Integrand size = 34, antiderivative size = 293 \[ \int \frac {1-2 x^4+x^8}{\sqrt [4]{-1+x^4} \left (1-2 x^4+2 x^8\right )} \, dx=\frac {1}{4} \arctan \left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{8} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \arctan \left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{-1+x^4}}{-x^2+\sqrt {-1+x^4}}\right )-\frac {1}{8} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \arctan \left (\frac {\left (\sqrt {\frac {2}{2-\sqrt {2}}}-\frac {2}{\sqrt {2-\sqrt {2}}}\right ) x \sqrt [4]{-1+x^4}}{-x^2+\sqrt {-1+x^4}}\right )+\frac {1}{4} \text {arctanh}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{8} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{-1+x^4}}{x^2+\sqrt {-1+x^4}}\right )+\frac {1}{8} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{-1+x^4}}{x^2+\sqrt {-1+x^4}}\right ) \]
1/4*arctan(x/(x^4-1)^(1/4))+1/16*(4-2*2^(1/2))^(1/2)*arctan((2+2^(1/2))^(1 /2)*x*(x^4-1)^(1/4)/(-x^2+(x^4-1)^(1/2)))-1/16*(4+2*2^(1/2))^(1/2)*arctan( (2^(1/2)/(2-2^(1/2))^(1/2)-2/(2-2^(1/2))^(1/2))*x*(x^4-1)^(1/4)/(-x^2+(x^4 -1)^(1/2)))+1/4*arctanh(x/(x^4-1)^(1/4))+1/16*(4+2*2^(1/2))^(1/2)*arctanh( (2-2^(1/2))^(1/2)*x*(x^4-1)^(1/4)/(x^2+(x^4-1)^(1/2)))+1/16*(4-2*2^(1/2))^ (1/2)*arctanh((2+2^(1/2))^(1/2)*x*(x^4-1)^(1/4)/(x^2+(x^4-1)^(1/2)))
Time = 0.94 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.85 \[ \int \frac {1-2 x^4+x^8}{\sqrt [4]{-1+x^4} \left (1-2 x^4+2 x^8\right )} \, dx=\frac {1}{16} \left (4 \arctan \left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\sqrt {2 \left (2+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{-1+x^4}}{-x^2+\sqrt {-1+x^4}}\right )+\sqrt {4-2 \sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{-1+x^4}}{-x^2+\sqrt {-1+x^4}}\right )+4 \text {arctanh}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\sqrt {2 \left (2+\sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{-1+x^4}}{x^2+\sqrt {-1+x^4}}\right )+\sqrt {4-2 \sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{-1+x^4}}{x^2+\sqrt {-1+x^4}}\right )\right ) \]
(4*ArcTan[x/(-1 + x^4)^(1/4)] + Sqrt[2*(2 + Sqrt[2])]*ArcTan[(Sqrt[2 - Sqr t[2]]*x*(-1 + x^4)^(1/4))/(-x^2 + Sqrt[-1 + x^4])] + Sqrt[4 - 2*Sqrt[2]]*A rcTan[(Sqrt[2 + Sqrt[2]]*x*(-1 + x^4)^(1/4))/(-x^2 + Sqrt[-1 + x^4])] + 4* ArcTanh[x/(-1 + x^4)^(1/4)] + Sqrt[2*(2 + Sqrt[2])]*ArcTanh[(Sqrt[2 - Sqrt [2]]*x*(-1 + x^4)^(1/4))/(x^2 + Sqrt[-1 + x^4])] + Sqrt[4 - 2*Sqrt[2]]*Arc Tanh[(Sqrt[2 + Sqrt[2]]*x*(-1 + x^4)^(1/4))/(x^2 + Sqrt[-1 + x^4])])/16
Result contains complex when optimal does not.
Time = 0.63 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.41, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.618, Rules used = {1380, 1758, 933, 1026, 770, 756, 216, 219, 902, 755, 27, 756, 216, 219, 1476, 1082, 217, 1479, 25, 27, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^8-2 x^4+1}{\sqrt [4]{x^4-1} \left (2 x^8-2 x^4+1\right )} \, dx\) |
| \(\Big \downarrow \) 1380 |
| \(\displaystyle \int \frac {\left (x^4-1\right )^{7/4}}{2 x^8-2 x^4+1}dx\) |
| \(\Big \downarrow \) 1758 |
| \(\displaystyle 2 i \int \frac {\left (x^4-1\right )^{7/4}}{4 x^4-(2-2 i)}dx-2 i \int \frac {\left (x^4-1\right )^{7/4}}{4 x^4-(2+2 i)}dx\) |
| \(\Big \downarrow \) 933 |
| \(\displaystyle 2 i \left (\frac {1}{16} \left (x^4-1\right )^{3/4} x+\frac {1}{16} \int \frac {(14+2 i)-(20+8 i) x^4}{\sqrt [4]{x^4-1} \left (4 x^4-(2-2 i)\right )}dx\right )-2 i \left (\frac {1}{16} \left (x^4-1\right )^{3/4} x+\frac {1}{16} \int \frac {(14-2 i)-(20-8 i) x^4}{\sqrt [4]{x^4-1} \left (4 x^4-(2+2 i)\right )}dx\right )\) |
| \(\Big \downarrow \) 1026 |
| \(\displaystyle 2 i \left (\frac {1}{16} \left (x^4-1\right )^{3/4} x+\frac {1}{16} \left (8 i \int \frac {1}{\sqrt [4]{x^4-1} \left (4 x^4-(2-2 i)\right )}dx-(5+2 i) \int \frac {1}{\sqrt [4]{x^4-1}}dx\right )\right )-2 i \left (\frac {1}{16} \left (x^4-1\right )^{3/4} x+\frac {1}{16} \left ((-5+2 i) \int \frac {1}{\sqrt [4]{x^4-1}}dx-8 i \int \frac {1}{\sqrt [4]{x^4-1} \left (4 x^4-(2+2 i)\right )}dx\right )\right )\) |
| \(\Big \downarrow \) 770 |
| \(\displaystyle 2 i \left (\frac {1}{16} \left (x^4-1\right )^{3/4} x+\frac {1}{16} \left (8 i \int \frac {1}{\sqrt [4]{x^4-1} \left (4 x^4-(2-2 i)\right )}dx-(5+2 i) \int \frac {1}{1-\frac {x^4}{x^4-1}}d\frac {x}{\sqrt [4]{x^4-1}}\right )\right )-2 i \left (\frac {1}{16} \left (x^4-1\right )^{3/4} x+\frac {1}{16} \left (-8 i \int \frac {1}{\sqrt [4]{x^4-1} \left (4 x^4-(2+2 i)\right )}dx-(5-2 i) \int \frac {1}{1-\frac {x^4}{x^4-1}}d\frac {x}{\sqrt [4]{x^4-1}}\right )\right )\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle 2 i \left (\frac {1}{16} \left (x^4-1\right )^{3/4} x+\frac {1}{16} \left (8 i \int \frac {1}{\sqrt [4]{x^4-1} \left (4 x^4-(2-2 i)\right )}dx-(5+2 i) \left (\frac {1}{2} \int \frac {1}{1-\frac {x^2}{\sqrt {x^4-1}}}d\frac {x}{\sqrt [4]{x^4-1}}+\frac {1}{2} \int \frac {1}{\frac {x^2}{\sqrt {x^4-1}}+1}d\frac {x}{\sqrt [4]{x^4-1}}\right )\right )\right )-2 i \left (\frac {1}{16} \left (x^4-1\right )^{3/4} x+\frac {1}{16} \left (-8 i \int \frac {1}{\sqrt [4]{x^4-1} \left (4 x^4-(2+2 i)\right )}dx-(5-2 i) \left (\frac {1}{2} \int \frac {1}{1-\frac {x^2}{\sqrt {x^4-1}}}d\frac {x}{\sqrt [4]{x^4-1}}+\frac {1}{2} \int \frac {1}{\frac {x^2}{\sqrt {x^4-1}}+1}d\frac {x}{\sqrt [4]{x^4-1}}\right )\right )\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle 2 i \left (\frac {1}{16} \left (x^4-1\right )^{3/4} x+\frac {1}{16} \left (8 i \int \frac {1}{\sqrt [4]{x^4-1} \left (4 x^4-(2-2 i)\right )}dx-(5+2 i) \left (\frac {1}{2} \int \frac {1}{1-\frac {x^2}{\sqrt {x^4-1}}}d\frac {x}{\sqrt [4]{x^4-1}}+\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4-1}}\right )\right )\right )\right )-2 i \left (\frac {1}{16} \left (x^4-1\right )^{3/4} x+\frac {1}{16} \left (-(5-2 i) \left (\frac {1}{2} \int \frac {1}{1-\frac {x^2}{\sqrt {x^4-1}}}d\frac {x}{\sqrt [4]{x^4-1}}+\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4-1}}\right )\right )-8 i \int \frac {1}{\sqrt [4]{x^4-1} \left (4 x^4-(2+2 i)\right )}dx\right )\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle 2 i \left (\frac {1}{16} \left (x^4-1\right )^{3/4} x+\frac {1}{16} \left (8 i \int \frac {1}{\sqrt [4]{x^4-1} \left (4 x^4-(2-2 i)\right )}dx-(5+2 i) \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4-1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )\right )\right )\right )-2 i \left (\frac {1}{16} \left (x^4-1\right )^{3/4} x+\frac {1}{16} \left ((-5+2 i) \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4-1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )\right )-8 i \int \frac {1}{\sqrt [4]{x^4-1} \left (4 x^4-(2+2 i)\right )}dx\right )\right )\) |
| \(\Big \downarrow \) 902 |
| \(\displaystyle 2 i \left (\frac {1}{16} \left (x^4-1\right )^{3/4} x+\frac {1}{16} \left (8 i \int \frac {1}{-\frac {(2+2 i) x^4}{x^4-1}-(2-2 i)}d\frac {x}{\sqrt [4]{x^4-1}}-(5+2 i) \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4-1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )\right )\right )\right )-2 i \left (\frac {1}{16} \left (x^4-1\right )^{3/4} x+\frac {1}{16} \left ((-5+2 i) \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4-1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )\right )-8 i \int \frac {1}{-\frac {(2-2 i) x^4}{x^4-1}-(2+2 i)}d\frac {x}{\sqrt [4]{x^4-1}}\right )\right )\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle 2 i \left (\frac {1}{16} \left (x^4-1\right )^{3/4} x+\frac {1}{16} \left (8 i \int \frac {1}{-\frac {(2+2 i) x^4}{x^4-1}-(2-2 i)}d\frac {x}{\sqrt [4]{x^4-1}}-(5+2 i) \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4-1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )\right )\right )\right )-2 i \left (\frac {1}{16} \left (x^4-1\right )^{3/4} x+\frac {1}{16} \left ((-5+2 i) \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4-1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )\right )-8 i \left (-\frac {1}{2} (-1)^{3/4} \int \frac {\frac {x^2}{\sqrt {x^4-1}}+\sqrt [4]{-1}}{-\frac {(2-2 i) x^4}{x^4-1}-(2+2 i)}d\frac {x}{\sqrt [4]{x^4-1}}-\frac {1}{2} (-1)^{3/4} \int \frac {\sqrt [4]{-1} \left (\frac {(-1)^{3/4} x^2}{\sqrt {x^4-1}}+1\right )}{-\frac {(2-2 i) x^4}{x^4-1}-(2+2 i)}d\frac {x}{\sqrt [4]{x^4-1}}\right )\right )\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 i \left (\frac {1}{16} \left (x^4-1\right )^{3/4} x+\frac {1}{16} \left (8 i \int \frac {1}{-\frac {(2+2 i) x^4}{x^4-1}-(2-2 i)}d\frac {x}{\sqrt [4]{x^4-1}}-(5+2 i) \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4-1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )\right )\right )\right )-2 i \left (\frac {1}{16} \left (x^4-1\right )^{3/4} x+\frac {1}{16} \left ((-5+2 i) \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4-1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )\right )-8 i \left (\frac {1}{2} \int \frac {\frac {(-1)^{3/4} x^2}{\sqrt {x^4-1}}+1}{-\frac {(2-2 i) x^4}{x^4-1}-(2+2 i)}d\frac {x}{\sqrt [4]{x^4-1}}-\frac {1}{2} (-1)^{3/4} \int \frac {\frac {x^2}{\sqrt {x^4-1}}+\sqrt [4]{-1}}{-\frac {(2-2 i) x^4}{x^4-1}-(2+2 i)}d\frac {x}{\sqrt [4]{x^4-1}}\right )\right )\right )\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle 2 i \left (\frac {1}{16} \left (x^4-1\right )^{3/4} x+\frac {1}{16} \left (8 i \left (-\frac {i \int \frac {1}{\sqrt [4]{-1}-\frac {x^2}{\sqrt {x^4-1}}}d\frac {x}{\sqrt [4]{x^4-1}}}{4 \sqrt {2}}-\frac {i \int \frac {1}{\frac {x^2}{\sqrt {x^4-1}}+\sqrt [4]{-1}}d\frac {x}{\sqrt [4]{x^4-1}}}{4 \sqrt {2}}\right )-(5+2 i) \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4-1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )\right )\right )\right )-2 i \left (\frac {1}{16} \left (x^4-1\right )^{3/4} x+\frac {1}{16} \left ((-5+2 i) \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4-1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )\right )-8 i \left (\frac {1}{2} \int \frac {\frac {(-1)^{3/4} x^2}{\sqrt {x^4-1}}+1}{-\frac {(2-2 i) x^4}{x^4-1}-(2+2 i)}d\frac {x}{\sqrt [4]{x^4-1}}-\frac {1}{2} (-1)^{3/4} \int \frac {\frac {x^2}{\sqrt {x^4-1}}+\sqrt [4]{-1}}{-\frac {(2-2 i) x^4}{x^4-1}-(2+2 i)}d\frac {x}{\sqrt [4]{x^4-1}}\right )\right )\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle 2 i \left (\frac {1}{16} \left (x^4-1\right )^{3/4} x+\frac {1}{16} \left (8 i \left (\frac {(-1)^{3/8} \arctan \left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4-1}}\right )}{4 \sqrt {2}}-\frac {i \int \frac {1}{\sqrt [4]{-1}-\frac {x^2}{\sqrt {x^4-1}}}d\frac {x}{\sqrt [4]{x^4-1}}}{4 \sqrt {2}}\right )-(5+2 i) \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4-1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )\right )\right )\right )-2 i \left (\frac {1}{16} \left (x^4-1\right )^{3/4} x+\frac {1}{16} \left ((-5+2 i) \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4-1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )\right )-8 i \left (\frac {1}{2} \int \frac {\frac {(-1)^{3/4} x^2}{\sqrt {x^4-1}}+1}{-\frac {(2-2 i) x^4}{x^4-1}-(2+2 i)}d\frac {x}{\sqrt [4]{x^4-1}}-\frac {1}{2} (-1)^{3/4} \int \frac {\frac {x^2}{\sqrt {x^4-1}}+\sqrt [4]{-1}}{-\frac {(2-2 i) x^4}{x^4-1}-(2+2 i)}d\frac {x}{\sqrt [4]{x^4-1}}\right )\right )\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle 2 i \left (\frac {1}{16} \left (x^4-1\right )^{3/4} x+\frac {1}{16} \left (8 i \left (\frac {(-1)^{3/8} \arctan \left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4-1}}\right )}{4 \sqrt {2}}+\frac {(-1)^{3/8} \text {arctanh}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4-1}}\right )}{4 \sqrt {2}}\right )-(5+2 i) \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4-1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )\right )\right )\right )-2 i \left (\frac {1}{16} \left (x^4-1\right )^{3/4} x+\frac {1}{16} \left ((-5+2 i) \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4-1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )\right )-8 i \left (\frac {1}{2} \int \frac {\frac {(-1)^{3/4} x^2}{\sqrt {x^4-1}}+1}{-\frac {(2-2 i) x^4}{x^4-1}-(2+2 i)}d\frac {x}{\sqrt [4]{x^4-1}}-\frac {1}{2} (-1)^{3/4} \int \frac {\frac {x^2}{\sqrt {x^4-1}}+\sqrt [4]{-1}}{-\frac {(2-2 i) x^4}{x^4-1}-(2+2 i)}d\frac {x}{\sqrt [4]{x^4-1}}\right )\right )\right )\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle 2 i \left (\frac {1}{16} \left (x^4-1\right )^{3/4} x+\frac {1}{16} \left (8 i \left (\frac {(-1)^{3/8} \arctan \left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4-1}}\right )}{4 \sqrt {2}}+\frac {(-1)^{3/8} \text {arctanh}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4-1}}\right )}{4 \sqrt {2}}\right )-(5+2 i) \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4-1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )\right )\right )\right )-2 i \left (\frac {1}{16} \left (x^4-1\right )^{3/4} x+\frac {1}{16} \left ((-5+2 i) \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4-1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )\right )-8 i \left (\frac {1}{2} \int \frac {\frac {(-1)^{3/4} x^2}{\sqrt {x^4-1}}+1}{-\frac {(2-2 i) x^4}{x^4-1}-(2+2 i)}d\frac {x}{\sqrt [4]{x^4-1}}-\frac {1}{2} (-1)^{3/4} \left (\left (-\frac {1}{8}-\frac {i}{8}\right ) \int \frac {1}{\frac {x^2}{\sqrt {x^4-1}}-\frac {\sqrt [8]{-1} \sqrt {2} x}{\sqrt [4]{x^4-1}}+\sqrt [4]{-1}}d\frac {x}{\sqrt [4]{x^4-1}}-\left (\frac {1}{8}+\frac {i}{8}\right ) \int \frac {1}{\frac {x^2}{\sqrt {x^4-1}}+\frac {\sqrt [8]{-1} \sqrt {2} x}{\sqrt [4]{x^4-1}}+\sqrt [4]{-1}}d\frac {x}{\sqrt [4]{x^4-1}}\right )\right )\right )\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle 2 i \left (\frac {1}{16} \left (x^4-1\right )^{3/4} x+\frac {1}{16} \left (8 i \left (\frac {(-1)^{3/8} \arctan \left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4-1}}\right )}{4 \sqrt {2}}+\frac {(-1)^{3/8} \text {arctanh}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4-1}}\right )}{4 \sqrt {2}}\right )-(5+2 i) \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4-1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )\right )\right )\right )-2 i \left (\frac {1}{16} \left (x^4-1\right )^{3/4} x+\frac {1}{16} \left ((-5+2 i) \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4-1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )\right )-8 i \left (\frac {1}{2} \int \frac {\frac {(-1)^{3/4} x^2}{\sqrt {x^4-1}}+1}{-\frac {(2-2 i) x^4}{x^4-1}-(2+2 i)}d\frac {x}{\sqrt [4]{x^4-1}}-\frac {1}{2} (-1)^{3/4} \left (\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (-1)^{7/8} \int \frac {1}{-\frac {x^2}{\sqrt {x^4-1}}-1}d\left (\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{x^4-1}}+1\right )}{\sqrt {2}}-\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (-1)^{7/8} \int \frac {1}{-\frac {x^2}{\sqrt {x^4-1}}-1}d\left (1-\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{x^4-1}}\right )}{\sqrt {2}}\right )\right )\right )\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 2 i \left (\frac {1}{16} \left (x^4-1\right )^{3/4} x+\frac {1}{16} \left (8 i \left (\frac {(-1)^{3/8} \arctan \left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4-1}}\right )}{4 \sqrt {2}}+\frac {(-1)^{3/8} \text {arctanh}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4-1}}\right )}{4 \sqrt {2}}\right )-(5+2 i) \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4-1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )\right )\right )\right )-2 i \left (\frac {1}{16} \left (x^4-1\right )^{3/4} x+\frac {1}{16} \left ((-5+2 i) \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4-1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )\right )-8 i \left (\frac {1}{2} \int \frac {\frac {(-1)^{3/4} x^2}{\sqrt {x^4-1}}+1}{-\frac {(2-2 i) x^4}{x^4-1}-(2+2 i)}d\frac {x}{\sqrt [4]{x^4-1}}-\frac {1}{2} (-1)^{3/4} \left (\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (-1)^{7/8} \arctan \left (1-\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{x^4-1}}\right )}{\sqrt {2}}-\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (-1)^{7/8} \arctan \left (\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{x^4-1}}+1\right )}{\sqrt {2}}\right )\right )\right )\right )\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle 2 i \left (\frac {1}{16} \left (x^4-1\right )^{3/4} x+\frac {1}{16} \left (8 i \left (\frac {(-1)^{3/8} \arctan \left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4-1}}\right )}{4 \sqrt {2}}+\frac {(-1)^{3/8} \text {arctanh}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4-1}}\right )}{4 \sqrt {2}}\right )-(5+2 i) \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4-1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )\right )\right )\right )-2 i \left (\frac {1}{16} \left (x^4-1\right )^{3/4} x+\frac {1}{16} \left ((-5+2 i) \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4-1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )\right )-8 i \left (\frac {1}{2} \left (-\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) (-1)^{5/8} \int -\frac {\sqrt [8]{-1} \sqrt {2}-\frac {2 x}{\sqrt [4]{x^4-1}}}{\frac {x^2}{\sqrt {x^4-1}}-\frac {\sqrt [8]{-1} \sqrt {2} x}{\sqrt [4]{x^4-1}}+\sqrt [4]{-1}}d\frac {x}{\sqrt [4]{x^4-1}}}{\sqrt {2}}-\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) (-1)^{5/8} \int -\frac {\sqrt {2} \left (\frac {\sqrt {2} x}{\sqrt [4]{x^4-1}}+\sqrt [8]{-1}\right )}{\frac {x^2}{\sqrt {x^4-1}}+\frac {\sqrt [8]{-1} \sqrt {2} x}{\sqrt [4]{x^4-1}}+\sqrt [4]{-1}}d\frac {x}{\sqrt [4]{x^4-1}}}{\sqrt {2}}\right )-\frac {1}{2} (-1)^{3/4} \left (\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (-1)^{7/8} \arctan \left (1-\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{x^4-1}}\right )}{\sqrt {2}}-\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (-1)^{7/8} \arctan \left (\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{x^4-1}}+1\right )}{\sqrt {2}}\right )\right )\right )\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 i \left (\frac {1}{16} \left (x^4-1\right )^{3/4} x+\frac {1}{16} \left (8 i \left (\frac {(-1)^{3/8} \arctan \left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4-1}}\right )}{4 \sqrt {2}}+\frac {(-1)^{3/8} \text {arctanh}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4-1}}\right )}{4 \sqrt {2}}\right )-(5+2 i) \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4-1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )\right )\right )\right )-2 i \left (\frac {1}{16} \left (x^4-1\right )^{3/4} x+\frac {1}{16} \left ((-5+2 i) \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4-1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )\right )-8 i \left (\frac {1}{2} \left (\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) (-1)^{5/8} \int \frac {\sqrt [8]{-1} \sqrt {2}-\frac {2 x}{\sqrt [4]{x^4-1}}}{\frac {x^2}{\sqrt {x^4-1}}-\frac {\sqrt [8]{-1} \sqrt {2} x}{\sqrt [4]{x^4-1}}+\sqrt [4]{-1}}d\frac {x}{\sqrt [4]{x^4-1}}}{\sqrt {2}}+\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) (-1)^{5/8} \int \frac {\sqrt {2} \left (\frac {\sqrt {2} x}{\sqrt [4]{x^4-1}}+\sqrt [8]{-1}\right )}{\frac {x^2}{\sqrt {x^4-1}}+\frac {\sqrt [8]{-1} \sqrt {2} x}{\sqrt [4]{x^4-1}}+\sqrt [4]{-1}}d\frac {x}{\sqrt [4]{x^4-1}}}{\sqrt {2}}\right )-\frac {1}{2} (-1)^{3/4} \left (\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (-1)^{7/8} \arctan \left (1-\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{x^4-1}}\right )}{\sqrt {2}}-\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (-1)^{7/8} \arctan \left (\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{x^4-1}}+1\right )}{\sqrt {2}}\right )\right )\right )\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 i \left (\frac {1}{16} \left (x^4-1\right )^{3/4} x+\frac {1}{16} \left (8 i \left (\frac {(-1)^{3/8} \arctan \left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4-1}}\right )}{4 \sqrt {2}}+\frac {(-1)^{3/8} \text {arctanh}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4-1}}\right )}{4 \sqrt {2}}\right )-(5+2 i) \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4-1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )\right )\right )\right )-2 i \left (\frac {1}{16} \left (x^4-1\right )^{3/4} x+\frac {1}{16} \left ((-5+2 i) \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4-1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )\right )-8 i \left (\frac {1}{2} \left (\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) (-1)^{5/8} \int \frac {\sqrt [8]{-1} \sqrt {2}-\frac {2 x}{\sqrt [4]{x^4-1}}}{\frac {x^2}{\sqrt {x^4-1}}-\frac {\sqrt [8]{-1} \sqrt {2} x}{\sqrt [4]{x^4-1}}+\sqrt [4]{-1}}d\frac {x}{\sqrt [4]{x^4-1}}}{\sqrt {2}}+\left (\frac {1}{8}+\frac {i}{8}\right ) (-1)^{5/8} \int \frac {\frac {\sqrt {2} x}{\sqrt [4]{x^4-1}}+\sqrt [8]{-1}}{\frac {x^2}{\sqrt {x^4-1}}+\frac {\sqrt [8]{-1} \sqrt {2} x}{\sqrt [4]{x^4-1}}+\sqrt [4]{-1}}d\frac {x}{\sqrt [4]{x^4-1}}\right )-\frac {1}{2} (-1)^{3/4} \left (\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (-1)^{7/8} \arctan \left (1-\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{x^4-1}}\right )}{\sqrt {2}}-\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (-1)^{7/8} \arctan \left (\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{x^4-1}}+1\right )}{\sqrt {2}}\right )\right )\right )\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 2 i \left (\frac {1}{16} \left (x^4-1\right )^{3/4} x+\frac {1}{16} \left (8 i \left (\frac {(-1)^{3/8} \arctan \left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4-1}}\right )}{4 \sqrt {2}}+\frac {(-1)^{3/8} \text {arctanh}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4-1}}\right )}{4 \sqrt {2}}\right )-(5+2 i) \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4-1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )\right )\right )\right )-2 i \left (\frac {1}{16} \left (x^4-1\right )^{3/4} x+\frac {1}{16} \left ((-5+2 i) \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4-1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )\right )-8 i \left (\frac {1}{2} \left (\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) (-1)^{5/8} \log \left (\frac {\sqrt [8]{-1} \sqrt {2} x}{\sqrt [4]{x^4-1}}+\frac {x^2}{\sqrt {x^4-1}}+\sqrt [4]{-1}\right )}{\sqrt {2}}-\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) (-1)^{5/8} \log \left (-\frac {\sqrt [8]{-1} \sqrt {2} x}{\sqrt [4]{x^4-1}}+\frac {x^2}{\sqrt {x^4-1}}+\sqrt [4]{-1}\right )}{\sqrt {2}}\right )-\frac {1}{2} (-1)^{3/4} \left (\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (-1)^{7/8} \arctan \left (1-\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{x^4-1}}\right )}{\sqrt {2}}-\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (-1)^{7/8} \arctan \left (\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{x^4-1}}+1\right )}{\sqrt {2}}\right )\right )\right )\right )\) |
(2*I)*((x*(-1 + x^4)^(3/4))/16 + ((-5 - 2*I)*(ArcTan[x/(-1 + x^4)^(1/4)]/2 + ArcTanh[x/(-1 + x^4)^(1/4)]/2) + (8*I)*(((-1)^(3/8)*ArcTan[((-1)^(7/8)* x)/(-1 + x^4)^(1/4)])/(4*Sqrt[2]) + ((-1)^(3/8)*ArcTanh[((-1)^(7/8)*x)/(-1 + x^4)^(1/4)])/(4*Sqrt[2])))/16) - (2*I)*((x*(-1 + x^4)^(3/4))/16 + ((-5 + 2*I)*(ArcTan[x/(-1 + x^4)^(1/4)]/2 + ArcTanh[x/(-1 + x^4)^(1/4)]/2) - (8 *I)*(-1/2*((-1)^(3/4)*(((1/4 + I/4)*(-1)^(7/8)*ArcTan[1 - ((-1)^(7/8)*Sqrt [2]*x)/(-1 + x^4)^(1/4)])/Sqrt[2] - ((1/4 + I/4)*(-1)^(7/8)*ArcTan[1 + ((- 1)^(7/8)*Sqrt[2]*x)/(-1 + x^4)^(1/4)])/Sqrt[2])) + (((-1/8 - I/8)*(-1)^(5/ 8)*Log[(-1)^(1/4) + x^2/Sqrt[-1 + x^4] - ((-1)^(1/8)*Sqrt[2]*x)/(-1 + x^4) ^(1/4)])/Sqrt[2] + ((1/8 + I/8)*(-1)^(5/8)*Log[(-1)^(1/4) + x^2/Sqrt[-1 + x^4] + ((-1)^(1/8)*Sqrt[2]*x)/(-1 + x^4)^(1/4)])/Sqrt[2])/2))/16)
3.29.49.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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])
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + 1/n) Subst[In t[1/(1 - b*x^n)^(p + 1/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p + 1 /n]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Su bst[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]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[d*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(b*(n*(p + q) + 1))), x] + Simp[1/(b*(n*(p + q) + 1)) Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Sim p[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d , 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] && !IGtQ[p, 1] && IntBinomialQ[ a, b, c, d, n, p, q, x]
Int[(((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)* (x_)^(n_)), x_Symbol] :> Simp[f/d Int[(a + b*x^n)^p, x], x] + Simp[(d*e - c*f)/d Int[(a + b*x^n)^p/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, p, n}, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[(u_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S imp[1/c^p Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_ )), x_Symbol] :> With[{r = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/r) Int[(d + e*x ^n)^q/(b - r + 2*c*x^n), x], x] - Simp[2*(c/r) Int[(d + e*x^n)^q/(b + r + 2*c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[n2, 2*n] && Ne Q[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[q]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 7.12 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.30
| method | result | size |
| pseudoelliptic | \(-\frac {\arctan \left (\frac {\left (x^{4}-1\right )^{\frac {1}{4}}}{x}\right )}{4}+\frac {\ln \left (\frac {\left (x^{4}-1\right )^{\frac {1}{4}}+x}{x}\right )}{8}-\frac {\ln \left (\frac {\left (x^{4}-1\right )^{\frac {1}{4}}-x}{x}\right )}{8}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )}{\sum }\frac {\left (\textit {\_R}^{4}+1\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{4}-1\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{5}}\right )}{16}\) | \(87\) |
| trager | \(\text {Expression too large to display}\) | \(1039\) |
-1/4*arctan((x^4-1)^(1/4)/x)+1/8*ln(((x^4-1)^(1/4)+x)/x)-1/8*ln(((x^4-1)^( 1/4)-x)/x)-1/16*sum(1/_R^5*(_R^4+1)*ln((-_R*x+(x^4-1)^(1/4))/x),_R=RootOf( _Z^8+1))
Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.17 \[ \int \frac {1-2 x^4+x^8}{\sqrt [4]{-1+x^4} \left (1-2 x^4+2 x^8\right )} \, dx=-\frac {1}{16} \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} \log \left (\frac {4 \, {\left (\sqrt {2} {\left (\left (-1\right )^{\frac {7}{8}} x + \left (-1\right )^{\frac {3}{8}} x\right )} + 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right )}}{x}\right ) + \frac {1}{16} \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} \log \left (-\frac {4 \, {\left (\sqrt {2} {\left (\left (-1\right )^{\frac {7}{8}} x + \left (-1\right )^{\frac {3}{8}} x\right )} - 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right )}}{x}\right ) - \frac {1}{16} i \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} \log \left (-\frac {4 \, {\left (\sqrt {2} {\left (i \, \left (-1\right )^{\frac {7}{8}} x + i \, \left (-1\right )^{\frac {3}{8}} x\right )} - 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right )}}{x}\right ) + \frac {1}{16} i \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} \log \left (-\frac {4 \, {\left (\sqrt {2} {\left (-i \, \left (-1\right )^{\frac {7}{8}} x - i \, \left (-1\right )^{\frac {3}{8}} x\right )} - 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right )}}{x}\right ) - \left (\frac {1}{16} i - \frac {1}{16}\right ) \, \left (-1\right )^{\frac {1}{8}} \log \left (-\frac {8 \, {\left (\left (i + 1\right ) \, \left (-1\right )^{\frac {7}{8}} x - \left (i + 1\right ) \, \left (-1\right )^{\frac {3}{8}} x - 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right )}}{x}\right ) + \left (\frac {1}{16} i + \frac {1}{16}\right ) \, \left (-1\right )^{\frac {1}{8}} \log \left (-\frac {8 \, {\left (-\left (i - 1\right ) \, \left (-1\right )^{\frac {7}{8}} x + \left (i - 1\right ) \, \left (-1\right )^{\frac {3}{8}} x - 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right )}}{x}\right ) - \left (\frac {1}{16} i + \frac {1}{16}\right ) \, \left (-1\right )^{\frac {1}{8}} \log \left (-\frac {8 \, {\left (\left (i - 1\right ) \, \left (-1\right )^{\frac {7}{8}} x - \left (i - 1\right ) \, \left (-1\right )^{\frac {3}{8}} x - 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right )}}{x}\right ) + \left (\frac {1}{16} i - \frac {1}{16}\right ) \, \left (-1\right )^{\frac {1}{8}} \log \left (-\frac {8 \, {\left (-\left (i + 1\right ) \, \left (-1\right )^{\frac {7}{8}} x + \left (i + 1\right ) \, \left (-1\right )^{\frac {3}{8}} x - 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right )}}{x}\right ) - \frac {1}{4} \, \arctan \left (\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{8} \, \log \left (\frac {x + {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{8} \, \log \left (-\frac {x - {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) \]
-1/16*sqrt(2)*(-1)^(1/8)*log(4*(sqrt(2)*((-1)^(7/8)*x + (-1)^(3/8)*x) + 2* (x^4 - 1)^(1/4))/x) + 1/16*sqrt(2)*(-1)^(1/8)*log(-4*(sqrt(2)*((-1)^(7/8)* x + (-1)^(3/8)*x) - 2*(x^4 - 1)^(1/4))/x) - 1/16*I*sqrt(2)*(-1)^(1/8)*log( -4*(sqrt(2)*(I*(-1)^(7/8)*x + I*(-1)^(3/8)*x) - 2*(x^4 - 1)^(1/4))/x) + 1/ 16*I*sqrt(2)*(-1)^(1/8)*log(-4*(sqrt(2)*(-I*(-1)^(7/8)*x - I*(-1)^(3/8)*x) - 2*(x^4 - 1)^(1/4))/x) - (1/16*I - 1/16)*(-1)^(1/8)*log(-8*((I + 1)*(-1) ^(7/8)*x - (I + 1)*(-1)^(3/8)*x - 2*(x^4 - 1)^(1/4))/x) + (1/16*I + 1/16)* (-1)^(1/8)*log(-8*(-(I - 1)*(-1)^(7/8)*x + (I - 1)*(-1)^(3/8)*x - 2*(x^4 - 1)^(1/4))/x) - (1/16*I + 1/16)*(-1)^(1/8)*log(-8*((I - 1)*(-1)^(7/8)*x - (I - 1)*(-1)^(3/8)*x - 2*(x^4 - 1)^(1/4))/x) + (1/16*I - 1/16)*(-1)^(1/8)* log(-8*(-(I + 1)*(-1)^(7/8)*x + (I + 1)*(-1)^(3/8)*x - 2*(x^4 - 1)^(1/4))/ x) - 1/4*arctan((x^4 - 1)^(1/4)/x) + 1/8*log((x + (x^4 - 1)^(1/4))/x) - 1/ 8*log(-(x - (x^4 - 1)^(1/4))/x)
Timed out. \[ \int \frac {1-2 x^4+x^8}{\sqrt [4]{-1+x^4} \left (1-2 x^4+2 x^8\right )} \, dx=\text {Timed out} \]
\[ \int \frac {1-2 x^4+x^8}{\sqrt [4]{-1+x^4} \left (1-2 x^4+2 x^8\right )} \, dx=\int { \frac {x^{8} - 2 \, x^{4} + 1}{{\left (2 \, x^{8} - 2 \, x^{4} + 1\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}} \,d x } \]
\[ \int \frac {1-2 x^4+x^8}{\sqrt [4]{-1+x^4} \left (1-2 x^4+2 x^8\right )} \, dx=\int { \frac {x^{8} - 2 \, x^{4} + 1}{{\left (2 \, x^{8} - 2 \, x^{4} + 1\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}} \,d x } \]
Timed out. \[ \int \frac {1-2 x^4+x^8}{\sqrt [4]{-1+x^4} \left (1-2 x^4+2 x^8\right )} \, dx=\int \frac {x^8-2\,x^4+1}{{\left (x^4-1\right )}^{1/4}\,\left (2\,x^8-2\,x^4+1\right )} \,d x \]