Integrand size = 51, antiderivative size = 69 \[ \int \frac {\left (-1+x^4\right ) \left (1+x^4\right ) \sqrt {-1-x^2+x^4}}{\left (-2-x^2+2 x^4\right )^2 \left (-2+x^2+2 x^4\right )} \, dx=-\frac {x \sqrt {-1-x^2+x^4}}{16 \left (-2-x^2+2 x^4\right )}-\frac {1}{16} \sqrt {\frac {3}{2}} \arctan \left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt {-1-x^2+x^4}}\right ) \] Output:
-x*(x^4-x^2-1)^(1/2)/(32*x^4-16*x^2-32)-1/32*6^(1/2)*arctan(1/2*6^(1/2)*x/ (x^4-x^2-1)^(1/2))
Time = 0.84 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.94 \[ \int \frac {\left (-1+x^4\right ) \left (1+x^4\right ) \sqrt {-1-x^2+x^4}}{\left (-2-x^2+2 x^4\right )^2 \left (-2+x^2+2 x^4\right )} \, dx=\frac {1}{32} \left (\frac {2 x \sqrt {-1-x^2+x^4}}{2+x^2-2 x^4}-\sqrt {6} \arctan \left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt {-1-x^2+x^4}}\right )\right ) \] Input:
Integrate[((-1 + x^4)*(1 + x^4)*Sqrt[-1 - x^2 + x^4])/((-2 - x^2 + 2*x^4)^ 2*(-2 + x^2 + 2*x^4)),x]
Output:
((2*x*Sqrt[-1 - x^2 + x^4])/(2 + x^2 - 2*x^4) - Sqrt[6]*ArcTan[(Sqrt[3/2]* x)/Sqrt[-1 - x^2 + x^4]])/32
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 9.81 (sec) , antiderivative size = 6101, normalized size of antiderivative = 88.42, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.039, Rules used = {7279, 2009}
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.
| \(\displaystyle \int \frac {\left (x^4-1\right ) \left (x^4+1\right ) \sqrt {x^4-x^2-1}}{\left (2 x^4-x^2-2\right )^2 \left (2 x^4+x^2-2\right )} \, dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {\sqrt {x^4-x^2-1} \left (-4 x^2-1\right )}{16 \left (2 x^4+x^2-2\right )}+\frac {\left (4 x^2+1\right ) \sqrt {x^4-x^2-1}}{16 \left (2 x^4-x^2-2\right )}+\frac {\left (x^2+4\right ) \sqrt {x^4-x^2-1}}{8 \left (2 x^4-x^2-2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 \sqrt {x^4-x^2-1} x}{17 \left (1-\sqrt {17}\right ) \left (-4 x^2-\sqrt {17}+1\right )}+\frac {\sqrt {x^4-x^2-1} x}{68 \left (-4 x^2-\sqrt {17}+1\right )}+\frac {4 \sqrt {x^4-x^2-1} x}{17 \left (1+\sqrt {17}\right ) \left (-4 x^2+\sqrt {17}+1\right )}+\frac {\sqrt {x^4-x^2-1} x}{68 \left (-4 x^2+\sqrt {17}+1\right )}-\frac {\left (17+2 \sqrt {17}\right ) \left (-2 x^2-\sqrt {5}+1\right ) x}{544 \sqrt {x^4-x^2-1}}-\frac {\left (-2 x^2-\sqrt {5}+1\right ) x}{34 \left (1+\sqrt {17}\right ) \sqrt {x^4-x^2-1}}-\frac {\left (-2 x^2-\sqrt {5}+1\right ) x}{34 \left (1-\sqrt {17}\right ) \sqrt {x^4-x^2-1}}-\frac {\left (17-2 \sqrt {17}\right ) \left (-2 x^2-\sqrt {5}+1\right ) x}{544 \sqrt {x^4-x^2-1}}+\frac {\left (-2 x^2-\sqrt {5}+1\right ) x}{17 \sqrt {x^4-x^2-1}}-\frac {\sqrt [4]{5} \left (17+2 \sqrt {17}\right ) \sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{544 \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}-\frac {\sqrt [4]{5} \sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{34 \left (1+\sqrt {17}\right ) \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}-\frac {\sqrt [4]{5} \sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{34 \left (1-\sqrt {17}\right ) \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}-\frac {\sqrt [4]{5} \left (17-2 \sqrt {17}\right ) \sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{544 \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}+\frac {\sqrt [4]{5} \sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{17 \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}+\frac {3 \left (1+\sqrt {5}\right ) \left (1+\sqrt {17}\right ) \sqrt {2 x^2+\sqrt {5}-1} \sqrt {1-\frac {2 x^2}{1+\sqrt {5}}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{64 \sqrt {2} \left (3+2 \sqrt {5}+\sqrt {17}\right ) \sqrt {x^4-x^2-1}}-\frac {\left (1+\sqrt {5}\right ) \left (1-\sqrt {17}\right ) \left (2-\sqrt {17}\right ) \sqrt {2 x^2+\sqrt {5}-1} \sqrt {1-\frac {2 x^2}{1+\sqrt {5}}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{64 \sqrt {34} \left (1+2 \sqrt {5}+\sqrt {17}\right ) \sqrt {x^4-x^2-1}}+\frac {3 \left (1+\sqrt {5}\right ) \left (1-\sqrt {17}\right ) \sqrt {2 x^2+\sqrt {5}-1} \sqrt {1-\frac {2 x^2}{1+\sqrt {5}}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{64 \sqrt {34} \left (1+2 \sqrt {5}+\sqrt {17}\right ) \sqrt {x^4-x^2-1}}+\frac {\left (1+\sqrt {5}\right ) \sqrt {2 x^2+\sqrt {5}-1} \sqrt {1-\frac {2 x^2}{1+\sqrt {5}}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{4 \sqrt {34} \left (1+2 \sqrt {5}+\sqrt {17}\right ) \sqrt {x^4-x^2-1}}+\frac {\left (1+\sqrt {5}\right ) \left (1+\sqrt {17}\right ) \left (2+\sqrt {17}\right ) \sqrt {2 x^2+\sqrt {5}-1} \sqrt {1-\frac {2 x^2}{1+\sqrt {5}}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{64 \sqrt {34} \left (1+2 \sqrt {5}-\sqrt {17}\right ) \sqrt {x^4-x^2-1}}-\frac {3 \left (1+\sqrt {5}\right ) \left (1+\sqrt {17}\right ) \sqrt {2 x^2+\sqrt {5}-1} \sqrt {1-\frac {2 x^2}{1+\sqrt {5}}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{64 \sqrt {34} \left (1+2 \sqrt {5}-\sqrt {17}\right ) \sqrt {x^4-x^2-1}}+\frac {3 \left (1+\sqrt {5}\right ) \left (1-\sqrt {17}\right ) \sqrt {2 x^2+\sqrt {5}-1} \sqrt {1-\frac {2 x^2}{1+\sqrt {5}}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{64 \sqrt {2} \left (3+2 \sqrt {5}-\sqrt {17}\right ) \sqrt {x^4-x^2-1}}-\frac {\left (1+\sqrt {5}\right ) \sqrt {2 x^2+\sqrt {5}-1} \sqrt {1-\frac {2 x^2}{1+\sqrt {5}}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{4 \sqrt {34} \left (1+2 \sqrt {5}-\sqrt {17}\right ) \sqrt {x^4-x^2-1}}-\frac {\left (17+\sqrt {17} \left (1-2 \sqrt {5}\right )\right ) \sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{1088 \sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}-\frac {\left (17-\sqrt {17} \left (1-2 \sqrt {5}\right )\right ) \sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{1088 \sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}-\frac {\left (51+19 \sqrt {17}\right ) \sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{1088 \sqrt [4]{5} \left (1+2 \sqrt {5}-\sqrt {17}\right ) \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}-\frac {\left (1-2 \sqrt {5}-\sqrt {17}\right ) \left (17+2 \sqrt {17}\right ) \sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{2176 \sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}+\frac {\left (3+2 \sqrt {5}+\sqrt {17}\right ) \sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{136 \sqrt [4]{5} \left (1+\sqrt {17}\right ) \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}+\frac {\left (3+2 \sqrt {5}+\sqrt {17}\right ) \sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{2176 \sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}-\frac {3 \left (1+\sqrt {17}\right ) \sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{64 \sqrt [4]{5} \left (3+2 \sqrt {5}+\sqrt {17}\right ) \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}+\frac {3 \left (17-\sqrt {17}\right ) \sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{1088 \sqrt [4]{5} \left (1+2 \sqrt {5}+\sqrt {17}\right ) \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}-\frac {\left (51-19 \sqrt {17}\right ) \sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{1088 \sqrt [4]{5} \left (1+2 \sqrt {5}+\sqrt {17}\right ) \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}-\frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{4 \sqrt [4]{5} \sqrt {17} \left (1+2 \sqrt {5}+\sqrt {17}\right ) \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}+\frac {\left (3-2 \sqrt {5}+\sqrt {17}\right ) \sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{128 \sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}-\frac {\left (17-2 \sqrt {17}\right ) \left (1-2 \sqrt {5}+\sqrt {17}\right ) \sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{2176 \sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}+\frac {3 \left (17+\sqrt {17}\right ) \sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{1088 \sqrt [4]{5} \left (1+2 \sqrt {5}-\sqrt {17}\right ) \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}+\frac {\left (3+2 \sqrt {5}-\sqrt {17}\right ) \sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{136 \sqrt [4]{5} \left (1-\sqrt {17}\right ) \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}+\frac {\left (3+2 \sqrt {5}-\sqrt {17}\right ) \sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{2176 \sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}-\frac {3 \left (1-\sqrt {17}\right ) \sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{64 \sqrt [4]{5} \left (3+2 \sqrt {5}-\sqrt {17}\right ) \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}+\frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{4 \sqrt [4]{5} \sqrt {17} \left (1+2 \sqrt {5}-\sqrt {17}\right ) \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}+\frac {\left (3-2 \sqrt {5}-\sqrt {17}\right ) \sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{128 \sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}-\frac {3 \left (1+\sqrt {5}\right ) \sqrt {2 x^2+\sqrt {5}-1} \sqrt {1-\frac {2 x^2}{1+\sqrt {5}}} \operatorname {EllipticPi}\left (-\frac {2 \left (1+\sqrt {5}\right )}{1-\sqrt {17}},\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{64 \sqrt {2} \sqrt {x^4-x^2-1}}-\frac {\left (1+\sqrt {5}\right ) \left (2-\sqrt {17}\right ) \sqrt {2 x^2+\sqrt {5}-1} \sqrt {1-\frac {2 x^2}{1+\sqrt {5}}} \operatorname {EllipticPi}\left (\frac {2 \left (1+\sqrt {5}\right )}{1-\sqrt {17}},\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{64 \sqrt {34} \sqrt {x^4-x^2-1}}+\frac {\left (1+\sqrt {5}\right ) \sqrt {2 x^2+\sqrt {5}-1} \sqrt {1-\frac {2 x^2}{1+\sqrt {5}}} \operatorname {EllipticPi}\left (\frac {2 \left (1+\sqrt {5}\right )}{1-\sqrt {17}},\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{4 \sqrt {34} \left (1-\sqrt {17}\right ) \sqrt {x^4-x^2-1}}+\frac {3 \left (1+\sqrt {5}\right ) \sqrt {2 x^2+\sqrt {5}-1} \sqrt {1-\frac {2 x^2}{1+\sqrt {5}}} \operatorname {EllipticPi}\left (\frac {2 \left (1+\sqrt {5}\right )}{1-\sqrt {17}},\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{64 \sqrt {34} \sqrt {x^4-x^2-1}}-\frac {3 \left (1+\sqrt {5}\right ) \sqrt {2 x^2+\sqrt {5}-1} \sqrt {1-\frac {2 x^2}{1+\sqrt {5}}} \operatorname {EllipticPi}\left (-\frac {2 \left (1+\sqrt {5}\right )}{1+\sqrt {17}},\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{64 \sqrt {2} \sqrt {x^4-x^2-1}}+\frac {\left (1+\sqrt {5}\right ) \left (2+\sqrt {17}\right ) \sqrt {2 x^2+\sqrt {5}-1} \sqrt {1-\frac {2 x^2}{1+\sqrt {5}}} \operatorname {EllipticPi}\left (\frac {2 \left (1+\sqrt {5}\right )}{1+\sqrt {17}},\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{64 \sqrt {34} \sqrt {x^4-x^2-1}}-\frac {\left (1+\sqrt {5}\right ) \sqrt {2 x^2+\sqrt {5}-1} \sqrt {1-\frac {2 x^2}{1+\sqrt {5}}} \operatorname {EllipticPi}\left (\frac {2 \left (1+\sqrt {5}\right )}{1+\sqrt {17}},\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{4 \sqrt {34} \left (1+\sqrt {17}\right ) \sqrt {x^4-x^2-1}}-\frac {3 \left (1+\sqrt {5}\right ) \sqrt {2 x^2+\sqrt {5}-1} \sqrt {1-\frac {2 x^2}{1+\sqrt {5}}} \operatorname {EllipticPi}\left (\frac {2 \left (1+\sqrt {5}\right )}{1+\sqrt {17}},\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{64 \sqrt {34} \sqrt {x^4-x^2-1}}\) |
Input:
Int[((-1 + x^4)*(1 + x^4)*Sqrt[-1 - x^2 + x^4])/((-2 - x^2 + 2*x^4)^2*(-2 + x^2 + 2*x^4)),x]
Output:
(x*(1 - Sqrt[5] - 2*x^2))/(17*Sqrt[-1 - x^2 + x^4]) - ((17 - 2*Sqrt[17])*x *(1 - Sqrt[5] - 2*x^2))/(544*Sqrt[-1 - x^2 + x^4]) - (x*(1 - Sqrt[5] - 2*x ^2))/(34*(1 - Sqrt[17])*Sqrt[-1 - x^2 + x^4]) - (x*(1 - Sqrt[5] - 2*x^2))/ (34*(1 + Sqrt[17])*Sqrt[-1 - x^2 + x^4]) - ((17 + 2*Sqrt[17])*x*(1 - Sqrt[ 5] - 2*x^2))/(544*Sqrt[-1 - x^2 + x^4]) + (x*Sqrt[-1 - x^2 + x^4])/(68*(1 - Sqrt[17] - 4*x^2)) + (4*x*Sqrt[-1 - x^2 + x^4])/(17*(1 - Sqrt[17])*(1 - Sqrt[17] - 4*x^2)) + (x*Sqrt[-1 - x^2 + x^4])/(68*(1 + Sqrt[17] - 4*x^2)) + (4*x*Sqrt[-1 - x^2 + x^4])/(17*(1 + Sqrt[17])*(1 + Sqrt[17] - 4*x^2)) + (5^(1/4)*Sqrt[-2 - (1 - Sqrt[5])*x^2]*Sqrt[(2 + (1 + Sqrt[5])*x^2)/(2 + (1 - Sqrt[5])*x^2)]*EllipticE[ArcSin[(Sqrt[2]*5^(1/4)*x)/Sqrt[-2 - (1 - Sqrt [5])*x^2]], (5 - Sqrt[5])/10])/(17*Sqrt[(2 + (1 - Sqrt[5])*x^2)^(-1)]*Sqrt [-1 - x^2 + x^4]) - (5^(1/4)*(17 - 2*Sqrt[17])*Sqrt[-2 - (1 - Sqrt[5])*x^2 ]*Sqrt[(2 + (1 + Sqrt[5])*x^2)/(2 + (1 - Sqrt[5])*x^2)]*EllipticE[ArcSin[( Sqrt[2]*5^(1/4)*x)/Sqrt[-2 - (1 - Sqrt[5])*x^2]], (5 - Sqrt[5])/10])/(544* Sqrt[(2 + (1 - Sqrt[5])*x^2)^(-1)]*Sqrt[-1 - x^2 + x^4]) - (5^(1/4)*Sqrt[- 2 - (1 - Sqrt[5])*x^2]*Sqrt[(2 + (1 + Sqrt[5])*x^2)/(2 + (1 - Sqrt[5])*x^2 )]*EllipticE[ArcSin[(Sqrt[2]*5^(1/4)*x)/Sqrt[-2 - (1 - Sqrt[5])*x^2]], (5 - Sqrt[5])/10])/(34*(1 - Sqrt[17])*Sqrt[(2 + (1 - Sqrt[5])*x^2)^(-1)]*Sqrt [-1 - x^2 + x^4]) - (5^(1/4)*Sqrt[-2 - (1 - Sqrt[5])*x^2]*Sqrt[(2 + (1 + S qrt[5])*x^2)/(2 + (1 - Sqrt[5])*x^2)]*EllipticE[ArcSin[(Sqrt[2]*5^(1/4)...
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] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 4.23 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.83
| method | result | size |
| risch | \(-\frac {x \sqrt {x^{4}-x^{2}-1}}{16 \left (2 x^{4}-x^{2}-2\right )}+\frac {\sqrt {6}\, \arctan \left (\frac {\sqrt {6}\, \sqrt {x^{4}-x^{2}-1}}{3 x}\right )}{32}\) | \(57\) |
| default | \(\frac {\sqrt {6}\, \left (2 x^{4}-x^{2}-2\right ) \arctan \left (\frac {\sqrt {6}\, \sqrt {x^{4}-x^{2}-1}}{3 x}\right )-2 x \sqrt {x^{4}-x^{2}-1}}{64 x^{4}-32 x^{2}-64}\) | \(69\) |
| pseudoelliptic | \(\frac {\sqrt {6}\, \left (2 x^{4}-x^{2}-2\right ) \arctan \left (\frac {\sqrt {6}\, \sqrt {x^{4}-x^{2}-1}}{3 x}\right )-2 x \sqrt {x^{4}-x^{2}-1}}{64 x^{4}-32 x^{2}-64}\) | \(69\) |
| elliptic | \(\frac {\left (\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {x^{4}-x^{2}-1}\, \sqrt {2}\, \sqrt {3}}{3 x}\right )}{16}-\frac {\sqrt {x^{4}-x^{2}-1}\, \sqrt {2}}{64 x \left (\frac {x^{4}-x^{2}-1}{2 x^{2}}+\frac {1}{4}\right )}\right ) \sqrt {2}}{2}\) | \(75\) |
| trager | \(-\frac {x \sqrt {x^{4}-x^{2}-1}}{16 \left (2 x^{4}-x^{2}-2\right )}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+6\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6\right ) x^{4}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6\right ) x^{2}+12 x \sqrt {x^{4}-x^{2}-1}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6\right )}{2 x^{4}+x^{2}-2}\right )}{64}\) | \(100\) |
Input:
int((x^4-1)*(x^4+1)*(x^4-x^2-1)^(1/2)/(2*x^4-x^2-2)^2/(2*x^4+x^2-2),x,meth od=_RETURNVERBOSE)
Output:
-1/16*x*(x^4-x^2-1)^(1/2)/(2*x^4-x^2-2)+1/32*6^(1/2)*arctan(1/3*6^(1/2)/x* (x^4-x^2-1)^(1/2))
Time = 0.16 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.17 \[ \int \frac {\left (-1+x^4\right ) \left (1+x^4\right ) \sqrt {-1-x^2+x^4}}{\left (-2-x^2+2 x^4\right )^2 \left (-2+x^2+2 x^4\right )} \, dx=-\frac {\sqrt {\frac {3}{2}} {\left (2 \, x^{4} - x^{2} - 2\right )} \arctan \left (\frac {4 \, \sqrt {\frac {3}{2}} \sqrt {x^{4} - x^{2} - 1} x}{2 \, x^{4} - 5 \, x^{2} - 2}\right ) + 2 \, \sqrt {x^{4} - x^{2} - 1} x}{32 \, {\left (2 \, x^{4} - x^{2} - 2\right )}} \] Input:
integrate((x^4-1)*(x^4+1)*(x^4-x^2-1)^(1/2)/(2*x^4-x^2-2)^2/(2*x^4+x^2-2), x, algorithm="fricas")
Output:
-1/32*(sqrt(3/2)*(2*x^4 - x^2 - 2)*arctan(4*sqrt(3/2)*sqrt(x^4 - x^2 - 1)* x/(2*x^4 - 5*x^2 - 2)) + 2*sqrt(x^4 - x^2 - 1)*x)/(2*x^4 - x^2 - 2)
Timed out. \[ \int \frac {\left (-1+x^4\right ) \left (1+x^4\right ) \sqrt {-1-x^2+x^4}}{\left (-2-x^2+2 x^4\right )^2 \left (-2+x^2+2 x^4\right )} \, dx=\text {Timed out} \] Input:
integrate((x**4-1)*(x**4+1)*(x**4-x**2-1)**(1/2)/(2*x**4-x**2-2)**2/(2*x** 4+x**2-2),x)
Output:
Timed out
\[ \int \frac {\left (-1+x^4\right ) \left (1+x^4\right ) \sqrt {-1-x^2+x^4}}{\left (-2-x^2+2 x^4\right )^2 \left (-2+x^2+2 x^4\right )} \, dx=\int { \frac {\sqrt {x^{4} - x^{2} - 1} {\left (x^{4} + 1\right )} {\left (x^{4} - 1\right )}}{{\left (2 \, x^{4} + x^{2} - 2\right )} {\left (2 \, x^{4} - x^{2} - 2\right )}^{2}} \,d x } \] Input:
integrate((x^4-1)*(x^4+1)*(x^4-x^2-1)^(1/2)/(2*x^4-x^2-2)^2/(2*x^4+x^2-2), x, algorithm="maxima")
Output:
integrate(sqrt(x^4 - x^2 - 1)*(x^4 + 1)*(x^4 - 1)/((2*x^4 + x^2 - 2)*(2*x^ 4 - x^2 - 2)^2), x)
\[ \int \frac {\left (-1+x^4\right ) \left (1+x^4\right ) \sqrt {-1-x^2+x^4}}{\left (-2-x^2+2 x^4\right )^2 \left (-2+x^2+2 x^4\right )} \, dx=\int { \frac {\sqrt {x^{4} - x^{2} - 1} {\left (x^{4} + 1\right )} {\left (x^{4} - 1\right )}}{{\left (2 \, x^{4} + x^{2} - 2\right )} {\left (2 \, x^{4} - x^{2} - 2\right )}^{2}} \,d x } \] Input:
integrate((x^4-1)*(x^4+1)*(x^4-x^2-1)^(1/2)/(2*x^4-x^2-2)^2/(2*x^4+x^2-2), x, algorithm="giac")
Output:
integrate(sqrt(x^4 - x^2 - 1)*(x^4 + 1)*(x^4 - 1)/((2*x^4 + x^2 - 2)*(2*x^ 4 - x^2 - 2)^2), x)
Timed out. \[ \int \frac {\left (-1+x^4\right ) \left (1+x^4\right ) \sqrt {-1-x^2+x^4}}{\left (-2-x^2+2 x^4\right )^2 \left (-2+x^2+2 x^4\right )} \, dx=\int \frac {\left (x^4-1\right )\,\left (x^4+1\right )\,\sqrt {x^4-x^2-1}}{{\left (-2\,x^4+x^2+2\right )}^2\,\left (2\,x^4+x^2-2\right )} \,d x \] Input:
int(((x^4 - 1)*(x^4 + 1)*(x^4 - x^2 - 1)^(1/2))/((x^2 - 2*x^4 + 2)^2*(x^2 + 2*x^4 - 2)),x)
Output:
int(((x^4 - 1)*(x^4 + 1)*(x^4 - x^2 - 1)^(1/2))/((x^2 - 2*x^4 + 2)^2*(x^2 + 2*x^4 - 2)), x)
\[ \int \frac {\left (-1+x^4\right ) \left (1+x^4\right ) \sqrt {-1-x^2+x^4}}{\left (-2-x^2+2 x^4\right )^2 \left (-2+x^2+2 x^4\right )} \, dx=\frac {-\sqrt {x^{4}-x^{2}-1}\, x +6 \left (\int \frac {\sqrt {x^{4}-x^{2}-1}}{8 x^{16}-12 x^{14}-30 x^{12}+39 x^{10}+43 x^{8}-39 x^{6}-30 x^{4}+12 x^{2}+8}d x \right ) x^{4}-3 \left (\int \frac {\sqrt {x^{4}-x^{2}-1}}{8 x^{16}-12 x^{14}-30 x^{12}+39 x^{10}+43 x^{8}-39 x^{6}-30 x^{4}+12 x^{2}+8}d x \right ) x^{2}-6 \left (\int \frac {\sqrt {x^{4}-x^{2}-1}}{8 x^{16}-12 x^{14}-30 x^{12}+39 x^{10}+43 x^{8}-39 x^{6}-30 x^{4}+12 x^{2}+8}d x \right )+6 \left (\int \frac {\sqrt {x^{4}-x^{2}-1}\, x^{12}}{8 x^{16}-12 x^{14}-30 x^{12}+39 x^{10}+43 x^{8}-39 x^{6}-30 x^{4}+12 x^{2}+8}d x \right ) x^{4}-3 \left (\int \frac {\sqrt {x^{4}-x^{2}-1}\, x^{12}}{8 x^{16}-12 x^{14}-30 x^{12}+39 x^{10}+43 x^{8}-39 x^{6}-30 x^{4}+12 x^{2}+8}d x \right ) x^{2}-6 \left (\int \frac {\sqrt {x^{4}-x^{2}-1}\, x^{12}}{8 x^{16}-12 x^{14}-30 x^{12}+39 x^{10}+43 x^{8}-39 x^{6}-30 x^{4}+12 x^{2}+8}d x \right )-6 \left (\int \frac {\sqrt {x^{4}-x^{2}-1}\, x^{10}}{8 x^{16}-12 x^{14}-30 x^{12}+39 x^{10}+43 x^{8}-39 x^{6}-30 x^{4}+12 x^{2}+8}d x \right ) x^{4}+3 \left (\int \frac {\sqrt {x^{4}-x^{2}-1}\, x^{10}}{8 x^{16}-12 x^{14}-30 x^{12}+39 x^{10}+43 x^{8}-39 x^{6}-30 x^{4}+12 x^{2}+8}d x \right ) x^{2}+6 \left (\int \frac {\sqrt {x^{4}-x^{2}-1}\, x^{10}}{8 x^{16}-12 x^{14}-30 x^{12}+39 x^{10}+43 x^{8}-39 x^{6}-30 x^{4}+12 x^{2}+8}d x \right )+6 \left (\int \frac {\sqrt {x^{4}-x^{2}-1}\, x^{2}}{8 x^{16}-12 x^{14}-30 x^{12}+39 x^{10}+43 x^{8}-39 x^{6}-30 x^{4}+12 x^{2}+8}d x \right ) x^{4}-3 \left (\int \frac {\sqrt {x^{4}-x^{2}-1}\, x^{2}}{8 x^{16}-12 x^{14}-30 x^{12}+39 x^{10}+43 x^{8}-39 x^{6}-30 x^{4}+12 x^{2}+8}d x \right ) x^{2}-6 \left (\int \frac {\sqrt {x^{4}-x^{2}-1}\, x^{2}}{8 x^{16}-12 x^{14}-30 x^{12}+39 x^{10}+43 x^{8}-39 x^{6}-30 x^{4}+12 x^{2}+8}d x \right )}{14 x^{4}-7 x^{2}-14} \] Input:
int((x^4-1)*(x^4+1)*(x^4-x^2-1)^(1/2)/(2*x^4-x^2-2)^2/(2*x^4+x^2-2),x)
Output:
( - sqrt(x**4 - x**2 - 1)*x + 6*int(sqrt(x**4 - x**2 - 1)/(8*x**16 - 12*x* *14 - 30*x**12 + 39*x**10 + 43*x**8 - 39*x**6 - 30*x**4 + 12*x**2 + 8),x)* x**4 - 3*int(sqrt(x**4 - x**2 - 1)/(8*x**16 - 12*x**14 - 30*x**12 + 39*x** 10 + 43*x**8 - 39*x**6 - 30*x**4 + 12*x**2 + 8),x)*x**2 - 6*int(sqrt(x**4 - x**2 - 1)/(8*x**16 - 12*x**14 - 30*x**12 + 39*x**10 + 43*x**8 - 39*x**6 - 30*x**4 + 12*x**2 + 8),x) + 6*int((sqrt(x**4 - x**2 - 1)*x**12)/(8*x**16 - 12*x**14 - 30*x**12 + 39*x**10 + 43*x**8 - 39*x**6 - 30*x**4 + 12*x**2 + 8),x)*x**4 - 3*int((sqrt(x**4 - x**2 - 1)*x**12)/(8*x**16 - 12*x**14 - 3 0*x**12 + 39*x**10 + 43*x**8 - 39*x**6 - 30*x**4 + 12*x**2 + 8),x)*x**2 - 6*int((sqrt(x**4 - x**2 - 1)*x**12)/(8*x**16 - 12*x**14 - 30*x**12 + 39*x* *10 + 43*x**8 - 39*x**6 - 30*x**4 + 12*x**2 + 8),x) - 6*int((sqrt(x**4 - x **2 - 1)*x**10)/(8*x**16 - 12*x**14 - 30*x**12 + 39*x**10 + 43*x**8 - 39*x **6 - 30*x**4 + 12*x**2 + 8),x)*x**4 + 3*int((sqrt(x**4 - x**2 - 1)*x**10) /(8*x**16 - 12*x**14 - 30*x**12 + 39*x**10 + 43*x**8 - 39*x**6 - 30*x**4 + 12*x**2 + 8),x)*x**2 + 6*int((sqrt(x**4 - x**2 - 1)*x**10)/(8*x**16 - 12* x**14 - 30*x**12 + 39*x**10 + 43*x**8 - 39*x**6 - 30*x**4 + 12*x**2 + 8),x ) + 6*int((sqrt(x**4 - x**2 - 1)*x**2)/(8*x**16 - 12*x**14 - 30*x**12 + 39 *x**10 + 43*x**8 - 39*x**6 - 30*x**4 + 12*x**2 + 8),x)*x**4 - 3*int((sqrt( x**4 - x**2 - 1)*x**2)/(8*x**16 - 12*x**14 - 30*x**12 + 39*x**10 + 43*x**8 - 39*x**6 - 30*x**4 + 12*x**2 + 8),x)*x**2 - 6*int((sqrt(x**4 - x**2 -...