Integrand size = 24, antiderivative size = 225 \[ \int \frac {\left (1+x^4\right )^{3/4}}{1+2 x^4+2 x^8} \, dx=\frac {1}{8} \sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{1+x^4}}{-x^2+\sqrt {1+x^4}}\right )+\frac {1}{8} \sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{1+x^4}}{-x^2+\sqrt {1+x^4}}\right )+\frac {1}{8} \sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{1+x^4}}{x^2+\sqrt {1+x^4}}\right )+\frac {1}{8} \sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{1+x^4}}{x^2+\sqrt {1+x^4}}\right ) \]
1/8*(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)))+1/8*(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)))+1/8*(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)))+1/8*(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 = 0.50 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.96 \[ \int \frac {\left (1+x^4\right )^{3/4}}{1+2 x^4+2 x^8} \, dx=\frac {1}{8} \left (\sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{1+x^4}}{-x^2+\sqrt {1+x^4}}\right )+\sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{1+x^4}}{-x^2+\sqrt {1+x^4}}\right )+\sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{1+x^4}}{x^2+\sqrt {1+x^4}}\right )+\sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{1+x^4}}{x^2+\sqrt {1+x^4}}\right )\right ) \]
(Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[2 - Sqrt[2]]*x*(1 + x^4)^(1/4))/(-x^2 + Sq rt[1 + x^4])] + Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[2 + Sqrt[2]]*x*(1 + x^4)^(1 /4))/(-x^2 + Sqrt[1 + x^4])] + Sqrt[2 - Sqrt[2]]*ArcTanh[(Sqrt[2 - Sqrt[2] ]*x*(1 + x^4)^(1/4))/(x^2 + Sqrt[1 + x^4])] + Sqrt[2 + Sqrt[2]]*ArcTanh[(S qrt[2 + Sqrt[2]]*x*(1 + x^4)^(1/4))/(x^2 + Sqrt[1 + x^4])])/8
Result contains complex when optimal does not.
Time = 0.55 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.71, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.792, Rules used = {1758, 916, 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.
| \(\displaystyle \int \frac {\left (x^4+1\right )^{3/4}}{2 x^8+2 x^4+1} \, dx\) |
| \(\Big \downarrow \) 1758 |
| \(\displaystyle 2 i \int \frac {\left (x^4+1\right )^{3/4}}{4 x^4+(2+2 i)}dx-2 i \int \frac {\left (x^4+1\right )^{3/4}}{4 x^4+(2-2 i)}dx\) |
| \(\Big \downarrow \) 916 |
| \(\displaystyle 2 i \left (\frac {1}{4} \int \frac {1}{\sqrt [4]{x^4+1}}dx+\left (\frac {1}{2}-\frac {i}{2}\right ) \int \frac {1}{\sqrt [4]{x^4+1} \left (4 x^4+(2+2 i)\right )}dx\right )-2 i \left (\frac {1}{4} \int \frac {1}{\sqrt [4]{x^4+1}}dx+\left (\frac {1}{2}+\frac {i}{2}\right ) \int \frac {1}{\sqrt [4]{x^4+1} \left (4 x^4+(2-2 i)\right )}dx\right )\) |
| \(\Big \downarrow \) 770 |
| \(\displaystyle 2 i \left (\frac {1}{4} \int \frac {1}{1-\frac {x^4}{x^4+1}}d\frac {x}{\sqrt [4]{x^4+1}}+\left (\frac {1}{2}-\frac {i}{2}\right ) \int \frac {1}{\sqrt [4]{x^4+1} \left (4 x^4+(2+2 i)\right )}dx\right )-2 i \left (\frac {1}{4} \int \frac {1}{1-\frac {x^4}{x^4+1}}d\frac {x}{\sqrt [4]{x^4+1}}+\left (\frac {1}{2}+\frac {i}{2}\right ) \int \frac {1}{\sqrt [4]{x^4+1} \left (4 x^4+(2-2 i)\right )}dx\right )\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle 2 i \left (\frac {1}{4} \left (\frac {1}{2} \int \frac {1}{1-\frac {x^2}{\sqrt {x^4+1}}}d\frac {x}{\sqrt [4]{x^4+1}}+\frac {1}{2} \int \frac {1}{\frac {x^2}{\sqrt {x^4+1}}+1}d\frac {x}{\sqrt [4]{x^4+1}}\right )+\left (\frac {1}{2}-\frac {i}{2}\right ) \int \frac {1}{\sqrt [4]{x^4+1} \left (4 x^4+(2+2 i)\right )}dx\right )-2 i \left (\frac {1}{4} \left (\frac {1}{2} \int \frac {1}{1-\frac {x^2}{\sqrt {x^4+1}}}d\frac {x}{\sqrt [4]{x^4+1}}+\frac {1}{2} \int \frac {1}{\frac {x^2}{\sqrt {x^4+1}}+1}d\frac {x}{\sqrt [4]{x^4+1}}\right )+\left (\frac {1}{2}+\frac {i}{2}\right ) \int \frac {1}{\sqrt [4]{x^4+1} \left (4 x^4+(2-2 i)\right )}dx\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle 2 i \left (\frac {1}{4} \left (\frac {1}{2} \int \frac {1}{1-\frac {x^2}{\sqrt {x^4+1}}}d\frac {x}{\sqrt [4]{x^4+1}}+\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4+1}}\right )\right )+\left (\frac {1}{2}-\frac {i}{2}\right ) \int \frac {1}{\sqrt [4]{x^4+1} \left (4 x^4+(2+2 i)\right )}dx\right )-2 i \left (\frac {1}{4} \left (\frac {1}{2} \int \frac {1}{1-\frac {x^2}{\sqrt {x^4+1}}}d\frac {x}{\sqrt [4]{x^4+1}}+\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4+1}}\right )\right )+\left (\frac {1}{2}+\frac {i}{2}\right ) \int \frac {1}{\sqrt [4]{x^4+1} \left (4 x^4+(2-2 i)\right )}dx\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle 2 i \left (\frac {1}{4} \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )\right )+\left (\frac {1}{2}-\frac {i}{2}\right ) \int \frac {1}{\sqrt [4]{x^4+1} \left (4 x^4+(2+2 i)\right )}dx\right )-2 i \left (\frac {1}{4} \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )\right )+\left (\frac {1}{2}+\frac {i}{2}\right ) \int \frac {1}{\sqrt [4]{x^4+1} \left (4 x^4+(2-2 i)\right )}dx\right )\) |
| \(\Big \downarrow \) 902 |
| \(\displaystyle 2 i \left (\frac {1}{4} \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )\right )+\left (\frac {1}{2}-\frac {i}{2}\right ) \int \frac {1}{\frac {(2-2 i) x^4}{x^4+1}+(2+2 i)}d\frac {x}{\sqrt [4]{x^4+1}}\right )-2 i \left (\frac {1}{4} \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )\right )+\left (\frac {1}{2}+\frac {i}{2}\right ) \int \frac {1}{\frac {(2+2 i) x^4}{x^4+1}+(2-2 i)}d\frac {x}{\sqrt [4]{x^4+1}}\right )\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle 2 i \left (\frac {1}{4} \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )\right )+\left (\frac {1}{2}-\frac {i}{2}\right ) \left (-\frac {1}{2} (-1)^{3/4} \int \frac {\frac {x^2}{\sqrt {x^4+1}}+\sqrt [4]{-1}}{\frac {(2-2 i) x^4}{x^4+1}+(2+2 i)}d\frac {x}{\sqrt [4]{x^4+1}}-\frac {1}{2} (-1)^{3/4} \int \frac {\sqrt [4]{-1} \left (\frac {(-1)^{3/4} x^2}{\sqrt {x^4+1}}+1\right )}{\frac {(2-2 i) x^4}{x^4+1}+(2+2 i)}d\frac {x}{\sqrt [4]{x^4+1}}\right )\right )-2 i \left (\frac {1}{4} \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )\right )+\left (\frac {1}{2}+\frac {i}{2}\right ) \int \frac {1}{\frac {(2+2 i) x^4}{x^4+1}+(2-2 i)}d\frac {x}{\sqrt [4]{x^4+1}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 i \left (\frac {1}{4} \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )\right )+\left (\frac {1}{2}-\frac {i}{2}\right ) \left (\frac {1}{2} \int \frac {\frac {(-1)^{3/4} x^2}{\sqrt {x^4+1}}+1}{\frac {(2-2 i) x^4}{x^4+1}+(2+2 i)}d\frac {x}{\sqrt [4]{x^4+1}}-\frac {1}{2} (-1)^{3/4} \int \frac {\frac {x^2}{\sqrt {x^4+1}}+\sqrt [4]{-1}}{\frac {(2-2 i) x^4}{x^4+1}+(2+2 i)}d\frac {x}{\sqrt [4]{x^4+1}}\right )\right )-2 i \left (\frac {1}{4} \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )\right )+\left (\frac {1}{2}+\frac {i}{2}\right ) \int \frac {1}{\frac {(2+2 i) x^4}{x^4+1}+(2-2 i)}d\frac {x}{\sqrt [4]{x^4+1}}\right )\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle 2 i \left (\frac {1}{4} \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )\right )+\left (\frac {1}{2}-\frac {i}{2}\right ) \left (\frac {1}{2} \int \frac {\frac {(-1)^{3/4} x^2}{\sqrt {x^4+1}}+1}{\frac {(2-2 i) x^4}{x^4+1}+(2+2 i)}d\frac {x}{\sqrt [4]{x^4+1}}-\frac {1}{2} (-1)^{3/4} \int \frac {\frac {x^2}{\sqrt {x^4+1}}+\sqrt [4]{-1}}{\frac {(2-2 i) x^4}{x^4+1}+(2+2 i)}d\frac {x}{\sqrt [4]{x^4+1}}\right )\right )-2 i \left (\frac {1}{4} \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )\right )+\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\frac {i \int \frac {1}{\sqrt [4]{-1}-\frac {x^2}{\sqrt {x^4+1}}}d\frac {x}{\sqrt [4]{x^4+1}}}{4 \sqrt {2}}+\frac {i \int \frac {1}{\frac {x^2}{\sqrt {x^4+1}}+\sqrt [4]{-1}}d\frac {x}{\sqrt [4]{x^4+1}}}{4 \sqrt {2}}\right )\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle 2 i \left (\frac {1}{4} \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )\right )+\left (\frac {1}{2}-\frac {i}{2}\right ) \left (\frac {1}{2} \int \frac {\frac {(-1)^{3/4} x^2}{\sqrt {x^4+1}}+1}{\frac {(2-2 i) x^4}{x^4+1}+(2+2 i)}d\frac {x}{\sqrt [4]{x^4+1}}-\frac {1}{2} (-1)^{3/4} \int \frac {\frac {x^2}{\sqrt {x^4+1}}+\sqrt [4]{-1}}{\frac {(2-2 i) x^4}{x^4+1}+(2+2 i)}d\frac {x}{\sqrt [4]{x^4+1}}\right )\right )-2 i \left (\frac {1}{4} \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )\right )+\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-\frac {(-1)^{3/8} \arctan \left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4+1}}\right )}{4 \sqrt {2}}+\frac {i \int \frac {1}{\sqrt [4]{-1}-\frac {x^2}{\sqrt {x^4+1}}}d\frac {x}{\sqrt [4]{x^4+1}}}{4 \sqrt {2}}\right )\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle 2 i \left (\frac {1}{4} \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )\right )+\left (\frac {1}{2}-\frac {i}{2}\right ) \left (\frac {1}{2} \int \frac {\frac {(-1)^{3/4} x^2}{\sqrt {x^4+1}}+1}{\frac {(2-2 i) x^4}{x^4+1}+(2+2 i)}d\frac {x}{\sqrt [4]{x^4+1}}-\frac {1}{2} (-1)^{3/4} \int \frac {\frac {x^2}{\sqrt {x^4+1}}+\sqrt [4]{-1}}{\frac {(2-2 i) x^4}{x^4+1}+(2+2 i)}d\frac {x}{\sqrt [4]{x^4+1}}\right )\right )-2 i \left (\frac {1}{4} \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )\right )+\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-\frac {(-1)^{3/8} \arctan \left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4+1}}\right )}{4 \sqrt {2}}-\frac {(-1)^{3/8} \text {arctanh}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4+1}}\right )}{4 \sqrt {2}}\right )\right )\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle 2 i \left (\frac {1}{4} \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )\right )+\left (\frac {1}{2}-\frac {i}{2}\right ) \left (\frac {1}{2} \int \frac {\frac {(-1)^{3/4} x^2}{\sqrt {x^4+1}}+1}{\frac {(2-2 i) x^4}{x^4+1}+(2+2 i)}d\frac {x}{\sqrt [4]{x^4+1}}-\frac {1}{2} (-1)^{3/4} \left (\left (\frac {1}{8}+\frac {i}{8}\right ) \int \frac {1}{\frac {x^2}{\sqrt {x^4+1}}-\frac {\sqrt [8]{-1} \sqrt {2} x}{\sqrt [4]{x^4+1}}+\sqrt [4]{-1}}d\frac {x}{\sqrt [4]{x^4+1}}+\left (\frac {1}{8}+\frac {i}{8}\right ) \int \frac {1}{\frac {x^2}{\sqrt {x^4+1}}+\frac {\sqrt [8]{-1} \sqrt {2} x}{\sqrt [4]{x^4+1}}+\sqrt [4]{-1}}d\frac {x}{\sqrt [4]{x^4+1}}\right )\right )\right )-2 i \left (\frac {1}{4} \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )\right )+\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-\frac {(-1)^{3/8} \arctan \left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4+1}}\right )}{4 \sqrt {2}}-\frac {(-1)^{3/8} \text {arctanh}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4+1}}\right )}{4 \sqrt {2}}\right )\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle 2 i \left (\frac {1}{4} \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )\right )+\left (\frac {1}{2}-\frac {i}{2}\right ) \left (\frac {1}{2} \int \frac {\frac {(-1)^{3/4} x^2}{\sqrt {x^4+1}}+1}{\frac {(2-2 i) x^4}{x^4+1}+(2+2 i)}d\frac {x}{\sqrt [4]{x^4+1}}-\frac {1}{2} (-1)^{3/4} \left (\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (-1)^{7/8} \int \frac {1}{-\frac {x^2}{\sqrt {x^4+1}}-1}d\left (1-\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{x^4+1}}\right )}{\sqrt {2}}-\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (-1)^{7/8} \int \frac {1}{-\frac {x^2}{\sqrt {x^4+1}}-1}d\left (\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{x^4+1}}+1\right )}{\sqrt {2}}\right )\right )\right )-2 i \left (\frac {1}{4} \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )\right )+\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-\frac {(-1)^{3/8} \arctan \left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4+1}}\right )}{4 \sqrt {2}}-\frac {(-1)^{3/8} \text {arctanh}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4+1}}\right )}{4 \sqrt {2}}\right )\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 2 i \left (\frac {1}{4} \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )\right )+\left (\frac {1}{2}-\frac {i}{2}\right ) \left (\frac {1}{2} \int \frac {\frac {(-1)^{3/4} x^2}{\sqrt {x^4+1}}+1}{\frac {(2-2 i) x^4}{x^4+1}+(2+2 i)}d\frac {x}{\sqrt [4]{x^4+1}}-\frac {1}{2} (-1)^{3/4} \left (\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (-1)^{7/8} \arctan \left (\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{x^4+1}}+1\right )}{\sqrt {2}}-\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (-1)^{7/8} \arctan \left (1-\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{x^4+1}}\right )}{\sqrt {2}}\right )\right )\right )-2 i \left (\frac {1}{4} \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )\right )+\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-\frac {(-1)^{3/8} \arctan \left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4+1}}\right )}{4 \sqrt {2}}-\frac {(-1)^{3/8} \text {arctanh}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4+1}}\right )}{4 \sqrt {2}}\right )\right )\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle 2 i \left (\frac {1}{4} \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )\right )+\left (\frac {1}{2}-\frac {i}{2}\right ) \left (\frac {1}{2} \left (\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) (-1)^{5/8} \int -\frac {\sqrt [8]{-1} \sqrt {2}-\frac {2 x}{\sqrt [4]{x^4+1}}}{\frac {x^2}{\sqrt {x^4+1}}-\frac {\sqrt [8]{-1} \sqrt {2} x}{\sqrt [4]{x^4+1}}+\sqrt [4]{-1}}d\frac {x}{\sqrt [4]{x^4+1}}}{\sqrt {2}}+\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) (-1)^{5/8} \int -\frac {\sqrt {2} \left (\frac {\sqrt {2} x}{\sqrt [4]{x^4+1}}+\sqrt [8]{-1}\right )}{\frac {x^2}{\sqrt {x^4+1}}+\frac {\sqrt [8]{-1} \sqrt {2} x}{\sqrt [4]{x^4+1}}+\sqrt [4]{-1}}d\frac {x}{\sqrt [4]{x^4+1}}}{\sqrt {2}}\right )-\frac {1}{2} (-1)^{3/4} \left (\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (-1)^{7/8} \arctan \left (\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{x^4+1}}+1\right )}{\sqrt {2}}-\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (-1)^{7/8} \arctan \left (1-\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{x^4+1}}\right )}{\sqrt {2}}\right )\right )\right )-2 i \left (\frac {1}{4} \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )\right )+\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-\frac {(-1)^{3/8} \arctan \left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4+1}}\right )}{4 \sqrt {2}}-\frac {(-1)^{3/8} \text {arctanh}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4+1}}\right )}{4 \sqrt {2}}\right )\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 i \left (\frac {1}{4} \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )\right )+\left (\frac {1}{2}-\frac {i}{2}\right ) \left (\frac {1}{2} \left (-\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) (-1)^{5/8} \int \frac {\sqrt [8]{-1} \sqrt {2}-\frac {2 x}{\sqrt [4]{x^4+1}}}{\frac {x^2}{\sqrt {x^4+1}}-\frac {\sqrt [8]{-1} \sqrt {2} x}{\sqrt [4]{x^4+1}}+\sqrt [4]{-1}}d\frac {x}{\sqrt [4]{x^4+1}}}{\sqrt {2}}-\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) (-1)^{5/8} \int \frac {\sqrt {2} \left (\frac {\sqrt {2} x}{\sqrt [4]{x^4+1}}+\sqrt [8]{-1}\right )}{\frac {x^2}{\sqrt {x^4+1}}+\frac {\sqrt [8]{-1} \sqrt {2} x}{\sqrt [4]{x^4+1}}+\sqrt [4]{-1}}d\frac {x}{\sqrt [4]{x^4+1}}}{\sqrt {2}}\right )-\frac {1}{2} (-1)^{3/4} \left (\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (-1)^{7/8} \arctan \left (\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{x^4+1}}+1\right )}{\sqrt {2}}-\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (-1)^{7/8} \arctan \left (1-\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{x^4+1}}\right )}{\sqrt {2}}\right )\right )\right )-2 i \left (\frac {1}{4} \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )\right )+\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-\frac {(-1)^{3/8} \arctan \left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4+1}}\right )}{4 \sqrt {2}}-\frac {(-1)^{3/8} \text {arctanh}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4+1}}\right )}{4 \sqrt {2}}\right )\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 i \left (\frac {1}{4} \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )\right )+\left (\frac {1}{2}-\frac {i}{2}\right ) \left (\frac {1}{2} \left (-\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) (-1)^{5/8} \int \frac {\sqrt [8]{-1} \sqrt {2}-\frac {2 x}{\sqrt [4]{x^4+1}}}{\frac {x^2}{\sqrt {x^4+1}}-\frac {\sqrt [8]{-1} \sqrt {2} x}{\sqrt [4]{x^4+1}}+\sqrt [4]{-1}}d\frac {x}{\sqrt [4]{x^4+1}}}{\sqrt {2}}-\left (\frac {1}{8}+\frac {i}{8}\right ) (-1)^{5/8} \int \frac {\frac {\sqrt {2} x}{\sqrt [4]{x^4+1}}+\sqrt [8]{-1}}{\frac {x^2}{\sqrt {x^4+1}}+\frac {\sqrt [8]{-1} \sqrt {2} x}{\sqrt [4]{x^4+1}}+\sqrt [4]{-1}}d\frac {x}{\sqrt [4]{x^4+1}}\right )-\frac {1}{2} (-1)^{3/4} \left (\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (-1)^{7/8} \arctan \left (\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{x^4+1}}+1\right )}{\sqrt {2}}-\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (-1)^{7/8} \arctan \left (1-\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{x^4+1}}\right )}{\sqrt {2}}\right )\right )\right )-2 i \left (\frac {1}{4} \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )\right )+\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-\frac {(-1)^{3/8} \arctan \left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4+1}}\right )}{4 \sqrt {2}}-\frac {(-1)^{3/8} \text {arctanh}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4+1}}\right )}{4 \sqrt {2}}\right )\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 2 i \left (\frac {1}{4} \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )\right )+\left (\frac {1}{2}-\frac {i}{2}\right ) \left (\frac {1}{2} \left (\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) (-1)^{5/8} \log \left (-\frac {\sqrt [8]{-1} \sqrt {2} x}{\sqrt [4]{x^4+1}}+\frac {x^2}{\sqrt {x^4+1}}+\sqrt [4]{-1}\right )}{\sqrt {2}}-\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) (-1)^{5/8} \log \left (\frac {\sqrt [8]{-1} \sqrt {2} x}{\sqrt [4]{x^4+1}}+\frac {x^2}{\sqrt {x^4+1}}+\sqrt [4]{-1}\right )}{\sqrt {2}}\right )-\frac {1}{2} (-1)^{3/4} \left (\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (-1)^{7/8} \arctan \left (\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{x^4+1}}+1\right )}{\sqrt {2}}-\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (-1)^{7/8} \arctan \left (1-\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{x^4+1}}\right )}{\sqrt {2}}\right )\right )\right )-2 i \left (\frac {1}{4} \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )\right )+\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-\frac {(-1)^{3/8} \arctan \left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4+1}}\right )}{4 \sqrt {2}}-\frac {(-1)^{3/8} \text {arctanh}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4+1}}\right )}{4 \sqrt {2}}\right )\right )\) |
(-2*I)*((ArcTan[x/(1 + x^4)^(1/4)]/2 + ArcTanh[x/(1 + x^4)^(1/4)]/2)/4 + ( 1/2 + I/2)*(-1/4*((-1)^(3/8)*ArcTan[((-1)^(7/8)*x)/(1 + x^4)^(1/4)])/Sqrt[ 2] - ((-1)^(3/8)*ArcTanh[((-1)^(7/8)*x)/(1 + x^4)^(1/4)])/(4*Sqrt[2]))) + (2*I)*((ArcTan[x/(1 + x^4)^(1/4)]/2 + ArcTanh[x/(1 + x^4)^(1/4)]/2)/4 + (1 /2 - I/2)*(-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))
3.26.100.3.1 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_)), x_Symbol] :> Si mp[b/d Int[(a + b*x^n)^(p - 1), x], x] - Simp[(b*c - a*d)/d Int[(a + b* x^n)^(p - 1)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p - 1) + 1, 0] && IntegerQ[n]
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[((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 = 2.68 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.15
| method | result | size |
| pseudoelliptic | \(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{4}+1\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}}\right )}{8}\) | \(33\) |
| trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{7} \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{11} x^{4}+\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{7} x^{4}+\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{7}-2 \left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{2} x^{3}-2 \sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right ) x^{2}-2 \left (x^{4}+1\right )^{\frac {3}{4}} x}{\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{4} x^{4}-x^{4}-1}\right )}{8}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{3} \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{7} x^{4}-2 \sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{5} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{3} x^{4}-2 \left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{2} x^{3}+2 \left (x^{4}+1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{3}}{\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{4} x^{4}-x^{4}-1}\right )}{8}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{5} \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{9} x^{4}+2 \left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{6} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{5} x^{4}+2 \sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{3} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{5}+2 \left (x^{4}+1\right )^{\frac {3}{4}} x}{\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{4} x^{4}+x^{4}+1}\right )}{8}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right ) \ln \left (-\frac {2 \sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{7} x^{2}-2 \left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{6} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{5} x^{4}-\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right ) x^{4}+2 \left (x^{4}+1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )}{\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{4} x^{4}+x^{4}+1}\right )}{8}\) | \(463\) |
Result contains complex when optimal does not.
Time = 5.43 (sec) , antiderivative size = 1115, normalized size of antiderivative = 4.96 \[ \int \frac {\left (1+x^4\right )^{3/4}}{1+2 x^4+2 x^8} \, dx=\text {Too large to display} \]
(1/32*I - 1/32)*sqrt(2)*(-1)^(1/8)*log(-(2*sqrt(2)*sqrt(x^4 + 1)*((I + 1)* (-1)^(3/8)*x^2 + (-1)^(7/8)*((2*I + 2)*x^6 + (I + 1)*x^2)) + 4*(2*x^5 - (I - 1)*x)*(x^4 + 1)^(3/4) - sqrt(2)*((-1)^(5/8)*(-(2*I - 2)*x^8 + I - 1) + (-1)^(1/8)*(-(2*I - 2)*x^8 - (4*I - 4)*x^4 - I + 1)) + 4*(x^4 + 1)^(1/4)*( -I*(-1)^(1/4)*x^3 + (-1)^(3/4)*(-2*I*x^7 - I*x^3)))/(2*x^8 + 2*x^4 + 1)) - (1/32*I + 1/32)*sqrt(2)*(-1)^(1/8)*log(-(2*sqrt(2)*sqrt(x^4 + 1)*(-(I - 1 )*(-1)^(3/8)*x^2 + (-1)^(7/8)*(-(2*I - 2)*x^6 - (I - 1)*x^2)) + 4*(2*x^5 - (I - 1)*x)*(x^4 + 1)^(3/4) - sqrt(2)*((-1)^(5/8)*((2*I + 2)*x^8 - I - 1) + (-1)^(1/8)*((2*I + 2)*x^8 + (4*I + 4)*x^4 + I + 1)) + 4*(x^4 + 1)^(1/4)* (I*(-1)^(1/4)*x^3 + (-1)^(3/4)*(2*I*x^7 + I*x^3)))/(2*x^8 + 2*x^4 + 1)) + (1/32*I + 1/32)*sqrt(2)*(-1)^(1/8)*log(-(2*sqrt(2)*sqrt(x^4 + 1)*((I - 1)* (-1)^(3/8)*x^2 + (-1)^(7/8)*((2*I - 2)*x^6 + (I - 1)*x^2)) + 4*(2*x^5 - (I - 1)*x)*(x^4 + 1)^(3/4) - sqrt(2)*((-1)^(5/8)*(-(2*I + 2)*x^8 + I + 1) + (-1)^(1/8)*(-(2*I + 2)*x^8 - (4*I + 4)*x^4 - I - 1)) + 4*(x^4 + 1)^(1/4)*( I*(-1)^(1/4)*x^3 + (-1)^(3/4)*(2*I*x^7 + I*x^3)))/(2*x^8 + 2*x^4 + 1)) - ( 1/32*I - 1/32)*sqrt(2)*(-1)^(1/8)*log(-(2*sqrt(2)*sqrt(x^4 + 1)*(-(I + 1)* (-1)^(3/8)*x^2 + (-1)^(7/8)*(-(2*I + 2)*x^6 - (I + 1)*x^2)) + 4*(2*x^5 - ( I - 1)*x)*(x^4 + 1)^(3/4) - sqrt(2)*((-1)^(5/8)*((2*I - 2)*x^8 - I + 1) + (-1)^(1/8)*((2*I - 2)*x^8 + (4*I - 4)*x^4 + I - 1)) + 4*(x^4 + 1)^(1/4)*(- I*(-1)^(1/4)*x^3 + (-1)^(3/4)*(-2*I*x^7 - I*x^3)))/(2*x^8 + 2*x^4 + 1))...
Timed out. \[ \int \frac {\left (1+x^4\right )^{3/4}}{1+2 x^4+2 x^8} \, dx=\text {Timed out} \]
\[ \int \frac {\left (1+x^4\right )^{3/4}}{1+2 x^4+2 x^8} \, dx=\int { \frac {{\left (x^{4} + 1\right )}^{\frac {3}{4}}}{2 \, x^{8} + 2 \, x^{4} + 1} \,d x } \]
\[ \int \frac {\left (1+x^4\right )^{3/4}}{1+2 x^4+2 x^8} \, dx=\int { \frac {{\left (x^{4} + 1\right )}^{\frac {3}{4}}}{2 \, x^{8} + 2 \, x^{4} + 1} \,d x } \]
Timed out. \[ \int \frac {\left (1+x^4\right )^{3/4}}{1+2 x^4+2 x^8} \, dx=\int \frac {{\left (x^4+1\right )}^{3/4}}{2\,x^8+2\,x^4+1} \,d x \]