Integrand size = 39, antiderivative size = 141 \[ \int \frac {\left (2+x^6\right ) \left (-1-x^4+x^6\right )}{\sqrt [4]{1-x^4-x^6} \left (-1+x^6\right )^2} \, dx=-\frac {x \left (1-x^4-x^6\right )^{3/4}}{2 \left (-1+x^6\right )}-\frac {5 \arctan \left (\frac {\sqrt {2} x \sqrt [4]{1-x^4-x^6}}{-x^2+\sqrt {1-x^4-x^6}}\right )}{4 \sqrt {2}}-\frac {5 \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{1-x^4-x^6}}{x^2+\sqrt {1-x^4-x^6}}\right )}{4 \sqrt {2}} \]
-x*(-x^6-x^4+1)^(3/4)/(2*x^6-2)-5/8*arctan(2^(1/2)*x*(-x^6-x^4+1)^(1/4)/(- x^2+(-x^6-x^4+1)^(1/2)))*2^(1/2)-5/8*arctanh(2^(1/2)*x*(-x^6-x^4+1)^(1/4)/ (x^2+(-x^6-x^4+1)^(1/2)))*2^(1/2)
Time = 1.72 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00 \[ \int \frac {\left (2+x^6\right ) \left (-1-x^4+x^6\right )}{\sqrt [4]{1-x^4-x^6} \left (-1+x^6\right )^2} \, dx=-\frac {x \left (1-x^4-x^6\right )^{3/4}}{2 \left (-1+x^6\right )}-\frac {5 \arctan \left (\frac {\sqrt {2} x \sqrt [4]{1-x^4-x^6}}{-x^2+\sqrt {1-x^4-x^6}}\right )}{4 \sqrt {2}}-\frac {5 \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{1-x^4-x^6}}{x^2+\sqrt {1-x^4-x^6}}\right )}{4 \sqrt {2}} \]
-1/2*(x*(1 - x^4 - x^6)^(3/4))/(-1 + x^6) - (5*ArcTan[(Sqrt[2]*x*(1 - x^4 - x^6)^(1/4))/(-x^2 + Sqrt[1 - x^4 - x^6])])/(4*Sqrt[2]) - (5*ArcTanh[(Sqr t[2]*x*(1 - x^4 - x^6)^(1/4))/(x^2 + Sqrt[1 - x^4 - x^6])])/(4*Sqrt[2])
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.
| \(\displaystyle \int \frac {\left (x^6+2\right ) \left (x^6-x^4-1\right )}{\sqrt [4]{-x^6-x^4+1} \left (x^6-1\right )^2} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {1}{12 (x-1)^2 \sqrt [4]{-x^6-x^4+1}}-\frac {1}{12 (x+1)^2 \sqrt [4]{-x^6-x^4+1}}+\frac {1}{\sqrt [4]{-x^6-x^4+1}}+\frac {5 (x-3)}{12 \left (x^2-x+1\right ) \sqrt [4]{-x^6-x^4+1}}+\frac {5}{6 \left (x^2-1\right ) \sqrt [4]{-x^6-x^4+1}}-\frac {5 (x+3)}{12 \left (x^2+x+1\right ) \sqrt [4]{-x^6-x^4+1}}+\frac {1}{4 \left (x^2-x+1\right )^2 \sqrt [4]{-x^6-x^4+1}}+\frac {1}{4 \left (x^2+x+1\right )^2 \sqrt [4]{-x^6-x^4+1}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \int \frac {1}{\sqrt [4]{-x^6-x^4+1}}dx-\frac {1}{3} \int \frac {1}{\left (-2 x+i \sqrt {3}-1\right )^2 \sqrt [4]{-x^6-x^4+1}}dx+\frac {i \int \frac {1}{\left (-2 x+i \sqrt {3}-1\right ) \sqrt [4]{-x^6-x^4+1}}dx}{3 \sqrt {3}}-\frac {1}{3} \int \frac {1}{\left (-2 x+i \sqrt {3}+1\right )^2 \sqrt [4]{-x^6-x^4+1}}dx+\frac {i \int \frac {1}{\left (-2 x+i \sqrt {3}+1\right ) \sqrt [4]{-x^6-x^4+1}}dx}{3 \sqrt {3}}-\frac {5}{12} \int \frac {1}{(1-x) \sqrt [4]{-x^6-x^4+1}}dx-\frac {1}{12} \int \frac {1}{(x-1)^2 \sqrt [4]{-x^6-x^4+1}}dx-\frac {1}{12} \int \frac {1}{(x+1)^2 \sqrt [4]{-x^6-x^4+1}}dx-\frac {5}{12} \int \frac {1}{(x+1) \sqrt [4]{-x^6-x^4+1}}dx+\frac {5}{36} \left (3+5 i \sqrt {3}\right ) \int \frac {1}{\left (2 x-i \sqrt {3}-1\right ) \sqrt [4]{-x^6-x^4+1}}dx-\frac {5}{36} \left (3-5 i \sqrt {3}\right ) \int \frac {1}{\left (2 x-i \sqrt {3}+1\right ) \sqrt [4]{-x^6-x^4+1}}dx-\frac {1}{3} \int \frac {1}{\left (2 x+i \sqrt {3}-1\right )^2 \sqrt [4]{-x^6-x^4+1}}dx+\frac {5}{36} \left (3-5 i \sqrt {3}\right ) \int \frac {1}{\left (2 x+i \sqrt {3}-1\right ) \sqrt [4]{-x^6-x^4+1}}dx+\frac {i \int \frac {1}{\left (2 x+i \sqrt {3}-1\right ) \sqrt [4]{-x^6-x^4+1}}dx}{3 \sqrt {3}}-\frac {1}{3} \int \frac {1}{\left (2 x+i \sqrt {3}+1\right )^2 \sqrt [4]{-x^6-x^4+1}}dx-\frac {5}{36} \left (3+5 i \sqrt {3}\right ) \int \frac {1}{\left (2 x+i \sqrt {3}+1\right ) \sqrt [4]{-x^6-x^4+1}}dx+\frac {i \int \frac {1}{\left (2 x+i \sqrt {3}+1\right ) \sqrt [4]{-x^6-x^4+1}}dx}{3 \sqrt {3}}\) |
3.21.2.3.1 Defintions of rubi rules used
Time = 14.01 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.24
| method | result | size |
| pseudoelliptic | \(\frac {5 \left (x^{6}-1\right ) \left (\ln \left (\frac {-\left (-x^{6}-x^{4}+1\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {-x^{6}-x^{4}+1}}{\left (-x^{6}-x^{4}+1\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {-x^{6}-x^{4}+1}}\right )+2 \arctan \left (\frac {\left (-x^{6}-x^{4}+1\right )^{\frac {1}{4}} \sqrt {2}+x}{x}\right )+2 \arctan \left (\frac {\left (-x^{6}-x^{4}+1\right )^{\frac {1}{4}} \sqrt {2}-x}{x}\right )\right ) \sqrt {2}-8 \left (-x^{6}-x^{4}+1\right )^{\frac {3}{4}} x}{16 x^{6}-16}\) | \(175\) |
| trager | \(-\frac {x \left (-x^{6}-x^{4}+1\right )^{\frac {3}{4}}}{2 \left (x^{6}-1\right )}-\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{6}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{4}+2 \left (-x^{6}-x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \sqrt {-x^{6}-x^{4}+1}\, x^{2}+2 \left (-x^{6}-x^{4}+1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}}{\left (-1+x \right ) \left (x^{2}+x +1\right ) \left (1+x \right ) \left (x^{2}-x +1\right )}\right )}{8}+\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \sqrt {-x^{6}-x^{4}+1}\, x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{6}-2 \left (-x^{6}-x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{4}+2 \left (-x^{6}-x^{4}+1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{\left (-1+x \right ) \left (x^{2}+x +1\right ) \left (1+x \right ) \left (x^{2}-x +1\right )}\right )}{8}\) | \(309\) |
| risch | \(\frac {x \left (x^{6}+x^{4}-1\right )}{2 \left (x^{6}-1\right ) \left (-x^{6}-x^{4}+1\right )^{\frac {1}{4}}}+\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{6}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{4}+2 \left (-x^{6}-x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \sqrt {-x^{6}-x^{4}+1}\, x^{2}+2 \left (-x^{6}-x^{4}+1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}}{\left (-1+x \right ) \left (x^{2}+x +1\right ) \left (1+x \right ) \left (x^{2}-x +1\right )}\right )}{8}-\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \sqrt {-x^{6}-x^{4}+1}\, x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{6}-2 \left (-x^{6}-x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{4}+2 \left (-x^{6}-x^{4}+1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{\left (-1+x \right ) \left (x^{2}+x +1\right ) \left (1+x \right ) \left (x^{2}-x +1\right )}\right )}{8}\) | \(317\) |
(5*(x^6-1)*(ln((-(-x^6-x^4+1)^(1/4)*2^(1/2)*x+x^2+(-x^6-x^4+1)^(1/2))/((-x ^6-x^4+1)^(1/4)*2^(1/2)*x+x^2+(-x^6-x^4+1)^(1/2)))+2*arctan(((-x^6-x^4+1)^ (1/4)*2^(1/2)+x)/x)+2*arctan(((-x^6-x^4+1)^(1/4)*2^(1/2)-x)/x))*2^(1/2)-8* (-x^6-x^4+1)^(3/4)*x)/(16*x^6-16)
Result contains complex when optimal does not.
Time = 6.23 (sec) , antiderivative size = 411, normalized size of antiderivative = 2.91 \[ \int \frac {\left (2+x^6\right ) \left (-1-x^4+x^6\right )}{\sqrt [4]{1-x^4-x^6} \left (-1+x^6\right )^2} \, dx=-\frac {5 \, \sqrt {2} {\left (-\left (i + 1\right ) \, x^{6} + i + 1\right )} \log \left (\frac {4 i \, {\left (-x^{6} - x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - \left (2 i - 2\right ) \, \sqrt {2} \sqrt {-x^{6} - x^{4} + 1} x^{2} - 4 \, {\left (-x^{6} - x^{4} + 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (-\left (i + 1\right ) \, x^{6} - \left (2 i + 2\right ) \, x^{4} + i + 1\right )}}{x^{6} - 1}\right ) + 5 \, \sqrt {2} {\left (\left (i + 1\right ) \, x^{6} - i - 1\right )} \log \left (\frac {4 i \, {\left (-x^{6} - x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + \left (2 i - 2\right ) \, \sqrt {2} \sqrt {-x^{6} - x^{4} + 1} x^{2} - 4 \, {\left (-x^{6} - x^{4} + 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (\left (i + 1\right ) \, x^{6} + \left (2 i + 2\right ) \, x^{4} - i - 1\right )}}{x^{6} - 1}\right ) + 5 \, \sqrt {2} {\left (\left (i - 1\right ) \, x^{6} - i + 1\right )} \log \left (\frac {-4 i \, {\left (-x^{6} - x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + \left (2 i + 2\right ) \, \sqrt {2} \sqrt {-x^{6} - x^{4} + 1} x^{2} - 4 \, {\left (-x^{6} - x^{4} + 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (\left (i - 1\right ) \, x^{6} + \left (2 i - 2\right ) \, x^{4} - i + 1\right )}}{x^{6} - 1}\right ) + 5 \, \sqrt {2} {\left (-\left (i - 1\right ) \, x^{6} + i - 1\right )} \log \left (\frac {-4 i \, {\left (-x^{6} - x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - \left (2 i + 2\right ) \, \sqrt {2} \sqrt {-x^{6} - x^{4} + 1} x^{2} - 4 \, {\left (-x^{6} - x^{4} + 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (-\left (i - 1\right ) \, x^{6} - \left (2 i - 2\right ) \, x^{4} + i - 1\right )}}{x^{6} - 1}\right ) + 16 \, {\left (-x^{6} - x^{4} + 1\right )}^{\frac {3}{4}} x}{32 \, {\left (x^{6} - 1\right )}} \]
-1/32*(5*sqrt(2)*(-(I + 1)*x^6 + I + 1)*log((4*I*(-x^6 - x^4 + 1)^(1/4)*x^ 3 - (2*I - 2)*sqrt(2)*sqrt(-x^6 - x^4 + 1)*x^2 - 4*(-x^6 - x^4 + 1)^(3/4)* x + sqrt(2)*(-(I + 1)*x^6 - (2*I + 2)*x^4 + I + 1))/(x^6 - 1)) + 5*sqrt(2) *((I + 1)*x^6 - I - 1)*log((4*I*(-x^6 - x^4 + 1)^(1/4)*x^3 + (2*I - 2)*sqr t(2)*sqrt(-x^6 - x^4 + 1)*x^2 - 4*(-x^6 - x^4 + 1)^(3/4)*x + sqrt(2)*((I + 1)*x^6 + (2*I + 2)*x^4 - I - 1))/(x^6 - 1)) + 5*sqrt(2)*((I - 1)*x^6 - I + 1)*log((-4*I*(-x^6 - x^4 + 1)^(1/4)*x^3 + (2*I + 2)*sqrt(2)*sqrt(-x^6 - x^4 + 1)*x^2 - 4*(-x^6 - x^4 + 1)^(3/4)*x + sqrt(2)*((I - 1)*x^6 + (2*I - 2)*x^4 - I + 1))/(x^6 - 1)) + 5*sqrt(2)*(-(I - 1)*x^6 + I - 1)*log((-4*I*( -x^6 - x^4 + 1)^(1/4)*x^3 - (2*I + 2)*sqrt(2)*sqrt(-x^6 - x^4 + 1)*x^2 - 4 *(-x^6 - x^4 + 1)^(3/4)*x + sqrt(2)*(-(I - 1)*x^6 - (2*I - 2)*x^4 + I - 1) )/(x^6 - 1)) + 16*(-x^6 - x^4 + 1)^(3/4)*x)/(x^6 - 1)
\[ \int \frac {\left (2+x^6\right ) \left (-1-x^4+x^6\right )}{\sqrt [4]{1-x^4-x^6} \left (-1+x^6\right )^2} \, dx=\int \frac {\left (x^{6} + 2\right ) \left (x^{6} - x^{4} - 1\right )}{\left (x - 1\right )^{2} \left (x + 1\right )^{2} \left (x^{2} - x + 1\right )^{2} \left (x^{2} + x + 1\right )^{2} \sqrt [4]{- x^{6} - x^{4} + 1}}\, dx \]
Integral((x**6 + 2)*(x**6 - x**4 - 1)/((x - 1)**2*(x + 1)**2*(x**2 - x + 1 )**2*(x**2 + x + 1)**2*(-x**6 - x**4 + 1)**(1/4)), x)
\[ \int \frac {\left (2+x^6\right ) \left (-1-x^4+x^6\right )}{\sqrt [4]{1-x^4-x^6} \left (-1+x^6\right )^2} \, dx=\int { \frac {{\left (x^{6} - x^{4} - 1\right )} {\left (x^{6} + 2\right )}}{{\left (x^{6} - 1\right )}^{2} {\left (-x^{6} - x^{4} + 1\right )}^{\frac {1}{4}}} \,d x } \]
\[ \int \frac {\left (2+x^6\right ) \left (-1-x^4+x^6\right )}{\sqrt [4]{1-x^4-x^6} \left (-1+x^6\right )^2} \, dx=\int { \frac {{\left (x^{6} - x^{4} - 1\right )} {\left (x^{6} + 2\right )}}{{\left (x^{6} - 1\right )}^{2} {\left (-x^{6} - x^{4} + 1\right )}^{\frac {1}{4}}} \,d x } \]
Timed out. \[ \int \frac {\left (2+x^6\right ) \left (-1-x^4+x^6\right )}{\sqrt [4]{1-x^4-x^6} \left (-1+x^6\right )^2} \, dx=\int -\frac {\left (x^6+2\right )\,\left (-x^6+x^4+1\right )}{{\left (x^6-1\right )}^2\,{\left (-x^6-x^4+1\right )}^{1/4}} \,d x \]