Integrand size = 35, antiderivative size = 107 \[ \int \frac {x^4 \left (-4+x^3\right )}{\sqrt [4]{-1+x^3} \left (-1+2 x^3-x^6+x^8\right )} \, dx=\arctan \left (\frac {\sqrt [4]{-1+x^3}}{x}\right )+\frac {\arctan \left (\frac {\sqrt {2} x \sqrt [4]{-1+x^3}}{-x^2+\sqrt {-1+x^3}}\right )}{\sqrt {2}}-\text {arctanh}\left (\frac {x}{\sqrt [4]{-1+x^3}}\right )+\frac {\text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-1+x^3}}{x^2+\sqrt {-1+x^3}}\right )}{\sqrt {2}} \]
arctan((x^3-1)^(1/4)/x)+1/2*2^(1/2)*arctan(2^(1/2)*x*(x^3-1)^(1/4)/(-x^2+( x^3-1)^(1/2)))-arctanh(x/(x^3-1)^(1/4))+1/2*2^(1/2)*arctanh(2^(1/2)*x*(x^3 -1)^(1/4)/(x^2+(x^3-1)^(1/2)))
Time = 1.15 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00 \[ \int \frac {x^4 \left (-4+x^3\right )}{\sqrt [4]{-1+x^3} \left (-1+2 x^3-x^6+x^8\right )} \, dx=\arctan \left (\frac {\sqrt [4]{-1+x^3}}{x}\right )+\frac {\arctan \left (\frac {\sqrt {2} x \sqrt [4]{-1+x^3}}{-x^2+\sqrt {-1+x^3}}\right )}{\sqrt {2}}-\text {arctanh}\left (\frac {x}{\sqrt [4]{-1+x^3}}\right )+\frac {\text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-1+x^3}}{x^2+\sqrt {-1+x^3}}\right )}{\sqrt {2}} \]
ArcTan[(-1 + x^3)^(1/4)/x] + ArcTan[(Sqrt[2]*x*(-1 + x^3)^(1/4))/(-x^2 + S qrt[-1 + x^3])]/Sqrt[2] - ArcTanh[x/(-1 + x^3)^(1/4)] + ArcTanh[(Sqrt[2]*x *(-1 + x^3)^(1/4))/(x^2 + Sqrt[-1 + x^3])]/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.
| \(\displaystyle \int \frac {x^4 \left (x^3-4\right )}{\sqrt [4]{x^3-1} \left (x^8-x^6+2 x^3-1\right )} \, dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {\left (-x^3+x^2-1\right ) \left (x^3-4\right ) x^4}{2 \sqrt [4]{x^3-1} \left (x^4-x^3+1\right )}+\frac {\left (x^3-4\right ) \left (x^3+x^2+1\right ) x^4}{2 \sqrt [4]{x^3-1} \left (x^4+x^3-1\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \int \frac {1}{\sqrt [4]{x^3-1} \left (x^4-x^3+1\right )}dx+\frac {1}{2} \int \frac {x^3}{\sqrt [4]{x^3-1} \left (x^4-x^3+1\right )}dx-2 \int \frac {1}{\sqrt [4]{x^3-1} \left (x^4+x^3-1\right )}dx+\frac {1}{2} \int \frac {x^3}{\sqrt [4]{x^3-1} \left (x^4+x^3-1\right )}dx\) |
3.16.68.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.89 (sec) , antiderivative size = 428, normalized size of antiderivative = 4.00
| method | result | size |
| trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{3}-1}\, x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \left (x^{3}-1\right )^{\frac {3}{4}} x -2 \left (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^{4}-x^{3}+1}\right )}{2}+\frac {\ln \left (\frac {2 \left (x^{3}-1\right )^{\frac {3}{4}} x -2 x^{2} \sqrt {x^{3}-1}+2 \left (x^{3}-1\right )^{\frac {1}{4}} x^{3}-x^{4}-x^{3}+1}{x^{4}-x^{3}+1}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) \sqrt {x^{3}-1}\, x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) x^{4}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{3}-1\right )^{\frac {1}{4}} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) x^{3}+2 \left (x^{3}-1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )}{x^{4}+x^{3}-1}\right )}{2}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) \sqrt {x^{3}-1}\, x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{3}-1\right )^{\frac {1}{4}} x^{3}-2 \left (x^{3}-1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )}{x^{4}+x^{3}-1}\right )}{2}\) | \(428\) |
1/2*RootOf(_Z^2+1)*ln((2*RootOf(_Z^2+1)*(x^3-1)^(1/2)*x^2-RootOf(_Z^2+1)*x ^4+2*(x^3-1)^(3/4)*x-2*(x^3-1)^(1/4)*x^3-RootOf(_Z^2+1)*x^3+RootOf(_Z^2+1) )/(x^4-x^3+1))+1/2*ln((2*(x^3-1)^(3/4)*x-2*x^2*(x^3-1)^(1/2)+2*(x^3-1)^(1/ 4)*x^3-x^4-x^3+1)/(x^4-x^3+1))+1/2*RootOf(_Z^2+RootOf(_Z^2+1))*ln(-(2*Root Of(_Z^2+1)*RootOf(_Z^2+RootOf(_Z^2+1))*(x^3-1)^(1/2)*x^2-RootOf(_Z^2+RootO f(_Z^2+1))*x^4+2*RootOf(_Z^2+1)*(x^3-1)^(1/4)*x^3+RootOf(_Z^2+RootOf(_Z^2+ 1))*x^3+2*(x^3-1)^(3/4)*x-RootOf(_Z^2+RootOf(_Z^2+1)))/(x^4+x^3-1))-1/2*Ro otOf(_Z^2+1)*RootOf(_Z^2+RootOf(_Z^2+1))*ln((-RootOf(_Z^2+RootOf(_Z^2+1))* RootOf(_Z^2+1)*x^4+2*RootOf(_Z^2+RootOf(_Z^2+1))*(x^3-1)^(1/2)*x^2+RootOf( _Z^2+RootOf(_Z^2+1))*RootOf(_Z^2+1)*x^3+2*RootOf(_Z^2+1)*(x^3-1)^(1/4)*x^3 -2*(x^3-1)^(3/4)*x-RootOf(_Z^2+1)*RootOf(_Z^2+RootOf(_Z^2+1)))/(x^4+x^3-1) )
Result contains complex when optimal does not.
Time = 34.74 (sec) , antiderivative size = 381, normalized size of antiderivative = 3.56 \[ \int \frac {x^4 \left (-4+x^3\right )}{\sqrt [4]{-1+x^3} \left (-1+2 x^3-x^6+x^8\right )} \, dx=-\left (\frac {1}{8} i + \frac {1}{8}\right ) \, \sqrt {2} \log \left (\frac {4 i \, {\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{3} - \left (2 i - 2\right ) \, \sqrt {2} \sqrt {x^{3} - 1} x^{2} - 4 \, {\left (x^{3} - 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (-\left (i + 1\right ) \, x^{4} + \left (i + 1\right ) \, x^{3} - i - 1\right )}}{x^{4} + x^{3} - 1}\right ) + \left (\frac {1}{8} i + \frac {1}{8}\right ) \, \sqrt {2} \log \left (\frac {4 i \, {\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{3} + \left (2 i - 2\right ) \, \sqrt {2} \sqrt {x^{3} - 1} x^{2} - 4 \, {\left (x^{3} - 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (\left (i + 1\right ) \, x^{4} - \left (i + 1\right ) \, x^{3} + i + 1\right )}}{x^{4} + x^{3} - 1}\right ) + \left (\frac {1}{8} i - \frac {1}{8}\right ) \, \sqrt {2} \log \left (\frac {-4 i \, {\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{3} + \left (2 i + 2\right ) \, \sqrt {2} \sqrt {x^{3} - 1} x^{2} - 4 \, {\left (x^{3} - 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (\left (i - 1\right ) \, x^{4} - \left (i - 1\right ) \, x^{3} + i - 1\right )}}{x^{4} + x^{3} - 1}\right ) - \left (\frac {1}{8} i - \frac {1}{8}\right ) \, \sqrt {2} \log \left (\frac {-4 i \, {\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{3} - \left (2 i + 2\right ) \, \sqrt {2} \sqrt {x^{3} - 1} x^{2} - 4 \, {\left (x^{3} - 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (-\left (i - 1\right ) \, x^{4} + \left (i - 1\right ) \, x^{3} - i + 1\right )}}{x^{4} + x^{3} - 1}\right ) + \frac {1}{2} \, \arctan \left (\frac {2 \, {\left ({\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{3} + {\left (x^{3} - 1\right )}^{\frac {3}{4}} x\right )}}{x^{4} - x^{3} + 1}\right ) + \frac {1}{2} \, \log \left (\frac {x^{4} - 2 \, {\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{3} + x^{3} + 2 \, \sqrt {x^{3} - 1} x^{2} - 2 \, {\left (x^{3} - 1\right )}^{\frac {3}{4}} x - 1}{x^{4} - x^{3} + 1}\right ) \]
-(1/8*I + 1/8)*sqrt(2)*log((4*I*(x^3 - 1)^(1/4)*x^3 - (2*I - 2)*sqrt(2)*sq rt(x^3 - 1)*x^2 - 4*(x^3 - 1)^(3/4)*x + sqrt(2)*(-(I + 1)*x^4 + (I + 1)*x^ 3 - I - 1))/(x^4 + x^3 - 1)) + (1/8*I + 1/8)*sqrt(2)*log((4*I*(x^3 - 1)^(1 /4)*x^3 + (2*I - 2)*sqrt(2)*sqrt(x^3 - 1)*x^2 - 4*(x^3 - 1)^(3/4)*x + sqrt (2)*((I + 1)*x^4 - (I + 1)*x^3 + I + 1))/(x^4 + x^3 - 1)) + (1/8*I - 1/8)* sqrt(2)*log((-4*I*(x^3 - 1)^(1/4)*x^3 + (2*I + 2)*sqrt(2)*sqrt(x^3 - 1)*x^ 2 - 4*(x^3 - 1)^(3/4)*x + sqrt(2)*((I - 1)*x^4 - (I - 1)*x^3 + I - 1))/(x^ 4 + x^3 - 1)) - (1/8*I - 1/8)*sqrt(2)*log((-4*I*(x^3 - 1)^(1/4)*x^3 - (2*I + 2)*sqrt(2)*sqrt(x^3 - 1)*x^2 - 4*(x^3 - 1)^(3/4)*x + sqrt(2)*(-(I - 1)* x^4 + (I - 1)*x^3 - I + 1))/(x^4 + x^3 - 1)) + 1/2*arctan(2*((x^3 - 1)^(1/ 4)*x^3 + (x^3 - 1)^(3/4)*x)/(x^4 - x^3 + 1)) + 1/2*log((x^4 - 2*(x^3 - 1)^ (1/4)*x^3 + x^3 + 2*sqrt(x^3 - 1)*x^2 - 2*(x^3 - 1)^(3/4)*x - 1)/(x^4 - x^ 3 + 1))
Timed out. \[ \int \frac {x^4 \left (-4+x^3\right )}{\sqrt [4]{-1+x^3} \left (-1+2 x^3-x^6+x^8\right )} \, dx=\text {Timed out} \]
\[ \int \frac {x^4 \left (-4+x^3\right )}{\sqrt [4]{-1+x^3} \left (-1+2 x^3-x^6+x^8\right )} \, dx=\int { \frac {{\left (x^{3} - 4\right )} x^{4}}{{\left (x^{8} - x^{6} + 2 \, x^{3} - 1\right )} {\left (x^{3} - 1\right )}^{\frac {1}{4}}} \,d x } \]
\[ \int \frac {x^4 \left (-4+x^3\right )}{\sqrt [4]{-1+x^3} \left (-1+2 x^3-x^6+x^8\right )} \, dx=\int { \frac {{\left (x^{3} - 4\right )} x^{4}}{{\left (x^{8} - x^{6} + 2 \, x^{3} - 1\right )} {\left (x^{3} - 1\right )}^{\frac {1}{4}}} \,d x } \]
Timed out. \[ \int \frac {x^4 \left (-4+x^3\right )}{\sqrt [4]{-1+x^3} \left (-1+2 x^3-x^6+x^8\right )} \, dx=\int \frac {x^4\,\left (x^3-4\right )}{{\left (x^3-1\right )}^{1/4}\,\left (x^8-x^6+2\,x^3-1\right )} \,d x \]