\(\int \frac {x^4 (-4+x^3)}{\sqrt [4]{-1+x^3} (-1+2 x^3-x^6+x^8)} \, dx\) [1568]

Optimal result

Integrand size = 35, antiderivative size = 107 \[ \int \frac {x^4 \left (-4+x^3\right )}{\sqrt [4]{-1+x^3} \left (-1+2 x^3-x^6+x^8\right )} \, dx=\arctan \left (\frac {\sqrt [4]{-1+x^3}}{x}\right )+\frac {\arctan \left (\frac {\sqrt {2} x \sqrt [4]{-1+x^3}}{-x^2+\sqrt {-1+x^3}}\right )}{\sqrt {2}}-\text {arctanh}\left (\frac {x}{\sqrt [4]{-1+x^3}}\right )+\frac {\text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-1+x^3}}{x^2+\sqrt {-1+x^3}}\right )}{\sqrt {2}} \] Output:

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)))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 1.14 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00 \[ \int \frac {x^4 \left (-4+x^3\right )}{\sqrt [4]{-1+x^3} \left (-1+2 x^3-x^6+x^8\right )} \, dx=\arctan \left (\frac {\sqrt [4]{-1+x^3}}{x}\right )+\frac {\arctan \left (\frac {\sqrt {2} x \sqrt [4]{-1+x^3}}{-x^2+\sqrt {-1+x^3}}\right )}{\sqrt {2}}-\text {arctanh}\left (\frac {x}{\sqrt [4]{-1+x^3}}\right )+\frac {\text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-1+x^3}}{x^2+\sqrt {-1+x^3}}\right )}{\sqrt {2}} \] Input:

Integrate[(x^4*(-4 + x^3))/((-1 + x^3)^(1/4)*(-1 + 2*x^3 - x^6 + x^8)),x]
 

Output:

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]
 

Rubi [F]

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

\(\displaystyle -2 \int \frac {1}{\sqrt [4]{x^3-1} \left (x^4-x^3+1\right )}dx+\frac {1}{2} \int \frac {x^3}{\sqrt [4]{x^3-1} \left (x^4-x^3+1\right )}dx-2 \int \frac {1}{\sqrt [4]{x^3-1} \left (x^4+x^3-1\right )}dx+\frac {1}{2} \int \frac {x^3}{\sqrt [4]{x^3-1} \left (x^4+x^3-1\right )}dx\)

Input:

Int[(x^4*(-4 + x^3))/((-1 + x^3)^(1/4)*(-1 + 2*x^3 - x^6 + x^8)),x]
 

Output:

$Aborted
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 3.11 (sec) , antiderivative size = 442, normalized size of antiderivative = 4.13

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}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+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^{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}-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^{4}+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^{3}+\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}+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 ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) \sqrt {x^{3}-1}\, x^{2}+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}-\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}\)	\(442\)

Input:

int(x^4*(x^3-4)/(x^3-1)^(1/4)/(x^8-x^6+2*x^3-1),x,method=_RETURNVERBOSE)
 

Output:

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-RootOf(_Z^2+1)*x^3+2*(x^3-1)^(3/4)*x-2*(x^3-1)^(1/4)*x^3+RootOf(_Z^2+1) 
)/(x^4-x^3+1))-1/2*RootOf(_Z^2-RootOf(_Z^2+1))*ln(-(2*RootOf(_Z^2+1)*RootO 
f(_Z^2-RootOf(_Z^2+1))*(x^3-1)^(1/2)*x^2-2*RootOf(_Z^2+1)*(x^3-1)^(1/4)*x^ 
3+RootOf(_Z^2-RootOf(_Z^2+1))*x^4+2*(x^3-1)^(3/4)*x-RootOf(_Z^2-RootOf(_Z^ 
2+1))*x^3+RootOf(_Z^2-RootOf(_Z^2+1)))/(x^4+x^3-1))-1/2*RootOf(_Z^2+1)*Roo 
tOf(_Z^2-RootOf(_Z^2+1))*ln(-(RootOf(_Z^2-RootOf(_Z^2+1))*RootOf(_Z^2+1)*x 
^4+2*RootOf(_Z^2+1)*(x^3-1)^(1/4)*x^3-RootOf(_Z^2-RootOf(_Z^2+1))*RootOf(_ 
Z^2+1)*x^3+2*RootOf(_Z^2-RootOf(_Z^2+1))*(x^3-1)^(1/2)*x^2+2*(x^3-1)^(3/4) 
*x+RootOf(_Z^2+1)*RootOf(_Z^2-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) 
)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 480 vs. \(2 (90) = 180\).

Time = 33.27 (sec) , antiderivative size = 480, normalized size of antiderivative = 4.49 \[ \int \frac {x^4 \left (-4+x^3\right )}{\sqrt [4]{-1+x^3} \left (-1+2 x^3-x^6+x^8\right )} \, dx=-\frac {1}{4} \, \sqrt {2} \arctan \left (\frac {x^{8} + 2 \, x^{7} + x^{6} - 2 \, x^{4} - 2 \, x^{3} + 2 \, \sqrt {2} {\left (3 \, x^{5} - x^{4} + x\right )} {\left (x^{3} - 1\right )}^{\frac {3}{4}} + 2 \, \sqrt {2} {\left (x^{7} - 3 \, x^{6} + 3 \, x^{3}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{4}} + 4 \, {\left (x^{6} + x^{5} - x^{2}\right )} \sqrt {x^{3} - 1} + 1}{x^{8} - 14 \, x^{7} + x^{6} + 14 \, x^{4} - 2 \, x^{3} + 1}\right ) - \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {x^{8} + 2 \, x^{7} + x^{6} - 2 \, x^{4} - 2 \, x^{3} - 2 \, \sqrt {2} {\left (3 \, x^{5} - x^{4} + x\right )} {\left (x^{3} - 1\right )}^{\frac {3}{4}} - 2 \, \sqrt {2} {\left (x^{7} - 3 \, x^{6} + 3 \, x^{3}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{4}} + 4 \, {\left (x^{6} + x^{5} - x^{2}\right )} \sqrt {x^{3} - 1} + 1}{x^{8} - 14 \, x^{7} + x^{6} + 14 \, x^{4} - 2 \, x^{3} + 1}\right ) + \frac {1}{8} \, \sqrt {2} \log \left (\frac {x^{4} + 2 \, \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{3} + x^{3} + 4 \, \sqrt {x^{3} - 1} x^{2} + 2 \, \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {3}{4}} x - 1}{x^{4} + x^{3} - 1}\right ) - \frac {1}{8} \, \sqrt {2} \log \left (\frac {x^{4} - 2 \, \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{3} + x^{3} + 4 \, \sqrt {x^{3} - 1} x^{2} - 2 \, \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {3}{4}} x - 1}{x^{4} + x^{3} - 1}\right ) + \frac {1}{2} \, \arctan \left (\frac {2 \, {\left ({\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{3} + {\left (x^{3} - 1\right )}^{\frac {3}{4}} x\right )}}{x^{4} - x^{3} + 1}\right ) + \frac {1}{2} \, \log \left (\frac {x^{4} - 2 \, {\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{3} + x^{3} + 2 \, \sqrt {x^{3} - 1} x^{2} - 2 \, {\left (x^{3} - 1\right )}^{\frac {3}{4}} x - 1}{x^{4} - x^{3} + 1}\right ) \] Input:

integrate(x^4*(x^3-4)/(x^3-1)^(1/4)/(x^8-x^6+2*x^3-1),x, algorithm="fricas 
")
 

Output:

-1/4*sqrt(2)*arctan((x^8 + 2*x^7 + x^6 - 2*x^4 - 2*x^3 + 2*sqrt(2)*(3*x^5 
- x^4 + x)*(x^3 - 1)^(3/4) + 2*sqrt(2)*(x^7 - 3*x^6 + 3*x^3)*(x^3 - 1)^(1/ 
4) + 4*(x^6 + x^5 - x^2)*sqrt(x^3 - 1) + 1)/(x^8 - 14*x^7 + x^6 + 14*x^4 - 
 2*x^3 + 1)) - 1/4*sqrt(2)*arctan(-(x^8 + 2*x^7 + x^6 - 2*x^4 - 2*x^3 - 2* 
sqrt(2)*(3*x^5 - x^4 + x)*(x^3 - 1)^(3/4) - 2*sqrt(2)*(x^7 - 3*x^6 + 3*x^3 
)*(x^3 - 1)^(1/4) + 4*(x^6 + x^5 - x^2)*sqrt(x^3 - 1) + 1)/(x^8 - 14*x^7 + 
 x^6 + 14*x^4 - 2*x^3 + 1)) + 1/8*sqrt(2)*log((x^4 + 2*sqrt(2)*(x^3 - 1)^( 
1/4)*x^3 + x^3 + 4*sqrt(x^3 - 1)*x^2 + 2*sqrt(2)*(x^3 - 1)^(3/4)*x - 1)/(x 
^4 + x^3 - 1)) - 1/8*sqrt(2)*log((x^4 - 2*sqrt(2)*(x^3 - 1)^(1/4)*x^3 + x^ 
3 + 4*sqrt(x^3 - 1)*x^2 - 2*sqrt(2)*(x^3 - 1)^(3/4)*x - 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))
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^4 \left (-4+x^3\right )}{\sqrt [4]{-1+x^3} \left (-1+2 x^3-x^6+x^8\right )} \, dx=\text {Timed out} \] Input:

integrate(x**4*(x**3-4)/(x**3-1)**(1/4)/(x**8-x**6+2*x**3-1),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^4 \left (-4+x^3\right )}{\sqrt [4]{-1+x^3} \left (-1+2 x^3-x^6+x^8\right )} \, dx=\int { \frac {{\left (x^{3} - 4\right )} x^{4}}{{\left (x^{8} - x^{6} + 2 \, x^{3} - 1\right )} {\left (x^{3} - 1\right )}^{\frac {1}{4}}} \,d x } \] Input:

integrate(x^4*(x^3-4)/(x^3-1)^(1/4)/(x^8-x^6+2*x^3-1),x, algorithm="maxima 
")
 

Output:

integrate((x^3 - 4)*x^4/((x^8 - x^6 + 2*x^3 - 1)*(x^3 - 1)^(1/4)), x)
 

Giac [F]

\[ \int \frac {x^4 \left (-4+x^3\right )}{\sqrt [4]{-1+x^3} \left (-1+2 x^3-x^6+x^8\right )} \, dx=\int { \frac {{\left (x^{3} - 4\right )} x^{4}}{{\left (x^{8} - x^{6} + 2 \, x^{3} - 1\right )} {\left (x^{3} - 1\right )}^{\frac {1}{4}}} \,d x } \] Input:

integrate(x^4*(x^3-4)/(x^3-1)^(1/4)/(x^8-x^6+2*x^3-1),x, algorithm="giac")
 

Output:

integrate((x^3 - 4)*x^4/((x^8 - x^6 + 2*x^3 - 1)*(x^3 - 1)^(1/4)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \left (-4+x^3\right )}{\sqrt [4]{-1+x^3} \left (-1+2 x^3-x^6+x^8\right )} \, dx=\int \frac {x^4\,\left (x^3-4\right )}{{\left (x^3-1\right )}^{1/4}\,\left (x^8-x^6+2\,x^3-1\right )} \,d x \] Input:

int((x^4*(x^3 - 4))/((x^3 - 1)^(1/4)*(2*x^3 - x^6 + x^8 - 1)),x)
 

Output:

int((x^4*(x^3 - 4))/((x^3 - 1)^(1/4)*(2*x^3 - x^6 + x^8 - 1)), x)
 

Reduce [F]

\[ \int \frac {x^4 \left (-4+x^3\right )}{\sqrt [4]{-1+x^3} \left (-1+2 x^3-x^6+x^8\right )} \, dx=\int \frac {x^{7}}{\left (x^{3}-1\right )^{\frac {1}{4}} x^{8}-\left (x^{3}-1\right )^{\frac {1}{4}} x^{6}+2 \left (x^{3}-1\right )^{\frac {1}{4}} x^{3}-\left (x^{3}-1\right )^{\frac {1}{4}}}d x -4 \left (\int \frac {x^{4}}{\left (x^{3}-1\right )^{\frac {1}{4}} x^{8}-\left (x^{3}-1\right )^{\frac {1}{4}} x^{6}+2 \left (x^{3}-1\right )^{\frac {1}{4}} x^{3}-\left (x^{3}-1\right )^{\frac {1}{4}}}d x \right ) \] Input:

int(x^4*(x^3-4)/(x^3-1)^(1/4)/(x^8-x^6+2*x^3-1),x)
 

Output:

int(x**7/((x**3 - 1)**(1/4)*x**8 - (x**3 - 1)**(1/4)*x**6 + 2*(x**3 - 1)** 
(1/4)*x**3 - (x**3 - 1)**(1/4)),x) - 4*int(x**4/((x**3 - 1)**(1/4)*x**8 - 
(x**3 - 1)**(1/4)*x**6 + 2*(x**3 - 1)**(1/4)*x**3 - (x**3 - 1)**(1/4)),x)