\(\int \frac {2+x^2}{x (2-2 x+x^2) \sqrt [3]{1-x+x^2}} \, dx\) [1706]

Optimal result

Integrand size = 31, antiderivative size = 115 \[ \int \frac {2+x^2}{x \left (2-2 x+x^2\right ) \sqrt [3]{1-x+x^2}} \, dx=-\sqrt {3} \arctan \left (\frac {\frac {2}{\sqrt {3}}-\frac {2 x}{\sqrt {3}}+\frac {\sqrt [3]{1-x+x^2}}{\sqrt {3}}}{\sqrt [3]{1-x+x^2}}\right )+\log \left (-1+x+\sqrt [3]{1-x+x^2}\right )-\frac {1}{2} \log \left (1-2 x+x^2+(1-x) \sqrt [3]{1-x+x^2}+\left (1-x+x^2\right )^{2/3}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.89 \[ \int \frac {2+x^2}{x \left (2-2 x+x^2\right ) \sqrt [3]{1-x+x^2}} \, dx=-\sqrt {3} \arctan \left (\frac {2-2 x+\sqrt [3]{1-x+x^2}}{\sqrt {3} \sqrt [3]{1-x+x^2}}\right )+\log \left (-1+x+\sqrt [3]{1-x+x^2}\right )-\frac {1}{2} \log \left (1-2 x+x^2-(-1+x) \sqrt [3]{1-x+x^2}+\left (1-x+x^2\right )^{2/3}\right ) \] Input:

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

Output:

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

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

\(\Big \downarrow \) 7279

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

\(\Big \downarrow \) 2009

\(\displaystyle 2 \int \frac {1}{\left (x^2-2 x+2\right ) \sqrt [3]{x^2-x+1}}dx-\frac {3 \sqrt [3]{-\frac {-2 x-i \sqrt {3}+1}{x}} \sqrt [3]{-\frac {-2 x+i \sqrt {3}+1}{x}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},\frac {1}{3},\frac {5}{3},\frac {1-i \sqrt {3}}{2 x},\frac {1+i \sqrt {3}}{2 x}\right )}{2\ 2^{2/3} \sqrt [3]{x^2-x+1}}\)

Input:

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

Output:

$Aborted
 

Maple [C] (verified)

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

Time = 1.41 (sec) , antiderivative size = 484, normalized size of antiderivative = 4.21

method	result	size
trager	\(\ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-3 \left (x^{2}-x +1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 x \left (x^{2}-x +1\right )^{\frac {2}{3}}+6 \left (x^{2}-x +1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x +10 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-2 x^{3}-3 \left (x^{2}-x +1\right )^{\frac {2}{3}}-3 \left (x^{2}-x +1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-10 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +8 x^{2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-8 x +4}{x \left (x^{2}-2 x +2\right )}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}-x +1\right )^{\frac {2}{3}} x +2 \left (x^{2}-x +1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}-x +1\right )^{\frac {2}{3}}-x \left (x^{2}-x +1\right )^{\frac {2}{3}}-4 \left (x^{2}-x +1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +\left (x^{2}-x +1\right )^{\frac {1}{3}} x^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x +2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+\left (x^{2}-x +1\right )^{\frac {2}{3}}+2 \left (x^{2}-x +1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2 \left (x^{2}-x +1\right )^{\frac {1}{3}} x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -x^{2}+\left (x^{2}-x +1\right )^{\frac {1}{3}}+x -1}{x \left (x^{2}-2 x +2\right )}\right )\)	\(484\)

Input:

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

Output:

ln((-RootOf(_Z^2+_Z+1)^2*x^3-3*(x^2-x+1)^(1/3)*RootOf(_Z^2+_Z+1)*x^2+3*Roo 
tOf(_Z^2+_Z+1)^2*x^2-3*RootOf(_Z^2+_Z+1)*x^3+3*x*(x^2-x+1)^(2/3)+6*(x^2-x+ 
1)^(1/3)*RootOf(_Z^2+_Z+1)*x-3*RootOf(_Z^2+_Z+1)^2*x+10*RootOf(_Z^2+_Z+1)* 
x^2-2*x^3-3*(x^2-x+1)^(2/3)-3*(x^2-x+1)^(1/3)*RootOf(_Z^2+_Z+1)+RootOf(_Z^ 
2+_Z+1)^2-10*RootOf(_Z^2+_Z+1)*x+8*x^2+4*RootOf(_Z^2+_Z+1)-8*x+4)/x/(x^2-2 
*x+2))+RootOf(_Z^2+_Z+1)*ln(-(-RootOf(_Z^2+_Z+1)^2*x^3+RootOf(_Z^2+_Z+1)*( 
x^2-x+1)^(2/3)*x+2*(x^2-x+1)^(1/3)*RootOf(_Z^2+_Z+1)*x^2+3*RootOf(_Z^2+_Z+ 
1)^2*x^2-RootOf(_Z^2+_Z+1)*x^3-RootOf(_Z^2+_Z+1)*(x^2-x+1)^(2/3)-x*(x^2-x+ 
1)^(2/3)-4*(x^2-x+1)^(1/3)*RootOf(_Z^2+_Z+1)*x+(x^2-x+1)^(1/3)*x^2-3*RootO 
f(_Z^2+_Z+1)^2*x+2*RootOf(_Z^2+_Z+1)*x^2+(x^2-x+1)^(2/3)+2*(x^2-x+1)^(1/3) 
*RootOf(_Z^2+_Z+1)-2*(x^2-x+1)^(1/3)*x+RootOf(_Z^2+_Z+1)^2-2*RootOf(_Z^2+_ 
Z+1)*x-x^2+(x^2-x+1)^(1/3)+x-1)/x/(x^2-2*x+2))
 

Fricas [A] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.28 \[ \int \frac {2+x^2}{x \left (2-2 x+x^2\right ) \sqrt [3]{1-x+x^2}} \, dx=-\sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} {\left (x^{2} - x + 1\right )}^{\frac {2}{3}} {\left (x - 1\right )} + 2 \, \sqrt {3} {\left (x^{2} - x + 1\right )}^{\frac {1}{3}} {\left (x^{2} - 2 \, x + 1\right )} + \sqrt {3} {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}}{x^{3} - 11 \, x^{2} + 11 \, x - 9}\right ) + \frac {1}{2} \, \log \left (\frac {x^{3} - 2 \, x^{2} + 3 \, {\left (x^{2} - x + 1\right )}^{\frac {2}{3}} {\left (x - 1\right )} + 3 \, {\left (x^{2} - x + 1\right )}^{\frac {1}{3}} {\left (x^{2} - 2 \, x + 1\right )} + 2 \, x}{x^{3} - 2 \, x^{2} + 2 \, x}\right ) \] Input:

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

Output:

-sqrt(3)*arctan((4*sqrt(3)*(x^2 - x + 1)^(2/3)*(x - 1) + 2*sqrt(3)*(x^2 - 
x + 1)^(1/3)*(x^2 - 2*x + 1) + sqrt(3)*(x^3 - 3*x^2 + 3*x - 1))/(x^3 - 11* 
x^2 + 11*x - 9)) + 1/2*log((x^3 - 2*x^2 + 3*(x^2 - x + 1)^(2/3)*(x - 1) + 
3*(x^2 - x + 1)^(1/3)*(x^2 - 2*x + 1) + 2*x)/(x^3 - 2*x^2 + 2*x))
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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