Integrand size = 41, antiderivative size = 109 \[ \int \frac {\left (2+x^6\right ) \left (1-2 x^6+x^8+x^{12}\right )}{x^8 \sqrt [4]{-1+x^6} \left (-1+x^4+x^6\right )} \, dx=\frac {2 \left (-1+x^6\right )^{3/4} \left (-3-7 x^4+3 x^6\right )}{21 x^7}-\sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt [4]{-1+x^6}}{-x^2+\sqrt {-1+x^6}}\right )-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-1+x^6}}{x^2+\sqrt {-1+x^6}}\right ) \]
2/21*(x^6-1)^(3/4)*(3*x^6-7*x^4-3)/x^7-2^(1/2)*arctan(2^(1/2)*x*(x^6-1)^(1 /4)/(-x^2+(x^6-1)^(1/2)))-2^(1/2)*arctanh(2^(1/2)*x*(x^6-1)^(1/4)/(x^2+(x^ 6-1)^(1/2)))
Time = 8.03 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00 \[ \int \frac {\left (2+x^6\right ) \left (1-2 x^6+x^8+x^{12}\right )}{x^8 \sqrt [4]{-1+x^6} \left (-1+x^4+x^6\right )} \, dx=\frac {2 \left (-1+x^6\right )^{3/4} \left (-3-7 x^4+3 x^6\right )}{21 x^7}-\sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt [4]{-1+x^6}}{-x^2+\sqrt {-1+x^6}}\right )-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-1+x^6}}{x^2+\sqrt {-1+x^6}}\right ) \]
(2*(-1 + x^6)^(3/4)*(-3 - 7*x^4 + 3*x^6))/(21*x^7) - Sqrt[2]*ArcTan[(Sqrt[ 2]*x*(-1 + x^6)^(1/4))/(-x^2 + Sqrt[-1 + x^6])] - Sqrt[2]*ArcTanh[(Sqrt[2] *x*(-1 + x^6)^(1/4))/(x^2 + Sqrt[-1 + x^6])]
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.
| \(\displaystyle \int \frac {\left (x^6+2\right ) \left (x^{12}+x^8-2 x^6+1\right )}{x^8 \sqrt [4]{x^6-1} \left (x^6+x^4-1\right )} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2}{\sqrt [4]{x^6-1}}-\frac {2}{\sqrt [4]{x^6-1} x^8}+\frac {x^4}{\sqrt [4]{x^6-1}}-\frac {2 \left (x^4-3\right )}{\sqrt [4]{x^6-1} \left (x^6+x^4-1\right )}-\frac {2}{\sqrt [4]{x^6-1} x^4}-\frac {x^2}{\sqrt [4]{x^6-1}}+\frac {1}{\sqrt [4]{x^6-1} x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 6 \int \frac {1}{\sqrt [4]{x^6-1} \left (x^6+x^4-1\right )}dx-2 \int \frac {x^4}{\sqrt [4]{x^6-1} \left (x^6+x^4-1\right )}dx+\frac {2 \sqrt [4]{1-x^6} x \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{4},\frac {7}{6},x^6\right )}{\sqrt [4]{x^6-1}}-\frac {\sqrt [4]{1-x^6} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{4},\frac {5}{6},x^6\right )}{\sqrt [4]{x^6-1} x}+\frac {2 \sqrt [4]{1-x^6} \operatorname {Hypergeometric2F1}\left (-\frac {7}{6},\frac {1}{4},-\frac {1}{6},x^6\right )}{7 \sqrt [4]{x^6-1} x^7}+\frac {\sqrt [4]{1-x^6} x^5 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {5}{6},\frac {11}{6},x^6\right )}{5 \sqrt [4]{x^6-1}}-\frac {2 \left (x^6-1\right )^{3/4}}{3 x^3}\) |
3.17.1.3.1 Defintions of rubi rules used
Time = 35.42 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.19
| method | result | size |
| pseudoelliptic | \(\frac {21 x^{7} \left (\ln \left (\frac {-\left (x^{6}-1\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{6}-1}}{\left (x^{6}-1\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{6}-1}}\right )+2 \arctan \left (\frac {\left (x^{6}-1\right )^{\frac {1}{4}} \sqrt {2}+x}{x}\right )+2 \arctan \left (\frac {\left (x^{6}-1\right )^{\frac {1}{4}} \sqrt {2}-x}{x}\right )\right ) \sqrt {2}+4 \left (x^{6}-1\right )^{\frac {3}{4}} \left (3 x^{6}-7 x^{4}-3\right )}{42 x^{7}}\) | \(130\) |
| trager | \(\frac {2 \left (x^{6}-1\right )^{\frac {3}{4}} \left (3 x^{6}-7 x^{4}-3\right )}{21 x^{7}}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{6}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{4}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \left (x^{6}-1\right )^{\frac {1}{4}} x^{3}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \sqrt {x^{6}-1}\, x^{2}-2 \left (x^{6}-1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}}{x^{6}+x^{4}-1}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \sqrt {x^{6}-1}\, x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{6}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \left (x^{6}-1\right )^{\frac {1}{4}} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{4}+2 \left (x^{6}-1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{x^{6}+x^{4}-1}\right )\) | \(229\) |
| risch | \(\frac {\frac {2}{7} x^{12}-\frac {2}{3} x^{10}-\frac {4}{7} x^{6}+\frac {2}{3} x^{4}+\frac {2}{7}}{x^{7} \left (x^{6}-1\right )^{\frac {1}{4}}}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \sqrt {x^{6}-1}\, x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{6}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \left (x^{6}-1\right )^{\frac {1}{4}} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{4}-2 \left (x^{6}-1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{x^{6}+x^{4}-1}\right )-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{6}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{6}-1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{4}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (x^{6}-1\right )^{\frac {1}{4}} x^{3}+2 x^{2} \sqrt {x^{6}-1}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}}{x^{6}+x^{4}-1}\right )\) | \(239\) |
1/42*(21*x^7*(ln((-(x^6-1)^(1/4)*2^(1/2)*x+x^2+(x^6-1)^(1/2))/((x^6-1)^(1/ 4)*2^(1/2)*x+x^2+(x^6-1)^(1/2)))+2*arctan(((x^6-1)^(1/4)*2^(1/2)+x)/x)+2*a rctan(((x^6-1)^(1/4)*2^(1/2)-x)/x))*2^(1/2)+4*(x^6-1)^(3/4)*(3*x^6-7*x^4-3 ))/x^7
Result contains complex when optimal does not.
Time = 219.36 (sec) , antiderivative size = 323, normalized size of antiderivative = 2.96 \[ \int \frac {\left (2+x^6\right ) \left (1-2 x^6+x^8+x^{12}\right )}{x^8 \sqrt [4]{-1+x^6} \left (-1+x^4+x^6\right )} \, dx=\frac {\left (21 i - 21\right ) \, \sqrt {2} x^{7} \log \left (\frac {4 i \, {\left (x^{6} - 1\right )}^{\frac {1}{4}} x^{3} + \left (2 i + 2\right ) \, \sqrt {2} \sqrt {x^{6} - 1} x^{2} + 4 \, {\left (x^{6} - 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (-\left (i - 1\right ) \, x^{6} + \left (i - 1\right ) \, x^{4} + i - 1\right )}}{x^{6} + x^{4} - 1}\right ) - \left (21 i - 21\right ) \, \sqrt {2} x^{7} \log \left (\frac {4 i \, {\left (x^{6} - 1\right )}^{\frac {1}{4}} x^{3} - \left (2 i + 2\right ) \, \sqrt {2} \sqrt {x^{6} - 1} x^{2} + 4 \, {\left (x^{6} - 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (\left (i - 1\right ) \, x^{6} - \left (i - 1\right ) \, x^{4} - i + 1\right )}}{x^{6} + x^{4} - 1}\right ) - \left (21 i + 21\right ) \, \sqrt {2} x^{7} \log \left (\frac {-4 i \, {\left (x^{6} - 1\right )}^{\frac {1}{4}} x^{3} - \left (2 i - 2\right ) \, \sqrt {2} \sqrt {x^{6} - 1} x^{2} + 4 \, {\left (x^{6} - 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (\left (i + 1\right ) \, x^{6} - \left (i + 1\right ) \, x^{4} - i - 1\right )}}{x^{6} + x^{4} - 1}\right ) + \left (21 i + 21\right ) \, \sqrt {2} x^{7} \log \left (\frac {-4 i \, {\left (x^{6} - 1\right )}^{\frac {1}{4}} x^{3} + \left (2 i - 2\right ) \, \sqrt {2} \sqrt {x^{6} - 1} x^{2} + 4 \, {\left (x^{6} - 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (-\left (i + 1\right ) \, x^{6} + \left (i + 1\right ) \, x^{4} + i + 1\right )}}{x^{6} + x^{4} - 1}\right ) + 8 \, {\left (3 \, x^{6} - 7 \, x^{4} - 3\right )} {\left (x^{6} - 1\right )}^{\frac {3}{4}}}{84 \, x^{7}} \]
1/84*((21*I - 21)*sqrt(2)*x^7*log((4*I*(x^6 - 1)^(1/4)*x^3 + (2*I + 2)*sqr t(2)*sqrt(x^6 - 1)*x^2 + 4*(x^6 - 1)^(3/4)*x + sqrt(2)*(-(I - 1)*x^6 + (I - 1)*x^4 + I - 1))/(x^6 + x^4 - 1)) - (21*I - 21)*sqrt(2)*x^7*log((4*I*(x^ 6 - 1)^(1/4)*x^3 - (2*I + 2)*sqrt(2)*sqrt(x^6 - 1)*x^2 + 4*(x^6 - 1)^(3/4) *x + sqrt(2)*((I - 1)*x^6 - (I - 1)*x^4 - I + 1))/(x^6 + x^4 - 1)) - (21*I + 21)*sqrt(2)*x^7*log((-4*I*(x^6 - 1)^(1/4)*x^3 - (2*I - 2)*sqrt(2)*sqrt( x^6 - 1)*x^2 + 4*(x^6 - 1)^(3/4)*x + sqrt(2)*((I + 1)*x^6 - (I + 1)*x^4 - I - 1))/(x^6 + x^4 - 1)) + (21*I + 21)*sqrt(2)*x^7*log((-4*I*(x^6 - 1)^(1/ 4)*x^3 + (2*I - 2)*sqrt(2)*sqrt(x^6 - 1)*x^2 + 4*(x^6 - 1)^(3/4)*x + sqrt( 2)*(-(I + 1)*x^6 + (I + 1)*x^4 + I + 1))/(x^6 + x^4 - 1)) + 8*(3*x^6 - 7*x ^4 - 3)*(x^6 - 1)^(3/4))/x^7
Timed out. \[ \int \frac {\left (2+x^6\right ) \left (1-2 x^6+x^8+x^{12}\right )}{x^8 \sqrt [4]{-1+x^6} \left (-1+x^4+x^6\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\left (2+x^6\right ) \left (1-2 x^6+x^8+x^{12}\right )}{x^8 \sqrt [4]{-1+x^6} \left (-1+x^4+x^6\right )} \, dx=\int { \frac {{\left (x^{12} + x^{8} - 2 \, x^{6} + 1\right )} {\left (x^{6} + 2\right )}}{{\left (x^{6} + x^{4} - 1\right )} {\left (x^{6} - 1\right )}^{\frac {1}{4}} x^{8}} \,d x } \]
\[ \int \frac {\left (2+x^6\right ) \left (1-2 x^6+x^8+x^{12}\right )}{x^8 \sqrt [4]{-1+x^6} \left (-1+x^4+x^6\right )} \, dx=\int { \frac {{\left (x^{12} + x^{8} - 2 \, x^{6} + 1\right )} {\left (x^{6} + 2\right )}}{{\left (x^{6} + x^{4} - 1\right )} {\left (x^{6} - 1\right )}^{\frac {1}{4}} x^{8}} \,d x } \]
Timed out. \[ \int \frac {\left (2+x^6\right ) \left (1-2 x^6+x^8+x^{12}\right )}{x^8 \sqrt [4]{-1+x^6} \left (-1+x^4+x^6\right )} \, dx=\int \frac {\left (x^6+2\right )\,\left (x^{12}+x^8-2\,x^6+1\right )}{x^8\,{\left (x^6-1\right )}^{1/4}\,\left (x^6+x^4-1\right )} \,d x \]