Integrand size = 44, antiderivative size = 106 \[ \int \frac {\left (1+x^6\right ) \left (-1+2 x^6\right ) \left (-1+x^4+2 x^6\right )^{5/4}}{x^{10} \left (-1-x^4+2 x^6\right )} \, dx=\frac {\sqrt [4]{-1+x^4+2 x^6} \left (5-19 x^4-20 x^6+104 x^8+38 x^{10}+20 x^{12}\right )}{45 x^9}+\sqrt [4]{2} \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4+2 x^6}}\right )-\sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4+2 x^6}}\right ) \]
1/45*(2*x^6+x^4-1)^(1/4)*(20*x^12+38*x^10+104*x^8-20*x^6-19*x^4+5)/x^9+2^( 1/4)*arctan(2^(1/4)*x/(2*x^6+x^4-1)^(1/4))-2^(1/4)*arctanh(2^(1/4)*x/(2*x^ 6+x^4-1)^(1/4))
Time = 1.67 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+x^6\right ) \left (-1+2 x^6\right ) \left (-1+x^4+2 x^6\right )^{5/4}}{x^{10} \left (-1-x^4+2 x^6\right )} \, dx=\frac {\sqrt [4]{-1+x^4+2 x^6} \left (5-19 x^4-20 x^6+104 x^8+38 x^{10}+20 x^{12}\right )}{45 x^9}+\sqrt [4]{2} \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4+2 x^6}}\right )-\sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4+2 x^6}}\right ) \]
((-1 + x^4 + 2*x^6)^(1/4)*(5 - 19*x^4 - 20*x^6 + 104*x^8 + 38*x^10 + 20*x^ 12))/(45*x^9) + 2^(1/4)*ArcTan[(2^(1/4)*x)/(-1 + x^4 + 2*x^6)^(1/4)] - 2^( 1/4)*ArcTanh[(2^(1/4)*x)/(-1 + x^4 + 2*x^6)^(1/4)]
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.
| \(\displaystyle \int \frac {\left (x^6+1\right ) \left (2 x^6-1\right ) \left (2 x^6+x^4-1\right )^{5/4}}{x^{10} \left (2 x^6-x^4-1\right )} \, dx\) |
| \(\Big \downarrow \) 2461 |
| \(\displaystyle \int \left (\frac {\left (x^6+1\right ) \left (2 x^6-1\right ) \left (2 x^6+x^4-1\right )^{5/4}}{4 x^{10} \left (x^2-1\right )}+\frac {\left (-2 x^2-3\right ) \left (x^6+1\right ) \left (2 x^6-1\right ) \left (2 x^6+x^4-1\right )^{5/4}}{4 x^{10} \left (2 x^4+x^2+1\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (3+i \sqrt {7}\right ) \int \frac {\left (2 x^6+x^4-1\right )^{5/4}}{\sqrt {-1-i \sqrt {7}}-2 x}dx}{2 \sqrt {-1-i \sqrt {7}}}+\frac {\left (3-i \sqrt {7}\right ) \int \frac {\left (2 x^6+x^4-1\right )^{5/4}}{\sqrt {-1+i \sqrt {7}}-2 x}dx}{2 \sqrt {-1+i \sqrt {7}}}+\frac {1}{4} \int \frac {\left (2 x^6+x^4-1\right )^{5/4}}{-x-1}dx+\frac {1}{4} \int \frac {\left (2 x^6+x^4-1\right )^{5/4}}{x-1}dx-\int \frac {\left (2 x^6+x^4-1\right )^{5/4}}{x^6}dx+\int \frac {\left (2 x^6+x^4-1\right )^{5/4}}{x^4}dx+\frac {\left (3+i \sqrt {7}\right ) \int \frac {\left (2 x^6+x^4-1\right )^{5/4}}{2 x+\sqrt {-1-i \sqrt {7}}}dx}{2 \sqrt {-1-i \sqrt {7}}}+\frac {\left (3-i \sqrt {7}\right ) \int \frac {\left (2 x^6+x^4-1\right )^{5/4}}{2 x+\sqrt {-1+i \sqrt {7}}}dx}{2 \sqrt {-1+i \sqrt {7}}}+\int \frac {\left (2 x^6+x^4-1\right )^{5/4}}{x^{10}}dx+\int \frac {\left (2 x^6+x^4-1\right )^{5/4}}{x^2}dx\) |
3.16.41.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px /. x -> Sqrt[x]]}, Int[ExpandIntegrand[u, (Qx /. x -> x^2)^p, x], x] /; !SumQ[NonfreeFactors[ Qx, x]]] /; PolyQ[Px, x^2] && GtQ[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 58.01 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.14
| method | result | size |
| pseudoelliptic | \(\frac {-45 x^{9} \left (\ln \left (\frac {2^{\frac {1}{4}} x +\left (2 x^{6}+x^{4}-1\right )^{\frac {1}{4}}}{-2^{\frac {1}{4}} x +\left (2 x^{6}+x^{4}-1\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\left (2 x^{6}+x^{4}-1\right )^{\frac {1}{4}} 2^{\frac {3}{4}}}{2 x}\right )\right ) 2^{\frac {1}{4}}+2 \left (2 x^{6}+x^{4}-1\right )^{\frac {1}{4}} \left (20 x^{12}+38 x^{10}+104 x^{8}-20 x^{6}-19 x^{4}+5\right )}{90 x^{9}}\) | \(121\) |
| trager | \(\frac {\left (2 x^{6}+x^{4}-1\right )^{\frac {1}{4}} \left (20 x^{12}+38 x^{10}+104 x^{8}-20 x^{6}-19 x^{4}+5\right )}{45 x^{9}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (-\frac {-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{6}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{4}+4 \left (2 x^{6}+x^{4}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{3}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \sqrt {2 x^{6}+x^{4}-1}\, x^{2}-4 \left (2 x^{6}+x^{4}-1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right )}{\left (-1+x \right ) \left (1+x \right ) \left (2 x^{4}+x^{2}+1\right )}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) \ln \left (\frac {-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x^{6}-3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x^{4}+4 \left (2 x^{6}+x^{4}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{3}-4 \sqrt {2 x^{6}+x^{4}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) x^{2}+4 \left (2 x^{6}+x^{4}-1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3}}{\left (-1+x \right ) \left (1+x \right ) \left (2 x^{4}+x^{2}+1\right )}\right )}{2}\) | \(361\) |
| risch | \(\text {Expression too large to display}\) | \(1583\) |
1/90*(-45*x^9*(ln((2^(1/4)*x+(2*x^6+x^4-1)^(1/4))/(-2^(1/4)*x+(2*x^6+x^4-1 )^(1/4)))+2*arctan(1/2*(2*x^6+x^4-1)^(1/4)/x*2^(3/4)))*2^(1/4)+2*(2*x^6+x^ 4-1)^(1/4)*(20*x^12+38*x^10+104*x^8-20*x^6-19*x^4+5))/x^9
Result contains complex when optimal does not.
Time = 141.00 (sec) , antiderivative size = 435, normalized size of antiderivative = 4.10 \[ \int \frac {\left (1+x^6\right ) \left (-1+2 x^6\right ) \left (-1+x^4+2 x^6\right )^{5/4}}{x^{10} \left (-1-x^4+2 x^6\right )} \, dx=-\frac {45 \cdot 8^{\frac {3}{4}} x^{9} \log \left (\frac {4 \, \sqrt {2} {\left (2 \, x^{6} + x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 8^{\frac {3}{4}} \sqrt {2 \, x^{6} + x^{4} - 1} x^{2} + 4 \, {\left (2 \, x^{6} + x^{4} - 1\right )}^{\frac {3}{4}} x + 8^{\frac {1}{4}} {\left (2 \, x^{6} + 3 \, x^{4} - 1\right )}}{2 \, x^{6} - x^{4} - 1}\right ) - 45 i \cdot 8^{\frac {3}{4}} x^{9} \log \left (-\frac {4 \, \sqrt {2} {\left (2 \, x^{6} + x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + i \cdot 8^{\frac {3}{4}} \sqrt {2 \, x^{6} + x^{4} - 1} x^{2} - 4 \, {\left (2 \, x^{6} + x^{4} - 1\right )}^{\frac {3}{4}} x - 8^{\frac {1}{4}} {\left (2 i \, x^{6} + 3 i \, x^{4} - i\right )}}{2 \, x^{6} - x^{4} - 1}\right ) + 45 i \cdot 8^{\frac {3}{4}} x^{9} \log \left (-\frac {4 \, \sqrt {2} {\left (2 \, x^{6} + x^{4} - 1\right )}^{\frac {1}{4}} x^{3} - i \cdot 8^{\frac {3}{4}} \sqrt {2 \, x^{6} + x^{4} - 1} x^{2} - 4 \, {\left (2 \, x^{6} + x^{4} - 1\right )}^{\frac {3}{4}} x - 8^{\frac {1}{4}} {\left (-2 i \, x^{6} - 3 i \, x^{4} + i\right )}}{2 \, x^{6} - x^{4} - 1}\right ) - 45 \cdot 8^{\frac {3}{4}} x^{9} \log \left (\frac {4 \, \sqrt {2} {\left (2 \, x^{6} + x^{4} - 1\right )}^{\frac {1}{4}} x^{3} - 8^{\frac {3}{4}} \sqrt {2 \, x^{6} + x^{4} - 1} x^{2} + 4 \, {\left (2 \, x^{6} + x^{4} - 1\right )}^{\frac {3}{4}} x - 8^{\frac {1}{4}} {\left (2 \, x^{6} + 3 \, x^{4} - 1\right )}}{2 \, x^{6} - x^{4} - 1}\right ) - 16 \, {\left (20 \, x^{12} + 38 \, x^{10} + 104 \, x^{8} - 20 \, x^{6} - 19 \, x^{4} + 5\right )} {\left (2 \, x^{6} + x^{4} - 1\right )}^{\frac {1}{4}}}{720 \, x^{9}} \]
-1/720*(45*8^(3/4)*x^9*log((4*sqrt(2)*(2*x^6 + x^4 - 1)^(1/4)*x^3 + 8^(3/4 )*sqrt(2*x^6 + x^4 - 1)*x^2 + 4*(2*x^6 + x^4 - 1)^(3/4)*x + 8^(1/4)*(2*x^6 + 3*x^4 - 1))/(2*x^6 - x^4 - 1)) - 45*I*8^(3/4)*x^9*log(-(4*sqrt(2)*(2*x^ 6 + x^4 - 1)^(1/4)*x^3 + I*8^(3/4)*sqrt(2*x^6 + x^4 - 1)*x^2 - 4*(2*x^6 + x^4 - 1)^(3/4)*x - 8^(1/4)*(2*I*x^6 + 3*I*x^4 - I))/(2*x^6 - x^4 - 1)) + 4 5*I*8^(3/4)*x^9*log(-(4*sqrt(2)*(2*x^6 + x^4 - 1)^(1/4)*x^3 - I*8^(3/4)*sq rt(2*x^6 + x^4 - 1)*x^2 - 4*(2*x^6 + x^4 - 1)^(3/4)*x - 8^(1/4)*(-2*I*x^6 - 3*I*x^4 + I))/(2*x^6 - x^4 - 1)) - 45*8^(3/4)*x^9*log((4*sqrt(2)*(2*x^6 + x^4 - 1)^(1/4)*x^3 - 8^(3/4)*sqrt(2*x^6 + x^4 - 1)*x^2 + 4*(2*x^6 + x^4 - 1)^(3/4)*x - 8^(1/4)*(2*x^6 + 3*x^4 - 1))/(2*x^6 - x^4 - 1)) - 16*(20*x^ 12 + 38*x^10 + 104*x^8 - 20*x^6 - 19*x^4 + 5)*(2*x^6 + x^4 - 1)^(1/4))/x^9
Timed out. \[ \int \frac {\left (1+x^6\right ) \left (-1+2 x^6\right ) \left (-1+x^4+2 x^6\right )^{5/4}}{x^{10} \left (-1-x^4+2 x^6\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\left (1+x^6\right ) \left (-1+2 x^6\right ) \left (-1+x^4+2 x^6\right )^{5/4}}{x^{10} \left (-1-x^4+2 x^6\right )} \, dx=\int { \frac {{\left (2 \, x^{6} + x^{4} - 1\right )}^{\frac {5}{4}} {\left (2 \, x^{6} - 1\right )} {\left (x^{6} + 1\right )}}{{\left (2 \, x^{6} - x^{4} - 1\right )} x^{10}} \,d x } \]
\[ \int \frac {\left (1+x^6\right ) \left (-1+2 x^6\right ) \left (-1+x^4+2 x^6\right )^{5/4}}{x^{10} \left (-1-x^4+2 x^6\right )} \, dx=\int { \frac {{\left (2 \, x^{6} + x^{4} - 1\right )}^{\frac {5}{4}} {\left (2 \, x^{6} - 1\right )} {\left (x^{6} + 1\right )}}{{\left (2 \, x^{6} - x^{4} - 1\right )} x^{10}} \,d x } \]
Timed out. \[ \int \frac {\left (1+x^6\right ) \left (-1+2 x^6\right ) \left (-1+x^4+2 x^6\right )^{5/4}}{x^{10} \left (-1-x^4+2 x^6\right )} \, dx=\int -\frac {\left (x^6+1\right )\,\left (2\,x^6-1\right )\,{\left (2\,x^6+x^4-1\right )}^{5/4}}{x^{10}\,\left (-2\,x^6+x^4+1\right )} \,d x \]