 + x^2)^(1/4)*ArcTan
h[(((-1)^(1/3)*2^(2/3) + (1 + Sqrt[5])^(2/3))^(1/4)*Sqrt[x])/((1 + Sqrt[5])^(1/6)*(-1 + x^2)^(1/4))])/(6*2^(11
/12)*((-1)^(1/3)*2^(2/3) + (1 + Sqrt[5])^(2/3))^(1/4)*(-x^3 + x^5)^(1/4)) - ((-1)^(11/12)*(1 + Sqrt[5])^(1/12)
*x^(3/4)*(-1 + x^2)^(1/4)*ArcTanh[((-1)^(11/12)*(2^(2/3) + (-1)^(1/3)*(1 + Sqrt[5])^(2/3))^(1/4)*Sqrt[x])/((1
+ Sqrt[5])^(1/6)*(-1 + x^2)^(1/4))])/(6*2^(11/12)*(2^(2/3) + (-1)^(1/3)*(1 + Sqrt[5])^(2/3))^(1/4)*(-x^3 + x^5
)^(1/4)) - ((1/12 - I/12)*(-1)^(1/6)*(1 + Sqrt[5])^(1/12)*x^(3/4)*(-1 + x^2)^(1/4)*ArcTanh[((-1)^(11/12)*(2^(2
/3) + (-1)^(1/3)*(1 + Sqrt[5])^(2/3))^(1/4)*Sqrt[x])/((1 + Sqrt[5])^(1/6)*(-1 + x^2)^(1/4))])/(2^(5/12)*(2^(2/
3) + (-1)^(1/3)*(1 + Sqrt[5])^(2/3))^(1/4)*(-x^3 + x^5)^(1/4)) + ((1 - I)^(1/3)*(-1 + Sqrt[5])^(1/12)*x^(3/4)*
(-1 + x^2)^(1/4)*ArcTanh[(2^(1/4)*(-1 + x^2)^(1/4))/(-2 - I*((-1 + I)*(1 - Sqrt[5]))^(2/3))^(1/4)])/(12*(-2 -
I*((-1 + I)*(1 - Sqrt[5]))^(2/3))^(1/4)*(-x^3 + x^5)^(1/4)) - ((-1 + I)^(1/3)*(-1 + Sqrt[5])^(1/12)*x^(3/4)*(-
1 + x^2)^(1/4)*ArcTanh[(2^(1/4)*(-1 + x^2)^(1/4))/(-2 + (-1)^(1/6)*((-1 + I)*(1 - Sqrt[5]))^(2/3))^(1/4)])/(12
*(-2 + (-1)^(1/6)*((-1 + I)*(1 - Sqrt[5]))^(2/3))^(1/4)*(-x^3 + x^5)^(1/4)) + ((-1)^(2/3)*(1 - I)^(1/3)*(-1 +
Sqrt[5])^(1/12)*x^(3/4)*(-1 + x^2)^(1/4)*ArcTanh[(2^(1/4)*(-1 + x^2)^(1/4))/(-2 + (-1)^(5/6)*((-1 + I)*(1 - Sq
rt[5]))^(2/3))^(1/4)])/(12*(-2 + (-1)^(5/6)*((-1 + I)*(1 - Sqrt[5]))^(2/3))^(1/4)*(-x^3 + x^5)^(1/4)) - ((I/12
)*(-1 + I)^(1/3)*(-1 + Sqrt[5])^(1/12)*((-I)/(2*I - ((1 - I)*(1 - Sqrt[5]))^(2/3)))^(1/4)*x^(3/4)*(-1 + x^2)^(
1/4)*ArcTanh[(2^(1/4)*(-1 + x^2)^(1/4))/(-2 - I*((1 - I)*(1 - Sqrt[5]))^(2/3))^(1/4)])/(-x^3 + x^5)^(1/4) - ((
-1)^(1/3)*(-1 + I)^(1/3)*(-1 + Sqrt[5])^(1/12)*x^(3/4)*(-1 + x^2)^(1/4)*ArcTanh[(2^(1/4)*(-1 + x^2)^(1/4))/(-2
+ (-1)^(1/6)*((1 - I)*(1 - Sqrt[5]))^(2/3))^(1/4)])/(12*(-2 + (-1)^(1/6)*((1 - I)*(1 - Sqrt[5]))^(2/3))^(1/4)
*(-x^3 + x^5)^(1/4)) + ((-1)^(2/3)*(-1 + I)^(1/3)*(-1 + Sqrt[5])^(1/12)*x^(3/4)*(-1 + x^2)^(1/4)*ArcTanh[(2^(1
/4)*(-1 + x^2)^(1/4))/(-2 + (-1)^(5/6)*((1 - I)*(1 - Sqrt[5]))^(2/3))^(1/4)])/(12*(-2 + (-1)^(5/6)*((1 - I)*(1
- Sqrt[5]))^(2/3))^(1/4)*(-x^3 + x^5)^(1/4)) + ((-1)^(5/6)*(-1 - Sqrt[5])^(1/3)*(1 + Sqrt[5])^(1/12)*(Sqrt[2]
- Sqrt[-2 + 2^(1/3)*(-1 - Sqrt[5])^(2/3)])*(-1 + x^2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1 + x
^2])*EllipticF[2*ArcTan[(-1 + x^2)^(1/4)], 1/2])/(12*2^(11/12)*(4 - 2^(1/3)*(-1 - Sqrt[5])^(2/3))*x^(1/4)*(-x^
3 + x^5)^(1/4)) + ((-1 - Sqrt[5])^(1/6)*(1 + Sqrt[5])^(1/4)*(Sqrt[2] - Sqrt[-2 + 2^(1/3)*(-1 - Sqrt[5])^(2/3)]
)*(-1 + x^2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1 + x^2])*EllipticF[2*ArcTan[(-1 + x^2)^(1/4)],
1/2])/(12*2^(11/12)*(4 - 2^(1/3)*(-1 - Sqrt[5])^(2/3))*x^(1/4)*(-x^3 + x^5)^(1/4)) + ((-1)^(5/6)*(-1 - Sqrt[5
])^(1/3)*(1 + Sqrt[5])^(1/12)*(Sqrt[2] + Sqrt[-2 + 2^(1/3)*(-1 - Sqrt[5])^(2/3)])*(-1 + x^2)^(1/4)*Sqrt[x^2/(1
+ Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1 + x^2])*EllipticF[2*ArcTan[(-1 + x^2)^(1/4)], 1/2])/(12*2^(11/12)*(4 - 2^(1
/3)*(-1 - Sqrt[5])^(2/3))*x^(1/4)*(-x^3 + x^5)^(1/4)) + ((-1 - Sqrt[5])^(1/6)*(1 + Sqrt[5])^(1/4)*(Sqrt[2] + S
qrt[-2 + 2^(1/3)*(-1 - Sqrt[5])^(2/3)])*(-1 + x^2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1 + x^2])
*EllipticF[2*ArcTan[(-1 + x^2)^(1/4)], 1/2])/(12*2^(11/12)*(4 - 2^(1/3)*(-1 - Sqrt[5])^(2/3))*x^(1/4)*(-x^3 +
x^5)^(1/4)) + ((-1)^(2/3)*(1 - I)^(5/3)*(-1 + Sqrt[5])^(5/12)*(2 - Sqrt[2*(-2 + (-1)^(1/6)*((-1 + I)*(1 - Sqrt
[5]))^(2/3))])*(-1 + x^2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1 + x^2])*EllipticF[2*ArcTan[(-1 +
x^2)^(1/4)], 1/2])/(48*2^(1/4)*(4 - (-1)^(1/6)*((-1 + I)*(1 - Sqrt[5]))^(2/3))*x^(1/4)*(-x^3 + x^5)^(1/4)) +
((-1)^(2/3)*(1 - I)^(5/3)*(-1 + Sqrt[5])^(5/12)*(2 + Sqrt[2*(-2 + (-1)^(1/6)*((-1 + I)*(1 - Sqrt[5]))^(2/3))])
*(-1 + x^2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1 + x^2])*EllipticF[2*ArcTan[(-1 + x^2)^(1/4)],
1/2])/(48*2^(1/4)*(4 - (-1)^(1/6)*((-1 + I)*(1 - Sqrt[5]))^(2/3))*x^(1/4)*(-x^3 + x^5)^(1/4)) + ((-1)^(5/6)*(-
1 + Sqrt[5])^(5/12)*(2 - Sqrt[2*(-2 + (-1)^(5/6)*((-1 + I)*(1 - Sqrt[5]))^(2/3))])*(-1 + x^2)^(1/4)*Sqrt[x^2/(
1 + Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1 + x^2])*EllipticF[2*ArcTan[(-1 + x^2)^(1/4)], 1/2])/(24*(1 - I)^(1/3)*2^(1
/4)*(4 - (-1)^(5/6)*((-1 + I)*(1 - Sqrt[5]))^(2/3))*x^(1/4)*(-x^3 + x^5)^(1/4)) + ((-1)^(5/6)*(-1 + Sqrt[5])^(
5/12)*(2 + Sqrt[2*(-2 + (-1)^(5/6)*((-1 + I)*(1 - Sqrt[5]))^(2/3))])*(-1 + x^2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1 +
x^2])^2]*(1 + Sqrt[-1 + x^2])*EllipticF[2*ArcTan[(-1 + x^2)^(1/4)], 1/2])/(24*(1 - I)^(1/3)*2^(1/4)*(4 - (-1)^
(5/6)*((-1 + I)*(1 - Sqrt[5]))^(2/3))*x^(1/4)*(-x^3 + x^5)^(1/4)) + ((-1)^(2/3)*(-1 + I)^(5/3)*(-1 + Sqrt[5])^
(5/12)*(2 - Sqrt[2*(-2 + (-1)^(1/6)*((1 - I)*(1 - Sqrt[5]))^(2/3))])*(-1 + x^2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1 +
x^2])^2]*(1 + Sqrt[-1 + x^2])*EllipticF[2*ArcTan[(-1 + x^2)^(1/4)], 1/2])/(48*2^(1/4)*(4 - (-1)^(1/6)*((1 - I)
*(1 - Sqrt[5]))^(2/3))*x^(1/4)*(-x^3 + x^5)^(1/4)) + ((-1)^(2/3)*(-1 + I)^(5/3)*(-1 + Sqrt[5])^(5/12)*(2 + Sqr
t[2*(-2 + (-1)^(1/6)*((1 - I)*(1 - Sqrt[5]))^(2/3))])*(-1 + x^2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1 + x^2])^2]*(1 + S
qrt[-1 + x^2])*EllipticF[2*ArcTan[(-1 + x^2)^(1/4)], 1/2])/(48*2^(1/4)*(4 - (-1)^(1/6)*((1 - I)*(1 - Sqrt[5]))
^(2/3))*x^(1/4)*(-x^3 + x^5)^(1/4)) - ((1 - I)^(5/3)*(-1 + Sqrt[5])^(5/12)*(2 - Sqrt[2*(-2 + (-1)^(5/6)*((1 -
I)*(1 - Sqrt[5]))^(2/3))])*(-1 + x^2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1 + x^2])*EllipticF[2*
ArcTan[(-1 + x^2)^(1/4)], 1/2])/(48*2^(1/4)*(4 - (-1)^(5/6)*((1 - I)*(1 - Sqrt[5]))^(2/3))*x^(1/4)*(-x^3 + x^5
)^(1/4)) - ((1 - I)^(5/3)*(-1 + Sqrt[5])^(5/12)*(2 + Sqrt[2*(-2 + (-1)^(5/6)*((1 - I)*(1 - Sqrt[5]))^(2/3))])*
(-1 + x^2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1 + x^2])*EllipticF[2*ArcTan[(-1 + x^2)^(1/4)], 1
/2])/(48*2^(1/4)*(4 - (-1)^(5/6)*((1 - I)*(1 - Sqrt[5]))^(2/3))*x^(1/4)*(-x^3 + x^5)^(1/4)) + ((-1)^(1/4)*(-1
+ Sqrt[5])^(5/12)*(2 - Sqrt[-4 + (-1 + I)^(8/3)*(-1 + Sqrt[5])^(2/3)])*(-1 + x^2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1
+ x^2])^2]*(1 + Sqrt[-1 + x^2])*EllipticF[2*ArcTan[(-1 + x^2)^(1/4)], 1/2])/(24*2^(3/4)*(2*(-1 + I)^(4/3) + (-
1 + Sqrt[5])^(2/3))*x^(1/4)*(-x^3 + x^5)^(1/4)) + ((-1)^(1/4)*(-1 + Sqrt[5])^(5/12)*(2 + Sqrt[-4 + (-1 + I)^(8
/3)*(-1 + Sqrt[5])^(2/3)])*(-1 + x^2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1 + x^2])*EllipticF[2*
ArcTan[(-1 + x^2)^(1/4)], 1/2])/(24*2^(3/4)*(2*(-1 + I)^(4/3) + (-1 + Sqrt[5])^(2/3))*x^(1/4)*(-x^3 + x^5)^(1/
4)) - ((-1)^(1/4)*(-1 + Sqrt[5])^(5/12)*(2 - Sqrt[-4 + (1 - I)^(8/3)*(-1 + Sqrt[5])^(2/3)])*(-1 + x^2)^(1/4)*S
qrt[x^2/(1 + Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1 + x^2])*EllipticF[2*ArcTan[(-1 + x^2)^(1/4)], 1/2])/(24*2^(3/4)*(
2*(1 - I)^(4/3) + (-1 + Sqrt[5])^(2/3))*x^(1/4)*(-x^3 + x^5)^(1/4)) - ((-1)^(1/4)*(-1 + Sqrt[5])^(5/12)*(2 + S
qrt[-4 + (1 - I)^(8/3)*(-1 + Sqrt[5])^(2/3)])*(-1 + x^2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1 +
x^2])*EllipticF[2*ArcTan[(-1 + x^2)^(1/4)], 1/2])/(24*2^(3/4)*(2*(1 - I)^(4/3) + (-1 + Sqrt[5])^(2/3))*x^(1/4
)*(-x^3 + x^5)^(1/4)) - ((-1)^(5/6)*(-1 - Sqrt[5])^(1/3)*(1 + Sqrt[5])^(1/12)*(Sqrt[2] + Sqrt[-2 + 2^(1/3)*(-1
- Sqrt[5])^(2/3)])^2*(-1 + x^2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1 + x^2])*EllipticPi[(4 - (
2^(5/6)*(-1 - Sqrt[5])^(2/3))/Sqrt[-2 + 2^(1/3)*(-1 - Sqrt[5])^(2/3)])/8, 2*ArcTan[(-1 + x^2)^(1/4)], 1/2])/(2
4*2^(11/12)*(4 - 2^(1/3)*(-1 - Sqrt[5])^(2/3))*Sqrt[-2 + 2^(1/3)*(-1 - Sqrt[5])^(2/3)]*x^(1/4)*(-x^3 + x^5)^(1
/4)) - ((-1 - Sqrt[5])^(1/6)*(1 + Sqrt[5])^(1/4)*(Sqrt[2] + Sqrt[-2 + 2^(1/3)*(-1 - Sqrt[5])^(2/3)])^2*(-1 + x
^2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1 + x^2])*EllipticPi[(4 - (2^(5/6)*(-1 - Sqrt[5])^(2/3))
/Sqrt[-2 + 2^(1/3)*(-1 - Sqrt[5])^(2/3)])/8, 2*ArcTan[(-1 + x^2)^(1/4)], 1/2])/(24*2^(11/12)*(4 - 2^(1/3)*(-1
- Sqrt[5])^(2/3))*Sqrt[-2 + 2^(1/3)*(-1 - Sqrt[5])^(2/3)]*x^(1/4)*(-x^3 + x^5)^(1/4)) + ((-1)^(5/6)*(-1 - Sqrt
[5])^(1/3)*(1 + Sqrt[5])^(1/12)*(Sqrt[2] - Sqrt[-2 + 2^(1/3)*(-1 - Sqrt[5])^(2/3)])^2*(-1 + x^2)^(1/4)*Sqrt[x^
2/(1 + Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1 + x^2])*EllipticPi[(4 + (2^(5/6)*(-1 - Sqrt[5])^(2/3))/Sqrt[-2 + 2^(1/3
)*(-1 - Sqrt[5])^(2/3)])/8, 2*ArcTan[(-1 + x^2)^(1/4)], 1/2])/(24*2^(11/12)*(4 - 2^(1/3)*(-1 - Sqrt[5])^(2/3))
*Sqrt[-2 + 2^(1/3)*(-1 - Sqrt[5])^(2/3)]*x^(1/4)*(-x^3 + x^5)^(1/4)) + ((-1 - Sqrt[5])^(1/6)*(1 + Sqrt[5])^(1/
4)*(Sqrt[2] - Sqrt[-2 + 2^(1/3)*(-1 - Sqrt[5])^(2/3)])^2*(-1 + x^2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1 + x^2])^2]*(1
+ Sqrt[-1 + x^2])*EllipticPi[(4 + (2^(5/6)*(-1 - Sqrt[5])^(2/3))/Sqrt[-2 + 2^(1/3)*(-1 - Sqrt[5])^(2/3)])/8, 2
*ArcTan[(-1 + x^2)^(1/4)], 1/2])/(24*2^(11/12)*(4 - 2^(1/3)*(-1 - Sqrt[5])^(2/3))*Sqrt[-2 + 2^(1/3)*(-1 - Sqrt
[5])^(2/3)]*x^(1/4)*(-x^3 + x^5)^(1/4)) - ((I + Sqrt[3])*(-1 + Sqrt[5])^(5/12)*(2 + Sqrt[-4 + (I + Sqrt[3])*((
-1 + I)*(1 - Sqrt[5]))^(2/3)])^2*(-1 + x^2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1 + x^2])*Ellipt
icPi[-1/4*(Sqrt[2] - Sqrt[-2 + (-1)^(1/6)*((-1 + I)*(1 - Sqrt[5]))^(2/3)])^2/Sqrt[2*(-2 + (-1)^(1/6)*((-1 + I)
*(1 - Sqrt[5]))^(2/3))], 2*ArcTan[(-1 + x^2)^(1/4)], 1/2])/(96*(1 - I)^(1/3)*2^(1/4)*(4 - (-1)^(1/6)*((-1 + I)
*(1 - Sqrt[5]))^(2/3))*Sqrt[-4 + (I + Sqrt[3])*((-1 + I)*(1 - Sqrt[5]))^(2/3)]*x^(1/4)*(-x^3 + x^5)^(1/4)) + (
(I + Sqrt[3])*(-1 + Sqrt[5])^(5/12)*(2 - Sqrt[-4 + (I + Sqrt[3])*((-1 + I)*(1 - Sqrt[5]))^(2/3)])^2*(-1 + x^2)
^(1/4)*Sqrt[x^2/(1 + Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1 + x^2])*EllipticPi[(Sqrt[2] + Sqrt[-2 + (-1)^(1/6)*((-1 +
I)*(1 - Sqrt[5]))^(2/3)])^2/(4*Sqrt[2*(-2 + (-1)^(1/6)*((-1 + I)*(1 - Sqrt[5]))^(2/3))]), 2*ArcTan[(-1 + x^2)
^(1/4)], 1/2])/(96*(1 - I)^(1/3)*2^(1/4)*(4 - (-1)^(1/6)*((-1 + I)*(1 - Sqrt[5]))^(2/3))*Sqrt[-4 + (I + Sqrt[3
])*((-1 + I)*(1 - Sqrt[5]))^(2/3)]*x^(1/4)*(-x^3 + x^5)^(1/4)) - ((-1)^(5/6)*(-1 + Sqrt[5])^(5/12)*(2 + Sqrt[-
4 + (I - Sqrt[3])*((-1 + I)*(1 - Sqrt[5]))^(2/3)])^2*(-1 + x^2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1 + x^2])^2]*(1 + Sq
rt[-1 + x^2])*EllipticPi[-1/4*(Sqrt[2] - Sqrt[-2 + (-1)^(5/6)*((-1 + I)*(1 - Sqrt[5]))^(2/3)])^2/Sqrt[2*(-2 +
(-1)^(5/6)*((-1 + I)*(1 - Sqrt[5]))^(2/3))], 2*ArcTan[(-1 + x^2)^(1/4)], 1/2])/(48*(1 - I)^(1/3)*2^(1/4)*(4 -
(-1)^(5/6)*((-1 + I)*(1 - Sqrt[5]))^(2/3))*Sqrt[-4 + (I - Sqrt[3])*((-1 + I)*(1 - Sqrt[5]))^(2/3)]*x^(1/4)*(-x
^3 + x^5)^(1/4)) + ((I - Sqrt[3])*(-1 + Sqrt[5])^(5/12)*(2 - Sqrt[-4 + (I - Sqrt[3])*((-1 + I)*(1 - Sqrt[5]))^
(2/3)])^2*(-1 + x^2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1 + x^2])*EllipticPi[(Sqrt[2] + Sqrt[-2
+ (-1)^(5/6)*((-1 + I)*(1 - Sqrt[5]))^(2/3)])^2/(4*Sqrt[2*(-2 + (-1)^(5/6)*((-1 + I)*(1 - Sqrt[5]))^(2/3))]),
2*ArcTan[(-1 + x^2)^(1/4)], 1/2])/(96*(1 - I)^(1/3)*2^(1/4)*(4 - (-1)^(5/6)*((-1 + I)*(1 - Sqrt[5]))^(2/3))*S
qrt[-4 + (I - Sqrt[3])*((-1 + I)*(1 - Sqrt[5]))^(2/3)]*x^(1/4)*(-x^3 + x^5)^(1/4)) + ((-1)^(1/3)*(-1 + I)^(1/3
)*(1 + I)^(2/3)*(I + Sqrt[3])*(-1 + Sqrt[5])^(5/12)*(2 + Sqrt[-4 + (I + Sqrt[3])*((1 - I)*(1 - Sqrt[5]))^(2/3)
])^2*(-1 + x^2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1 + x^2])*EllipticPi[-1/4*(Sqrt[2] - Sqrt[-2
+ (-1)^(1/6)*((1 - I)*(1 - Sqrt[5]))^(2/3)])^2/Sqrt[2*(-2 + (-1)^(1/6)*((1 - I)*(1 - Sqrt[5]))^(2/3))], 2*Arc
Tan[(-1 + x^2)^(1/4)], 1/2])/(96*2^(11/12)*(4 - (-1)^(1/6)*((1 - I)*(1 - Sqrt[5]))^(2/3))*Sqrt[-4 + (I + Sqrt[
3])*((1 - I)*(1 - Sqrt[5]))^(2/3)]*x^(1/4)*(-x^3 + x^5)^(1/4)) - ((-1)^(1/3)*(-1 + I)^(1/3)*(1 + I)^(2/3)*(I +
Sqrt[3])*(-1 + Sqrt[5])^(5/12)*(2 - Sqrt[-4 + (I + Sqrt[3])*((1 - I)*(1 - Sqrt[5]))^(2/3)])^2*(-1 + x^2)^(1/4
)*Sqrt[x^2/(1 + Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1 + x^2])*EllipticPi[(Sqrt[2] + Sqrt[-2 + (-1)^(1/6)*((1 - I)*(1
- Sqrt[5]))^(2/3)])^2/(4*Sqrt[2*(-2 + (-1)^(1/6)*((1 - I)*(1 - Sqrt[5]))^(2/3))]), 2*ArcTan[(-1 + x^2)^(1/4)]
, 1/2])/(96*2^(11/12)*(4 - (-1)^(1/6)*((1 - I)*(1 - Sqrt[5]))^(2/3))*Sqrt[-4 + (I + Sqrt[3])*((1 - I)*(1 - Sqr
t[5]))^(2/3)]*x^(1/4)*(-x^3 + x^5)^(1/4)) + ((1 - I)^(5/3)*(-1 + Sqrt[5])^(5/12)*(2 + Sqrt[-4 + (I - Sqrt[3])*
((1 - I)*(1 - Sqrt[5]))^(2/3)])^2*(-1 + x^2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1 + x^2])*Ellip
ticPi[-1/4*(Sqrt[2] - Sqrt[-2 + (-1)^(5/6)*((1 - I)*(1 - Sqrt[5]))^(2/3)])^2/Sqrt[2*(-2 + (-1)^(5/6)*((1 - I)*
(1 - Sqrt[5]))^(2/3))], 2*ArcTan[(-1 + x^2)^(1/4)], 1/2])/(96*2^(1/4)*(4 - (-1)^(5/6)*((1 - I)*(1 - Sqrt[5]))^
(2/3))*Sqrt[-4 + (I - Sqrt[3])*((1 - I)*(1 - Sqrt[5]))^(2/3)]*x^(1/4)*(-x^3 + x^5)^(1/4)) + ((I - Sqrt[3])*(-1
+ Sqrt[5])^(5/12)*(2 - Sqrt[-4 + (I - Sqrt[3])*((1 - I)*(1 - Sqrt[5]))^(2/3)])^2*(-1 + x^2)^(1/4)*Sqrt[x^2/(1
+ Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1 + x^2])*EllipticPi[(Sqrt[2] + Sqrt[-2 + (-1)^(5/6)*((1 - I)*(1 - Sqrt[5]))^
(2/3)])^2/(4*Sqrt[2*(-2 + (-1)^(5/6)*((1 - I)*(1 - Sqrt[5]))^(2/3))]), 2*ArcTan[(-1 + x^2)^(1/4)], 1/2])/(96*(
-1 + I)^(1/3)*2^(1/4)*(4 - (-1)^(5/6)*((1 - I)*(1 - Sqrt[5]))^(2/3))*Sqrt[-4 + (I - Sqrt[3])*((1 - I)*(1 - Sqr
t[5]))^(2/3)]*x^(1/4)*(-x^3 + x^5)^(1/4)) - ((-1 + Sqrt[5])^(5/12)*(4 + (I*((1 - I)*(1 - Sqrt[5]))^(2/3))/Sqrt
[-1 - (I/2)*((1 - I)*(1 - Sqrt[5]))^(2/3)])*(-1 + x^2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1 + x
^2])*EllipticPi[(2 - (I*((1 - I)*(1 - Sqrt[5]))^(2/3))/Sqrt[-4 + (-1 + I)^(8/3)*(-1 + Sqrt[5])^(2/3)])/4, 2*Ar
cTan[(-1 + x^2)^(1/4)], 1/2])/(48*(-1 + I)^(1/3)*2^(1/4)*(4*I - ((1 - I)*(1 - Sqrt[5]))^(2/3))*x^(1/4)*(-x^3 +
x^5)^(1/4)) - ((-1 + Sqrt[5])^(5/12)*(4 - (I*((1 - I)*(1 - Sqrt[5]))^(2/3))/Sqrt[-1 - (I/2)*((1 - I)*(1 - Sqr
t[5]))^(2/3)])*(-1 + x^2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1 + x^2])*EllipticPi[(2 + (I*((1 -
I)*(1 - Sqrt[5]))^(2/3))/Sqrt[-4 + (-1 + I)^(8/3)*(-1 + Sqrt[5])^(2/3)])/4, 2*ArcTan[(-1 + x^2)^(1/4)], 1/2])
/(48*(-1 + I)^(1/3)*2^(1/4)*(4*I - ((1 - I)*(1 - Sqrt[5]))^(2/3))*x^(1/4)*(-x^3 + x^5)^(1/4)) - ((-1 + Sqrt[5]
)^(5/12)*(4 + (I*((-1 + I)*(1 - Sqrt[5]))^(2/3))/Sqrt[-1 - (I/2)*((-1 + I)*(1 - Sqrt[5]))^(2/3)])*(-1 + x^2)^(
1/4)*Sqrt[x^2/(1 + Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1 + x^2])*EllipticPi[(2 - (I*((-1 + I)*(1 - Sqrt[5]))^(2/3))/
Sqrt[-4 + (1 - I)^(8/3)*(-1 + Sqrt[5])^(2/3)])/4, 2*ArcTan[(-1 + x^2)^(1/4)], 1/2])/(48*(1 - I)^(1/3)*2^(1/4)*
(4*I - ((-1 + I)*(1 - Sqrt[5]))^(2/3))*x^(1/4)*(-x^3 + x^5)^(1/4)) - ((-1 + Sqrt[5])^(5/12)*(4 - (I*((-1 + I)*
(1 - Sqrt[5]))^(2/3))/Sqrt[-1 - (I/2)*((-1 + I)*(1 - Sqrt[5]))^(2/3)])*(-1 + x^2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1
+ x^2])^2]*(1 + Sqrt[-1 + x^2])*EllipticPi[(2 + (I*((-1 + I)*(1 - Sqrt[5]))^(2/3))/Sqrt[-4 + (1 - I)^(8/3)*(-1
+ Sqrt[5])^(2/3)])/4, 2*ArcTan[(-1 + x^2)^(1/4)], 1/2])/(48*(1 - I)^(1/3)*2^(1/4)*(4*I - ((-1 + I)*(1 - Sqrt[
5]))^(2/3))*x^(1/4)*(-x^3 + x^5)^(1/4)) + ((-1)^(1/6)*(-1 - Sqrt[5])^(1/3)*(1 + Sqrt[5])^(1/12)*(1 + Sqrt[2/(-
2 + 2^(1/3)*(1 + Sqrt[5])^(2/3))])*(Sqrt[2] + Sqrt[-2 + 2^(1/3)*(1 + Sqrt[5])^(2/3)])*(-1 + x^2)^(1/4)*Sqrt[x^
2/(1 + Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1 + x^2])*EllipticPi[(4 - (2^(5/6)*(1 + Sqrt[5])^(2/3))/Sqrt[-2 + 2^(1/3)
*(1 + Sqrt[5])^(2/3)])/8, 2*ArcTan[(-1 + x^2)^(1/4)], 1/2])/(24*2^(11/12)*(4 - 2^(1/3)*(1 + Sqrt[5])^(2/3))*x^
(1/4)*(-x^3 + x^5)^(1/4)) - ((I/24)*(1 + Sqrt[5])^(5/12)*(1 + Sqrt[2/(-2 + 2^(1/3)*(1 + Sqrt[5])^(2/3))])*(Sqr
t[2] + Sqrt[-2 + 2^(1/3)*(1 + Sqrt[5])^(2/3)])*(-1 + x^2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1
+ x^2])*EllipticPi[(4 - (2^(5/6)*(1 + Sqrt[5])^(2/3))/Sqrt[-2 + 2^(1/3)*(1 + Sqrt[5])^(2/3)])/8, 2*ArcTan[(-1
+ x^2)^(1/4)], 1/2])/(2^(11/12)*(4 - 2^(1/3)*(1 + Sqrt[5])^(2/3))*x^(1/4)*(-x^3 + x^5)^(1/4)) + ((-1)^(1/6)*(-
1 - Sqrt[5])^(1/3)*(1 + Sqrt[5])^(1/12)*(1 - Sqrt[2/(-2 + 2^(1/3)*(1 + Sqrt[5])^(2/3))])*(Sqrt[2] - Sqrt[-2 +
2^(1/3)*(1 + Sqrt[5])^(2/3)])*(-1 + x^2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1 + x^2])*EllipticP
i[(4 + (2^(5/6)*(1 + Sqrt[5])^(2/3))/Sqrt[-2 + 2^(1/3)*(1 + Sqrt[5])^(2/3)])/8, 2*ArcTan[(-1 + x^2)^(1/4)], 1/
2])/(24*2^(11/12)*(4 - 2^(1/3)*(1 + Sqrt[5])^(2/3))*x^(1/4)*(-x^3 + x^5)^(1/4)) - ((I/24)*(1 + Sqrt[5])^(5/12)
*(1 - Sqrt[2/(-2 + 2^(1/3)*(1 + Sqrt[5])^(2/3))])*(Sqrt[2] - Sqrt[-2 + 2^(1/3)*(1 + Sqrt[5])^(2/3)])*(-1 + x^2
)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1 + x^2])*EllipticPi[(4 + (2^(5/6)*(1 + Sqrt[5])^(2/3))/Sq
rt[-2 + 2^(1/3)*(1 + Sqrt[5])^(2/3)])/8, 2*ArcTan[(-1 + x^2)^(1/4)], 1/2])/(2^(11/12)*(4 - 2^(1/3)*(1 + Sqrt[5
])^(2/3))*x^(1/4)*(-x^3 + x^5)^(1/4)) - (4*x*(1 - x^2)^(1/4)*Hypergeometric2F1[1/8, 1/4, 9/8, x^2])/(-x^3 + x^
5)^(1/4)

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[a/b, 0]

Rule 226

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 251

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)],
x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p
] || GtQ[a, 0])

Rule 252

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^Fra
cPart[p]), Int[(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILt
Q[Simplify[1/n + p], 0] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 304

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}
, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !
GtQ[a/b, 0]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 408

Int[1/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Dist[2*(Sqrt[(-b)*(x^2/a)]/x), Subst[I
nt[x^2/(Sqrt[1 - x^4/a]*(b*c - a*d + d*x^4)), x], x, (a + b*x^2)^(1/4)], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b
*c - a*d, 0]

Rule 440

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 441

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^F
racPart[p]/(1 + b*(x^n/a))^FracPart[p]), Int[(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n,
p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 455

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 504

Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s
= Denominator[Rt[-a/b, 2]]}, Dist[s/(2*b), Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Dist[s/(2*b), Int[1/
((r - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 524

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*
((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 525

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPar
t[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]), Int[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x
] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] &&  !(IntegerQ[
p] || GtQ[a, 0])

Rule 760

Int[1/(((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(1/4)), x_Symbol] :> Dist[d, Int[1/((d^2 - e^2*x^2)*(a + c*x^
2)^(1/4)), x], x] - Dist[e, Int[x/((d^2 - e^2*x^2)*(a + c*x^2)^(1/4)), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ
[c*d^2 + a*e^2, 0]

Rule 1231

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[c/a, 2]}, Dist[(c*d + a*e*q
)/(c*d^2 - a*e^2), Int[1/Sqrt[a + c*x^4], x], x] - Dist[(a*e*(e + d*q))/(c*d^2 - a*e^2), Int[(1 + q*x^2)/((d +
e*x^2)*Sqrt[a + c*x^4]), x], x]] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0]
&& PosQ[c/a]

Rule 1254

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + c*x^4)^p, (d/
(d^2 - e^2*x^4) - e*(x^2/(d^2 - e^2*x^4)))^(-q), x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[c*d^2 + a*e^2, 0]
&&  !IntegerQ[p] && ILtQ[q, 0]

Rule 1452

Int[((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + c*x^(2
*n))^p, (d/(d^2 - e^2*x^(2*n)) - e*(x^n/(d^2 - e^2*x^(2*n))))^(-q), x], x] /; FreeQ[{a, c, d, e, n, p}, x] &&
EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[q, 0]

Rule 1483

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[I
nt[(d + e*x)^q*(a + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[Sim
plify[m - n + 1], 0]

Rule 1505

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{k = GCD[m +
1, n]}, Dist[1/k, Subst[Int[x^((m + 1)/k - 1)*(d + e*x^(n/k))^q*(a + c*x^(2*(n/k)))^p, x], x, x^k], x] /; k !=
1] /; FreeQ[{a, c, d, e, p, q}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && IntegerQ[m]

Rule 1576

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Dist[(f*x)^m
/x^m, Int[ExpandIntegrand[x^m*(a + c*x^(2*n))^p, (d/(d^2 - e^2*x^(2*n)) - e*(x^n/(d^2 - e^2*x^(2*n))))^(-q), x
], x], x] /; FreeQ[{a, c, d, e, f, m, n, p}, x] && EqQ[n2, 2*n] &&  !IntegerQ[p] && ILtQ[q, 0]

Rule 1721

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[B/A, 2]
}, Simp[(-(B*d - A*e))*(ArcTan[Rt[c*(d/e) + a*(e/d), 2]*(x/Sqrt[a + c*x^4])]/(2*d*e*Rt[c*(d/e) + a*(e/d), 2]))
, x] + Simp[(B*d + A*e)*(A + B*x^2)*(Sqrt[A^2*((a + c*x^4)/(a*(A + B*x^2)^2))]/(4*d*e*A*q*Sqrt[a + c*x^4]))*El
lipticPi[Cancel[-(B*d - A*e)^2/(4*d*e*A*B)], 2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c
*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0]

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 2184

Int[((c_) + (d_.)*(x_)^(n_.))^(q_)*((a_) + (b_.)*(x_)^(nn_.))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^
nn)^p, (c/(c^2 - d^2*x^(2*n)) - d*(x^n/(c^2 - d^2*x^(2*n))))^(-q), x], x] /; FreeQ[{a, b, c, d, n, nn, p}, x]
&&  !IntegerQ[p] && ILtQ[q, 0] && IGtQ[Log[2, nn/n], 0]

Rule 2185

Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_)^(n_.))^(q_)*((a_) + (b_.)*(x_)^(nn_.))^(p_), x_Symbol] :> Dist[(e*x)
^m/x^m, Int[ExpandIntegrand[x^m*(a + b*x^nn)^p, (c/(c^2 - d^2*x^(2*n)) - d*(x^n/(c^2 - d^2*x^(2*n))))^(-q), x]
, x], x] /; FreeQ[{a, b, c, d, e, m, n, nn, p}, x] &&  !IntegerQ[p] && ILtQ[q, 0] && IGtQ[Log[2, nn/n], 0]

Rule 6847

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

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*}

Mathematica [F]

\[ \int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1+x^3-x^6\right )} \, dx=\int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1+x^3-x^6\right )} \, dx \]

[In]

Integrate[(1 + x^6)/((-x^3 + x^5)^(1/4)*(1 + x^3 - x^6)),x]

[Out]

Integrate[(1 + x^6)/((-x^3 + x^5)^(1/4)*(1 + x^3 - x^6)), x]

Maple [F(-1)]

Timed out.

\[\int \frac {x^{6}+1}{\left (x^{5}-x^{3}\right )^{\frac {1}{4}} \left (-x^{6}+x^{3}+1\right )}d x\]

[In]

int((x^6+1)/(x^5-x^3)^(1/4)/(-x^6+x^3+1),x)

[Out]

int((x^6+1)/(x^5-x^3)^(1/4)/(-x^6+x^3+1),x)

Fricas [F(-1)]

Timed out. \[ \int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1+x^3-x^6\right )} \, dx=\text {Timed out} \]

[In]

integrate((x^6+1)/(x^5-x^3)^(1/4)/(-x^6+x^3+1),x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1+x^3-x^6\right )} \, dx=\text {Timed out} \]

[In]

integrate((x**6+1)/(x**5-x**3)**(1/4)/(-x**6+x**3+1),x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.67 \[ \int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1+x^3-x^6\right )} \, dx=\int { -\frac {x^{6} + 1}{{\left (x^{6} - x^{3} - 1\right )} {\left (x^{5} - x^{3}\right )}^{\frac {1}{4}}} \,d x } \]

[In]

integrate((x^6+1)/(x^5-x^3)^(1/4)/(-x^6+x^3+1),x, algorithm="maxima")

[Out]

-integrate((x^6 + 1)/((x^6 - x^3 - 1)*(x^5 - x^3)^(1/4)), x)

Giac [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.65 \[ \int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1+x^3-x^6\right )} \, dx=\int { -\frac {x^{6} + 1}{{\left (x^{6} - x^{3} - 1\right )} {\left (x^{5} - x^{3}\right )}^{\frac {1}{4}}} \,d x } \]

[In]

integrate((x^6+1)/(x^5-x^3)^(1/4)/(-x^6+x^3+1),x, algorithm="giac")

[Out]

integrate(-(x^6 + 1)/((x^6 - x^3 - 1)*(x^5 - x^3)^(1/4)), x)

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.63 \[ \int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1+x^3-x^6\right )} \, dx=\int \frac {x^6+1}{{\left (x^5-x^3\right )}^{1/4}\,\left (-x^6+x^3+1\right )} \,d x \]

[In]

int((x^6 + 1)/((x^5 - x^3)^(1/4)*(x^3 - x^6 + 1)),x)

[Out]

int((x^6 + 1)/((x^5 - x^3)^(1/4)*(x^3 - x^6 + 1)), x)