Integrand size = 41, antiderivative size = 51 \[ \int \frac {\sqrt {-1-x^2+x^6} \left (1+2 x^6\right )}{\left (-1+x^6\right ) \left (-2+x^2+2 x^6\right )} \, dx=\arctan \left (\frac {x}{\sqrt {-1-x^2+x^6}}\right )-\sqrt {\frac {3}{2}} \arctan \left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt {-1-x^2+x^6}}\right ) \]
Time = 1.53 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {-1-x^2+x^6} \left (1+2 x^6\right )}{\left (-1+x^6\right ) \left (-2+x^2+2 x^6\right )} \, dx=\arctan \left (\frac {x}{\sqrt {-1-x^2+x^6}}\right )-\sqrt {\frac {3}{2}} \arctan \left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt {-1-x^2+x^6}}\right ) \]
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.
\(\displaystyle \int \frac {\sqrt {x^6-x^2-1} \left (2 x^6+1\right )}{\left (x^6-1\right ) \left (2 x^6+x^2-2\right )} \, dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (\frac {\sqrt {x^6-x^2-1} (1-x)}{2 \left (x^2+x+1\right )}+\frac {\sqrt {x^6-x^2-1}}{x^2-1}+\frac {(x+1) \sqrt {x^6-x^2-1}}{2 \left (x^2-x+1\right )}+\frac {\left (-6 x^4-1\right ) \sqrt {x^6-x^2-1}}{2 x^6+x^2-2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{2} \int \frac {\sqrt {x^6-x^2-1}}{1-x}dx-\frac {1}{2} \int \frac {\sqrt {x^6-x^2-1}}{x+1}dx+\frac {1}{2} \left (1-i \sqrt {3}\right ) \int \frac {\sqrt {x^6-x^2-1}}{2 x-i \sqrt {3}-1}dx-\frac {1}{2} \left (1+i \sqrt {3}\right ) \int \frac {\sqrt {x^6-x^2-1}}{2 x-i \sqrt {3}+1}dx+\frac {1}{2} \left (1+i \sqrt {3}\right ) \int \frac {\sqrt {x^6-x^2-1}}{2 x+i \sqrt {3}-1}dx-\frac {1}{2} \left (1-i \sqrt {3}\right ) \int \frac {\sqrt {x^6-x^2-1}}{2 x+i \sqrt {3}+1}dx+\int \frac {\sqrt {x^6-x^2-1}}{-2 x^6-x^2+2}dx-6 \int \frac {x^4 \sqrt {x^6-x^2-1}}{2 x^6+x^2-2}dx\) |
3.7.47.3.1 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 = 4.18 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.92
method | result | size |
pseudoelliptic | \(-\arctan \left (\frac {\sqrt {x^{6}-x^{2}-1}}{x}\right )+\frac {\sqrt {6}\, \arctan \left (\frac {\sqrt {6}\, \sqrt {x^{6}-x^{2}-1}}{3 x}\right )}{2}\) | \(47\) |
trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+6\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6\right ) x^{6}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6\right ) x^{2}-12 \sqrt {x^{6}-x^{2}-1}\, x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6\right )}{2 x^{6}+x^{2}-2}\right )}{4}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{6}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \sqrt {x^{6}-x^{2}-1}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{\left (x -1\right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )}\right )}{2}\) | \(154\) |
Time = 0.32 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.47 \[ \int \frac {\sqrt {-1-x^2+x^6} \left (1+2 x^6\right )}{\left (-1+x^6\right ) \left (-2+x^2+2 x^6\right )} \, dx=-\frac {1}{4} \, \sqrt {3} \sqrt {2} \arctan \left (\frac {2 \, \sqrt {3} \sqrt {2} \sqrt {x^{6} - x^{2} - 1} x}{2 \, x^{6} - 5 \, x^{2} - 2}\right ) + \frac {1}{2} \, \arctan \left (\frac {2 \, \sqrt {x^{6} - x^{2} - 1} x}{x^{6} - 2 \, x^{2} - 1}\right ) \]
-1/4*sqrt(3)*sqrt(2)*arctan(2*sqrt(3)*sqrt(2)*sqrt(x^6 - x^2 - 1)*x/(2*x^6 - 5*x^2 - 2)) + 1/2*arctan(2*sqrt(x^6 - x^2 - 1)*x/(x^6 - 2*x^2 - 1))
Timed out. \[ \int \frac {\sqrt {-1-x^2+x^6} \left (1+2 x^6\right )}{\left (-1+x^6\right ) \left (-2+x^2+2 x^6\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {-1-x^2+x^6} \left (1+2 x^6\right )}{\left (-1+x^6\right ) \left (-2+x^2+2 x^6\right )} \, dx=\int { \frac {{\left (2 \, x^{6} + 1\right )} \sqrt {x^{6} - x^{2} - 1}}{{\left (2 \, x^{6} + x^{2} - 2\right )} {\left (x^{6} - 1\right )}} \,d x } \]
\[ \int \frac {\sqrt {-1-x^2+x^6} \left (1+2 x^6\right )}{\left (-1+x^6\right ) \left (-2+x^2+2 x^6\right )} \, dx=\int { \frac {{\left (2 \, x^{6} + 1\right )} \sqrt {x^{6} - x^{2} - 1}}{{\left (2 \, x^{6} + x^{2} - 2\right )} {\left (x^{6} - 1\right )}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {-1-x^2+x^6} \left (1+2 x^6\right )}{\left (-1+x^6\right ) \left (-2+x^2+2 x^6\right )} \, dx=\int \frac {\left (2\,x^6+1\right )\,\sqrt {x^6-x^2-1}}{\left (x^6-1\right )\,\left (2\,x^6+x^2-2\right )} \,d x \]