Integrand size = 61, antiderivative size = 163 \[ \int \frac {\sqrt {1+2 x^2-x^4} \left (-1+x^4\right ) \left (1+x^4\right )}{\left (-1-x^2+x^4\right ) \left (1+3 x^2-x^4-3 x^6+x^8\right )} \, dx=-\sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \arctan \left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x \sqrt {1+2 x^2-x^4}}{-1-2 x^2+x^4}\right )-\text {arctanh}\left (\frac {x \sqrt {1+2 x^2-x^4}}{-1-2 x^2+x^4}\right )+\sqrt {\frac {1}{10} \left (-1+\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x \sqrt {1+2 x^2-x^4}}{-1-2 x^2+x^4}\right ) \] Output:
-1/10*(10+10*5^(1/2))^(1/2)*arctan(1/2*(-2+2*5^(1/2))^(1/2)*x*(-x^4+2*x^2+ 1)^(1/2)/(x^4-2*x^2-1))-arctanh(x*(-x^4+2*x^2+1)^(1/2)/(x^4-2*x^2-1))+1/10 *(-10+10*5^(1/2))^(1/2)*arctanh(1/2*(2+2*5^(1/2))^(1/2)*x*(-x^4+2*x^2+1)^( 1/2)/(x^4-2*x^2-1))
Time = 0.81 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt {1+2 x^2-x^4} \left (-1+x^4\right ) \left (1+x^4\right )}{\left (-1-x^2+x^4\right ) \left (1+3 x^2-x^4-3 x^6+x^8\right )} \, dx=\sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \arctan \left (\frac {\sqrt {-1+\sqrt {5}} x}{\sqrt {2+4 x^2-2 x^4}}\right )-\sqrt {\frac {1}{10} \left (-1+\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {1+\sqrt {5}} x}{\sqrt {2+4 x^2-2 x^4}}\right )+\text {arctanh}\left (\frac {x}{\sqrt {1+2 x^2-x^4}}\right ) \] Input:
Integrate[(Sqrt[1 + 2*x^2 - x^4]*(-1 + x^4)*(1 + x^4))/((-1 - x^2 + x^4)*( 1 + 3*x^2 - x^4 - 3*x^6 + x^8)),x]
Output:
Sqrt[(1 + Sqrt[5])/10]*ArcTan[(Sqrt[-1 + Sqrt[5]]*x)/Sqrt[2 + 4*x^2 - 2*x^ 4]] - Sqrt[(-1 + Sqrt[5])/10]*ArcTanh[(Sqrt[1 + Sqrt[5]]*x)/Sqrt[2 + 4*x^2 - 2*x^4]] + ArcTanh[x/Sqrt[1 + 2*x^2 - x^4]]
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.
| \(\displaystyle \int \frac {\sqrt {-x^4+2 x^2+1} \left (x^4-1\right ) \left (x^4+1\right )}{\left (x^4-x^2-1\right ) \left (x^8-3 x^6-x^4+3 x^2+1\right )} \, dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {\sqrt {-x^4+2 x^2+1} \left (1-2 x^2\right )}{x^4-x^2-1}+\frac {\sqrt {-x^4+2 x^2+1} \left (2 x^6-4 x^4-x^2+2\right )}{x^8-3 x^6-x^4+3 x^2+1}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \int \frac {\sqrt {-x^4+2 x^2+1}}{x^8-3 x^6-x^4+3 x^2+1}dx-\int \frac {x^2 \sqrt {-x^4+2 x^2+1}}{x^8-3 x^6-x^4+3 x^2+1}dx-4 \int \frac {x^4 \sqrt {-x^4+2 x^2+1}}{x^8-3 x^6-x^4+3 x^2+1}dx+2 \int \frac {x^6 \sqrt {-x^4+2 x^2+1}}{x^8-3 x^6-x^4+3 x^2+1}dx-\frac {\left (1+2 \sqrt {2}+\sqrt {5}\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {-1+\sqrt {2}} x\right ),-3-2 \sqrt {2}\right )}{2 \sqrt {\sqrt {2}-1}}-\frac {\left (1+2 \sqrt {2}-\sqrt {5}\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {-1+\sqrt {2}} x\right ),-3-2 \sqrt {2}\right )}{2 \sqrt {\sqrt {2}-1}}+\frac {2 E\left (\arcsin \left (\sqrt {-1+\sqrt {2}} x\right )|-3-2 \sqrt {2}\right )}{\sqrt {1+\sqrt {2}}}+\frac {\operatorname {EllipticPi}\left (\frac {2 \left (1+\sqrt {2}\right )}{1-\sqrt {5}},\arcsin \left (\sqrt {-1+\sqrt {2}} x\right ),-3-2 \sqrt {2}\right )}{\sqrt {\sqrt {2}-1}}+\frac {\operatorname {EllipticPi}\left (\frac {2 \left (1+\sqrt {2}\right )}{1+\sqrt {5}},\arcsin \left (\sqrt {-1+\sqrt {2}} x\right ),-3-2 \sqrt {2}\right )}{\sqrt {\sqrt {2}-1}}\) |
Input:
Int[(Sqrt[1 + 2*x^2 - x^4]*(-1 + x^4)*(1 + x^4))/((-1 - x^2 + x^4)*(1 + 3* x^2 - x^4 - 3*x^6 + x^8)),x]
Output:
$Aborted
Time = 12.27 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.69
| method | result | size |
| elliptic | \(\frac {\left (-\frac {2 \sqrt {5}\, \arctan \left (\frac {\sqrt {-x^{4}+2 x^{2}+1}\, \sqrt {2}}{x \sqrt {\sqrt {5}-1}}\right )}{5 \sqrt {\sqrt {5}-1}}-\frac {2 \sqrt {5}\, \operatorname {arctanh}\left (\frac {\sqrt {-x^{4}+2 x^{2}+1}\, \sqrt {2}}{x \sqrt {\sqrt {5}+1}}\right )}{5 \sqrt {\sqrt {5}+1}}+\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-x^{4}+2 x^{2}+1}}{x}\right )\right ) \sqrt {2}}{2}\) | \(112\) |
| default | \(\frac {\left (-\frac {1}{50}-\frac {3 i}{50}\right ) \left (\left (\left (-10+30 i\right ) \sqrt {2+2 i}\, \sqrt {-x^{4}+2 x^{2}+1}+\left (80-40 i\right ) x \right ) \operatorname {arctanh}\left (\frac {\sqrt {-x^{4}+2 x^{2}+1}}{x}\right )+\left (\frac {\arctan \left (\frac {-15 \sqrt {-x^{4}+2 x^{2}+1}\, \sqrt {2+2 i}\, x +3 \left (\left (-\frac {1}{3}+i\right ) x^{4}+\left (-2-\frac {2 i}{3}\right ) x^{2}+\frac {1}{3}-i\right ) \sqrt {5}+3 \left (-5+\frac {5 i}{3}\right ) x^{4}+20 x^{2}+15-5 i}{\sqrt {65+105 i+\left (15+55 i\right ) \sqrt {5}}\, \left (x^{2}+i\right )^{2}}\right ) \left (3+i+\left (1+3 i\right ) \sqrt {5}\right ) \sqrt {65+105 i+\left (-15-55 i\right ) \sqrt {5}}}{3}+80-40 i+\frac {\arctan \left (\frac {15 \sqrt {-x^{4}+2 x^{2}+1}\, \sqrt {2+2 i}\, x +3 \left (\left (-\frac {1}{3}+i\right ) x^{4}+\left (-2-\frac {2 i}{3}\right ) x^{2}+\frac {1}{3}-i\right ) \sqrt {5}+3 \left (5-\frac {5 i}{3}\right ) x^{4}-20 x^{2}-15+5 i}{\sqrt {65+105 i+\left (-15-55 i\right ) \sqrt {5}}\, \left (x^{2}+i\right )^{2}}\right ) \left (-3-i+\left (1+3 i\right ) \sqrt {5}\right ) \sqrt {65+105 i+\left (15+55 i\right ) \sqrt {5}}}{3}\right ) \left (\sqrt {2+2 i}\, x -\sqrt {-x^{4}+2 x^{2}+1}\right )\right ) x}{\left (i x^{2}-1+\sqrt {2+2 i}\, x -\sqrt {-x^{4}+2 x^{2}+1}\right ) \left (i x^{2}-1-\sqrt {2+2 i}\, x +\sqrt {-x^{4}+2 x^{2}+1}\right )}\) | \(370\) |
| pseudoelliptic | \(\frac {\left (-\frac {1}{50}-\frac {3 i}{50}\right ) \left (\left (\left (-10+30 i\right ) \sqrt {2+2 i}\, \sqrt {-x^{4}+2 x^{2}+1}+\left (80-40 i\right ) x \right ) \operatorname {arctanh}\left (\frac {\sqrt {-x^{4}+2 x^{2}+1}}{x}\right )+\left (\frac {\arctan \left (\frac {-15 \sqrt {-x^{4}+2 x^{2}+1}\, \sqrt {2+2 i}\, x +3 \left (\left (-\frac {1}{3}+i\right ) x^{4}+\left (-2-\frac {2 i}{3}\right ) x^{2}+\frac {1}{3}-i\right ) \sqrt {5}+3 \left (-5+\frac {5 i}{3}\right ) x^{4}+20 x^{2}+15-5 i}{\sqrt {65+105 i+\left (15+55 i\right ) \sqrt {5}}\, \left (x^{2}+i\right )^{2}}\right ) \left (3+i+\left (1+3 i\right ) \sqrt {5}\right ) \sqrt {65+105 i+\left (-15-55 i\right ) \sqrt {5}}}{3}+80-40 i+\frac {\arctan \left (\frac {15 \sqrt {-x^{4}+2 x^{2}+1}\, \sqrt {2+2 i}\, x +3 \left (\left (-\frac {1}{3}+i\right ) x^{4}+\left (-2-\frac {2 i}{3}\right ) x^{2}+\frac {1}{3}-i\right ) \sqrt {5}+3 \left (5-\frac {5 i}{3}\right ) x^{4}-20 x^{2}-15+5 i}{\sqrt {65+105 i+\left (-15-55 i\right ) \sqrt {5}}\, \left (x^{2}+i\right )^{2}}\right ) \left (-3-i+\left (1+3 i\right ) \sqrt {5}\right ) \sqrt {65+105 i+\left (15+55 i\right ) \sqrt {5}}}{3}\right ) \left (\sqrt {2+2 i}\, x -\sqrt {-x^{4}+2 x^{2}+1}\right )\right ) x}{\left (i x^{2}-1+\sqrt {2+2 i}\, x -\sqrt {-x^{4}+2 x^{2}+1}\right ) \left (i x^{2}-1-\sqrt {2+2 i}\, x +\sqrt {-x^{4}+2 x^{2}+1}\right )}\) | \(370\) |
| trager | \(-\frac {\ln \left (-\frac {x^{4}+2 \sqrt {-x^{4}+2 x^{2}+1}\, x -3 x^{2}-1}{x^{4}-x^{2}-1}\right )}{2}-\operatorname {RootOf}\left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right ) \ln \left (\frac {600 x^{2} \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )^{5}-30 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )^{3} x^{4}+130 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )^{3} x^{2}-2 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right ) x^{4}+10 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )^{2} \sqrt {-x^{4}+2 x^{2}+1}\, x +30 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )^{3}+6 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right ) x^{2}+\sqrt {-x^{4}+2 x^{2}+1}\, x +2 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )}{20 x^{2} \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )^{2}+x^{4}-x^{2}-1}\right )-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+100 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )^{2}+5\right ) \ln \left (\frac {1200 \operatorname {RootOf}\left (\textit {\_Z}^{2}+100 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )^{2}+5\right ) \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )^{4} x^{2}+60 \operatorname {RootOf}\left (\textit {\_Z}^{2}+100 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )^{2}+5\right ) \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )^{2} x^{4}-140 \operatorname {RootOf}\left (\textit {\_Z}^{2}+100 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )^{2}+5\right ) \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )^{2} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+100 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )^{2}+5\right ) x^{4}-200 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )^{2} \sqrt {-x^{4}+2 x^{2}+1}\, x -60 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+100 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )^{2}+5\right )+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+100 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )^{2}+5\right ) x^{2}+10 \sqrt {-x^{4}+2 x^{2}+1}\, x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+100 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )^{2}+5\right )}{20 x^{2} \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )^{2}-x^{4}+2 x^{2}+1}\right )}{10}\) | \(631\) |
Input:
int((-x^4+2*x^2+1)^(1/2)*(x^4-1)*(x^4+1)/(x^4-x^2-1)/(x^8-3*x^6-x^4+3*x^2+ 1),x,method=_RETURNVERBOSE)
Output:
1/2*(-2/5*5^(1/2)/(5^(1/2)-1)^(1/2)*arctan((-x^4+2*x^2+1)^(1/2)*2^(1/2)/x/ (5^(1/2)-1)^(1/2))-2/5*5^(1/2)/(5^(1/2)+1)^(1/2)*arctanh((-x^4+2*x^2+1)^(1 /2)*2^(1/2)/x/(5^(1/2)+1)^(1/2))+2^(1/2)*arctanh((-x^4+2*x^2+1)^(1/2)/x))* 2^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 411 vs. \(2 (132) = 264\).
Time = 0.26 (sec) , antiderivative size = 411, normalized size of antiderivative = 2.52 \[ \int \frac {\sqrt {1+2 x^2-x^4} \left (-1+x^4\right ) \left (1+x^4\right )}{\left (-1-x^2+x^4\right ) \left (1+3 x^2-x^4-3 x^6+x^8\right )} \, dx=\frac {1}{2} \, \sqrt {\frac {1}{10} \, \sqrt {5} + \frac {1}{10}} \arctan \left (-\frac {{\left (5 \, x^{5} - 10 \, x^{3} - \sqrt {5} {\left (x^{5} - x\right )} - 5 \, x\right )} \sqrt {-x^{4} + 2 \, x^{2} + 1} \sqrt {\frac {1}{10} \, \sqrt {5} + \frac {1}{10}}}{x^{8} - 5 \, x^{6} + 3 \, x^{4} + 5 \, x^{2} + 1}\right ) + \frac {1}{4} \, \sqrt {\frac {1}{10} \, \sqrt {5} - \frac {1}{10}} \log \left (\frac {2 \, {\left (3 \, x^{5} - 7 \, x^{3} + \sqrt {5} {\left (x^{5} - 3 \, x^{3} - x\right )} - 3 \, x\right )} \sqrt {-x^{4} + 2 \, x^{2} + 1} + {\left (5 \, x^{8} - 35 \, x^{6} + 45 \, x^{4} + 35 \, x^{2} + \sqrt {5} {\left (3 \, x^{8} - 17 \, x^{6} + 19 \, x^{4} + 17 \, x^{2} + 3\right )} + 5\right )} \sqrt {\frac {1}{10} \, \sqrt {5} - \frac {1}{10}}}{x^{8} - 3 \, x^{6} - x^{4} + 3 \, x^{2} + 1}\right ) - \frac {1}{4} \, \sqrt {\frac {1}{10} \, \sqrt {5} - \frac {1}{10}} \log \left (\frac {2 \, {\left (3 \, x^{5} - 7 \, x^{3} + \sqrt {5} {\left (x^{5} - 3 \, x^{3} - x\right )} - 3 \, x\right )} \sqrt {-x^{4} + 2 \, x^{2} + 1} - {\left (5 \, x^{8} - 35 \, x^{6} + 45 \, x^{4} + 35 \, x^{2} + \sqrt {5} {\left (3 \, x^{8} - 17 \, x^{6} + 19 \, x^{4} + 17 \, x^{2} + 3\right )} + 5\right )} \sqrt {\frac {1}{10} \, \sqrt {5} - \frac {1}{10}}}{x^{8} - 3 \, x^{6} - x^{4} + 3 \, x^{2} + 1}\right ) + \frac {1}{2} \, \log \left (-\frac {x^{4} - 3 \, x^{2} - 2 \, \sqrt {-x^{4} + 2 \, x^{2} + 1} x - 1}{x^{4} - x^{2} - 1}\right ) \] Input:
integrate((-x^4+2*x^2+1)^(1/2)*(x^4-1)*(x^4+1)/(x^4-x^2-1)/(x^8-3*x^6-x^4+ 3*x^2+1),x, algorithm="fricas")
Output:
1/2*sqrt(1/10*sqrt(5) + 1/10)*arctan(-(5*x^5 - 10*x^3 - sqrt(5)*(x^5 - x) - 5*x)*sqrt(-x^4 + 2*x^2 + 1)*sqrt(1/10*sqrt(5) + 1/10)/(x^8 - 5*x^6 + 3*x ^4 + 5*x^2 + 1)) + 1/4*sqrt(1/10*sqrt(5) - 1/10)*log((2*(3*x^5 - 7*x^3 + s qrt(5)*(x^5 - 3*x^3 - x) - 3*x)*sqrt(-x^4 + 2*x^2 + 1) + (5*x^8 - 35*x^6 + 45*x^4 + 35*x^2 + sqrt(5)*(3*x^8 - 17*x^6 + 19*x^4 + 17*x^2 + 3) + 5)*sqr t(1/10*sqrt(5) - 1/10))/(x^8 - 3*x^6 - x^4 + 3*x^2 + 1)) - 1/4*sqrt(1/10*s qrt(5) - 1/10)*log((2*(3*x^5 - 7*x^3 + sqrt(5)*(x^5 - 3*x^3 - x) - 3*x)*sq rt(-x^4 + 2*x^2 + 1) - (5*x^8 - 35*x^6 + 45*x^4 + 35*x^2 + sqrt(5)*(3*x^8 - 17*x^6 + 19*x^4 + 17*x^2 + 3) + 5)*sqrt(1/10*sqrt(5) - 1/10))/(x^8 - 3*x ^6 - x^4 + 3*x^2 + 1)) + 1/2*log(-(x^4 - 3*x^2 - 2*sqrt(-x^4 + 2*x^2 + 1)* x - 1)/(x^4 - x^2 - 1))
Timed out. \[ \int \frac {\sqrt {1+2 x^2-x^4} \left (-1+x^4\right ) \left (1+x^4\right )}{\left (-1-x^2+x^4\right ) \left (1+3 x^2-x^4-3 x^6+x^8\right )} \, dx=\text {Timed out} \] Input:
integrate((-x**4+2*x**2+1)**(1/2)*(x**4-1)*(x**4+1)/(x**4-x**2-1)/(x**8-3* x**6-x**4+3*x**2+1),x)
Output:
Timed out
\[ \int \frac {\sqrt {1+2 x^2-x^4} \left (-1+x^4\right ) \left (1+x^4\right )}{\left (-1-x^2+x^4\right ) \left (1+3 x^2-x^4-3 x^6+x^8\right )} \, dx=\int { \frac {{\left (x^{4} + 1\right )} {\left (x^{4} - 1\right )} \sqrt {-x^{4} + 2 \, x^{2} + 1}}{{\left (x^{8} - 3 \, x^{6} - x^{4} + 3 \, x^{2} + 1\right )} {\left (x^{4} - x^{2} - 1\right )}} \,d x } \] Input:
integrate((-x^4+2*x^2+1)^(1/2)*(x^4-1)*(x^4+1)/(x^4-x^2-1)/(x^8-3*x^6-x^4+ 3*x^2+1),x, algorithm="maxima")
Output:
integrate((x^4 + 1)*(x^4 - 1)*sqrt(-x^4 + 2*x^2 + 1)/((x^8 - 3*x^6 - x^4 + 3*x^2 + 1)*(x^4 - x^2 - 1)), x)
\[ \int \frac {\sqrt {1+2 x^2-x^4} \left (-1+x^4\right ) \left (1+x^4\right )}{\left (-1-x^2+x^4\right ) \left (1+3 x^2-x^4-3 x^6+x^8\right )} \, dx=\int { \frac {{\left (x^{4} + 1\right )} {\left (x^{4} - 1\right )} \sqrt {-x^{4} + 2 \, x^{2} + 1}}{{\left (x^{8} - 3 \, x^{6} - x^{4} + 3 \, x^{2} + 1\right )} {\left (x^{4} - x^{2} - 1\right )}} \,d x } \] Input:
integrate((-x^4+2*x^2+1)^(1/2)*(x^4-1)*(x^4+1)/(x^4-x^2-1)/(x^8-3*x^6-x^4+ 3*x^2+1),x, algorithm="giac")
Output:
integrate((x^4 + 1)*(x^4 - 1)*sqrt(-x^4 + 2*x^2 + 1)/((x^8 - 3*x^6 - x^4 + 3*x^2 + 1)*(x^4 - x^2 - 1)), x)
Timed out. \[ \int \frac {\sqrt {1+2 x^2-x^4} \left (-1+x^4\right ) \left (1+x^4\right )}{\left (-1-x^2+x^4\right ) \left (1+3 x^2-x^4-3 x^6+x^8\right )} \, dx=-\int \frac {\left (x^4-1\right )\,\left (x^4+1\right )\,\sqrt {-x^4+2\,x^2+1}}{\left (-x^4+x^2+1\right )\,\left (x^8-3\,x^6-x^4+3\,x^2+1\right )} \,d x \] Input:
int(-((x^4 - 1)*(x^4 + 1)*(2*x^2 - x^4 + 1)^(1/2))/((x^2 - x^4 + 1)*(3*x^2 - x^4 - 3*x^6 + x^8 + 1)),x)
Output:
-int(((x^4 - 1)*(x^4 + 1)*(2*x^2 - x^4 + 1)^(1/2))/((x^2 - x^4 + 1)*(3*x^2 - x^4 - 3*x^6 + x^8 + 1)), x)
\[ \int \frac {\sqrt {1+2 x^2-x^4} \left (-1+x^4\right ) \left (1+x^4\right )}{\left (-1-x^2+x^4\right ) \left (1+3 x^2-x^4-3 x^6+x^8\right )} \, dx=-\left (\int \frac {\sqrt {-x^{4}+2 x^{2}+1}}{x^{12}-4 x^{10}+x^{8}+7 x^{6}-x^{4}-4 x^{2}-1}d x \right )+\int \frac {\sqrt {-x^{4}+2 x^{2}+1}\, x^{8}}{x^{12}-4 x^{10}+x^{8}+7 x^{6}-x^{4}-4 x^{2}-1}d x \] Input:
int((-x^4+2*x^2+1)^(1/2)*(x^4-1)*(x^4+1)/(x^4-x^2-1)/(x^8-3*x^6-x^4+3*x^2+ 1),x)
Output:
- int(sqrt( - x**4 + 2*x**2 + 1)/(x**12 - 4*x**10 + x**8 + 7*x**6 - x**4 - 4*x**2 - 1),x) + int((sqrt( - x**4 + 2*x**2 + 1)*x**8)/(x**12 - 4*x**10 + x**8 + 7*x**6 - x**4 - 4*x**2 - 1),x)