\(\int \frac {1+x^2}{(-1+x+x^2) \sqrt [3]{-1+x^6}} \, dx\) [2517]

Optimal result

Integrand size = 23, antiderivative size = 210 \[ \int \frac {1+x^2}{\left (-1+x+x^2\right ) \sqrt [3]{-1+x^6}} \, dx=-\frac {\arctan \left (\frac {\sqrt {3} \sqrt [3]{-1+x^6}}{-2^{2/3}-2^{2/3} x+2^{2/3} x^2+\sqrt [3]{-1+x^6}}\right )}{2^{2/3} \sqrt {3}}-\frac {\log \left (-2^{2/3}-2^{2/3} x+2^{2/3} x^2-2 \sqrt [3]{-1+x^6}\right )}{3\ 2^{2/3}}+\frac {\log \left (\sqrt [3]{2}+2 \sqrt [3]{2} x-\sqrt [3]{2} x^2-2 \sqrt [3]{2} x^3+\sqrt [3]{2} x^4+\left (-2^{2/3}-2^{2/3} x+2^{2/3} x^2\right ) \sqrt [3]{-1+x^6}+2 \left (-1+x^6\right )^{2/3}\right )}{6\ 2^{2/3}} \] Output:

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

Mathematica [A] (verified)

Time = 5.16 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.89 \[ \int \frac {1+x^2}{\left (-1+x+x^2\right ) \sqrt [3]{-1+x^6}} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-1+x^6}}{2^{2/3}+2^{2/3} x-2^{2/3} x^2-\sqrt [3]{-1+x^6}}\right )-2 \log \left (-2^{2/3}-2^{2/3} x+2^{2/3} x^2-2 \sqrt [3]{-1+x^6}\right )+\log \left (\sqrt [3]{2}+2 \sqrt [3]{2} x-\sqrt [3]{2} x^2-2 \sqrt [3]{2} x^3+\sqrt [3]{2} x^4+2^{2/3} \left (-1-x+x^2\right ) \sqrt [3]{-1+x^6}+2 \left (-1+x^6\right )^{2/3}\right )}{6\ 2^{2/3}} \] Input:

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

Output:

(2*Sqrt[3]*ArcTan[(Sqrt[3]*(-1 + x^6)^(1/3))/(2^(2/3) + 2^(2/3)*x - 2^(2/3 
)*x^2 - (-1 + x^6)^(1/3))] - 2*Log[-2^(2/3) - 2^(2/3)*x + 2^(2/3)*x^2 - 2* 
(-1 + x^6)^(1/3)] + Log[2^(1/3) + 2*2^(1/3)*x - 2^(1/3)*x^2 - 2*2^(1/3)*x^ 
3 + 2^(1/3)*x^4 + 2^(2/3)*(-1 - x + x^2)*(-1 + x^6)^(1/3) + 2*(-1 + x^6)^( 
2/3)])/(6*2^(2/3))
 

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.

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

\(\Big \downarrow \) 7279

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

\(\Big \downarrow \) 2009

\(\displaystyle -\left (1-\sqrt {5}\right ) \int \frac {1}{\left (2 x-\sqrt {5}+1\right ) \sqrt [3]{x^6-1}}dx-\left (1+\sqrt {5}\right ) \int \frac {1}{\left (2 x+\sqrt {5}+1\right ) \sqrt [3]{x^6-1}}dx+\frac {x \sqrt [3]{1-x^6} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},x^6\right )}{\sqrt [3]{x^6-1}}\)

Input:

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

Output:

$Aborted
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 42.20 (sec) , antiderivative size = 2693, normalized size of antiderivative = 12.82

method	result	size
trager	\(\text {Expression too large to display}\)	\(2693\)

Input:

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

Output:

1/6*RootOf(_Z^3+2)*ln((-18*x^2*(x^6-1)^(2/3)+48*RootOf(RootOf(_Z^3+2)^2+2* 
_Z*RootOf(_Z^3+2)+4*_Z^2)^2*RootOf(_Z^3+2)^2*x+30*RootOf(RootOf(_Z^3+2)^2+ 
2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^3*x-15*RootOf(_Z^3+2)^2*(x^6-1) 
^(1/3)*x^4+30*RootOf(_Z^3+2)^2*(x^6-1)^(1/3)*x^3-48*RootOf(RootOf(_Z^3+2)^ 
2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^2*(x^6-1)^(2/3)+15*RootOf(_Z^ 
3+2)^2*(x^6-1)^(1/3)*x^2-30*RootOf(_Z^3+2)^2*(x^6-1)^(1/3)*x+18*RootOf(Roo 
tOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)*(x^6-1)^(1/3)-56* 
RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x^6+24*RootOf(RootOf(_ 
Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x^5-25*RootOf(_Z^3+2)*x^3+15*RootOf(_ 
Z^3+2)*x+48*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^ 
3+2)^2*(x^6-1)^(2/3)*x^2-35*RootOf(_Z^3+2)*x^6+15*RootOf(_Z^3+2)*x^5-40*Ro 
otOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x^3+24*RootOf(RootOf(_Z^ 
3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x+18*x*(x^6-1)^(2/3)+18*(x^6-1)^(2/3)+4 
8*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)^2*RootOf(_Z^3+2)^2*x 
^5+30*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^3 
*x^5+35*RootOf(_Z^3+2)-15*RootOf(_Z^3+2)^2*(x^6-1)^(1/3)+56*RootOf(RootOf( 
_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)-50*RootOf(RootOf(_Z^3+2)^2+2*_Z*Root 
Of(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^3*x^3-80*RootOf(RootOf(_Z^3+2)^2+2*_Z*Ro 
otOf(_Z^3+2)+4*_Z^2)^2*RootOf(_Z^3+2)^2*x^3+18*RootOf(RootOf(_Z^3+2)^2+2*_ 
Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)*(x^6-1)^(1/3)*x^4-48*RootOf(Roo...
 

Fricas [A] (verification not implemented)

Time = 11.63 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.08 \[ \int \frac {1+x^2}{\left (-1+x+x^2\right ) \sqrt [3]{-1+x^6}} \, dx=\frac {1}{6} \cdot 4^{\frac {1}{6}} \sqrt {3} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (2 \cdot 4^{\frac {2}{3}} {\left (x^{6} - 1\right )}^{\frac {2}{3}} {\left (x^{2} - x - 1\right )} - 4^{\frac {1}{3}} {\left (x^{6} + 3 \, x^{5} - 5 \, x^{3} + 3 \, x - 1\right )} - 4 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} {\left (x^{4} - 2 \, x^{3} - x^{2} + 2 \, x + 1\right )}\right )}}{6 \, {\left (3 \, x^{6} - 3 \, x^{5} + 5 \, x^{3} - 3 \, x - 3\right )}}\right ) + \frac {1}{24} \cdot 4^{\frac {2}{3}} \log \left (\frac {4^{\frac {2}{3}} {\left (x^{6} - 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (x^{4} - 2 \, x^{3} - x^{2} + 2 \, x + 1\right )} + 2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} {\left (x^{2} - x - 1\right )}}{x^{4} + 2 \, x^{3} - x^{2} - 2 \, x + 1}\right ) - \frac {1}{12} \cdot 4^{\frac {2}{3}} \log \left (-\frac {4^{\frac {1}{3}} {\left (x^{2} - x - 1\right )} - 2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2} + x - 1}\right ) \] Input:

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

Output:

1/6*4^(1/6)*sqrt(3)*arctan(1/6*4^(1/6)*sqrt(3)*(2*4^(2/3)*(x^6 - 1)^(2/3)* 
(x^2 - x - 1) - 4^(1/3)*(x^6 + 3*x^5 - 5*x^3 + 3*x - 1) - 4*(x^6 - 1)^(1/3 
)*(x^4 - 2*x^3 - x^2 + 2*x + 1))/(3*x^6 - 3*x^5 + 5*x^3 - 3*x - 3)) + 1/24 
*4^(2/3)*log((4^(2/3)*(x^6 - 1)^(2/3) + 4^(1/3)*(x^4 - 2*x^3 - x^2 + 2*x + 
 1) + 2*(x^6 - 1)^(1/3)*(x^2 - x - 1))/(x^4 + 2*x^3 - x^2 - 2*x + 1)) - 1/ 
12*4^(2/3)*log(-(4^(1/3)*(x^2 - x - 1) - 2*(x^6 - 1)^(1/3))/(x^2 + x - 1))
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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