Integrand size = 58, antiderivative size = 87 \[ \int \frac {\left (1-x^2+2 x^4\right ) \sqrt {1-x^2-x^4-x^6}}{\left (-1+x^2\right ) \left (1+x^2\right ) \left (-1+x^4+x^6\right )} \, dx=\arctan \left (\frac {x \sqrt {1-x^2-x^4-x^6}}{-1+x^2+x^4+x^6}\right )-\sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt {1-x^2-x^4-x^6}}{-1+x^2+x^4+x^6}\right ) \]
arctan(x*(-x^6-x^4-x^2+1)^(1/2)/(x^6+x^4+x^2-1))-2^(1/2)*arctan(2^(1/2)*x* (-x^6-x^4-x^2+1)^(1/2)/(x^6+x^4+x^2-1))
Time = 0.21 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.71 \[ \int \frac {\left (1-x^2+2 x^4\right ) \sqrt {1-x^2-x^4-x^6}}{\left (-1+x^2\right ) \left (1+x^2\right ) \left (-1+x^4+x^6\right )} \, dx=-\arctan \left (\frac {x}{\sqrt {1-x^2-x^4-x^6}}\right )+\sqrt {2} \arctan \left (\frac {\sqrt {2} x}{\sqrt {1-x^2-x^4-x^6}}\right ) \]
Integrate[((1 - x^2 + 2*x^4)*Sqrt[1 - x^2 - x^4 - x^6])/((-1 + x^2)*(1 + x ^2)*(-1 + x^4 + x^6)),x]
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.
\(\displaystyle \int \frac {\left (2 x^4-x^2+1\right ) \sqrt {-x^6-x^4-x^2+1}}{\left (x^2-1\right ) \left (x^2+1\right ) \left (x^6+x^4-1\right )} \, dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (-\frac {\left (3 x^2+2\right ) \sqrt {-x^6-x^4-x^2+1} x^2}{x^6+x^4-1}+\frac {\sqrt {-x^6-x^4-x^2+1}}{x^2-1}+\frac {2 \sqrt {-x^6-x^4-x^2+1}}{x^2+1}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle i \int \frac {\sqrt {-x^6-x^4-x^2+1}}{i-x}dx-\frac {1}{2} \int \frac {\sqrt {-x^6-x^4-x^2+1}}{1-x}dx+i \int \frac {\sqrt {-x^6-x^4-x^2+1}}{x+i}dx-\frac {1}{2} \int \frac {\sqrt {-x^6-x^4-x^2+1}}{x+1}dx-2 \int \frac {x^2 \sqrt {-x^6-x^4-x^2+1}}{x^6+x^4-1}dx-3 \int \frac {x^4 \sqrt {-x^6-x^4-x^2+1}}{x^6+x^4-1}dx\) |
3.12.91.3.1 Defintions of rubi rules used
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 6.24 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.68
method | result | size |
pseudoelliptic | \(\arctan \left (\frac {\sqrt {-x^{6}-x^{4}-x^{2}+1}}{x}\right )-\sqrt {2}\, \arctan \left (\frac {\sqrt {-x^{6}-x^{4}-x^{2}+1}\, \sqrt {2}}{2 x}\right )\) | \(59\) |
trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{6}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{4}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{2}+4 \sqrt {-x^{6}-x^{4}-x^{2}+1}\, x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )}{\left (x -1\right ) \left (1+x \right ) \left (x^{2}+1\right )^{2}}\right )}{2}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{6}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \sqrt {-x^{6}-x^{4}-x^{2}+1}\, x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{6}+x^{4}-1}\right )}{2}\) | \(175\) |
int((2*x^4-x^2+1)*(-x^6-x^4-x^2+1)^(1/2)/(x^2-1)/(x^2+1)/(x^6+x^4-1),x,met hod=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (79) = 158\).
Time = 0.35 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.11 \[ \int \frac {\left (1-x^2+2 x^4\right ) \sqrt {1-x^2-x^4-x^6}}{\left (-1+x^2\right ) \left (1+x^2\right ) \left (-1+x^4+x^6\right )} \, dx=-\frac {1}{10} \, \sqrt {2} \arctan \left (\frac {2 \, \sqrt {2} {\left (6 \, x^{7} + x^{5} - 4 \, x^{3} + x\right )} \sqrt {-x^{6} - x^{4} - x^{2} + 1}}{17 \, x^{10} + 11 \, x^{8} - 2 \, x^{6} - 18 \, x^{4} + 9 \, x^{2} - 1}\right ) + \frac {1}{5} \, \sqrt {2} \arctan \left (\frac {2 \, \sqrt {2} {\left (x^{7} + x^{3} - 2 \, x\right )} \sqrt {-x^{6} - x^{4} - x^{2} + 1}}{3 \, x^{10} + 3 \, x^{8} + 10 \, x^{6} + 6 \, x^{4} + 11 \, x^{2} - 1}\right ) + \frac {1}{2} \, \arctan \left (\frac {2 \, \sqrt {-x^{6} - x^{4} - x^{2} + 1} x}{x^{6} + x^{4} + 2 \, x^{2} - 1}\right ) \]
integrate((2*x^4-x^2+1)*(-x^6-x^4-x^2+1)^(1/2)/(x^2-1)/(x^2+1)/(x^6+x^4-1) ,x, algorithm="fricas")
-1/10*sqrt(2)*arctan(2*sqrt(2)*(6*x^7 + x^5 - 4*x^3 + x)*sqrt(-x^6 - x^4 - x^2 + 1)/(17*x^10 + 11*x^8 - 2*x^6 - 18*x^4 + 9*x^2 - 1)) + 1/5*sqrt(2)*a rctan(2*sqrt(2)*(x^7 + x^3 - 2*x)*sqrt(-x^6 - x^4 - x^2 + 1)/(3*x^10 + 3*x ^8 + 10*x^6 + 6*x^4 + 11*x^2 - 1)) + 1/2*arctan(2*sqrt(-x^6 - x^4 - x^2 + 1)*x/(x^6 + x^4 + 2*x^2 - 1))
Timed out. \[ \int \frac {\left (1-x^2+2 x^4\right ) \sqrt {1-x^2-x^4-x^6}}{\left (-1+x^2\right ) \left (1+x^2\right ) \left (-1+x^4+x^6\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\left (1-x^2+2 x^4\right ) \sqrt {1-x^2-x^4-x^6}}{\left (-1+x^2\right ) \left (1+x^2\right ) \left (-1+x^4+x^6\right )} \, dx=\int { \frac {\sqrt {-x^{6} - x^{4} - x^{2} + 1} {\left (2 \, x^{4} - x^{2} + 1\right )}}{{\left (x^{6} + x^{4} - 1\right )} {\left (x^{2} + 1\right )} {\left (x^{2} - 1\right )}} \,d x } \]
integrate((2*x^4-x^2+1)*(-x^6-x^4-x^2+1)^(1/2)/(x^2-1)/(x^2+1)/(x^6+x^4-1) ,x, algorithm="maxima")
integrate(sqrt(-x^6 - x^4 - x^2 + 1)*(2*x^4 - x^2 + 1)/((x^6 + x^4 - 1)*(x ^2 + 1)*(x^2 - 1)), x)
\[ \int \frac {\left (1-x^2+2 x^4\right ) \sqrt {1-x^2-x^4-x^6}}{\left (-1+x^2\right ) \left (1+x^2\right ) \left (-1+x^4+x^6\right )} \, dx=\int { \frac {\sqrt {-x^{6} - x^{4} - x^{2} + 1} {\left (2 \, x^{4} - x^{2} + 1\right )}}{{\left (x^{6} + x^{4} - 1\right )} {\left (x^{2} + 1\right )} {\left (x^{2} - 1\right )}} \,d x } \]
integrate((2*x^4-x^2+1)*(-x^6-x^4-x^2+1)^(1/2)/(x^2-1)/(x^2+1)/(x^6+x^4-1) ,x, algorithm="giac")
integrate(sqrt(-x^6 - x^4 - x^2 + 1)*(2*x^4 - x^2 + 1)/((x^6 + x^4 - 1)*(x ^2 + 1)*(x^2 - 1)), x)
Timed out. \[ \int \frac {\left (1-x^2+2 x^4\right ) \sqrt {1-x^2-x^4-x^6}}{\left (-1+x^2\right ) \left (1+x^2\right ) \left (-1+x^4+x^6\right )} \, dx=\int \frac {\left (2\,x^4-x^2+1\right )\,\sqrt {-x^6-x^4-x^2+1}}{\left (x^2-1\right )\,\left (x^2+1\right )\,\left (x^6+x^4-1\right )} \,d x \]