\(\int \frac {-1+x^4}{(1+x^4) \sqrt {1-x-x^2+x^3+x^4}} \, dx\) [1782]

Optimal result

Integrand size = 33, antiderivative size = 120 \[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx=\frac {1}{2} \text {RootSum}\left [3-8 \text {$\#$1}+6 \text {$\#$1}^2+\text {$\#$1}^4\&,\frac {\log (x)-\log \left (1-x^2+\sqrt {1-x-x^2+x^3+x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}^2+\log \left (1-x^2+\sqrt {1-x-x^2+x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-2+3 \text {$\#$1}+\text {$\#$1}^3}\&\right ] \] Output:

Unintegrable
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx=\frac {1}{2} \text {RootSum}\left [3-8 \text {$\#$1}+6 \text {$\#$1}^2+\text {$\#$1}^4\&,\frac {\log (x)-\log \left (1-x^2+\sqrt {1-x-x^2+x^3+x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}^2+\log \left (1-x^2+\sqrt {1-x-x^2+x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-2+3 \text {$\#$1}+\text {$\#$1}^3}\&\right ] \] Input:

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

Output:

RootSum[3 - 8*#1 + 6*#1^2 + #1^4 & , (Log[x] - Log[1 - x^2 + Sqrt[1 - x - 
x^2 + x^3 + x^4] - x*#1] - Log[x]*#1^2 + Log[1 - x^2 + Sqrt[1 - x - x^2 + 
x^3 + x^4] - x*#1]*#1^2)/(-2 + 3*#1 + #1^3) & ]/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.

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

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

Input:

Int[(-1 + x^4)/((1 + x^4)*Sqrt[1 - x - x^2 + x^3 + x^4]),x]
 

Output:

$Aborted
 

Maple [N/A] (verified)

Time = 5.43 (sec) , antiderivative size = 256, normalized size of antiderivative = 2.13

method	result	size
default	\(-\frac {\left (x \sqrt {3}+x^{2}-x -1\right ) \left (\left (3-2 \sqrt {3}\right ) \sqrt {1+\sqrt {3}}\, \operatorname {arctanh}\left (\frac {2 \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (x \sqrt {3}+x^{2}-x -1\right )^{2}}}}{\sqrt {1+\sqrt {3}}}\right )+\sqrt {-3+3 \sqrt {3}}\, \arctan \left (\frac {\sqrt {3}\, \sqrt {-3+3 \sqrt {3}}\, \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (x \sqrt {3}+x^{2}-x -1\right )^{2}}}\, \left (-x \sqrt {3}+x^{2}-x -1\right ) \left (x \sqrt {3}+x^{2}-x -1\right )}{6 x^{4}+6 x^{3}-6 x^{2}-6 x +6}\right )\right ) \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (x \sqrt {3}+x^{2}-x -1\right )^{2}}}}{3 \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{x^{2}}}\, \left (1+\sqrt {3}\right ) \left (-2+\sqrt {3}\right ) x}\)	\(256\)
pseudoelliptic	\(-\frac {\left (x \sqrt {3}+x^{2}-x -1\right ) \left (\left (3-2 \sqrt {3}\right ) \sqrt {1+\sqrt {3}}\, \operatorname {arctanh}\left (\frac {2 \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (x \sqrt {3}+x^{2}-x -1\right )^{2}}}}{\sqrt {1+\sqrt {3}}}\right )+\sqrt {-3+3 \sqrt {3}}\, \arctan \left (\frac {\sqrt {3}\, \sqrt {-3+3 \sqrt {3}}\, \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (x \sqrt {3}+x^{2}-x -1\right )^{2}}}\, \left (-x \sqrt {3}+x^{2}-x -1\right ) \left (x \sqrt {3}+x^{2}-x -1\right )}{6 x^{4}+6 x^{3}-6 x^{2}-6 x +6}\right )\right ) \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (x \sqrt {3}+x^{2}-x -1\right )^{2}}}}{3 \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{x^{2}}}\, \left (1+\sqrt {3}\right ) \left (-2+\sqrt {3}\right ) x}\)	\(256\)
trager	\(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right ) \ln \left (\frac {12 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right ) x +48 \sqrt {x^{4}+x^{3}-x^{2}-x +1}\, \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right ) x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right ) x +2 \sqrt {x^{4}+x^{3}-x^{2}-x +1}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right )}{12 x \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}-x^{2}+x +1}\right )}{6}-\operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right ) \ln \left (-\frac {12 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{3} x +8 \sqrt {x^{4}+x^{3}-x^{2}-x +1}\, \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+3 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right ) x^{2}+3 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right ) x -3 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )+\sqrt {x^{4}+x^{3}-x^{2}-x +1}}{12 x \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+x^{2}+x -1}\right )\)	\(398\)
elliptic	\(\text {Expression too large to display}\)	\(3623\)

Input:

int((x^4-1)/(x^4+1)/(x^4+x^3-x^2-x+1)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.15 (sec) , antiderivative size = 725, normalized size of antiderivative = 6.04 \[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx =\text {Too large to display} \] Input:

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

Output:

1/2*sqrt(1/6*sqrt(3) + 1/6)*arctan(3*(6*(51*x^6 + 78*x^5 - 159*x^4 - 216*x 
^3 + 159*x^2 - sqrt(3)*(7*x^6 + 80*x^5 + 29*x^4 - 144*x^3 - 29*x^2 + 80*x 
- 7) + 78*x - 51)*sqrt(x^4 + x^3 - x^2 - x + 1) + 11*(37*x^8 + 36*x^7 - 36 
*x^6 - 36*x^5 + 74*x^4 + 36*x^3 - 36*x^2 + 9*sqrt(3)*(3*x^8 + 4*x^7 - 4*x^ 
6 - 4*x^5 + 6*x^4 + 4*x^3 - 4*x^2 - 4*x + 3) - 36*x + 37)*sqrt(1/6*sqrt(3) 
 - 1/6))*sqrt(1/6*sqrt(3) + 1/6)/(409*x^8 + 1584*x^7 - 288*x^6 - 4176*x^5 
- 478*x^4 + 4176*x^3 - 288*x^2 - 1584*x + 409)) - 1/2*sqrt(1/6*sqrt(3) + 1 
/6)*arctan(-3*(6*(51*x^6 + 78*x^5 - 159*x^4 - 216*x^3 + 159*x^2 - sqrt(3)* 
(7*x^6 + 80*x^5 + 29*x^4 - 144*x^3 - 29*x^2 + 80*x - 7) + 78*x - 51)*sqrt( 
x^4 + x^3 - x^2 - x + 1) - 11*(37*x^8 + 36*x^7 - 36*x^6 - 36*x^5 + 74*x^4 
+ 36*x^3 - 36*x^2 + 9*sqrt(3)*(3*x^8 + 4*x^7 - 4*x^6 - 4*x^5 + 6*x^4 + 4*x 
^3 - 4*x^2 - 4*x + 3) - 36*x + 37)*sqrt(1/6*sqrt(3) - 1/6))*sqrt(1/6*sqrt( 
3) + 1/6)/(409*x^8 + 1584*x^7 - 288*x^6 - 4176*x^5 - 478*x^4 + 4176*x^3 - 
288*x^2 - 1584*x + 409)) - 1/4*sqrt(1/6*sqrt(3) - 1/6)*log((4*x^4 + 4*x^3 
- 4*x^2 + 4*sqrt(x^4 + x^3 - x^2 - x + 1)*(3*x^2 + sqrt(3)*(2*x^2 + x - 2) 
 + 3*x - 3)*sqrt(1/6*sqrt(3) - 1/6) + sqrt(3)*(3*x^4 + 4*x^3 - 4*x^2 - 4*x 
 + 3) - 4*x + 4)/(x^4 + 1)) + 1/4*sqrt(1/6*sqrt(3) - 1/6)*log((4*x^4 + 4*x 
^3 - 4*x^2 - 4*sqrt(x^4 + x^3 - x^2 - x + 1)*(3*x^2 + sqrt(3)*(2*x^2 + x - 
 2) + 3*x - 3)*sqrt(1/6*sqrt(3) - 1/6) + sqrt(3)*(3*x^4 + 4*x^3 - 4*x^2 - 
4*x + 3) - 4*x + 4)/(x^4 + 1))
 

Sympy [N/A]

Not integrable

Time = 2.38 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.28 \[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{\left (x^{4} + 1\right ) \sqrt {x^{4} + x^{3} - x^{2} - x + 1}}\, dx \] Input:

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

Output:

Integral((x - 1)*(x + 1)*(x**2 + 1)/((x**4 + 1)*sqrt(x**4 + x**3 - x**2 - 
x + 1)), x)
 

Maxima [N/A]

Not integrable

Time = 0.17 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.28 \[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx=\int { \frac {x^{4} - 1}{\sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (x^{4} + 1\right )}} \,d x } \] Input:

integrate((x^4-1)/(x^4+1)/(x^4+x^3-x^2-x+1)^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [N/A]

Not integrable

Time = 0.21 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.28 \[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx=\int { \frac {x^{4} - 1}{\sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (x^{4} + 1\right )}} \,d x } \] Input:

integrate((x^4-1)/(x^4+1)/(x^4+x^3-x^2-x+1)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [N/A]

Not integrable

Time = 8.45 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.28 \[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx=\int \frac {x^4-1}{\left (x^4+1\right )\,\sqrt {x^4+x^3-x^2-x+1}} \,d x \] Input:

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

Output:

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

Reduce [N/A]

Not integrable

Time = 200.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.28 \[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx=\int \frac {x^{4}-1}{\left (x^{4}+1\right ) \sqrt {x^{4}+x^{3}-x^{2}-x +1}}d x \] Input:

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

Output:

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