\(\int \frac {x^6 (-4+x^3)}{(-1+x^3)^{3/4} (1-2 x^3+x^6+x^8)} \, dx\) [2698]

Optimal result

Integrand size = 33, antiderivative size = 245 \[ \int \frac {x^6 \left (-4+x^3\right )}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )} \, dx=\frac {1}{2} \sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{-1+x^3}}{-x^2+\sqrt {-1+x^3}}\right )-\frac {1}{2} \sqrt {2-\sqrt {2}} \arctan \left (\frac {\left (\sqrt {\frac {2}{2-\sqrt {2}}}-\frac {2}{\sqrt {2-\sqrt {2}}}\right ) x \sqrt [4]{-1+x^3}}{-x^2+\sqrt {-1+x^3}}\right )-\frac {1}{2} \sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{-1+x^3}}{x^2+\sqrt {-1+x^3}}\right )-\frac {1}{2} \sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{-1+x^3}}{x^2+\sqrt {-1+x^3}}\right ) \] Output:

1/2*(2+2^(1/2))^(1/2)*arctan((2+2^(1/2))^(1/2)*x*(x^3-1)^(1/4)/(-x^2+(x^3- 
1)^(1/2)))-1/2*(2-2^(1/2))^(1/2)*arctan((2^(1/2)/(2-2^(1/2))^(1/2)-2/(2-2^ 
(1/2))^(1/2))*x*(x^3-1)^(1/4)/(-x^2+(x^3-1)^(1/2)))-1/2*(2-2^(1/2))^(1/2)* 
arctanh((2-2^(1/2))^(1/2)*x*(x^3-1)^(1/4)/(x^2+(x^3-1)^(1/2)))-1/2*(2+2^(1 
/2))^(1/2)*arctanh((2+2^(1/2))^(1/2)*x*(x^3-1)^(1/4)/(x^2+(x^3-1)^(1/2)))
 

Mathematica [A] (verified)

Time = 3.50 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.89 \[ \int \frac {x^6 \left (-4+x^3\right )}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )} \, dx=\frac {1}{2} \left (\sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{-1+x^3}}{-x^2+\sqrt {-1+x^3}}\right )+\sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{-1+x^3}}{-x^2+\sqrt {-1+x^3}}\right )-\sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{-1+x^3}}{x^2+\sqrt {-1+x^3}}\right )-\sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{-1+x^3}}{x^2+\sqrt {-1+x^3}}\right )\right ) \] Input:

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

Output:

(Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[2 - Sqrt[2]]*x*(-1 + x^3)^(1/4))/(-x^2 + S 
qrt[-1 + x^3])] + Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[2 + Sqrt[2]]*x*(-1 + x^3) 
^(1/4))/(-x^2 + Sqrt[-1 + x^3])] - Sqrt[2 - Sqrt[2]]*ArcTanh[(Sqrt[2 - Sqr 
t[2]]*x*(-1 + x^3)^(1/4))/(x^2 + Sqrt[-1 + x^3])] - Sqrt[2 + Sqrt[2]]*ArcT 
anh[(Sqrt[2 + Sqrt[2]]*x*(-1 + x^3)^(1/4))/(x^2 + Sqrt[-1 + x^3])])/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.

\(\displaystyle \int \frac {x^6 \left (x^3-4\right )}{\left (x^3-1\right )^{3/4} \left (x^8+x^6-2 x^3+1\right )} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {x}{\left (x^3-1\right )^{3/4}}+\frac {\left (-x^6-4 x^5+2 x^3-1\right ) x}{\left (x^3-1\right )^{3/4} \left (x^8+x^6-2 x^3+1\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\int \frac {x}{\left (x^3-1\right )^{3/4} \left (x^8+x^6-2 x^3+1\right )}dx-4 \int \frac {x^6}{\left (x^3-1\right )^{3/4} \left (x^8+x^6-2 x^3+1\right )}dx-\int \frac {x^7}{\left (x^3-1\right )^{3/4} \left (x^8+x^6-2 x^3+1\right )}dx+2 \int \frac {x^4}{\left (x^3-1\right )^{3/4} \left (x^8+x^6-2 x^3+1\right )}dx+\frac {\left (1-x^3\right )^{3/4} x^2 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {3}{4},\frac {5}{3},x^3\right )}{2 \left (x^3-1\right )^{3/4}}\)

Input:

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

Output:

$Aborted
 

Maple [C] (verified)

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

Time = 2.44 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.89

method	result	size
trager	\(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{7} \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{7} \sqrt {x^{3}-1}\, x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{6} \left (x^{3}-1\right )^{\frac {1}{4}} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{5} x^{4}-\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right ) x^{3}+2 \left (x^{3}-1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )}{\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{4} x^{4}+x^{3}-1}\right )}{2}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{11} x^{4}+\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{7} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{7}-2 \left (x^{3}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{2} x^{3}-2 \sqrt {x^{3}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right ) x^{2}-2 \left (x^{3}-1\right )^{\frac {3}{4}} x}{\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{4} x^{4}-x^{3}+1}\right )}{2}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{3} \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{9} x^{4}+2 \operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{6} \left (x^{3}-1\right )^{\frac {1}{4}} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{5} x^{3}+2 \operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{3} \sqrt {x^{3}-1}\, x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{5}+2 \left (x^{3}-1\right )^{\frac {3}{4}} x}{\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{4} x^{4}+x^{3}-1}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{5} \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{7} x^{4}-2 \operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{5} \sqrt {x^{3}-1}\, x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{3} x^{3}-2 \left (x^{3}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{2} x^{3}+2 \left (x^{3}-1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{3}}{\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{4} x^{4}-x^{3}+1}\right )}{2}\)	\(463\)

Input:

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

Output:

-1/2*RootOf(_Z^8+1)^7*ln(-(2*RootOf(_Z^8+1)^7*(x^3-1)^(1/2)*x^2-2*RootOf(_ 
Z^8+1)^6*(x^3-1)^(1/4)*x^3+RootOf(_Z^8+1)^5*x^4-RootOf(_Z^8+1)*x^3+2*(x^3- 
1)^(3/4)*x+RootOf(_Z^8+1))/(RootOf(_Z^8+1)^4*x^4+x^3-1))-1/2*RootOf(_Z^8+1 
)*ln((RootOf(_Z^8+1)^11*x^4+RootOf(_Z^8+1)^7*x^3-RootOf(_Z^8+1)^7-2*(x^3-1 
)^(1/4)*RootOf(_Z^8+1)^2*x^3-2*(x^3-1)^(1/2)*RootOf(_Z^8+1)*x^2-2*(x^3-1)^ 
(3/4)*x)/(RootOf(_Z^8+1)^4*x^4-x^3+1))-1/2*RootOf(_Z^8+1)^3*ln(-(RootOf(_Z 
^8+1)^9*x^4+2*RootOf(_Z^8+1)^6*(x^3-1)^(1/4)*x^3-RootOf(_Z^8+1)^5*x^3+2*Ro 
otOf(_Z^8+1)^3*(x^3-1)^(1/2)*x^2+RootOf(_Z^8+1)^5+2*(x^3-1)^(3/4)*x)/(Root 
Of(_Z^8+1)^4*x^4+x^3-1))+1/2*RootOf(_Z^8+1)^5*ln(-(RootOf(_Z^8+1)^7*x^4-2* 
RootOf(_Z^8+1)^5*(x^3-1)^(1/2)*x^2+RootOf(_Z^8+1)^3*x^3-2*(x^3-1)^(1/4)*Ro 
otOf(_Z^8+1)^2*x^3+2*(x^3-1)^(3/4)*x-RootOf(_Z^8+1)^3)/(RootOf(_Z^8+1)^4*x 
^4-x^3+1))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.92 \[ \int \frac {x^6 \left (-4+x^3\right )}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )} \, dx=-\left (\frac {1}{4} i + \frac {1}{4}\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} \log \left (\frac {\left (i + 1\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} x + 2 \, {\left (x^{3} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \left (\frac {1}{4} i - \frac {1}{4}\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} \log \left (\frac {-\left (i - 1\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} x + 2 \, {\left (x^{3} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \left (\frac {1}{4} i - \frac {1}{4}\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} \log \left (\frac {\left (i - 1\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} x + 2 \, {\left (x^{3} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \left (\frac {1}{4} i + \frac {1}{4}\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} \log \left (\frac {-\left (i + 1\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} x + 2 \, {\left (x^{3} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \left (-1\right )^{\frac {1}{8}} \log \left (\frac {\left (-1\right )^{\frac {1}{8}} x + {\left (x^{3} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \left (-1\right )^{\frac {1}{8}} \log \left (-\frac {\left (-1\right )^{\frac {1}{8}} x - {\left (x^{3} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} i \, \left (-1\right )^{\frac {1}{8}} \log \left (\frac {i \, \left (-1\right )^{\frac {1}{8}} x + {\left (x^{3} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} i \, \left (-1\right )^{\frac {1}{8}} \log \left (\frac {-i \, \left (-1\right )^{\frac {1}{8}} x + {\left (x^{3} - 1\right )}^{\frac {1}{4}}}{x}\right ) \] Input:

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

Output:

-(1/4*I + 1/4)*sqrt(2)*(-1)^(1/8)*log(((I + 1)*sqrt(2)*(-1)^(1/8)*x + 2*(x 
^3 - 1)^(1/4))/x) + (1/4*I - 1/4)*sqrt(2)*(-1)^(1/8)*log((-(I - 1)*sqrt(2) 
*(-1)^(1/8)*x + 2*(x^3 - 1)^(1/4))/x) - (1/4*I - 1/4)*sqrt(2)*(-1)^(1/8)*l 
og(((I - 1)*sqrt(2)*(-1)^(1/8)*x + 2*(x^3 - 1)^(1/4))/x) + (1/4*I + 1/4)*s 
qrt(2)*(-1)^(1/8)*log((-(I + 1)*sqrt(2)*(-1)^(1/8)*x + 2*(x^3 - 1)^(1/4))/ 
x) - 1/2*(-1)^(1/8)*log(((-1)^(1/8)*x + (x^3 - 1)^(1/4))/x) + 1/2*(-1)^(1/ 
8)*log(-((-1)^(1/8)*x - (x^3 - 1)^(1/4))/x) - 1/2*I*(-1)^(1/8)*log((I*(-1) 
^(1/8)*x + (x^3 - 1)^(1/4))/x) + 1/2*I*(-1)^(1/8)*log((-I*(-1)^(1/8)*x + ( 
x^3 - 1)^(1/4))/x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^6 \left (-4+x^3\right )}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )} \, dx=\int \frac {x^{9}}{\left (x^{3}-1\right )^{\frac {3}{4}} x^{8}+\left (x^{3}-1\right )^{\frac {3}{4}} x^{6}-2 \left (x^{3}-1\right )^{\frac {3}{4}} x^{3}+\left (x^{3}-1\right )^{\frac {3}{4}}}d x -4 \left (\int \frac {x^{6}}{\left (x^{3}-1\right )^{\frac {3}{4}} x^{8}+\left (x^{3}-1\right )^{\frac {3}{4}} x^{6}-2 \left (x^{3}-1\right )^{\frac {3}{4}} x^{3}+\left (x^{3}-1\right )^{\frac {3}{4}}}d x \right ) \] Input:

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

Output:

int(x**9/((x**3 - 1)**(3/4)*x**8 + (x**3 - 1)**(3/4)*x**6 - 2*(x**3 - 1)** 
(3/4)*x**3 + (x**3 - 1)**(3/4)),x) - 4*int(x**6/((x**3 - 1)**(3/4)*x**8 + 
(x**3 - 1)**(3/4)*x**6 - 2*(x**3 - 1)**(3/4)*x**3 + (x**3 - 1)**(3/4)),x)