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

3.11.55.1 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)
 

3.11.55.2 Mathematica [A] (verified)

Time = 3.59 (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]
 

3.11.55.3 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
 

3.11.55.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.11.55.4 Maple [A] (verified)

Time = 7.02 (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	\(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{6}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{2}+2 \sqrt {-x^{6}+1}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{-x^{6}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+1}\right )}{4}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{6}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{5} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}+2 \sqrt {-x^{6}+1}\, x}{x^{6}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}-1}\right )}{4}\)	\(155\)

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))
 

3.11.55.5 Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.36 (sec) , antiderivative size = 309, normalized size of antiderivative = 3.91 \[ \int \frac {\sqrt {1-x^6} \left (1+2 x^6\right )}{1+x^4-2 x^6+x^{12}} \, dx=\left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (\left (i + 1\right ) \, x^{12} + \left (2 i - 2\right ) \, x^{8} - \left (2 i + 2\right ) \, x^{6} - \left (i + 1\right ) \, x^{4} - \left (2 i - 2\right ) \, x^{2} + i + 1\right )} - 4 \, {\left (x^{7} + i \, x^{3} - x\right )} \sqrt {-x^{6} + 1}}{x^{12} - 2 \, x^{6} + x^{4} + 1}\right ) - \left (\frac {1}{16} i - \frac {1}{16}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (-\left (i - 1\right ) \, x^{12} - \left (2 i + 2\right ) \, x^{8} + \left (2 i - 2\right ) \, x^{6} + \left (i - 1\right ) \, x^{4} + \left (2 i + 2\right ) \, x^{2} - i + 1\right )} - 4 \, {\left (x^{7} - i \, x^{3} - x\right )} \sqrt {-x^{6} + 1}}{x^{12} - 2 \, x^{6} + x^{4} + 1}\right ) + \left (\frac {1}{16} i - \frac {1}{16}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (\left (i - 1\right ) \, x^{12} + \left (2 i + 2\right ) \, x^{8} - \left (2 i - 2\right ) \, x^{6} - \left (i - 1\right ) \, x^{4} - \left (2 i + 2\right ) \, x^{2} + i - 1\right )} - 4 \, {\left (x^{7} - i \, x^{3} - x\right )} \sqrt {-x^{6} + 1}}{x^{12} - 2 \, x^{6} + x^{4} + 1}\right ) - \left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (-\left (i + 1\right ) \, x^{12} - \left (2 i - 2\right ) \, x^{8} + \left (2 i + 2\right ) \, x^{6} + \left (i + 1\right ) \, x^{4} + \left (2 i - 2\right ) \, x^{2} - i - 1\right )} - 4 \, {\left (x^{7} + i \, x^{3} - x\right )} \sqrt {-x^{6} + 1}}{x^{12} - 2 \, x^{6} + x^{4} + 1}\right ) \]

input

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

output

(1/16*I + 1/16)*sqrt(2)*log((sqrt(2)*((I + 1)*x^12 + (2*I - 2)*x^8 - (2*I 
+ 2)*x^6 - (I + 1)*x^4 - (2*I - 2)*x^2 + I + 1) - 4*(x^7 + I*x^3 - x)*sqrt 
(-x^6 + 1))/(x^12 - 2*x^6 + x^4 + 1)) - (1/16*I - 1/16)*sqrt(2)*log((sqrt( 
2)*(-(I - 1)*x^12 - (2*I + 2)*x^8 + (2*I - 2)*x^6 + (I - 1)*x^4 + (2*I + 2 
)*x^2 - I + 1) - 4*(x^7 - I*x^3 - x)*sqrt(-x^6 + 1))/(x^12 - 2*x^6 + x^4 + 
 1)) + (1/16*I - 1/16)*sqrt(2)*log((sqrt(2)*((I - 1)*x^12 + (2*I + 2)*x^8 
- (2*I - 2)*x^6 - (I - 1)*x^4 - (2*I + 2)*x^2 + I - 1) - 4*(x^7 - I*x^3 - 
x)*sqrt(-x^6 + 1))/(x^12 - 2*x^6 + x^4 + 1)) - (1/16*I + 1/16)*sqrt(2)*log 
((sqrt(2)*(-(I + 1)*x^12 - (2*I - 2)*x^8 + (2*I + 2)*x^6 + (I + 1)*x^4 + ( 
2*I - 2)*x^2 - I - 1) - 4*(x^7 + I*x^3 - x)*sqrt(-x^6 + 1))/(x^12 - 2*x^6 
+ x^4 + 1))
 

3.11.55.6 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)
 

3.11.55.7 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)
 

3.11.55.8 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)
 

3.11.55.9 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)