Integrand size = 40, antiderivative size = 81 \[ \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4} \left (-4+x^3+2 x^6\right )}{\left (1-x^3+x^6\right )^2} \, dx=-\frac {x \left (1-x^3+x^4+x^6\right )^{3/4}}{1-x^3+x^6}-\frac {3}{2} \arctan \left (\frac {x}{\sqrt [4]{1-x^3+x^4+x^6}}\right )-\frac {3}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{1-x^3+x^4+x^6}}\right ) \]
-x*(x^6+x^4-x^3+1)^(3/4)/(x^6-x^3+1)-3/2*arctan(x/(x^6+x^4-x^3+1)^(1/4))-3 /2*arctanh(x/(x^6+x^4-x^3+1)^(1/4))
Time = 0.14 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4} \left (-4+x^3+2 x^6\right )}{\left (1-x^3+x^6\right )^2} \, dx=-\frac {x \left (1-x^3+x^4+x^6\right )^{3/4}}{1-x^3+x^6}-\frac {3}{2} \arctan \left (\frac {x}{\sqrt [4]{1-x^3+x^4+x^6}}\right )-\frac {3}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{1-x^3+x^4+x^6}}\right ) \]
-((x*(1 - x^3 + x^4 + x^6)^(3/4))/(1 - x^3 + x^6)) - (3*ArcTan[x/(1 - x^3 + x^4 + x^6)^(1/4)])/2 - (3*ArcTanh[x/(1 - x^3 + x^4 + x^6)^(1/4)])/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.
| \(\displaystyle \int \frac {\left (x^6+x^4-x^3+1\right )^{3/4} \left (2 x^6+x^3-4\right )}{\left (x^6-x^3+1\right )^2} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {3 \left (x^6+x^4-x^3+1\right )^{3/4} \left (x^3-2\right )}{\left (x^6-x^3+1\right )^2}+\frac {2 \left (x^6+x^4-x^3+1\right )^{3/4}}{x^6-x^3+1}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (x^6+x^4-x^3+1\right )^{3/4} \left (2 x^6+x^3-4\right )}{\left (x^6-x^3+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {3 \left (x^6+x^4-x^3+1\right )^{3/4} \left (x^3-2\right )}{\left (x^6-x^3+1\right )^2}+\frac {2 \left (x^6+x^4-x^3+1\right )^{3/4}}{x^6-x^3+1}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (x^6+x^4-x^3+1\right )^{3/4} \left (2 x^6+x^3-4\right )}{\left (x^6-x^3+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {3 \left (x^6+x^4-x^3+1\right )^{3/4} \left (x^3-2\right )}{\left (x^6-x^3+1\right )^2}+\frac {2 \left (x^6+x^4-x^3+1\right )^{3/4}}{x^6-x^3+1}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (x^6+x^4-x^3+1\right )^{3/4} \left (2 x^6+x^3-4\right )}{\left (x^6-x^3+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {3 \left (x^6+x^4-x^3+1\right )^{3/4} \left (x^3-2\right )}{\left (x^6-x^3+1\right )^2}+\frac {2 \left (x^6+x^4-x^3+1\right )^{3/4}}{x^6-x^3+1}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (x^6+x^4-x^3+1\right )^{3/4} \left (2 x^6+x^3-4\right )}{\left (x^6-x^3+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {3 \left (x^6+x^4-x^3+1\right )^{3/4} \left (x^3-2\right )}{\left (x^6-x^3+1\right )^2}+\frac {2 \left (x^6+x^4-x^3+1\right )^{3/4}}{x^6-x^3+1}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (x^6+x^4-x^3+1\right )^{3/4} \left (2 x^6+x^3-4\right )}{\left (x^6-x^3+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {3 \left (x^6+x^4-x^3+1\right )^{3/4} \left (x^3-2\right )}{\left (x^6-x^3+1\right )^2}+\frac {2 \left (x^6+x^4-x^3+1\right )^{3/4}}{x^6-x^3+1}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (x^6+x^4-x^3+1\right )^{3/4} \left (2 x^6+x^3-4\right )}{\left (x^6-x^3+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {3 \left (x^6+x^4-x^3+1\right )^{3/4} \left (x^3-2\right )}{\left (x^6-x^3+1\right )^2}+\frac {2 \left (x^6+x^4-x^3+1\right )^{3/4}}{x^6-x^3+1}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (x^6+x^4-x^3+1\right )^{3/4} \left (2 x^6+x^3-4\right )}{\left (x^6-x^3+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {3 \left (x^6+x^4-x^3+1\right )^{3/4} \left (x^3-2\right )}{\left (x^6-x^3+1\right )^2}+\frac {2 \left (x^6+x^4-x^3+1\right )^{3/4}}{x^6-x^3+1}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (x^6+x^4-x^3+1\right )^{3/4} \left (2 x^6+x^3-4\right )}{\left (x^6-x^3+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {3 \left (x^6+x^4-x^3+1\right )^{3/4} \left (x^3-2\right )}{\left (x^6-x^3+1\right )^2}+\frac {2 \left (x^6+x^4-x^3+1\right )^{3/4}}{x^6-x^3+1}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (x^6+x^4-x^3+1\right )^{3/4} \left (2 x^6+x^3-4\right )}{\left (x^6-x^3+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {3 \left (x^6+x^4-x^3+1\right )^{3/4} \left (x^3-2\right )}{\left (x^6-x^3+1\right )^2}+\frac {2 \left (x^6+x^4-x^3+1\right )^{3/4}}{x^6-x^3+1}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (x^6+x^4-x^3+1\right )^{3/4} \left (2 x^6+x^3-4\right )}{\left (x^6-x^3+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {3 \left (x^6+x^4-x^3+1\right )^{3/4} \left (x^3-2\right )}{\left (x^6-x^3+1\right )^2}+\frac {2 \left (x^6+x^4-x^3+1\right )^{3/4}}{x^6-x^3+1}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (x^6+x^4-x^3+1\right )^{3/4} \left (2 x^6+x^3-4\right )}{\left (x^6-x^3+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {3 \left (x^6+x^4-x^3+1\right )^{3/4} \left (x^3-2\right )}{\left (x^6-x^3+1\right )^2}+\frac {2 \left (x^6+x^4-x^3+1\right )^{3/4}}{x^6-x^3+1}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (x^6+x^4-x^3+1\right )^{3/4} \left (2 x^6+x^3-4\right )}{\left (x^6-x^3+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {3 \left (x^6+x^4-x^3+1\right )^{3/4} \left (x^3-2\right )}{\left (x^6-x^3+1\right )^2}+\frac {2 \left (x^6+x^4-x^3+1\right )^{3/4}}{x^6-x^3+1}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (x^6+x^4-x^3+1\right )^{3/4} \left (2 x^6+x^3-4\right )}{\left (x^6-x^3+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {3 \left (x^6+x^4-x^3+1\right )^{3/4} \left (x^3-2\right )}{\left (x^6-x^3+1\right )^2}+\frac {2 \left (x^6+x^4-x^3+1\right )^{3/4}}{x^6-x^3+1}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (x^6+x^4-x^3+1\right )^{3/4} \left (2 x^6+x^3-4\right )}{\left (x^6-x^3+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {3 \left (x^6+x^4-x^3+1\right )^{3/4} \left (x^3-2\right )}{\left (x^6-x^3+1\right )^2}+\frac {2 \left (x^6+x^4-x^3+1\right )^{3/4}}{x^6-x^3+1}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (x^6+x^4-x^3+1\right )^{3/4} \left (2 x^6+x^3-4\right )}{\left (x^6-x^3+1\right )^2}dx\) |
3.11.90.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 10.62 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.73
| method | result | size |
| pseudoelliptic | \(\frac {\left (3 x^{6}-3 x^{3}+3\right ) \ln \left (\frac {\left (x^{6}+x^{4}-x^{3}+1\right )^{\frac {1}{4}}-x}{x}\right )+\left (-3 x^{6}+3 x^{3}-3\right ) \ln \left (\frac {\left (x^{6}+x^{4}-x^{3}+1\right )^{\frac {1}{4}}+x}{x}\right )+\left (6 x^{6}-6 x^{3}+6\right ) \arctan \left (\frac {\left (x^{6}+x^{4}-x^{3}+1\right )^{\frac {1}{4}}}{x}\right )-4 \left (x^{6}+x^{4}-x^{3}+1\right )^{\frac {3}{4}} x}{4 x^{6}-4 x^{3}+4}\) | \(140\) |
| trager | \(-\frac {x \left (x^{6}+x^{4}-x^{3}+1\right )^{\frac {3}{4}}}{x^{6}-x^{3}+1}-\frac {3 \ln \left (-\frac {x^{6}+2 \left (x^{6}+x^{4}-x^{3}+1\right )^{\frac {3}{4}} x +2 \sqrt {x^{6}+x^{4}-x^{3}+1}\, x^{2}+2 \left (x^{6}+x^{4}-x^{3}+1\right )^{\frac {1}{4}} x^{3}+2 x^{4}-x^{3}+1}{x^{6}-x^{3}+1}\right )}{4}-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{6}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{6}+x^{4}-x^{3}+1}\, x^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \left (x^{6}+x^{4}-x^{3}+1\right )^{\frac {3}{4}} x -2 \left (x^{6}+x^{4}-x^{3}+1\right )^{\frac {1}{4}} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{6}-x^{3}+1}\right )}{4}\) | \(247\) |
| risch | \(-\frac {x \left (x^{6}+x^{4}-x^{3}+1\right )^{\frac {3}{4}}}{x^{6}-x^{3}+1}+\frac {3 \ln \left (\frac {-x^{6}+2 \left (x^{6}+x^{4}-x^{3}+1\right )^{\frac {3}{4}} x -2 \sqrt {x^{6}+x^{4}-x^{3}+1}\, x^{2}+2 \left (x^{6}+x^{4}-x^{3}+1\right )^{\frac {1}{4}} x^{3}-2 x^{4}+x^{3}-1}{x^{6}-x^{3}+1}\right )}{4}+\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{6}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{6}+x^{4}-x^{3}+1}\, x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \left (x^{6}+x^{4}-x^{3}+1\right )^{\frac {3}{4}} x -2 \left (x^{6}+x^{4}-x^{3}+1\right )^{\frac {1}{4}} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{6}-x^{3}+1}\right )}{4}\) | \(248\) |
((3*x^6-3*x^3+3)*ln(((x^6+x^4-x^3+1)^(1/4)-x)/x)+(-3*x^6+3*x^3-3)*ln(((x^6 +x^4-x^3+1)^(1/4)+x)/x)+(6*x^6-6*x^3+6)*arctan(1/x*(x^6+x^4-x^3+1)^(1/4))- 4*(x^6+x^4-x^3+1)^(3/4)*x)/(4*x^6-4*x^3+4)
Leaf count of result is larger than twice the leaf count of optimal. 196 vs. \(2 (71) = 142\).
Time = 29.65 (sec) , antiderivative size = 196, normalized size of antiderivative = 2.42 \[ \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4} \left (-4+x^3+2 x^6\right )}{\left (1-x^3+x^6\right )^2} \, dx=-\frac {3 \, {\left (x^{6} - x^{3} + 1\right )} \arctan \left (\frac {2 \, {\left ({\left (x^{6} + x^{4} - x^{3} + 1\right )}^{\frac {1}{4}} x^{3} + {\left (x^{6} + x^{4} - x^{3} + 1\right )}^{\frac {3}{4}} x\right )}}{x^{6} - x^{3} + 1}\right ) - 3 \, {\left (x^{6} - x^{3} + 1\right )} \log \left (\frac {x^{6} + 2 \, x^{4} - 2 \, {\left (x^{6} + x^{4} - x^{3} + 1\right )}^{\frac {1}{4}} x^{3} - x^{3} + 2 \, \sqrt {x^{6} + x^{4} - x^{3} + 1} x^{2} - 2 \, {\left (x^{6} + x^{4} - x^{3} + 1\right )}^{\frac {3}{4}} x + 1}{x^{6} - x^{3} + 1}\right ) + 4 \, {\left (x^{6} + x^{4} - x^{3} + 1\right )}^{\frac {3}{4}} x}{4 \, {\left (x^{6} - x^{3} + 1\right )}} \]
-1/4*(3*(x^6 - x^3 + 1)*arctan(2*((x^6 + x^4 - x^3 + 1)^(1/4)*x^3 + (x^6 + x^4 - x^3 + 1)^(3/4)*x)/(x^6 - x^3 + 1)) - 3*(x^6 - x^3 + 1)*log((x^6 + 2 *x^4 - 2*(x^6 + x^4 - x^3 + 1)^(1/4)*x^3 - x^3 + 2*sqrt(x^6 + x^4 - x^3 + 1)*x^2 - 2*(x^6 + x^4 - x^3 + 1)^(3/4)*x + 1)/(x^6 - x^3 + 1)) + 4*(x^6 + x^4 - x^3 + 1)^(3/4)*x)/(x^6 - x^3 + 1)
Timed out. \[ \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4} \left (-4+x^3+2 x^6\right )}{\left (1-x^3+x^6\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4} \left (-4+x^3+2 x^6\right )}{\left (1-x^3+x^6\right )^2} \, dx=\int { \frac {{\left (2 \, x^{6} + x^{3} - 4\right )} {\left (x^{6} + x^{4} - x^{3} + 1\right )}^{\frac {3}{4}}}{{\left (x^{6} - x^{3} + 1\right )}^{2}} \,d x } \]
\[ \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4} \left (-4+x^3+2 x^6\right )}{\left (1-x^3+x^6\right )^2} \, dx=\int { \frac {{\left (2 \, x^{6} + x^{3} - 4\right )} {\left (x^{6} + x^{4} - x^{3} + 1\right )}^{\frac {3}{4}}}{{\left (x^{6} - x^{3} + 1\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4} \left (-4+x^3+2 x^6\right )}{\left (1-x^3+x^6\right )^2} \, dx=\int \frac {\left (2\,x^6+x^3-4\right )\,{\left (x^6+x^4-x^3+1\right )}^{3/4}}{{\left (x^6-x^3+1\right )}^2} \,d x \]