\(\int \frac {\sqrt {-1-x-x^2+x^4} (2+x+2 x^4)}{(-1-x+x^4) (-1-x+x^2+x^4)} \, dx\) [710]

Optimal result

Integrand size = 49, antiderivative size = 55 \[ \int \frac {\sqrt {-1-x-x^2+x^4} \left (2+x+2 x^4\right )}{\left (-1-x+x^4\right ) \left (-1-x+x^2+x^4\right )} \, dx=2 \arctan \left (\frac {x}{\sqrt {-1-x-x^2+x^4}}\right )-2 \sqrt {2} \arctan \left (\frac {\sqrt {2} x}{\sqrt {-1-x-x^2+x^4}}\right ) \] Output:

2*arctan(x/(x^4-x^2-x-1)^(1/2))-2*2^(1/2)*arctan(2^(1/2)*x/(x^4-x^2-x-1)^( 
1/2))
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {-1-x-x^2+x^4} \left (2+x+2 x^4\right )}{\left (-1-x+x^4\right ) \left (-1-x+x^2+x^4\right )} \, dx=2 \arctan \left (\frac {x}{\sqrt {-1-x-x^2+x^4}}\right )-2 \sqrt {2} \arctan \left (\frac {\sqrt {2} x}{\sqrt {-1-x-x^2+x^4}}\right ) \] Input:

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

Output:

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

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.

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

\(\Big \downarrow \) 2463

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

\(\Big \downarrow \) 2009

\(\displaystyle -\int \frac {\sqrt {x^4-x^2-x-1}}{x-1}dx+\int \frac {\sqrt {x^4-x^2-x-1}}{x^4-x-1}dx+4 \int \frac {x^2 \sqrt {x^4-x^2-x-1}}{x^4-x-1}dx+2 \int \frac {\sqrt {x^4-x^2-x-1}}{x^3+x^2+2 x+1}dx-\int \frac {x \sqrt {x^4-x^2-x-1}}{x^3+x^2+2 x+1}dx+2 \int \frac {x^2 \sqrt {x^4-x^2-x-1}}{x^3+x^2+2 x+1}dx-\int \frac {x^3 \sqrt {x^4-x^2-x-1}}{x^4-x-1}dx\)

Input:

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

Output:

$Aborted
 

Maple [A] (verified)

Time = 15.36 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.96

method	result	size
default	\(2 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x^{4}-x^{2}-x -1}}{2 x}\right )-2 \arctan \left (\frac {\sqrt {x^{4}-x^{2}-x -1}}{x}\right )\)	\(53\)
pseudoelliptic	\(2 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x^{4}-x^{2}-x -1}}{2 x}\right )-2 \arctan \left (\frac {\sqrt {x^{4}-x^{2}-x -1}}{x}\right )\)	\(53\)
trager	\(\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{4}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x -4 \sqrt {x^{4}-x^{2}-x -1}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )}{\left (-1+x \right ) \left (x^{3}+x^{2}+2 x +1\right )}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \sqrt {x^{4}-x^{2}-x -1}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{4}-x -1}\right )\)	\(163\)
elliptic	\(\text {Expression too large to display}\)	\(782916\)

Input:

int((x^4-x^2-x-1)^(1/2)*(2*x^4+x+2)/(x^4-x-1)/(x^4+x^2-x-1),x,method=_RETU 
RNVERBOSE)
 

Output:

2*2^(1/2)*arctan(1/2*2^(1/2)/x*(x^4-x^2-x-1)^(1/2))-2*arctan(1/x*(x^4-x^2- 
x-1)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.40 \[ \int \frac {\sqrt {-1-x-x^2+x^4} \left (2+x+2 x^4\right )}{\left (-1-x+x^4\right ) \left (-1-x+x^2+x^4\right )} \, dx=-\sqrt {2} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {x^{4} - x^{2} - x - 1} x}{x^{4} - 3 \, x^{2} - x - 1}\right ) + \arctan \left (\frac {2 \, \sqrt {x^{4} - x^{2} - x - 1} x}{x^{4} - 2 \, x^{2} - x - 1}\right ) \] Input:

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {-1-x-x^2+x^4} \left (2+x+2 x^4\right )}{\left (-1-x+x^4\right ) \left (-1-x+x^2+x^4\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

(844*sqrt(x**4 - x**2 - x - 1) + 237*int(sqrt(x**4 - x**2 - x - 1)/(x**12 
- 3*x**9 - 4*x**8 + 3*x**6 + 7*x**5 + 4*x**4 - x**3 - 3*x**2 - 3*x - 1),x) 
 + 5*int(sqrt(x**4 - x**2 - x - 1)/(x**11 - x**10 + x**9 - 4*x**8 + 3*x**5 
 + 4*x**4 - x**2 - 2*x - 1),x) - 1668*int((sqrt(x**4 - x**2 - x - 1)*x**11 
)/(x**12 - 3*x**9 - 4*x**8 + 3*x**6 + 7*x**5 + 4*x**4 - x**3 - 3*x**2 - 3* 
x - 1),x) - 20*int((sqrt(x**4 - x**2 - x - 1)*x**10)/(x**11 - x**10 + x**9 
 - 4*x**8 + 3*x**5 + 4*x**4 - x**2 - 2*x - 1),x) - 834*int((sqrt(x**4 - x* 
*2 - x - 1)*x**9)/(x**12 - 3*x**9 - 4*x**8 + 3*x**6 + 7*x**5 + 4*x**4 - x* 
*3 - 3*x**2 - 3*x - 1),x) + 20*int((sqrt(x**4 - x**2 - x - 1)*x**9)/(x**11 
 - x**10 + x**9 - 4*x**8 + 3*x**5 + 4*x**4 - x**2 - 2*x - 1),x) + 3933*int 
((sqrt(x**4 - x**2 - x - 1)*x**8)/(x**12 - 3*x**9 - 4*x**8 + 3*x**6 + 7*x* 
*5 + 4*x**4 - x**3 - 3*x**2 - 3*x - 1),x) - 30*int((sqrt(x**4 - x**2 - x - 
 1)*x**8)/(x**11 - x**10 + x**9 - 4*x**8 + 3*x**5 + 4*x**4 - x**2 - 2*x - 
1),x) + 4170*int((sqrt(x**4 - x**2 - x - 1)*x**7)/(x**12 - 3*x**9 - 4*x**8 
 + 3*x**6 + 7*x**5 + 4*x**4 - x**3 - 3*x**2 - 3*x - 1),x) + 75*int((sqrt(x 
**4 - x**2 - x - 1)*x**7)/(x**11 - x**10 + x**9 - 4*x**8 + 3*x**5 + 4*x**4 
 - x**2 - 2*x - 1),x) + 237*int((sqrt(x**4 - x**2 - x - 1)*x**6)/(x**12 - 
3*x**9 - 4*x**8 + 3*x**6 + 7*x**5 + 4*x**4 - x**3 - 3*x**2 - 3*x - 1),x) - 
 25*int((sqrt(x**4 - x**2 - x - 1)*x**6)/(x**11 - x**10 + x**9 - 4*x**8 + 
3*x**5 + 4*x**4 - x**2 - 2*x - 1),x) - 2622*int((sqrt(x**4 - x**2 - x -...