\(\int \frac {x^6 (4+x^3)}{(1+x^3)^{3/4} (-1-2 x^3-x^6+x^8)} \, dx\) [1694]

Optimal result

Integrand size = 35, antiderivative size = 113 \[ \int \frac {x^6 \left (4+x^3\right )}{\left (1+x^3\right )^{3/4} \left (-1-2 x^3-x^6+x^8\right )} \, dx=\arctan \left (\frac {x}{\sqrt [4]{1+x^3}}\right )-\frac {\arctan \left (\frac {-\frac {x^2}{\sqrt {2}}+\frac {\sqrt {1+x^3}}{\sqrt {2}}}{x \sqrt [4]{1+x^3}}\right )}{\sqrt {2}}-\text {arctanh}\left (\frac {x}{\sqrt [4]{1+x^3}}\right )-\frac {\text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{1+x^3}}{x^2+\sqrt {1+x^3}}\right )}{\sqrt {2}} \] Output:

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

Mathematica [A] (verified)

Time = 2.97 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.95 \[ \int \frac {x^6 \left (4+x^3\right )}{\left (1+x^3\right )^{3/4} \left (-1-2 x^3-x^6+x^8\right )} \, dx=\arctan \left (\frac {x}{\sqrt [4]{1+x^3}}\right )-\frac {\arctan \left (\frac {-x^2+\sqrt {1+x^3}}{\sqrt {2} x \sqrt [4]{1+x^3}}\right )}{\sqrt {2}}-\text {arctanh}\left (\frac {x}{\sqrt [4]{1+x^3}}\right )-\frac {\text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{1+x^3}}{x^2+\sqrt {1+x^3}}\right )}{\sqrt {2}} \] Input:

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

Output:

ArcTan[x/(1 + x^3)^(1/4)] - ArcTan[(-x^2 + Sqrt[1 + x^3])/(Sqrt[2]*x*(1 + 
x^3)^(1/4))]/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.

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

\(\Big \downarrow \) 2463

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

\(\Big \downarrow \) 2009

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

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] (warning: unable to verify)

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

Time = 2.22 (sec) , antiderivative size = 426, normalized size of antiderivative = 3.77

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 {\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 {\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 {\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 {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}\)	\(426\)

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.42 \[ \int \frac {x^6 \left (4+x^3\right )}{\left (1+x^3\right )^{3/4} \left (-1-2 x^3-x^6+x^8\right )} \, dx=-\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {x + \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \sqrt {2} \arctan \left (-\frac {x - \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{4} \, \sqrt {2} \log \left (\frac {x^{2} + \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {1}{4}} x + \sqrt {x^{3} + 1}}{x^{2}}\right ) + \frac {1}{4} \, \sqrt {2} \log \left (\frac {x^{2} - \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {1}{4}} x + \sqrt {x^{3} + 1}}{x^{2}}\right ) + \arctan \left (\frac {x}{{\left (x^{3} + 1\right )}^{\frac {1}{4}}}\right ) - \frac {1}{2} \, \log \left (\frac {x + {\left (x^{3} + 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \log \left (-\frac {x - {\left (x^{3} + 1\right )}^{\frac {1}{4}}}{x}\right ) \] Input:

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

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

Output:

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