\(\int \frac {(-2+x^6) (1-x^4+x^6)}{x^4 \sqrt [4]{1+x^6} (1+x^4+x^6)} \, dx\) [1321]

Optimal result

Integrand size = 38, antiderivative size = 95 \[ \int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{x^4 \sqrt [4]{1+x^6} \left (1+x^4+x^6\right )} \, dx=\frac {2 \left (1+x^6\right )^{3/4}}{3 x^3}+\sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt [4]{1+x^6}}{-x^2+\sqrt {1+x^6}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{1+x^6}}{x^2+\sqrt {1+x^6}}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 4.54 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{x^4 \sqrt [4]{1+x^6} \left (1+x^4+x^6\right )} \, dx=\frac {2 \left (1+x^6\right )^{3/4}}{3 x^3}+\sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt [4]{1+x^6}}{-x^2+\sqrt {1+x^6}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{1+x^6}}{x^2+\sqrt {1+x^6}}\right ) \] Input:

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

Output:

(2*(1 + x^6)^(3/4))/(3*x^3) + Sqrt[2]*ArcTan[(Sqrt[2]*x*(1 + x^6)^(1/4))/( 
-x^2 + Sqrt[1 + x^6])] + Sqrt[2]*ArcTanh[(Sqrt[2]*x*(1 + x^6)^(1/4))/(x^2 
+ Sqrt[1 + x^6])]
 

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.

\(\displaystyle \int \frac {\left (x^6-2\right ) \left (x^6-x^4+1\right )}{x^4 \sqrt [4]{x^6+1} \left (x^6+x^4+1\right )} \, dx\)

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

$Aborted
 

Maple [A] (verified)

Time = 22.09 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.36

method	result	size
pseudoelliptic	\(\frac {-3 \ln \left (\frac {-\left (x^{6}+1\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{6}+1}}{\left (x^{6}+1\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{6}+1}}\right ) \sqrt {2}\, x^{3}-6 \arctan \left (\frac {\left (x^{6}+1\right )^{\frac {1}{4}} \sqrt {2}+x}{x}\right ) \sqrt {2}\, x^{3}-6 \arctan \left (\frac {\left (x^{6}+1\right )^{\frac {1}{4}} \sqrt {2}-x}{x}\right ) \sqrt {2}\, x^{3}+4 \left (x^{6}+1\right )^{\frac {3}{4}}}{6 x^{3}}\)	\(129\)
trager	\(\frac {2 \left (x^{6}+1\right )^{\frac {3}{4}}}{3 x^{3}}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{6}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{4}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \left (x^{6}+1\right )^{\frac {1}{4}} x^{3}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \sqrt {x^{6}+1}\, x^{2}-2 \left (x^{6}+1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}}{x^{6}+x^{4}+1}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \sqrt {x^{6}+1}\, x^{2}-x^{6} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \left (x^{6}+1\right )^{\frac {1}{4}} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{4}-2 \left (x^{6}+1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{x^{6}+x^{4}+1}\right )\)	\(217\)
risch	\(\frac {2 \left (x^{6}+1\right )^{\frac {3}{4}}}{3 x^{3}}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{6}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{4}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \left (x^{6}+1\right )^{\frac {1}{4}} x^{3}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \sqrt {x^{6}+1}\, x^{2}+2 \left (x^{6}+1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}}{x^{6}+x^{4}+1}\right )-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \sqrt {x^{6}+1}\, x^{2}-x^{6} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \left (x^{6}+1\right )^{\frac {1}{4}} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{4}+2 \left (x^{6}+1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{x^{6}+x^{4}+1}\right )\)	\(220\)

Input:

int((x^6-2)*(x^6-x^4+1)/x^4/(x^6+1)^(1/4)/(x^6+x^4+1),x,method=_RETURNVERB 
OSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 406 vs. \(2 (76) = 152\).

Time = 117.24 (sec) , antiderivative size = 406, normalized size of antiderivative = 4.27 \[ \int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{x^4 \sqrt [4]{1+x^6} \left (1+x^4+x^6\right )} \, dx=\frac {6 \, \sqrt {2} x^{3} \arctan \left (\frac {x^{12} + 2 \, x^{10} + x^{8} + 2 \, x^{6} + 2 \, x^{4} + 2 \, \sqrt {2} {\left (x^{7} - 3 \, x^{5} + x\right )} {\left (x^{6} + 1\right )}^{\frac {3}{4}} + 2 \, \sqrt {2} {\left (3 \, x^{9} - x^{7} + 3 \, x^{3}\right )} {\left (x^{6} + 1\right )}^{\frac {1}{4}} + 4 \, {\left (x^{8} + x^{6} + x^{2}\right )} \sqrt {x^{6} + 1} + 1}{x^{12} - 14 \, x^{10} + x^{8} + 2 \, x^{6} - 14 \, x^{4} + 1}\right ) + 6 \, \sqrt {2} x^{3} \arctan \left (-\frac {x^{12} + 2 \, x^{10} + x^{8} + 2 \, x^{6} + 2 \, x^{4} - 2 \, \sqrt {2} {\left (x^{7} - 3 \, x^{5} + x\right )} {\left (x^{6} + 1\right )}^{\frac {3}{4}} - 2 \, \sqrt {2} {\left (3 \, x^{9} - x^{7} + 3 \, x^{3}\right )} {\left (x^{6} + 1\right )}^{\frac {1}{4}} + 4 \, {\left (x^{8} + x^{6} + x^{2}\right )} \sqrt {x^{6} + 1} + 1}{x^{12} - 14 \, x^{10} + x^{8} + 2 \, x^{6} - 14 \, x^{4} + 1}\right ) + 3 \, \sqrt {2} x^{3} \log \left (\frac {x^{6} + x^{4} + 2 \, \sqrt {2} {\left (x^{6} + 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{6} + 1} x^{2} + 2 \, \sqrt {2} {\left (x^{6} + 1\right )}^{\frac {3}{4}} x + 1}{x^{6} + x^{4} + 1}\right ) - 3 \, \sqrt {2} x^{3} \log \left (\frac {x^{6} + x^{4} - 2 \, \sqrt {2} {\left (x^{6} + 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{6} + 1} x^{2} - 2 \, \sqrt {2} {\left (x^{6} + 1\right )}^{\frac {3}{4}} x + 1}{x^{6} + x^{4} + 1}\right ) + 8 \, {\left (x^{6} + 1\right )}^{\frac {3}{4}}}{12 \, x^{3}} \] Input:

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

Output:

1/12*(6*sqrt(2)*x^3*arctan((x^12 + 2*x^10 + x^8 + 2*x^6 + 2*x^4 + 2*sqrt(2 
)*(x^7 - 3*x^5 + x)*(x^6 + 1)^(3/4) + 2*sqrt(2)*(3*x^9 - x^7 + 3*x^3)*(x^6 
 + 1)^(1/4) + 4*(x^8 + x^6 + x^2)*sqrt(x^6 + 1) + 1)/(x^12 - 14*x^10 + x^8 
 + 2*x^6 - 14*x^4 + 1)) + 6*sqrt(2)*x^3*arctan(-(x^12 + 2*x^10 + x^8 + 2*x 
^6 + 2*x^4 - 2*sqrt(2)*(x^7 - 3*x^5 + x)*(x^6 + 1)^(3/4) - 2*sqrt(2)*(3*x^ 
9 - x^7 + 3*x^3)*(x^6 + 1)^(1/4) + 4*(x^8 + x^6 + x^2)*sqrt(x^6 + 1) + 1)/ 
(x^12 - 14*x^10 + x^8 + 2*x^6 - 14*x^4 + 1)) + 3*sqrt(2)*x^3*log((x^6 + x^ 
4 + 2*sqrt(2)*(x^6 + 1)^(1/4)*x^3 + 4*sqrt(x^6 + 1)*x^2 + 2*sqrt(2)*(x^6 + 
 1)^(3/4)*x + 1)/(x^6 + x^4 + 1)) - 3*sqrt(2)*x^3*log((x^6 + x^4 - 2*sqrt( 
2)*(x^6 + 1)^(1/4)*x^3 + 4*sqrt(x^6 + 1)*x^2 - 2*sqrt(2)*(x^6 + 1)^(3/4)*x 
 + 1)/(x^6 + x^4 + 1)) + 8*(x^6 + 1)^(3/4))/x^3
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{x^4 \sqrt [4]{1+x^6} \left (1+x^4+x^6\right )} \, dx=\int \frac {x^{8}}{\left (x^{6}+1\right )^{\frac {1}{4}} x^{6}+\left (x^{6}+1\right )^{\frac {1}{4}} x^{4}+\left (x^{6}+1\right )^{\frac {1}{4}}}d x -\left (\int \frac {x^{6}}{\left (x^{6}+1\right )^{\frac {1}{4}} x^{6}+\left (x^{6}+1\right )^{\frac {1}{4}} x^{4}+\left (x^{6}+1\right )^{\frac {1}{4}}}d x \right )-\left (\int \frac {x^{2}}{\left (x^{6}+1\right )^{\frac {1}{4}} x^{6}+\left (x^{6}+1\right )^{\frac {1}{4}} x^{4}+\left (x^{6}+1\right )^{\frac {1}{4}}}d x \right )-2 \left (\int \frac {1}{\left (x^{6}+1\right )^{\frac {1}{4}} x^{10}+\left (x^{6}+1\right )^{\frac {1}{4}} x^{8}+\left (x^{6}+1\right )^{\frac {1}{4}} x^{4}}d x \right )+2 \left (\int \frac {1}{\left (x^{6}+1\right )^{\frac {1}{4}} x^{6}+\left (x^{6}+1\right )^{\frac {1}{4}} x^{4}+\left (x^{6}+1\right )^{\frac {1}{4}}}d x \right ) \] Input:

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

Output:

int(x**8/((x**6 + 1)**(1/4)*x**6 + (x**6 + 1)**(1/4)*x**4 + (x**6 + 1)**(1 
/4)),x) - int(x**6/((x**6 + 1)**(1/4)*x**6 + (x**6 + 1)**(1/4)*x**4 + (x** 
6 + 1)**(1/4)),x) - int(x**2/((x**6 + 1)**(1/4)*x**6 + (x**6 + 1)**(1/4)*x 
**4 + (x**6 + 1)**(1/4)),x) - 2*int(1/((x**6 + 1)**(1/4)*x**10 + (x**6 + 1 
)**(1/4)*x**8 + (x**6 + 1)**(1/4)*x**4),x) + 2*int(1/((x**6 + 1)**(1/4)*x* 
*6 + (x**6 + 1)**(1/4)*x**4 + (x**6 + 1)**(1/4)),x)