Integrand size = 44, antiderivative size = 117 \[ \int \frac {-3-x^4+3 x^6}{\left (1-x^4+x^6\right ) \sqrt [3]{1-x^3-x^4+x^6}} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1-x^3-x^4+x^6}}\right )-\log \left (x+\sqrt [3]{1-x^3-x^4+x^6}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{1-x^3-x^4+x^6}+\left (1-x^3-x^4+x^6\right )^{2/3}\right ) \]
-3^(1/2)*arctan(3^(1/2)*x/(-x+2*(x^6-x^4-x^3+1)^(1/3)))-ln(x+(x^6-x^4-x^3+ 1)^(1/3))+1/2*ln(x^2-x*(x^6-x^4-x^3+1)^(1/3)+(x^6-x^4-x^3+1)^(2/3))
Time = 0.29 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.97 \[ \int \frac {-3-x^4+3 x^6}{\left (1-x^4+x^6\right ) \sqrt [3]{1-x^3-x^4+x^6}} \, dx=\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x-2 \sqrt [3]{1-x^3-x^4+x^6}}\right )-\log \left (x+\sqrt [3]{1-x^3-x^4+x^6}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{1-x^3-x^4+x^6}+\left (1-x^3-x^4+x^6\right )^{2/3}\right ) \]
Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x - 2*(1 - x^3 - x^4 + x^6)^(1/3))] - Log[x + (1 - x^3 - x^4 + x^6)^(1/3)] + Log[x^2 - x*(1 - x^3 - x^4 + x^6)^(1/3) + ( 1 - x^3 - x^4 + x^6)^(2/3)]/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.
\(\displaystyle \int \frac {3 x^6-x^4-3}{\left (x^6-x^4+1\right ) \sqrt [3]{x^6-x^4-x^3+1}} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {3}{\sqrt [3]{x^6-x^4-x^3+1}}-\frac {2 \left (3-x^4\right )}{\left (x^6-x^4+1\right ) \sqrt [3]{x^6-x^4-x^3+1}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \int \frac {1}{\sqrt [3]{x^6-x^4-x^3+1}}dx-6 \int \frac {1}{\left (x^6-x^4+1\right ) \sqrt [3]{x^6-x^4-x^3+1}}dx+2 \int \frac {x^4}{\left (x^6-x^4+1\right ) \sqrt [3]{x^6-x^4-x^3+1}}dx\) |
3.18.37.3.1 Defintions of rubi rules used
Time = 2.36 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.95
method | result | size |
pseudoelliptic | \(\frac {\ln \left (\frac {x^{2}-x \left (x^{6}-x^{4}-x^{3}+1\right )^{\frac {1}{3}}+\left (x^{6}-x^{4}-x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x -2 \left (x^{6}-x^{4}-x^{3}+1\right )^{\frac {1}{3}}\right )}{3 x}\right )-\ln \left (\frac {x +\left (x^{6}-x^{4}-x^{3}+1\right )^{\frac {1}{3}}}{x}\right )\) | \(111\) |
trager | \(-\ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{6}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{4}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-x^{4}-x^{3}+1\right )^{\frac {2}{3}} x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-x^{4}-x^{3}+1\right )^{\frac {1}{3}} x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-\left (x^{6}-x^{4}-x^{3}+1\right )^{\frac {2}{3}} x -2 \left (x^{6}-x^{4}-x^{3}+1\right )^{\frac {1}{3}} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{x^{6}-x^{4}+1}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\frac {x^{6}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-x^{4}-x^{3}+1\right )^{\frac {2}{3}} x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-x^{4}-x^{3}+1\right )^{\frac {1}{3}} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-x^{4}-\left (x^{6}-x^{4}-x^{3}+1\right )^{\frac {2}{3}} x -2 \left (x^{6}-x^{4}-x^{3}+1\right )^{\frac {1}{3}} x^{2}-x^{3}+1}{x^{6}-x^{4}+1}\right )\) | \(340\) |
1/2*ln((x^2-x*(x^6-x^4-x^3+1)^(1/3)+(x^6-x^4-x^3+1)^(2/3))/x^2)-3^(1/2)*ar ctan(1/3*3^(1/2)*(x-2*(x^6-x^4-x^3+1)^(1/3))/x)-ln((x+(x^6-x^4-x^3+1)^(1/3 ))/x)
Time = 1.98 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.34 \[ \int \frac {-3-x^4+3 x^6}{\left (1-x^4+x^6\right ) \sqrt [3]{1-x^3-x^4+x^6}} \, dx=-\sqrt {3} \arctan \left (\frac {2 \, \sqrt {3} {\left (x^{6} - x^{4} - x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 2 \, \sqrt {3} {\left (x^{6} - x^{4} - x^{3} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (x^{6} - x^{4} + 1\right )}}{3 \, {\left (x^{6} - x^{4} - 2 \, x^{3} + 1\right )}}\right ) - \frac {1}{2} \, \log \left (\frac {x^{6} - x^{4} + 3 \, {\left (x^{6} - x^{4} - x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{6} - x^{4} - x^{3} + 1\right )}^{\frac {2}{3}} x + 1}{x^{6} - x^{4} + 1}\right ) \]
-sqrt(3)*arctan(1/3*(2*sqrt(3)*(x^6 - x^4 - x^3 + 1)^(1/3)*x^2 + 2*sqrt(3) *(x^6 - x^4 - x^3 + 1)^(2/3)*x + sqrt(3)*(x^6 - x^4 + 1))/(x^6 - x^4 - 2*x ^3 + 1)) - 1/2*log((x^6 - x^4 + 3*(x^6 - x^4 - x^3 + 1)^(1/3)*x^2 + 3*(x^6 - x^4 - x^3 + 1)^(2/3)*x + 1)/(x^6 - x^4 + 1))
Timed out. \[ \int \frac {-3-x^4+3 x^6}{\left (1-x^4+x^6\right ) \sqrt [3]{1-x^3-x^4+x^6}} \, dx=\text {Timed out} \]
\[ \int \frac {-3-x^4+3 x^6}{\left (1-x^4+x^6\right ) \sqrt [3]{1-x^3-x^4+x^6}} \, dx=\int { \frac {3 \, x^{6} - x^{4} - 3}{{\left (x^{6} - x^{4} - x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{6} - x^{4} + 1\right )}} \,d x } \]
\[ \int \frac {-3-x^4+3 x^6}{\left (1-x^4+x^6\right ) \sqrt [3]{1-x^3-x^4+x^6}} \, dx=\int { \frac {3 \, x^{6} - x^{4} - 3}{{\left (x^{6} - x^{4} - x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{6} - x^{4} + 1\right )}} \,d x } \]
Timed out. \[ \int \frac {-3-x^4+3 x^6}{\left (1-x^4+x^6\right ) \sqrt [3]{1-x^3-x^4+x^6}} \, dx=\int -\frac {-3\,x^6+x^4+3}{\left (x^6-x^4+1\right )\,{\left (x^6-x^4-x^3+1\right )}^{1/3}} \,d x \]