\(\int \frac {-1+x^6}{\sqrt [3]{-x^2+x^4} (1+x^6)} \, dx\) [2805]

Optimal result

Integrand size = 26, antiderivative size = 274 \[ \int \frac {-1+x^6}{\sqrt [3]{-x^2+x^4} \left (1+x^6\right )} \, dx=-\frac {2}{3} \arctan \left (\frac {x}{\sqrt [3]{-x^2+x^4}}\right )-\frac {\arctan \left (\frac {\sqrt [3]{2} x}{\sqrt [3]{-x^2+x^4}}\right )}{3 \sqrt [3]{2}}-\frac {1}{3} \arctan \left (\frac {x \sqrt [3]{-x^2+x^4}}{-x^2+\left (-x^2+x^4\right )^{2/3}}\right )-\frac {\arctan \left (\frac {2^{2/3} x \sqrt [3]{-x^2+x^4}}{-2 x^2+\sqrt [3]{2} \left (-x^2+x^4\right )^{2/3}}\right )}{6 \sqrt [3]{2}}-\frac {\text {arctanh}\left (\frac {\frac {x^2}{\sqrt {3}}+\frac {\left (-x^2+x^4\right )^{2/3}}{\sqrt {3}}}{x \sqrt [3]{-x^2+x^4}}\right )}{\sqrt {3}}-\frac {\text {arctanh}\left (\frac {\frac {\sqrt [3]{2} x^2}{\sqrt {3}}+\frac {\left (-x^2+x^4\right )^{2/3}}{\sqrt [3]{2} \sqrt {3}}}{x \sqrt [3]{-x^2+x^4}}\right )}{2 \sqrt [3]{2} \sqrt {3}} \] Output:

-2/3*arctan(x/(x^4-x^2)^(1/3))-1/6*arctan(2^(1/3)*x/(x^4-x^2)^(1/3))*2^(2/ 
3)-1/3*arctan(x*(x^4-x^2)^(1/3)/(-x^2+(x^4-x^2)^(2/3)))-1/12*arctan(2^(2/3 
)*x*(x^4-x^2)^(1/3)/(-2*x^2+2^(1/3)*(x^4-x^2)^(2/3)))*2^(2/3)-1/3*arctanh( 
(1/3*3^(1/2)*x^2+1/3*(x^4-x^2)^(2/3)*3^(1/2))/x/(x^4-x^2)^(1/3))*3^(1/2)-1 
/12*3^(1/2)*arctanh((1/3*2^(1/3)*x^2*3^(1/2)+1/6*(x^4-x^2)^(2/3)*2^(2/3)*3 
^(1/2))/x/(x^4-x^2)^(1/3))*2^(2/3)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.95 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.22 \[ \int \frac {-1+x^6}{\sqrt [3]{-x^2+x^4} \left (1+x^6\right )} \, dx=-\frac {x^{2/3} \sqrt [3]{-1+x^2} \left (8 \arctan \left (\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x^2}}\right )+2\ 2^{2/3} \arctan \left (\frac {\sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{-1+x^2}}\right )+4 \arctan \left (\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{x}}{2 \sqrt [3]{-1+x^2}}\right )+4 i \sqrt {3} \arctan \left (\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{x}}{2 \sqrt [3]{-1+x^2}}\right )+4 \arctan \left (\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{x}}{2 \sqrt [3]{-1+x^2}}\right )-4 i \sqrt {3} \arctan \left (\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{x}}{2 \sqrt [3]{-1+x^2}}\right )+2^{2/3} \arctan \left (\frac {2^{2/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}{-2 x^{2/3}+\sqrt [3]{2} \left (-1+x^2\right )^{2/3}}\right )+2^{2/3} \sqrt {3} \text {arctanh}\left (\frac {2^{2/3} \sqrt {3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}{2 x^{2/3}+\sqrt [3]{2} \left (-1+x^2\right )^{2/3}}\right )\right )}{12 \sqrt [3]{x^2 \left (-1+x^2\right )}} \] Input:

Integrate[(-1 + x^6)/((-x^2 + x^4)^(1/3)*(1 + x^6)),x]
 

Output:

-1/12*(x^(2/3)*(-1 + x^2)^(1/3)*(8*ArcTan[x^(1/3)/(-1 + x^2)^(1/3)] + 2*2^ 
(2/3)*ArcTan[(2^(1/3)*x^(1/3))/(-1 + x^2)^(1/3)] + 4*ArcTan[((1 - I*Sqrt[3 
])*x^(1/3))/(2*(-1 + x^2)^(1/3))] + (4*I)*Sqrt[3]*ArcTan[((1 - I*Sqrt[3])* 
x^(1/3))/(2*(-1 + x^2)^(1/3))] + 4*ArcTan[((1 + I*Sqrt[3])*x^(1/3))/(2*(-1 
 + x^2)^(1/3))] - (4*I)*Sqrt[3]*ArcTan[((1 + I*Sqrt[3])*x^(1/3))/(2*(-1 + 
x^2)^(1/3))] + 2^(2/3)*ArcTan[(2^(2/3)*x^(1/3)*(-1 + x^2)^(1/3))/(-2*x^(2/ 
3) + 2^(1/3)*(-1 + x^2)^(2/3))] + 2^(2/3)*Sqrt[3]*ArcTanh[(2^(2/3)*Sqrt[3] 
*x^(1/3)*(-1 + x^2)^(1/3))/(2*x^(2/3) + 2^(1/3)*(-1 + x^2)^(2/3))]))/(x^2* 
(-1 + x^2))^(1/3)
 

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.

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

\(\Big \downarrow \) 2467

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2019

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

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{x^2-1} \int \left (\frac {\left (\sqrt [18]{-1}+(-1)^{7/18}+(-1)^{13/18}\right ) \left (x^2-1\right )^{2/3}}{18 \left (\sqrt [18]{-1}-\sqrt [3]{x}\right )}+\frac {\left (\sqrt [18]{-1}+(-1)^{7/18}+(-1)^{13/18}\right ) \left (x^2-1\right )^{2/3}}{18 \left (\sqrt [3]{x}+\sqrt [18]{-1}\right )}+\frac {\left (\sqrt [18]{-1}-(-1)^{7/18}-(-1)^{13/18}\right ) \left (x^2-1\right )^{2/3}}{18 \left (\sqrt [18]{-1}-\sqrt [9]{-1} \sqrt [3]{x}\right )}+\frac {\left (\sqrt [18]{-1}-(-1)^{7/18}-(-1)^{13/18}\right ) \left (x^2-1\right )^{2/3}}{18 \left (\sqrt [9]{-1} \sqrt [3]{x}+\sqrt [18]{-1}\right )}+\frac {\sqrt [18]{-1} \left (x^2-1\right )^{2/3}}{18 \left (\sqrt [18]{-1}-(-1)^{2/9} \sqrt [3]{x}\right )}+\frac {\sqrt [18]{-1} \left (x^2-1\right )^{2/3}}{18 \left ((-1)^{2/9} \sqrt [3]{x}+\sqrt [18]{-1}\right )}+\frac {\left (\sqrt [18]{-1}+(-1)^{7/18}+(-1)^{13/18}\right ) \left (x^2-1\right )^{2/3}}{18 \left (\sqrt [18]{-1}-\sqrt [3]{-1} \sqrt [3]{x}\right )}+\frac {\left (\sqrt [18]{-1}+(-1)^{7/18}+(-1)^{13/18}\right ) \left (x^2-1\right )^{2/3}}{18 \left (\sqrt [3]{-1} \sqrt [3]{x}+\sqrt [18]{-1}\right )}+\frac {\left (\sqrt [18]{-1}-(-1)^{7/18}-(-1)^{13/18}\right ) \left (x^2-1\right )^{2/3}}{18 \left (\sqrt [18]{-1}-(-1)^{4/9} \sqrt [3]{x}\right )}+\frac {\left (\sqrt [18]{-1}-(-1)^{7/18}-(-1)^{13/18}\right ) \left (x^2-1\right )^{2/3}}{18 \left ((-1)^{4/9} \sqrt [3]{x}+\sqrt [18]{-1}\right )}+\frac {\sqrt [18]{-1} \left (x^2-1\right )^{2/3}}{18 \left (\sqrt [18]{-1}-(-1)^{5/9} \sqrt [3]{x}\right )}+\frac {\sqrt [18]{-1} \left (x^2-1\right )^{2/3}}{18 \left ((-1)^{5/9} \sqrt [3]{x}+\sqrt [18]{-1}\right )}+\frac {\left (\sqrt [18]{-1}+(-1)^{7/18}+(-1)^{13/18}\right ) \left (x^2-1\right )^{2/3}}{18 \left (\sqrt [18]{-1}-(-1)^{2/3} \sqrt [3]{x}\right )}+\frac {\left (\sqrt [18]{-1}+(-1)^{7/18}+(-1)^{13/18}\right ) \left (x^2-1\right )^{2/3}}{18 \left ((-1)^{2/3} \sqrt [3]{x}+\sqrt [18]{-1}\right )}+\frac {\left (\sqrt [18]{-1}-(-1)^{7/18}-(-1)^{13/18}\right ) \left (x^2-1\right )^{2/3}}{18 \left (\sqrt [18]{-1}-(-1)^{7/9} \sqrt [3]{x}\right )}+\frac {\left (\sqrt [18]{-1}-(-1)^{7/18}-(-1)^{13/18}\right ) \left (x^2-1\right )^{2/3}}{18 \left ((-1)^{7/9} \sqrt [3]{x}+\sqrt [18]{-1}\right )}+\frac {\sqrt [18]{-1} \left (x^2-1\right )^{2/3}}{18 \left (\sqrt [18]{-1}-(-1)^{8/9} \sqrt [3]{x}\right )}+\frac {\sqrt [18]{-1} \left (x^2-1\right )^{2/3}}{18 \left ((-1)^{8/9} \sqrt [3]{x}+\sqrt [18]{-1}\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x^4-x^2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{x^2-1} \left (-\frac {1}{18} \sqrt [18]{-1} \left (1+i \sqrt {3}\right ) \int \frac {\left (x^2-1\right )^{2/3}}{\sqrt [18]{-1}-\sqrt [3]{x}}d\sqrt [3]{x}-\frac {1}{18} \sqrt [18]{-1} \left (1+i \sqrt {3}\right ) \int \frac {\left (x^2-1\right )^{2/3}}{\sqrt [3]{x}+\sqrt [18]{-1}}d\sqrt [3]{x}-\frac {1}{18} \sqrt [18]{-1} \left (1-i \sqrt {3}\right ) \int \frac {\left (x^2-1\right )^{2/3}}{\sqrt [18]{-1}-\sqrt [9]{-1} \sqrt [3]{x}}d\sqrt [3]{x}-\frac {1}{18} \sqrt [18]{-1} \left (1-i \sqrt {3}\right ) \int \frac {\left (x^2-1\right )^{2/3}}{\sqrt [9]{-1} \sqrt [3]{x}+\sqrt [18]{-1}}d\sqrt [3]{x}-\frac {1}{18} \sqrt [18]{-1} \int \frac {\left (x^2-1\right )^{2/3}}{\sqrt [18]{-1}-(-1)^{2/9} \sqrt [3]{x}}d\sqrt [3]{x}-\frac {1}{18} \sqrt [18]{-1} \int \frac {\left (x^2-1\right )^{2/3}}{(-1)^{2/9} \sqrt [3]{x}+\sqrt [18]{-1}}d\sqrt [3]{x}-\frac {1}{18} \sqrt [18]{-1} \left (1+i \sqrt {3}\right ) \int \frac {\left (x^2-1\right )^{2/3}}{\sqrt [18]{-1}-\sqrt [3]{-1} \sqrt [3]{x}}d\sqrt [3]{x}-\frac {1}{18} \sqrt [18]{-1} \left (1+i \sqrt {3}\right ) \int \frac {\left (x^2-1\right )^{2/3}}{\sqrt [3]{-1} \sqrt [3]{x}+\sqrt [18]{-1}}d\sqrt [3]{x}-\frac {1}{18} \sqrt [18]{-1} \left (1-i \sqrt {3}\right ) \int \frac {\left (x^2-1\right )^{2/3}}{\sqrt [18]{-1}-(-1)^{4/9} \sqrt [3]{x}}d\sqrt [3]{x}-\frac {1}{18} \sqrt [18]{-1} \left (1-i \sqrt {3}\right ) \int \frac {\left (x^2-1\right )^{2/3}}{(-1)^{4/9} \sqrt [3]{x}+\sqrt [18]{-1}}d\sqrt [3]{x}-\frac {1}{18} \sqrt [18]{-1} \int \frac {\left (x^2-1\right )^{2/3}}{\sqrt [18]{-1}-(-1)^{5/9} \sqrt [3]{x}}d\sqrt [3]{x}-\frac {1}{18} \sqrt [18]{-1} \int \frac {\left (x^2-1\right )^{2/3}}{(-1)^{5/9} \sqrt [3]{x}+\sqrt [18]{-1}}d\sqrt [3]{x}-\frac {1}{18} \sqrt [18]{-1} \left (1+i \sqrt {3}\right ) \int \frac {\left (x^2-1\right )^{2/3}}{\sqrt [18]{-1}-(-1)^{2/3} \sqrt [3]{x}}d\sqrt [3]{x}-\frac {1}{18} \sqrt [18]{-1} \left (1+i \sqrt {3}\right ) \int \frac {\left (x^2-1\right )^{2/3}}{(-1)^{2/3} \sqrt [3]{x}+\sqrt [18]{-1}}d\sqrt [3]{x}-\frac {1}{18} \sqrt [18]{-1} \left (1-i \sqrt {3}\right ) \int \frac {\left (x^2-1\right )^{2/3}}{\sqrt [18]{-1}-(-1)^{7/9} \sqrt [3]{x}}d\sqrt [3]{x}-\frac {1}{18} \sqrt [18]{-1} \left (1-i \sqrt {3}\right ) \int \frac {\left (x^2-1\right )^{2/3}}{(-1)^{7/9} \sqrt [3]{x}+\sqrt [18]{-1}}d\sqrt [3]{x}-\frac {1}{18} \sqrt [18]{-1} \int \frac {\left (x^2-1\right )^{2/3}}{\sqrt [18]{-1}-(-1)^{8/9} \sqrt [3]{x}}d\sqrt [3]{x}-\frac {1}{18} \sqrt [18]{-1} \int \frac {\left (x^2-1\right )^{2/3}}{(-1)^{8/9} \sqrt [3]{x}+\sqrt [18]{-1}}d\sqrt [3]{x}\right )}{\sqrt [3]{x^4-x^2}}\)

Input:

Int[(-1 + x^6)/((-x^2 + x^4)^(1/3)*(1 + x^6)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 73.53 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.22

method	result	size
pseudoelliptic	\(\frac {\left (\left (-\ln \left (\frac {\sqrt {3}\, 2^{\frac {1}{3}} \left (x^{4}-x^{2}\right )^{\frac {1}{3}} x +2^{\frac {2}{3}} x^{2}+\left (x^{4}-x^{2}\right )^{\frac {2}{3}}}{x^{2}}\right )+\ln \left (\frac {-\sqrt {3}\, 2^{\frac {1}{3}} \left (x^{4}-x^{2}\right )^{\frac {1}{3}} x +2^{\frac {2}{3}} x^{2}+\left (x^{4}-x^{2}\right )^{\frac {2}{3}}}{x^{2}}\right )\right ) 2^{\frac {2}{3}}-4 \ln \left (\frac {\sqrt {3}\, \left (x^{4}-x^{2}\right )^{\frac {1}{3}} x +\left (x^{4}-x^{2}\right )^{\frac {2}{3}}+x^{2}}{x^{2}}\right )+4 \ln \left (\frac {-\sqrt {3}\, \left (x^{4}-x^{2}\right )^{\frac {1}{3}} x +\left (x^{4}-x^{2}\right )^{\frac {2}{3}}+x^{2}}{x^{2}}\right )\right ) \sqrt {3}}{24}+\frac {\left (2 \arctan \left (\frac {\left (x^{4}-x^{2}\right )^{\frac {1}{3}} 2^{\frac {2}{3}}+x \sqrt {3}}{x}\right )+2 \arctan \left (\frac {\left (x^{4}-x^{2}\right )^{\frac {1}{3}} 2^{\frac {2}{3}}-x \sqrt {3}}{x}\right )+4 \arctan \left (\frac {\left (x^{4}-x^{2}\right )^{\frac {1}{3}} 2^{\frac {2}{3}}}{2 x}\right )\right ) 2^{\frac {2}{3}}}{24}+\frac {2 \arctan \left (\frac {\left (x^{4}-x^{2}\right )^{\frac {1}{3}}}{x}\right )}{3}+\frac {\arctan \left (\frac {x \sqrt {3}+2 \left (x^{4}-x^{2}\right )^{\frac {1}{3}}}{x}\right )}{3}+\frac {\arctan \left (\frac {-x \sqrt {3}+2 \left (x^{4}-x^{2}\right )^{\frac {1}{3}}}{x}\right )}{3}\)	\(335\)

Input:

int((x^6-1)/(x^4-x^2)^(1/3)/(x^6+1),x,method=_RETURNVERBOSE)
 

Output:

1/24*((-ln((3^(1/2)*2^(1/3)*(x^4-x^2)^(1/3)*x+2^(2/3)*x^2+(x^4-x^2)^(2/3)) 
/x^2)+ln((-3^(1/2)*2^(1/3)*(x^4-x^2)^(1/3)*x+2^(2/3)*x^2+(x^4-x^2)^(2/3))/ 
x^2))*2^(2/3)-4*ln((3^(1/2)*(x^4-x^2)^(1/3)*x+(x^4-x^2)^(2/3)+x^2)/x^2)+4* 
ln((-3^(1/2)*(x^4-x^2)^(1/3)*x+(x^4-x^2)^(2/3)+x^2)/x^2))*3^(1/2)+1/24*(2* 
arctan(((x^4-x^2)^(1/3)*2^(2/3)+x*3^(1/2))/x)+2*arctan(((x^4-x^2)^(1/3)*2^ 
(2/3)-x*3^(1/2))/x)+4*arctan(1/2*(x^4-x^2)^(1/3)/x*2^(2/3)))*2^(2/3)+2/3*a 
rctan((x^4-x^2)^(1/3)/x)+1/3*arctan((x*3^(1/2)+2*(x^4-x^2)^(1/3))/x)+1/3*a 
rctan((-x*3^(1/2)+2*(x^4-x^2)^(1/3))/x)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 8.68 (sec) , antiderivative size = 2222, normalized size of antiderivative = 8.11 \[ \int \frac {-1+x^6}{\sqrt [3]{-x^2+x^4} \left (1+x^6\right )} \, dx=\text {Too large to display} \] Input:

integrate((x^6-1)/(x^4-x^2)^(1/3)/(x^6+1),x, algorithm="fricas")
 

Output:

-1/24*(-1/4)^(1/6)*(sqrt(-3) + 1)*log(-(8*(-1/4)^(2/3)*(128*x^5 - 59*x^4 - 
 768*x^3 + 59*x^2 + sqrt(-3)*(128*x^5 - 59*x^4 - 768*x^3 + 59*x^2 + 128*x) 
 + 128*x) + 4*(x^4 - x^2)^(2/3)*((512*I - 59)*x^2 - (118*I + 1024)*x - 512 
*I + 59) + 8*(x^4 - x^2)^(1/3)*((-1/4)^(5/6)*(59*x^3 + 1024*x^2 - sqrt(-3) 
*(59*x^3 + 1024*x^2 - 59*x) - 59*x) + (-1/4)^(1/3)*(256*x^3 - 59*x^2 - sqr 
t(-3)*(256*x^3 - 59*x^2 - 256*x) - 256*x)) - (-1/4)^(1/6)*(59*x^5 + 2048*x 
^4 - 354*x^3 - 2048*x^2 + sqrt(-3)*(59*x^5 + 2048*x^4 - 354*x^3 - 2048*x^2 
 + 59*x) + 59*x))/(x^5 + 2*x^3 + x)) + 1/24*(-1/4)^(1/6)*(sqrt(-3) + 1)*lo 
g(-(8*(-1/4)^(2/3)*(128*x^5 - 59*x^4 - 768*x^3 + 59*x^2 + sqrt(-3)*(128*x^ 
5 - 59*x^4 - 768*x^3 + 59*x^2 + 128*x) + 128*x) + 4*(x^4 - x^2)^(2/3)*(-(5 
12*I + 59)*x^2 + (118*I - 1024)*x + 512*I + 59) - 8*(x^4 - x^2)^(1/3)*((-1 
/4)^(5/6)*(59*x^3 + 1024*x^2 - sqrt(-3)*(59*x^3 + 1024*x^2 - 59*x) - 59*x) 
 - (-1/4)^(1/3)*(256*x^3 - 59*x^2 - sqrt(-3)*(256*x^3 - 59*x^2 - 256*x) - 
256*x)) + (-1/4)^(1/6)*(59*x^5 + 2048*x^4 - 354*x^3 - 2048*x^2 + sqrt(-3)* 
(59*x^5 + 2048*x^4 - 354*x^3 - 2048*x^2 + 59*x) + 59*x))/(x^5 + 2*x^3 + x) 
) + 1/24*(-1/4)^(1/6)*(sqrt(-3) - 1)*log(-(8*(-1/4)^(2/3)*(128*x^5 - 59*x^ 
4 - 768*x^3 + 59*x^2 - sqrt(-3)*(128*x^5 - 59*x^4 - 768*x^3 + 59*x^2 + 128 
*x) + 128*x) + 4*(x^4 - x^2)^(2/3)*((512*I - 59)*x^2 - (118*I + 1024)*x - 
512*I + 59) + 8*(x^4 - x^2)^(1/3)*((-1/4)^(5/6)*(59*x^3 + 1024*x^2 + sqrt( 
-3)*(59*x^3 + 1024*x^2 - 59*x) - 59*x) + (-1/4)^(1/3)*(256*x^3 - 59*x^2...
 

Sympy [F]

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

integrate((x**6-1)/(x**4-x**2)**(1/3)/(x**6+1),x)
 

Output:

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

Maxima [F]

\[ \int \frac {-1+x^6}{\sqrt [3]{-x^2+x^4} \left (1+x^6\right )} \, dx=\int { \frac {x^{6} - 1}{{\left (x^{6} + 1\right )} {\left (x^{4} - x^{2}\right )}^{\frac {1}{3}}} \,d x } \] Input:

integrate((x^6-1)/(x^4-x^2)^(1/3)/(x^6+1),x, algorithm="maxima")
 

Output:

integrate((x^6 - 1)/((x^6 + 1)*(x^4 - x^2)^(1/3)), x)
 

Giac [F]

\[ \int \frac {-1+x^6}{\sqrt [3]{-x^2+x^4} \left (1+x^6\right )} \, dx=\int { \frac {x^{6} - 1}{{\left (x^{6} + 1\right )} {\left (x^{4} - x^{2}\right )}^{\frac {1}{3}}} \,d x } \] Input:

integrate((x^6-1)/(x^4-x^2)^(1/3)/(x^6+1),x, algorithm="giac")
 

Output:

integrate((x^6 - 1)/((x^6 + 1)*(x^4 - x^2)^(1/3)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-1+x^6}{\sqrt [3]{-x^2+x^4} \left (1+x^6\right )} \, dx=\int \frac {x^6-1}{\left (x^6+1\right )\,{\left (x^4-x^2\right )}^{1/3}} \,d x \] Input:

int((x^6 - 1)/((x^6 + 1)*(x^4 - x^2)^(1/3)),x)
 

Output:

int((x^6 - 1)/((x^6 + 1)*(x^4 - x^2)^(1/3)), x)
 

Reduce [F]

\[ \int \frac {-1+x^6}{\sqrt [3]{-x^2+x^4} \left (1+x^6\right )} \, dx=\int \frac {x^{6}}{x^{\frac {20}{3}} \left (x^{2}-1\right )^{\frac {1}{3}}+x^{\frac {2}{3}} \left (x^{2}-1\right )^{\frac {1}{3}}}d x -\left (\int \frac {1}{x^{\frac {20}{3}} \left (x^{2}-1\right )^{\frac {1}{3}}+x^{\frac {2}{3}} \left (x^{2}-1\right )^{\frac {1}{3}}}d x \right ) \] Input:

int((x^6-1)/(x^4-x^2)^(1/3)/(x^6+1),x)
 

Output:

int(x**6/(x**(2/3)*(x**2 - 1)**(1/3)*x**6 + x**(2/3)*(x**2 - 1)**(1/3)),x) 
 - int(1/(x**(2/3)*(x**2 - 1)**(1/3)*x**6 + x**(2/3)*(x**2 - 1)**(1/3)),x)