Integrand size = 56, antiderivative size = 123 \[ \int \frac {\left (1+x^6\right )^2 \left (-1+2 x^6\right )}{\left (1-x^2+x^6\right )^{3/2} \left (1-x^2-x^4+2 x^6-x^8+x^{12}\right )} \, dx=\frac {x}{\sqrt {1-x^2+x^6}}-\sqrt {\frac {2}{145+65 \sqrt {5}}} \arctan \left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt {1-x^2+x^6}}\right )-\sqrt {\frac {1}{10} \left (29+13 \sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt {1-x^2+x^6}}\right ) \] Output:
x/(x^6-x^2+1)^(1/2)-2^(1/2)/(145+65*5^(1/2))^(1/2)*arctan(1/2*(2+2*5^(1/2) )^(1/2)*x/(x^6-x^2+1)^(1/2))-1/10*(290+130*5^(1/2))^(1/2)*arctanh(1/2*(-2+ 2*5^(1/2))^(1/2)*x/(x^6-x^2+1)^(1/2))
Time = 2.09 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.97 \[ \int \frac {\left (1+x^6\right )^2 \left (-1+2 x^6\right )}{\left (1-x^2+x^6\right )^{3/2} \left (1-x^2-x^4+2 x^6-x^8+x^{12}\right )} \, dx=\frac {x}{\sqrt {1-x^2+x^6}}-\sqrt {\frac {2}{145+65 \sqrt {5}}} \arctan \left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} x}{\sqrt {1-x^2+x^6}}\right )-\sqrt {\frac {1}{10} \left (29+13 \sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} x}{\sqrt {1-x^2+x^6}}\right ) \] Input:
Integrate[((1 + x^6)^2*(-1 + 2*x^6))/((1 - x^2 + x^6)^(3/2)*(1 - x^2 - x^4 + 2*x^6 - x^8 + x^12)),x]
Output:
x/Sqrt[1 - x^2 + x^6] - Sqrt[2/(145 + 65*Sqrt[5])]*ArcTan[(Sqrt[(1 + Sqrt[ 5])/2]*x)/Sqrt[1 - x^2 + x^6]] - Sqrt[(29 + 13*Sqrt[5])/10]*ArcTanh[(Sqrt[ (-1 + Sqrt[5])/2]*x)/Sqrt[1 - x^2 + x^6]]
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.
| \(\displaystyle \int \frac {\left (x^6+1\right )^2 \left (2 x^6-1\right )}{\left (x^6-x^2+1\right )^{3/2} \left (x^{12}-x^8+2 x^6-x^4-x^2+1\right )} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 x^6}{\left (x^6-x^2+1\right )^{3/2}}+\frac {2 x^2}{\left (x^6-x^2+1\right )^{3/2}}-\frac {1}{\left (x^6-x^2+1\right )^{3/2}}+\frac {\left (4 x^8-3 x^6+2 x^4+x^2-3\right ) x^2}{\left (x^6-x^2+1\right )^{3/2} \left (x^{12}-x^8+2 x^6-x^4-x^2+1\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \int \frac {x^2}{\left (x^6-x^2+1\right )^{3/2}}dx-3 \int \frac {x^2}{\left (x^6-x^2+1\right )^{3/2} \left (x^{12}-x^8+2 x^6-x^4-x^2+1\right )}dx+\int \frac {x^4}{\left (x^6-x^2+1\right )^{3/2} \left (x^{12}-x^8+2 x^6-x^4-x^2+1\right )}dx+2 \int \frac {x^6}{\left (x^6-x^2+1\right )^{3/2} \left (x^{12}-x^8+2 x^6-x^4-x^2+1\right )}dx-3 \int \frac {x^8}{\left (x^6-x^2+1\right )^{3/2} \left (x^{12}-x^8+2 x^6-x^4-x^2+1\right )}dx+4 \int \frac {x^{10}}{\left (x^6-x^2+1\right )^{3/2} \left (x^{12}-x^8+2 x^6-x^4-x^2+1\right )}dx-\frac {x}{\sqrt {x^6-x^2+1}}\) |
Input:
Int[((1 + x^6)^2*(-1 + 2*x^6))/((1 - x^2 + x^6)^(3/2)*(1 - x^2 - x^4 + 2*x ^6 - x^8 + x^12)),x]
Output:
$Aborted
Time = 6.61 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.02
| method | result | size |
| pseudoelliptic | \(\frac {-\frac {\sqrt {x^{6}-x^{2}+1}\, \sqrt {2+2 \sqrt {5}}\, \left (\sqrt {5}+\frac {5}{2}\right ) \operatorname {arctanh}\left (\frac {2 \sqrt {x^{6}-x^{2}+1}}{x \sqrt {-2+2 \sqrt {5}}}\right )}{5}-\frac {\sqrt {-2+2 \sqrt {5}}\, \sqrt {x^{6}-x^{2}+1}\, \left (\sqrt {5}-\frac {5}{2}\right ) \arctan \left (\frac {2 \sqrt {x^{6}-x^{2}+1}}{\sqrt {2+2 \sqrt {5}}\, x}\right )}{5}+x}{\sqrt {x^{6}-x^{2}+1}}\) | \(126\) |
| risch | \(\frac {x}{\sqrt {x^{6}-x^{2}+1}}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+100 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2}-145\right ) \ln \left (\frac {-2860 \operatorname {RootOf}\left (\textit {\_Z}^{2}+100 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2}-145\right ) \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2} x^{6}+4400 \operatorname {RootOf}\left (\textit {\_Z}^{2}+100 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2}-145\right ) \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{4} x^{2}-39 \operatorname {RootOf}\left (\textit {\_Z}^{2}+100 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2}-145\right ) x^{6}+1160 \operatorname {RootOf}\left (\textit {\_Z}^{2}+100 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2}-145\right ) \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2} x^{2}-2600 \sqrt {x^{6}-x^{2}+1}\, \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2} x -2860 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+100 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2}-145\right )+15 \operatorname {RootOf}\left (\textit {\_Z}^{2}+100 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2}-145\right ) x^{2}-650 \sqrt {x^{6}-x^{2}+1}\, x -39 \operatorname {RootOf}\left (\textit {\_Z}^{2}+100 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2}-145\right )}{13 x^{6}+20 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2} x^{2}-21 x^{2}+13}\right )}{10}+\operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right ) \ln \left (\frac {-1430 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{3} x^{6}-2200 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{5} x^{2}+2093 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right ) x^{6}+6960 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{3} x^{2}+130 \sqrt {x^{6}-x^{2}+1}\, \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2} x -1430 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{3}-5474 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right ) x^{2}-221 \sqrt {x^{6}-x^{2}+1}\, x +2093 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )}{-13 x^{6}+20 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2} x^{2}-8 x^{2}-13}\right )\) | \(597\) |
| trager | \(\frac {x}{\sqrt {x^{6}-x^{2}+1}}+\operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right ) \ln \left (\frac {-1430 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{3} x^{6}-2200 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{5} x^{2}+2093 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right ) x^{6}+6960 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{3} x^{2}+130 \sqrt {x^{6}-x^{2}+1}\, \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2} x -1430 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{3}-5474 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right ) x^{2}-221 \sqrt {x^{6}-x^{2}+1}\, x +2093 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )}{-13 x^{6}+20 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2} x^{2}-8 x^{2}-13}\right )+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+100 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2}-145\right ) \ln \left (-\frac {-2860 \operatorname {RootOf}\left (\textit {\_Z}^{2}+100 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2}-145\right ) \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2} x^{6}+4400 \operatorname {RootOf}\left (\textit {\_Z}^{2}+100 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2}-145\right ) \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{4} x^{2}-39 \operatorname {RootOf}\left (\textit {\_Z}^{2}+100 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2}-145\right ) x^{6}+1160 \operatorname {RootOf}\left (\textit {\_Z}^{2}+100 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2}-145\right ) \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2} x^{2}+2600 \sqrt {x^{6}-x^{2}+1}\, \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2} x -2860 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+100 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2}-145\right )+15 \operatorname {RootOf}\left (\textit {\_Z}^{2}+100 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2}-145\right ) x^{2}+650 \sqrt {x^{6}-x^{2}+1}\, x -39 \operatorname {RootOf}\left (\textit {\_Z}^{2}+100 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2}-145\right )}{13 x^{6}+20 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}-580 \textit {\_Z}^{2}-1\right )^{2} x^{2}-21 x^{2}+13}\right )}{10}\) | \(598\) |
Input:
int((x^6+1)^2*(2*x^6-1)/(x^6-x^2+1)^(3/2)/(x^12-x^8+2*x^6-x^4-x^2+1),x,met hod=_RETURNVERBOSE)
Output:
(-1/5*(x^6-x^2+1)^(1/2)*(2+2*5^(1/2))^(1/2)*(5^(1/2)+5/2)*arctanh(2*(x^6-x ^2+1)^(1/2)/x/(-2+2*5^(1/2))^(1/2))-1/5*(-2+2*5^(1/2))^(1/2)*(x^6-x^2+1)^( 1/2)*(5^(1/2)-5/2)*arctan(2/(2+2*5^(1/2))^(1/2)/x*(x^6-x^2+1)^(1/2))+x)/(x ^6-x^2+1)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 459 vs. \(2 (90) = 180\).
Time = 0.34 (sec) , antiderivative size = 459, normalized size of antiderivative = 3.73 \[ \int \frac {\left (1+x^6\right )^2 \left (-1+2 x^6\right )}{\left (1-x^2+x^6\right )^{3/2} \left (1-x^2-x^4+2 x^6-x^8+x^{12}\right )} \, dx=-\frac {2 \, {\left (x^{6} - x^{2} + 1\right )} \sqrt {\frac {13}{10} \, \sqrt {5} - \frac {29}{10}} \arctan \left (\frac {{\left (15 \, x^{7} - 5 \, x^{3} + \sqrt {5} {\left (7 \, x^{7} - 3 \, x^{3} + 7 \, x\right )} + 15 \, x\right )} \sqrt {x^{6} - x^{2} + 1} \sqrt {\frac {13}{10} \, \sqrt {5} - \frac {29}{10}}}{x^{12} - 3 \, x^{8} + 2 \, x^{6} + x^{4} - 3 \, x^{2} + 1}\right ) + {\left (x^{6} - x^{2} + 1\right )} \sqrt {\frac {13}{10} \, \sqrt {5} + \frac {29}{10}} \log \left (-\frac {2 \, {\left (3 \, x^{7} - 4 \, x^{3} - \sqrt {5} {\left (x^{7} - 2 \, x^{3} + x\right )} + 3 \, x\right )} \sqrt {x^{6} - x^{2} + 1} + {\left (25 \, x^{12} - 105 \, x^{8} + 50 \, x^{6} + 105 \, x^{4} - 105 \, x^{2} - \sqrt {5} {\left (11 \, x^{12} - 47 \, x^{8} + 22 \, x^{6} + 47 \, x^{4} - 47 \, x^{2} + 11\right )} + 25\right )} \sqrt {\frac {13}{10} \, \sqrt {5} + \frac {29}{10}}}{x^{12} - x^{8} + 2 \, x^{6} - x^{4} - x^{2} + 1}\right ) - {\left (x^{6} - x^{2} + 1\right )} \sqrt {\frac {13}{10} \, \sqrt {5} + \frac {29}{10}} \log \left (-\frac {2 \, {\left (3 \, x^{7} - 4 \, x^{3} - \sqrt {5} {\left (x^{7} - 2 \, x^{3} + x\right )} + 3 \, x\right )} \sqrt {x^{6} - x^{2} + 1} - {\left (25 \, x^{12} - 105 \, x^{8} + 50 \, x^{6} + 105 \, x^{4} - 105 \, x^{2} - \sqrt {5} {\left (11 \, x^{12} - 47 \, x^{8} + 22 \, x^{6} + 47 \, x^{4} - 47 \, x^{2} + 11\right )} + 25\right )} \sqrt {\frac {13}{10} \, \sqrt {5} + \frac {29}{10}}}{x^{12} - x^{8} + 2 \, x^{6} - x^{4} - x^{2} + 1}\right ) - 4 \, \sqrt {x^{6} - x^{2} + 1} x}{4 \, {\left (x^{6} - x^{2} + 1\right )}} \] Input:
integrate((x^6+1)^2*(2*x^6-1)/(x^6-x^2+1)^(3/2)/(x^12-x^8+2*x^6-x^4-x^2+1) ,x, algorithm="fricas")
Output:
-1/4*(2*(x^6 - x^2 + 1)*sqrt(13/10*sqrt(5) - 29/10)*arctan((15*x^7 - 5*x^3 + sqrt(5)*(7*x^7 - 3*x^3 + 7*x) + 15*x)*sqrt(x^6 - x^2 + 1)*sqrt(13/10*sq rt(5) - 29/10)/(x^12 - 3*x^8 + 2*x^6 + x^4 - 3*x^2 + 1)) + (x^6 - x^2 + 1) *sqrt(13/10*sqrt(5) + 29/10)*log(-(2*(3*x^7 - 4*x^3 - sqrt(5)*(x^7 - 2*x^3 + x) + 3*x)*sqrt(x^6 - x^2 + 1) + (25*x^12 - 105*x^8 + 50*x^6 + 105*x^4 - 105*x^2 - sqrt(5)*(11*x^12 - 47*x^8 + 22*x^6 + 47*x^4 - 47*x^2 + 11) + 25 )*sqrt(13/10*sqrt(5) + 29/10))/(x^12 - x^8 + 2*x^6 - x^4 - x^2 + 1)) - (x^ 6 - x^2 + 1)*sqrt(13/10*sqrt(5) + 29/10)*log(-(2*(3*x^7 - 4*x^3 - sqrt(5)* (x^7 - 2*x^3 + x) + 3*x)*sqrt(x^6 - x^2 + 1) - (25*x^12 - 105*x^8 + 50*x^6 + 105*x^4 - 105*x^2 - sqrt(5)*(11*x^12 - 47*x^8 + 22*x^6 + 47*x^4 - 47*x^ 2 + 11) + 25)*sqrt(13/10*sqrt(5) + 29/10))/(x^12 - x^8 + 2*x^6 - x^4 - x^2 + 1)) - 4*sqrt(x^6 - x^2 + 1)*x)/(x^6 - x^2 + 1)
Timed out. \[ \int \frac {\left (1+x^6\right )^2 \left (-1+2 x^6\right )}{\left (1-x^2+x^6\right )^{3/2} \left (1-x^2-x^4+2 x^6-x^8+x^{12}\right )} \, dx=\text {Timed out} \] Input:
integrate((x**6+1)**2*(2*x**6-1)/(x**6-x**2+1)**(3/2)/(x**12-x**8+2*x**6-x **4-x**2+1),x)
Output:
Timed out
\[ \int \frac {\left (1+x^6\right )^2 \left (-1+2 x^6\right )}{\left (1-x^2+x^6\right )^{3/2} \left (1-x^2-x^4+2 x^6-x^8+x^{12}\right )} \, dx=\int { \frac {{\left (2 \, x^{6} - 1\right )} {\left (x^{6} + 1\right )}^{2}}{{\left (x^{12} - x^{8} + 2 \, x^{6} - x^{4} - x^{2} + 1\right )} {\left (x^{6} - x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((x^6+1)^2*(2*x^6-1)/(x^6-x^2+1)^(3/2)/(x^12-x^8+2*x^6-x^4-x^2+1) ,x, algorithm="maxima")
Output:
integrate((2*x^6 - 1)*(x^6 + 1)^2/((x^12 - x^8 + 2*x^6 - x^4 - x^2 + 1)*(x ^6 - x^2 + 1)^(3/2)), x)
\[ \int \frac {\left (1+x^6\right )^2 \left (-1+2 x^6\right )}{\left (1-x^2+x^6\right )^{3/2} \left (1-x^2-x^4+2 x^6-x^8+x^{12}\right )} \, dx=\int { \frac {{\left (2 \, x^{6} - 1\right )} {\left (x^{6} + 1\right )}^{2}}{{\left (x^{12} - x^{8} + 2 \, x^{6} - x^{4} - x^{2} + 1\right )} {\left (x^{6} - x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((x^6+1)^2*(2*x^6-1)/(x^6-x^2+1)^(3/2)/(x^12-x^8+2*x^6-x^4-x^2+1) ,x, algorithm="giac")
Output:
integrate((2*x^6 - 1)*(x^6 + 1)^2/((x^12 - x^8 + 2*x^6 - x^4 - x^2 + 1)*(x ^6 - x^2 + 1)^(3/2)), x)
Timed out. \[ \int \frac {\left (1+x^6\right )^2 \left (-1+2 x^6\right )}{\left (1-x^2+x^6\right )^{3/2} \left (1-x^2-x^4+2 x^6-x^8+x^{12}\right )} \, dx=\int -\frac {{\left (x^6+1\right )}^2\,\left (2\,x^6-1\right )}{{\left (x^6-x^2+1\right )}^{3/2}\,\left (-x^{12}+x^8-2\,x^6+x^4+x^2-1\right )} \,d x \] Input:
int(-((x^6 + 1)^2*(2*x^6 - 1))/((x^6 - x^2 + 1)^(3/2)*(x^2 + x^4 - 2*x^6 + x^8 - x^12 - 1)),x)
Output:
int(-((x^6 + 1)^2*(2*x^6 - 1))/((x^6 - x^2 + 1)^(3/2)*(x^2 + x^4 - 2*x^6 + x^8 - x^12 - 1)), x)
\[ \int \frac {\left (1+x^6\right )^2 \left (-1+2 x^6\right )}{\left (1-x^2+x^6\right )^{3/2} \left (1-x^2-x^4+2 x^6-x^8+x^{12}\right )} \, dx =\text {Too large to display} \] Input:
int((x^6+1)^2*(2*x^6-1)/(x^6-x^2+1)^(3/2)/(x^12-x^8+2*x^6-x^4-x^2+1),x)
Output:
( - sqrt(x**6 - x**2 + 1)*x + 2*int((sqrt(x**6 - x**2 + 1)*x**14)/(x**24 - 3*x**20 + 4*x**18 + 2*x**16 - 9*x**14 + 7*x**12 + 4*x**10 - 10*x**8 + 5*x **6 + 2*x**4 - 3*x**2 + 1),x)*x**6 - 2*int((sqrt(x**6 - x**2 + 1)*x**14)/( x**24 - 3*x**20 + 4*x**18 + 2*x**16 - 9*x**14 + 7*x**12 + 4*x**10 - 10*x** 8 + 5*x**6 + 2*x**4 - 3*x**2 + 1),x)*x**2 + 2*int((sqrt(x**6 - x**2 + 1)*x **14)/(x**24 - 3*x**20 + 4*x**18 + 2*x**16 - 9*x**14 + 7*x**12 + 4*x**10 - 10*x**8 + 5*x**6 + 2*x**4 - 3*x**2 + 1),x) + 2*int((sqrt(x**6 - x**2 + 1) *x**10)/(x**24 - 3*x**20 + 4*x**18 + 2*x**16 - 9*x**14 + 7*x**12 + 4*x**10 - 10*x**8 + 5*x**6 + 2*x**4 - 3*x**2 + 1),x)*x**6 - 2*int((sqrt(x**6 - x* *2 + 1)*x**10)/(x**24 - 3*x**20 + 4*x**18 + 2*x**16 - 9*x**14 + 7*x**12 + 4*x**10 - 10*x**8 + 5*x**6 + 2*x**4 - 3*x**2 + 1),x)*x**2 + 2*int((sqrt(x* *6 - x**2 + 1)*x**10)/(x**24 - 3*x**20 + 4*x**18 + 2*x**16 - 9*x**14 + 7*x **12 + 4*x**10 - 10*x**8 + 5*x**6 + 2*x**4 - 3*x**2 + 1),x) + int((sqrt(x* *6 - x**2 + 1)*x**8)/(x**24 - 3*x**20 + 4*x**18 + 2*x**16 - 9*x**14 + 7*x* *12 + 4*x**10 - 10*x**8 + 5*x**6 + 2*x**4 - 3*x**2 + 1),x)*x**6 - int((sqr t(x**6 - x**2 + 1)*x**8)/(x**24 - 3*x**20 + 4*x**18 + 2*x**16 - 9*x**14 + 7*x**12 + 4*x**10 - 10*x**8 + 5*x**6 + 2*x**4 - 3*x**2 + 1),x)*x**2 + int( (sqrt(x**6 - x**2 + 1)*x**8)/(x**24 - 3*x**20 + 4*x**18 + 2*x**16 - 9*x**1 4 + 7*x**12 + 4*x**10 - 10*x**8 + 5*x**6 + 2*x**4 - 3*x**2 + 1),x) - int(( sqrt(x**6 - x**2 + 1)*x**4)/(x**24 - 3*x**20 + 4*x**18 + 2*x**16 - 9*x*...