\(\int \frac {-1+3 x^4}{(1-x+x^4) \sqrt [3]{x^2+x^6}} \, dx\) [1104]

Optimal result

Integrand size = 29, antiderivative size = 82 \[ \int \frac {-1+3 x^4}{\left (1-x+x^4\right ) \sqrt [3]{x^2+x^6}} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x^2+x^6}}\right )+\log \left (-x+\sqrt [3]{x^2+x^6}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{x^2+x^6}+\left (x^2+x^6\right )^{2/3}\right ) \] Output:

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

Mathematica [F]

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

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

Output:

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

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.

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

\(\Big \downarrow \) 2467

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{x^4+1} \left (4 \int \frac {1}{\sqrt [3]{x^4+1} \left (x^4-x+1\right )}d\sqrt [3]{x}-3 \int \frac {x}{\sqrt [3]{x^4+1} \left (x^4-x+1\right )}d\sqrt [3]{x}-3 \sqrt [3]{x} \operatorname {Hypergeometric2F1}\left (\frac {1}{12},\frac {1}{3},\frac {13}{12},-x^4\right )\right )}{\sqrt [3]{x^6+x^2}}\)

Input:

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

Output:

$Aborted
 

Maple [A] (verified)

Time = 3.37 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.04

method	result	size
pseudoelliptic	\(-\frac {\ln \left (\frac {\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {2}{3}}+\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )+\ln \left (\frac {\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{3}}-x}{x}\right )\)	\(85\)
trager	\(\ln \left (-\frac {4747 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{5}-6570 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{5}-5848 x^{5}-9494 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+12039 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}+x^{2}\right )^{\frac {2}{3}}+9873 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}+x^{2}\right )^{\frac {1}{3}} x +4747 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x -20089 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+21912 \left (x^{6}+x^{2}\right )^{\frac {2}{3}}-12039 x \left (x^{6}+x^{2}\right )^{\frac {1}{3}}-6570 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -8772 x^{2}-5848 x}{x \left (x^{4}-x +1\right )}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {2924 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{5}+13519 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{5}+14241 x^{5}-5848 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+12039 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}+x^{2}\right )^{\frac {2}{3}}-21912 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}+x^{2}\right )^{\frac {1}{3}} x +2924 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x -6570 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-9873 \left (x^{6}+x^{2}\right )^{\frac {2}{3}}-12039 x \left (x^{6}+x^{2}\right )^{\frac {1}{3}}+13519 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +4747 x^{2}+14241 x}{x \left (x^{4}-x +1\right )}\right )\)	\(337\)

Input:

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

Output:

-1/2*ln(((x^2*(x^4+1))^(2/3)+(x^2*(x^4+1))^(1/3)*x+x^2)/x^2)+3^(1/2)*arcta 
n(1/3*(2*(x^2*(x^4+1))^(1/3)+x)*3^(1/2)/x)+ln(((x^2*(x^4+1))^(1/3)-x)/x)
 

Fricas [A] (verification not implemented)

Time = 0.93 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.37 \[ \int \frac {-1+3 x^4}{\left (1-x+x^4\right ) \sqrt [3]{x^2+x^6}} \, dx=-\sqrt {3} \arctan \left (-\frac {2 \, \sqrt {3} {\left (x^{6} + x^{2}\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (x^{5} - x^{2} + x\right )} - 2 \, \sqrt {3} {\left (x^{6} + x^{2}\right )}^{\frac {2}{3}}}{3 \, {\left (x^{5} + x^{2} + x\right )}}\right ) + \frac {1}{2} \, \log \left (\frac {x^{5} - x^{2} + 3 \, {\left (x^{6} + x^{2}\right )}^{\frac {1}{3}} x + x - 3 \, {\left (x^{6} + x^{2}\right )}^{\frac {2}{3}}}{x^{5} - x^{2} + x}\right ) \] Input:

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {-1+3 x^4}{\left (1-x+x^4\right ) \sqrt [3]{x^2+x^6}} \, dx=3 \left (\int \frac {x^{4}}{x^{\frac {14}{3}} \left (x^{4}+1\right )^{\frac {1}{3}}-x^{\frac {5}{3}} \left (x^{4}+1\right )^{\frac {1}{3}}+x^{\frac {2}{3}} \left (x^{4}+1\right )^{\frac {1}{3}}}d x \right )-\left (\int \frac {1}{x^{\frac {14}{3}} \left (x^{4}+1\right )^{\frac {1}{3}}-x^{\frac {5}{3}} \left (x^{4}+1\right )^{\frac {1}{3}}+x^{\frac {2}{3}} \left (x^{4}+1\right )^{\frac {1}{3}}}d x \right ) \] Input:

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

Output:

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