Integrand size = 37, antiderivative size = 139 \[ \int \frac {\left (-1+x^8\right ) \left (1+x^8\right )}{\sqrt [4]{-1-x^4+x^8} \left (1-3 x^8+x^{16}\right )} \, dx=-\frac {x}{2 \sqrt [4]{-1-x^4+x^8}}+\frac {\arctan \left (\frac {2^{3/4} x \sqrt [4]{-1-x^4+x^8}}{\sqrt {2} x^2-\sqrt {-1-x^4+x^8}}\right )}{4\ 2^{3/4}}-\frac {\text {arctanh}\left (\frac {2 \sqrt [4]{2} x \sqrt [4]{-1-x^4+x^8}}{2 x^2+\sqrt {2} \sqrt {-1-x^4+x^8}}\right )}{4\ 2^{3/4}} \]
-1/2*x/(x^8-x^4-1)^(1/4)+1/8*arctan(2^(3/4)*x*(x^8-x^4-1)^(1/4)/(2^(1/2)*x ^2-(x^8-x^4-1)^(1/2)))*2^(1/4)-1/8*arctanh(2*2^(1/4)*x*(x^8-x^4-1)^(1/4)/( 2*x^2+2^(1/2)*(x^8-x^4-1)^(1/2)))*2^(1/4)
Time = 1.63 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.98 \[ \int \frac {\left (-1+x^8\right ) \left (1+x^8\right )}{\sqrt [4]{-1-x^4+x^8} \left (1-3 x^8+x^{16}\right )} \, dx=\frac {1}{8} \left (-\frac {4 x}{\sqrt [4]{-1-x^4+x^8}}+\sqrt [4]{2} \arctan \left (\frac {2^{3/4} x \sqrt [4]{-1-x^4+x^8}}{\sqrt {2} x^2-\sqrt {-1-x^4+x^8}}\right )-\sqrt [4]{2} \text {arctanh}\left (\frac {2 \sqrt [4]{2} x \sqrt [4]{-1-x^4+x^8}}{2 x^2+\sqrt {2} \sqrt {-1-x^4+x^8}}\right )\right ) \]
((-4*x)/(-1 - x^4 + x^8)^(1/4) + 2^(1/4)*ArcTan[(2^(3/4)*x*(-1 - x^4 + x^8 )^(1/4))/(Sqrt[2]*x^2 - Sqrt[-1 - x^4 + x^8])] - 2^(1/4)*ArcTanh[(2*2^(1/4 )*x*(-1 - x^4 + x^8)^(1/4))/(2*x^2 + Sqrt[2]*Sqrt[-1 - x^4 + x^8])])/8
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.
| \(\displaystyle \int \frac {\left (x^8-1\right ) \left (x^8+1\right )}{\sqrt [4]{x^8-x^4-1} \left (x^{16}-3 x^8+1\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x^{16}-1}{\sqrt [4]{x^8-x^4-1} \left (x^{16}-3 x^8+1\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {1}{\sqrt [4]{x^8-x^4-1}}-\frac {2-3 x^8}{\sqrt [4]{x^8-x^4-1} \left (x^{16}-3 x^8+1\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{2} \sqrt {3-\sqrt {5}} \int \frac {1}{\left (\sqrt {3-\sqrt {5}}-\sqrt {2} x^4\right ) \sqrt [4]{x^8-x^4-1}}dx-\frac {1}{2} \sqrt {3+\sqrt {5}} \int \frac {1}{\left (\sqrt {2} x^4+\sqrt {3+\sqrt {5}}\right ) \sqrt [4]{x^8-x^4-1}}dx+\frac {x \sqrt [4]{1-\frac {2 x^4}{1-\sqrt {5}}} \sqrt [4]{1-\frac {2 x^4}{1+\sqrt {5}}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{4},\frac {1}{4},\frac {5}{4},\frac {2 x^4}{1+\sqrt {5}},\frac {2 x^4}{1-\sqrt {5}}\right )}{\sqrt [4]{x^8-x^4-1}}-\frac {x \sqrt [4]{\sqrt {3+\sqrt {5}}-\sqrt {2} x^4} \sqrt [4]{\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x^4+1} \operatorname {AppellF1}\left (\frac {1}{4},\frac {5}{4},\frac {1}{4},\frac {5}{4},\sqrt {\frac {2}{3+\sqrt {5}}} x^4,-\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x^4\right )}{2 \sqrt [8]{3+\sqrt {5}} \sqrt [4]{x^8-x^4-1}}-\frac {x \sqrt [4]{\sqrt {2} x^4+\sqrt {3-\sqrt {5}}} \sqrt [4]{1-\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )} x^4} \operatorname {AppellF1}\left (\frac {1}{4},\frac {5}{4},\frac {1}{4},\frac {5}{4},-\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x^4,\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )} x^4\right )}{2 \sqrt [8]{3-\sqrt {5}} \sqrt [4]{x^8-x^4-1}}\) |
3.20.69.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 6.22 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.40
| method | result | size |
| pseudoelliptic | \(\frac {\frac {\ln \left (\frac {-\left (x^{8}-x^{4}-1\right )^{\frac {1}{4}} x 2^{\frac {3}{4}}+\sqrt {2}\, x^{2}+\sqrt {x^{8}-x^{4}-1}}{\left (x^{8}-x^{4}-1\right )^{\frac {1}{4}} x 2^{\frac {3}{4}}+\sqrt {2}\, x^{2}+\sqrt {x^{8}-x^{4}-1}}\right ) 2^{\frac {1}{4}} \left (x^{8}-x^{4}-1\right )^{\frac {1}{4}}}{2}+\arctan \left (\frac {2^{\frac {1}{4}} \left (x^{8}-x^{4}-1\right )^{\frac {1}{4}}-x}{x}\right ) 2^{\frac {1}{4}} \left (x^{8}-x^{4}-1\right )^{\frac {1}{4}}+\arctan \left (\frac {2^{\frac {1}{4}} \left (x^{8}-x^{4}-1\right )^{\frac {1}{4}}+x}{x}\right ) 2^{\frac {1}{4}} \left (x^{8}-x^{4}-1\right )^{\frac {1}{4}}-4 x}{8 \left (x^{8}-x^{4}-1\right )^{\frac {1}{4}}}\) | \(195\) |
| trager | \(-\frac {x}{2 \left (x^{8}-x^{4}-1\right )^{\frac {1}{4}}}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right ) x^{8}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{3} \sqrt {x^{8}-x^{4}-1}\, x^{2}+2 \left (x^{8}-x^{4}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right ) x^{4}-4 \left (x^{8}-x^{4}-1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )}{x^{8}+x^{4}-1}\right )}{16}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right ) \ln \left (-\frac {x^{8} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} \sqrt {x^{8}-x^{4}-1}\, x^{2}+2 \left (x^{8}-x^{4}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} x^{3}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right ) x^{4}+4 \left (x^{8}-x^{4}-1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right )}{x^{8}+x^{4}-1}\right )}{16}\) | \(286\) |
| risch | \(-\frac {x}{2 \left (x^{8}-x^{4}-1\right )^{\frac {1}{4}}}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right ) \ln \left (-\frac {x^{8} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} \sqrt {x^{8}-x^{4}-1}\, x^{2}+2 \left (x^{8}-x^{4}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} x^{3}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right ) x^{4}+4 \left (x^{8}-x^{4}-1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right )}{x^{8}+x^{4}-1}\right )}{16}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right ) x^{8}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{3} \sqrt {x^{8}-x^{4}-1}\, x^{2}-2 \left (x^{8}-x^{4}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right ) x^{4}+4 \left (x^{8}-x^{4}-1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )}{x^{8}+x^{4}-1}\right )}{16}\) | \(287\) |
1/8/(x^8-x^4-1)^(1/4)*(1/2*ln((-(x^8-x^4-1)^(1/4)*x*2^(3/4)+2^(1/2)*x^2+(x ^8-x^4-1)^(1/2))/((x^8-x^4-1)^(1/4)*x*2^(3/4)+2^(1/2)*x^2+(x^8-x^4-1)^(1/2 )))*2^(1/4)*(x^8-x^4-1)^(1/4)+arctan((2^(1/4)*(x^8-x^4-1)^(1/4)-x)/x)*2^(1 /4)*(x^8-x^4-1)^(1/4)+arctan((2^(1/4)*(x^8-x^4-1)^(1/4)+x)/x)*2^(1/4)*(x^8 -x^4-1)^(1/4)-4*x)
Result contains complex when optimal does not.
Time = 14.11 (sec) , antiderivative size = 430, normalized size of antiderivative = 3.09 \[ \int \frac {\left (-1+x^8\right ) \left (1+x^8\right )}{\sqrt [4]{-1-x^4+x^8} \left (1-3 x^8+x^{16}\right )} \, dx=\frac {2^{\frac {1}{4}} {\left (\left (i - 1\right ) \, x^{8} - \left (i - 1\right ) \, x^{4} - i + 1\right )} \log \left (\frac {\left (2 i + 2\right ) \cdot 2^{\frac {3}{4}} {\left (x^{8} - x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{8} - x^{4} - 1} x^{2} - \left (2 i - 2\right ) \cdot 2^{\frac {1}{4}} {\left (x^{8} - x^{4} - 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (-i \, x^{8} + 3 i \, x^{4} + i\right )}}{x^{8} + x^{4} - 1}\right ) + 2^{\frac {1}{4}} {\left (-\left (i + 1\right ) \, x^{8} + \left (i + 1\right ) \, x^{4} + i + 1\right )} \log \left (\frac {-\left (2 i - 2\right ) \cdot 2^{\frac {3}{4}} {\left (x^{8} - x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{8} - x^{4} - 1} x^{2} + \left (2 i + 2\right ) \cdot 2^{\frac {1}{4}} {\left (x^{8} - x^{4} - 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (i \, x^{8} - 3 i \, x^{4} - i\right )}}{x^{8} + x^{4} - 1}\right ) + 2^{\frac {1}{4}} {\left (\left (i + 1\right ) \, x^{8} - \left (i + 1\right ) \, x^{4} - i - 1\right )} \log \left (\frac {\left (2 i - 2\right ) \cdot 2^{\frac {3}{4}} {\left (x^{8} - x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{8} - x^{4} - 1} x^{2} - \left (2 i + 2\right ) \cdot 2^{\frac {1}{4}} {\left (x^{8} - x^{4} - 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (i \, x^{8} - 3 i \, x^{4} - i\right )}}{x^{8} + x^{4} - 1}\right ) + 2^{\frac {1}{4}} {\left (-\left (i - 1\right ) \, x^{8} + \left (i - 1\right ) \, x^{4} + i - 1\right )} \log \left (\frac {-\left (2 i + 2\right ) \cdot 2^{\frac {3}{4}} {\left (x^{8} - x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{8} - x^{4} - 1} x^{2} + \left (2 i - 2\right ) \cdot 2^{\frac {1}{4}} {\left (x^{8} - x^{4} - 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (-i \, x^{8} + 3 i \, x^{4} + i\right )}}{x^{8} + x^{4} - 1}\right ) - 16 \, {\left (x^{8} - x^{4} - 1\right )}^{\frac {3}{4}} x}{32 \, {\left (x^{8} - x^{4} - 1\right )}} \]
1/32*(2^(1/4)*((I - 1)*x^8 - (I - 1)*x^4 - I + 1)*log(((2*I + 2)*2^(3/4)*( x^8 - x^4 - 1)^(1/4)*x^3 + 4*sqrt(x^8 - x^4 - 1)*x^2 - (2*I - 2)*2^(1/4)*( x^8 - x^4 - 1)^(3/4)*x + sqrt(2)*(-I*x^8 + 3*I*x^4 + I))/(x^8 + x^4 - 1)) + 2^(1/4)*(-(I + 1)*x^8 + (I + 1)*x^4 + I + 1)*log((-(2*I - 2)*2^(3/4)*(x^ 8 - x^4 - 1)^(1/4)*x^3 + 4*sqrt(x^8 - x^4 - 1)*x^2 + (2*I + 2)*2^(1/4)*(x^ 8 - x^4 - 1)^(3/4)*x + sqrt(2)*(I*x^8 - 3*I*x^4 - I))/(x^8 + x^4 - 1)) + 2 ^(1/4)*((I + 1)*x^8 - (I + 1)*x^4 - I - 1)*log(((2*I - 2)*2^(3/4)*(x^8 - x ^4 - 1)^(1/4)*x^3 + 4*sqrt(x^8 - x^4 - 1)*x^2 - (2*I + 2)*2^(1/4)*(x^8 - x ^4 - 1)^(3/4)*x + sqrt(2)*(I*x^8 - 3*I*x^4 - I))/(x^8 + x^4 - 1)) + 2^(1/4 )*(-(I - 1)*x^8 + (I - 1)*x^4 + I - 1)*log((-(2*I + 2)*2^(3/4)*(x^8 - x^4 - 1)^(1/4)*x^3 + 4*sqrt(x^8 - x^4 - 1)*x^2 + (2*I - 2)*2^(1/4)*(x^8 - x^4 - 1)^(3/4)*x + sqrt(2)*(-I*x^8 + 3*I*x^4 + I))/(x^8 + x^4 - 1)) - 16*(x^8 - x^4 - 1)^(3/4)*x)/(x^8 - x^4 - 1)
Timed out. \[ \int \frac {\left (-1+x^8\right ) \left (1+x^8\right )}{\sqrt [4]{-1-x^4+x^8} \left (1-3 x^8+x^{16}\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\left (-1+x^8\right ) \left (1+x^8\right )}{\sqrt [4]{-1-x^4+x^8} \left (1-3 x^8+x^{16}\right )} \, dx=\int { \frac {{\left (x^{8} + 1\right )} {\left (x^{8} - 1\right )}}{{\left (x^{16} - 3 \, x^{8} + 1\right )} {\left (x^{8} - x^{4} - 1\right )}^{\frac {1}{4}}} \,d x } \]
\[ \int \frac {\left (-1+x^8\right ) \left (1+x^8\right )}{\sqrt [4]{-1-x^4+x^8} \left (1-3 x^8+x^{16}\right )} \, dx=\int { \frac {{\left (x^{8} + 1\right )} {\left (x^{8} - 1\right )}}{{\left (x^{16} - 3 \, x^{8} + 1\right )} {\left (x^{8} - x^{4} - 1\right )}^{\frac {1}{4}}} \,d x } \]
Timed out. \[ \int \frac {\left (-1+x^8\right ) \left (1+x^8\right )}{\sqrt [4]{-1-x^4+x^8} \left (1-3 x^8+x^{16}\right )} \, dx=\int \frac {\left (x^8-1\right )\,\left (x^8+1\right )}{{\left (x^8-x^4-1\right )}^{1/4}\,\left (x^{16}-3\,x^8+1\right )} \,d x \]