Integrand size = 38, antiderivative size = 95 \[ \int \frac {\left (-4+x^3\right ) \left (1-x^3+x^4\right )}{x^2 \left (-1+x^3\right )^{3/4} \left (-1+x^3+x^4\right )} \, dx=\frac {4 \sqrt [4]{-1+x^3}}{x}+2 \sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt [4]{-1+x^3}}{-x^2+\sqrt {-1+x^3}}\right )-2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-1+x^3}}{x^2+\sqrt {-1+x^3}}\right ) \]
4*(x^3-1)^(1/4)/x+2*2^(1/2)*arctan(2^(1/2)*x*(x^3-1)^(1/4)/(-x^2+(x^3-1)^( 1/2)))-2*2^(1/2)*arctanh(2^(1/2)*x*(x^3-1)^(1/4)/(x^2+(x^3-1)^(1/2)))
Time = 3.40 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-4+x^3\right ) \left (1-x^3+x^4\right )}{x^2 \left (-1+x^3\right )^{3/4} \left (-1+x^3+x^4\right )} \, dx=\frac {4 \sqrt [4]{-1+x^3}}{x}+2 \sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt [4]{-1+x^3}}{-x^2+\sqrt {-1+x^3}}\right )-2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-1+x^3}}{x^2+\sqrt {-1+x^3}}\right ) \]
(4*(-1 + x^3)^(1/4))/x + 2*Sqrt[2]*ArcTan[(Sqrt[2]*x*(-1 + x^3)^(1/4))/(-x ^2 + Sqrt[-1 + x^3])] - 2*Sqrt[2]*ArcTanh[(Sqrt[2]*x*(-1 + x^3)^(1/4))/(x^ 2 + Sqrt[-1 + x^3])]
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.
| \(\displaystyle \int \frac {\left (x^3-4\right ) \left (x^4-x^3+1\right )}{x^2 \left (x^3-1\right )^{3/4} \left (x^4+x^3-1\right )} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x}{\left (x^3-1\right )^{3/4}}-\frac {2}{\left (x^3-1\right )^{3/4}}+\frac {4}{\left (x^3-1\right )^{3/4} x^2}+\frac {2 \left (x^3-4 x^2+x-1\right )}{\left (x^3-1\right )^{3/4} \left (x^4+x^3-1\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \int \frac {1}{\left (x^3-1\right )^{3/4} \left (-x^4-x^3+1\right )}dx+2 \int \frac {x}{\left (x^3-1\right )^{3/4} \left (x^4+x^3-1\right )}dx+2 \int \frac {x^3}{\left (x^3-1\right )^{3/4} \left (x^4+x^3-1\right )}dx-8 \int \frac {x^2}{\left (x^3-1\right )^{3/4} \left (x^4+x^3-1\right )}dx-\frac {2 \left (1-x^3\right )^{3/4} x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {3}{4},\frac {4}{3},x^3\right )}{\left (x^3-1\right )^{3/4}}-\frac {4 \left (1-x^3\right )^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {3}{4},\frac {2}{3},x^3\right )}{\left (x^3-1\right )^{3/4} x}+\frac {\left (1-x^3\right )^{3/4} x^2 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {3}{4},\frac {5}{3},x^3\right )}{2 \left (x^3-1\right )^{3/4}}\) |
3.14.18.3.1 Defintions of rubi rules used
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 7.16 (sec) , antiderivative size = 219, normalized size of antiderivative = 2.31
| method | result | size |
| trager | \(\frac {4 \left (x^{3}-1\right )^{\frac {1}{4}}}{x}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{4}+2 \left (x^{3}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{3}-2 \sqrt {x^{3}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{2}+2 \left (x^{3}-1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}}{x^{4}+x^{3}-1}\right )-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {2 \sqrt {x^{3}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{2}-2 \left (x^{3}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{4}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{3}+2 \left (x^{3}-1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{x^{4}+x^{3}-1}\right )\) | \(219\) |
| risch | \(\frac {4 \left (x^{3}-1\right )^{\frac {1}{4}}}{x}+\frac {\left (-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{10}+2 \left (x^{9}-3 x^{6}+3 x^{3}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{7}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{9}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{7}-4 \left (x^{9}-3 x^{6}+3 x^{3}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{4}-3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{6}-2 \left (x^{9}-3 x^{6}+3 x^{3}-1\right )^{\frac {3}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{3}-2 \sqrt {x^{9}-3 x^{6}+3 x^{3}-1}\, x^{5}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{4}+2 \left (x^{9}-3 x^{6}+3 x^{3}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x +3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+2 \sqrt {x^{9}-3 x^{6}+3 x^{3}-1}\, x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}}{\left (x -1\right )^{2} \left (x^{2}+x +1\right )^{2} \left (x^{4}+x^{3}-1\right )}\right )+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{10}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{9}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{9}-3 x^{6}+3 x^{3}-1\right )^{\frac {3}{4}} x^{3}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{7}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (x^{9}-3 x^{6}+3 x^{3}-1\right )^{\frac {1}{4}} x^{7}+3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{6}-2 \sqrt {x^{9}-3 x^{6}+3 x^{3}-1}\, x^{5}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{4}+4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (x^{9}-3 x^{6}+3 x^{3}-1\right )^{\frac {1}{4}} x^{4}-3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+2 \sqrt {x^{9}-3 x^{6}+3 x^{3}-1}\, x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (x^{9}-3 x^{6}+3 x^{3}-1\right )^{\frac {1}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}}{\left (x -1\right )^{2} \left (x^{2}+x +1\right )^{2} \left (x^{4}+x^{3}-1\right )}\right )\right ) {\left (\left (x^{3}-1\right )^{3}\right )}^{\frac {1}{4}}}{\left (x^{3}-1\right )^{\frac {3}{4}}}\) | \(589\) |
4*(x^3-1)^(1/4)/x+2*RootOf(_Z^4+1)*ln(-(-RootOf(_Z^4+1)^3*x^4+2*(x^3-1)^(1 /4)*RootOf(_Z^4+1)^2*x^3+RootOf(_Z^4+1)^3*x^3-2*(x^3-1)^(1/2)*RootOf(_Z^4+ 1)*x^2+2*(x^3-1)^(3/4)*x-RootOf(_Z^4+1)^3)/(x^4+x^3-1))-2*RootOf(_Z^4+1)^3 *ln((2*(x^3-1)^(1/2)*RootOf(_Z^4+1)^3*x^2-2*(x^3-1)^(1/4)*RootOf(_Z^4+1)^2 *x^3+RootOf(_Z^4+1)*x^4-RootOf(_Z^4+1)*x^3+2*(x^3-1)^(3/4)*x+RootOf(_Z^4+1 ))/(x^4+x^3-1))
Result contains complex when optimal does not.
Time = 15.43 (sec) , antiderivative size = 295, normalized size of antiderivative = 3.11 \[ \int \frac {\left (-4+x^3\right ) \left (1-x^3+x^4\right )}{x^2 \left (-1+x^3\right )^{3/4} \left (-1+x^3+x^4\right )} \, dx=\frac {-\left (i + 1\right ) \, \sqrt {2} x \log \left (\frac {i \, x^{4} + \left (i + 1\right ) \, \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{3} - i \, x^{3} + 2 \, \sqrt {x^{3} - 1} x^{2} - \left (i - 1\right ) \, \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {3}{4}} x + i}{x^{4} + x^{3} - 1}\right ) + \left (i + 1\right ) \, \sqrt {2} x \log \left (\frac {i \, x^{4} - \left (i + 1\right ) \, \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{3} - i \, x^{3} + 2 \, \sqrt {x^{3} - 1} x^{2} + \left (i - 1\right ) \, \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {3}{4}} x + i}{x^{4} + x^{3} - 1}\right ) + \left (i - 1\right ) \, \sqrt {2} x \log \left (\frac {-i \, x^{4} - \left (i - 1\right ) \, \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{3} + i \, x^{3} + 2 \, \sqrt {x^{3} - 1} x^{2} + \left (i + 1\right ) \, \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {3}{4}} x - i}{x^{4} + x^{3} - 1}\right ) - \left (i - 1\right ) \, \sqrt {2} x \log \left (\frac {-i \, x^{4} + \left (i - 1\right ) \, \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{3} + i \, x^{3} + 2 \, \sqrt {x^{3} - 1} x^{2} - \left (i + 1\right ) \, \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {3}{4}} x - i}{x^{4} + x^{3} - 1}\right ) + 8 \, {\left (x^{3} - 1\right )}^{\frac {1}{4}}}{2 \, x} \]
1/2*(-(I + 1)*sqrt(2)*x*log((I*x^4 + (I + 1)*sqrt(2)*(x^3 - 1)^(1/4)*x^3 - I*x^3 + 2*sqrt(x^3 - 1)*x^2 - (I - 1)*sqrt(2)*(x^3 - 1)^(3/4)*x + I)/(x^4 + x^3 - 1)) + (I + 1)*sqrt(2)*x*log((I*x^4 - (I + 1)*sqrt(2)*(x^3 - 1)^(1 /4)*x^3 - I*x^3 + 2*sqrt(x^3 - 1)*x^2 + (I - 1)*sqrt(2)*(x^3 - 1)^(3/4)*x + I)/(x^4 + x^3 - 1)) + (I - 1)*sqrt(2)*x*log((-I*x^4 - (I - 1)*sqrt(2)*(x ^3 - 1)^(1/4)*x^3 + I*x^3 + 2*sqrt(x^3 - 1)*x^2 + (I + 1)*sqrt(2)*(x^3 - 1 )^(3/4)*x - I)/(x^4 + x^3 - 1)) - (I - 1)*sqrt(2)*x*log((-I*x^4 + (I - 1)* sqrt(2)*(x^3 - 1)^(1/4)*x^3 + I*x^3 + 2*sqrt(x^3 - 1)*x^2 - (I + 1)*sqrt(2 )*(x^3 - 1)^(3/4)*x - I)/(x^4 + x^3 - 1)) + 8*(x^3 - 1)^(1/4))/x
Timed out. \[ \int \frac {\left (-4+x^3\right ) \left (1-x^3+x^4\right )}{x^2 \left (-1+x^3\right )^{3/4} \left (-1+x^3+x^4\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\left (-4+x^3\right ) \left (1-x^3+x^4\right )}{x^2 \left (-1+x^3\right )^{3/4} \left (-1+x^3+x^4\right )} \, dx=\int { \frac {{\left (x^{4} - x^{3} + 1\right )} {\left (x^{3} - 4\right )}}{{\left (x^{4} + x^{3} - 1\right )} {\left (x^{3} - 1\right )}^{\frac {3}{4}} x^{2}} \,d x } \]
\[ \int \frac {\left (-4+x^3\right ) \left (1-x^3+x^4\right )}{x^2 \left (-1+x^3\right )^{3/4} \left (-1+x^3+x^4\right )} \, dx=\int { \frac {{\left (x^{4} - x^{3} + 1\right )} {\left (x^{3} - 4\right )}}{{\left (x^{4} + x^{3} - 1\right )} {\left (x^{3} - 1\right )}^{\frac {3}{4}} x^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (-4+x^3\right ) \left (1-x^3+x^4\right )}{x^2 \left (-1+x^3\right )^{3/4} \left (-1+x^3+x^4\right )} \, dx=\int \frac {\left (x^3-4\right )\,\left (x^4-x^3+1\right )}{x^2\,{\left (x^3-1\right )}^{3/4}\,\left (x^4+x^3-1\right )} \,d x \]