Integrand size = 49, antiderivative size = 55 \[ \int \frac {\sqrt {-1-x-x^2+x^4} \left (2+x+2 x^4\right )}{\left (-1-x+x^4\right ) \left (-1-x+x^2+x^4\right )} \, dx=2 \arctan \left (\frac {x}{\sqrt {-1-x-x^2+x^4}}\right )-2 \sqrt {2} \arctan \left (\frac {\sqrt {2} x}{\sqrt {-1-x-x^2+x^4}}\right ) \]
Time = 0.18 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {-1-x-x^2+x^4} \left (2+x+2 x^4\right )}{\left (-1-x+x^4\right ) \left (-1-x+x^2+x^4\right )} \, dx=2 \arctan \left (\frac {x}{\sqrt {-1-x-x^2+x^4}}\right )-2 \sqrt {2} \arctan \left (\frac {\sqrt {2} x}{\sqrt {-1-x-x^2+x^4}}\right ) \]
2*ArcTan[x/Sqrt[-1 - x - x^2 + x^4]] - 2*Sqrt[2]*ArcTan[(Sqrt[2]*x)/Sqrt[- 1 - x - 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.
\(\displaystyle \int \frac {\sqrt {x^4-x^2-x-1} \left (2 x^4+x+2\right )}{\left (x^4-x-1\right ) \left (x^4+x^2-x-1\right )} \, dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \left (\frac {\sqrt {x^4-x^2-x-1} \left (2 x^4+x+2\right )}{5 (x-1) \left (x^4-x-1\right )}+\frac {\left (-x^2-2 x-4\right ) \sqrt {x^4-x^2-x-1} \left (2 x^4+x+2\right )}{5 \left (x^3+x^2+2 x+1\right ) \left (x^4-x-1\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\int \frac {\sqrt {x^4-x^2-x-1}}{x-1}dx+\int \frac {\sqrt {x^4-x^2-x-1}}{x^4-x-1}dx+4 \int \frac {x^2 \sqrt {x^4-x^2-x-1}}{x^4-x-1}dx+2 \int \frac {\sqrt {x^4-x^2-x-1}}{x^3+x^2+2 x+1}dx-\int \frac {x \sqrt {x^4-x^2-x-1}}{x^3+x^2+2 x+1}dx+2 \int \frac {x^2 \sqrt {x^4-x^2-x-1}}{x^3+x^2+2 x+1}dx-\int \frac {x^3 \sqrt {x^4-x^2-x-1}}{x^4-x-1}dx\) |
3.8.10.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 21.19 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.96
method | result | size |
default | \(2 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x^{4}-x^{2}-x -1}}{2 x}\right )-2 \arctan \left (\frac {\sqrt {x^{4}-x^{2}-x -1}}{x}\right )\) | \(53\) |
pseudoelliptic | \(2 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x^{4}-x^{2}-x -1}}{2 x}\right )-2 \arctan \left (\frac {\sqrt {x^{4}-x^{2}-x -1}}{x}\right )\) | \(53\) |
trager | \(\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \sqrt {x^{4}-x^{2}-x -1}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{4}-x -1}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{4}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{2}+4 \sqrt {x^{4}-x^{2}-x -1}\, x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )}{\left (x -1\right ) \left (x^{3}+x^{2}+2 x +1\right )}\right )\) | \(162\) |
elliptic | \(\text {Expression too large to display}\) | \(788632\) |
Time = 0.34 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.40 \[ \int \frac {\sqrt {-1-x-x^2+x^4} \left (2+x+2 x^4\right )}{\left (-1-x+x^4\right ) \left (-1-x+x^2+x^4\right )} \, dx=-\sqrt {2} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {x^{4} - x^{2} - x - 1} x}{x^{4} - 3 \, x^{2} - x - 1}\right ) + \arctan \left (\frac {2 \, \sqrt {x^{4} - x^{2} - x - 1} x}{x^{4} - 2 \, x^{2} - x - 1}\right ) \]
-sqrt(2)*arctan(2*sqrt(2)*sqrt(x^4 - x^2 - x - 1)*x/(x^4 - 3*x^2 - x - 1)) + arctan(2*sqrt(x^4 - x^2 - x - 1)*x/(x^4 - 2*x^2 - x - 1))
Timed out. \[ \int \frac {\sqrt {-1-x-x^2+x^4} \left (2+x+2 x^4\right )}{\left (-1-x+x^4\right ) \left (-1-x+x^2+x^4\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {-1-x-x^2+x^4} \left (2+x+2 x^4\right )}{\left (-1-x+x^4\right ) \left (-1-x+x^2+x^4\right )} \, dx=\int { \frac {{\left (2 \, x^{4} + x + 2\right )} \sqrt {x^{4} - x^{2} - x - 1}}{{\left (x^{4} + x^{2} - x - 1\right )} {\left (x^{4} - x - 1\right )}} \,d x } \]
\[ \int \frac {\sqrt {-1-x-x^2+x^4} \left (2+x+2 x^4\right )}{\left (-1-x+x^4\right ) \left (-1-x+x^2+x^4\right )} \, dx=\int { \frac {{\left (2 \, x^{4} + x + 2\right )} \sqrt {x^{4} - x^{2} - x - 1}}{{\left (x^{4} + x^{2} - x - 1\right )} {\left (x^{4} - x - 1\right )}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {-1-x-x^2+x^4} \left (2+x+2 x^4\right )}{\left (-1-x+x^4\right ) \left (-1-x+x^2+x^4\right )} \, dx=\int \frac {\left (2\,x^4+x+2\right )\,\sqrt {x^4-x^2-x-1}}{\left (-x^4+x+1\right )\,\left (-x^4-x^2+x+1\right )} \,d x \]