\(\int \frac {\sqrt {1-x^6} (1+2 x^6)}{1+x^4-2 x^6+x^{12}} \, dx\) [1055]

Optimal result

Integrand size = 34, antiderivative size = 79 \[ \int \frac {\sqrt {1-x^6} \left (1+2 x^6\right )}{1+x^4-2 x^6+x^{12}} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} x \sqrt {1-x^6}}{-1+x^2+x^6}\right )}{2 \sqrt {2}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} x \sqrt {1-x^6}}{-1-x^2+x^6}\right )}{2 \sqrt {2}} \] Output:

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

Mathematica [A] (verified)

Time = 3.86 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {1-x^6} \left (1+2 x^6\right )}{1+x^4-2 x^6+x^{12}} \, dx=-\frac {\arctan \left (\frac {x \sqrt {2-2 x^6}}{-1+x^2+x^6}\right )+\text {arctanh}\left (\frac {-1-x^2+x^6}{x \sqrt {2-2 x^6}}\right )}{2 \sqrt {2}} \] Input:

Integrate[(Sqrt[1 - x^6]*(1 + 2*x^6))/(1 + x^4 - 2*x^6 + x^12),x]
 

Output:

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

\(\displaystyle \int \frac {\sqrt {1-x^6} \left (2 x^6+1\right )}{x^{12}-2 x^6+x^4+1} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 \sqrt {1-x^6} x^6}{x^{12}-2 x^6+x^4+1}+\frac {\sqrt {1-x^6}}{x^{12}-2 x^6+x^4+1}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \int \frac {\sqrt {1-x^6}}{x^{12}-2 x^6+x^4+1}dx+2 \int \frac {x^6 \sqrt {1-x^6}}{x^{12}-2 x^6+x^4+1}dx\)

Input:

Int[(Sqrt[1 - x^6]*(1 + 2*x^6))/(1 + x^4 - 2*x^6 + x^12),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 4.11 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.34

method	result	size
pseudoelliptic	\(-\frac {\sqrt {2}\, \left (\ln \left (\frac {-x^{6}-\sqrt {-x^{6}+1}\, \sqrt {2}\, x +x^{2}+1}{-x^{6}+\sqrt {-x^{6}+1}\, \sqrt {2}\, x +x^{2}+1}\right )+2 \arctan \left (\frac {\sqrt {-x^{6}+1}\, \sqrt {2}+x}{x}\right )+2 \arctan \left (\frac {\sqrt {-x^{6}+1}\, \sqrt {2}-x}{x}\right )\right )}{8}\)	\(106\)
trager	\(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{6}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{2}+2 \sqrt {-x^{6}+1}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{-x^{6}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+1}\right )}{4}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{6}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{5} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}-2 \sqrt {-x^{6}+1}\, x}{x^{6}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}-1}\right )}{4}\)	\(154\)

Input:

int((-x^6+1)^(1/2)*(2*x^6+1)/(x^12-2*x^6+x^4+1),x,method=_RETURNVERBOSE)
 

Output:

-1/8*2^(1/2)*(ln((-x^6-(-x^6+1)^(1/2)*2^(1/2)*x+x^2+1)/(-x^6+(-x^6+1)^(1/2 
)*2^(1/2)*x+x^2+1))+2*arctan(((-x^6+1)^(1/2)*2^(1/2)+x)/x)+2*arctan(((-x^6 
+1)^(1/2)*2^(1/2)-x)/x))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 282 vs. \(2 (63) = 126\).

Time = 0.23 (sec) , antiderivative size = 282, normalized size of antiderivative = 3.57 \[ \int \frac {\sqrt {1-x^6} \left (1+2 x^6\right )}{1+x^4-2 x^6+x^{12}} \, dx=-\frac {1}{8} \, \sqrt {2} \arctan \left (\frac {x^{12} - 2 \, x^{6} + x^{4} + 2 \, \sqrt {2} {\left (x^{7} + x^{3} - x\right )} \sqrt {-x^{6} + 1} + 1}{x^{12} + 4 \, x^{8} - 2 \, x^{6} + x^{4} - 4 \, x^{2} + 1}\right ) - \frac {1}{8} \, \sqrt {2} \arctan \left (-\frac {x^{12} - 2 \, x^{6} + x^{4} - 2 \, \sqrt {2} {\left (x^{7} + x^{3} - x\right )} \sqrt {-x^{6} + 1} + 1}{x^{12} + 4 \, x^{8} - 2 \, x^{6} + x^{4} - 4 \, x^{2} + 1}\right ) - \frac {1}{16} \, \sqrt {2} \log \left (\frac {x^{12} - 4 \, x^{8} - 2 \, x^{6} + x^{4} + 2 \, \sqrt {2} {\left (x^{7} - x^{3} - x\right )} \sqrt {-x^{6} + 1} + 4 \, x^{2} + 1}{x^{12} - 2 \, x^{6} + x^{4} + 1}\right ) + \frac {1}{16} \, \sqrt {2} \log \left (\frac {x^{12} - 4 \, x^{8} - 2 \, x^{6} + x^{4} - 2 \, \sqrt {2} {\left (x^{7} - x^{3} - x\right )} \sqrt {-x^{6} + 1} + 4 \, x^{2} + 1}{x^{12} - 2 \, x^{6} + x^{4} + 1}\right ) \] Input:

integrate((-x^6+1)^(1/2)*(2*x^6+1)/(x^12-2*x^6+x^4+1),x, algorithm="fricas 
")
 

Output:

-1/8*sqrt(2)*arctan((x^12 - 2*x^6 + x^4 + 2*sqrt(2)*(x^7 + x^3 - x)*sqrt(- 
x^6 + 1) + 1)/(x^12 + 4*x^8 - 2*x^6 + x^4 - 4*x^2 + 1)) - 1/8*sqrt(2)*arct 
an(-(x^12 - 2*x^6 + x^4 - 2*sqrt(2)*(x^7 + x^3 - x)*sqrt(-x^6 + 1) + 1)/(x 
^12 + 4*x^8 - 2*x^6 + x^4 - 4*x^2 + 1)) - 1/16*sqrt(2)*log((x^12 - 4*x^8 - 
 2*x^6 + x^4 + 2*sqrt(2)*(x^7 - x^3 - x)*sqrt(-x^6 + 1) + 4*x^2 + 1)/(x^12 
 - 2*x^6 + x^4 + 1)) + 1/16*sqrt(2)*log((x^12 - 4*x^8 - 2*x^6 + x^4 - 2*sq 
rt(2)*(x^7 - x^3 - x)*sqrt(-x^6 + 1) + 4*x^2 + 1)/(x^12 - 2*x^6 + x^4 + 1) 
)
 

Sympy [F]

\[ \int \frac {\sqrt {1-x^6} \left (1+2 x^6\right )}{1+x^4-2 x^6+x^{12}} \, dx=\int \frac {\sqrt {- \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (2 x^{6} + 1\right )}{x^{12} - 2 x^{6} + x^{4} + 1}\, dx \] Input:

integrate((-x**6+1)**(1/2)*(2*x**6+1)/(x**12-2*x**6+x**4+1),x)
 

Output:

Integral(sqrt(-(x - 1)*(x + 1)*(x**2 - x + 1)*(x**2 + x + 1))*(2*x**6 + 1) 
/(x**12 - 2*x**6 + x**4 + 1), x)
 

Maxima [F]

\[ \int \frac {\sqrt {1-x^6} \left (1+2 x^6\right )}{1+x^4-2 x^6+x^{12}} \, dx=\int { \frac {{\left (2 \, x^{6} + 1\right )} \sqrt {-x^{6} + 1}}{x^{12} - 2 \, x^{6} + x^{4} + 1} \,d x } \] Input:

integrate((-x^6+1)^(1/2)*(2*x^6+1)/(x^12-2*x^6+x^4+1),x, algorithm="maxima 
")
 

Output:

integrate((2*x^6 + 1)*sqrt(-x^6 + 1)/(x^12 - 2*x^6 + x^4 + 1), x)
 

Giac [F]

\[ \int \frac {\sqrt {1-x^6} \left (1+2 x^6\right )}{1+x^4-2 x^6+x^{12}} \, dx=\int { \frac {{\left (2 \, x^{6} + 1\right )} \sqrt {-x^{6} + 1}}{x^{12} - 2 \, x^{6} + x^{4} + 1} \,d x } \] Input:

integrate((-x^6+1)^(1/2)*(2*x^6+1)/(x^12-2*x^6+x^4+1),x, algorithm="giac")
 

Output:

integrate((2*x^6 + 1)*sqrt(-x^6 + 1)/(x^12 - 2*x^6 + x^4 + 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {1-x^6} \left (1+2 x^6\right )}{1+x^4-2 x^6+x^{12}} \, dx=\int \frac {\sqrt {1-x^6}\,\left (2\,x^6+1\right )}{x^{12}-2\,x^6+x^4+1} \,d x \] Input:

int(((1 - x^6)^(1/2)*(2*x^6 + 1))/(x^4 - 2*x^6 + x^12 + 1),x)
 

Output:

int(((1 - x^6)^(1/2)*(2*x^6 + 1))/(x^4 - 2*x^6 + x^12 + 1), x)
 

Reduce [F]

\[ \int \frac {\sqrt {1-x^6} \left (1+2 x^6\right )}{1+x^4-2 x^6+x^{12}} \, dx=\int \frac {\sqrt {-x^{6}+1}}{x^{12}-2 x^{6}+x^{4}+1}d x +2 \left (\int \frac {\sqrt {-x^{6}+1}\, x^{6}}{x^{12}-2 x^{6}+x^{4}+1}d x \right ) \] Input:

int((-x^6+1)^(1/2)*(2*x^6+1)/(x^12-2*x^6+x^4+1),x)
 

Output:

int(sqrt( - x**6 + 1)/(x**12 - 2*x**6 + x**4 + 1),x) + 2*int((sqrt( - x**6 
 + 1)*x**6)/(x**12 - 2*x**6 + x**4 + 1),x)