\(\int \frac {-1+x^6}{(1+x^6) \sqrt [3]{1+a^3 x^3+x^6}} \, dx\) [1826]

Optimal result

Integrand size = 29, antiderivative size = 124 \[ \int \frac {-1+x^6}{\left (1+x^6\right ) \sqrt [3]{1+a^3 x^3+x^6}} \, dx=-\frac {\arctan \left (\frac {\sqrt {3} a x}{a x+2 \sqrt [3]{1+a^3 x^3+x^6}}\right )}{\sqrt {3} a}+\frac {\log \left (-a x+\sqrt [3]{1+a^3 x^3+x^6}\right )}{3 a}-\frac {\log \left (a^2 x^2+a x \sqrt [3]{1+a^3 x^3+x^6}+\left (1+a^3 x^3+x^6\right )^{2/3}\right )}{6 a} \] Output:

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

Mathematica [A] (verified)

Time = 0.80 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.96 \[ \int \frac {-1+x^6}{\left (1+x^6\right ) \sqrt [3]{1+a^3 x^3+x^6}} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} a x}{a x+2 \sqrt [3]{1+a^3 x^3+x^6}}\right )-2 \log \left (a \left (a x-\sqrt [3]{1+a^3 x^3+x^6}\right )\right )+\log \left (a^2 x^2+a x \sqrt [3]{1+a^3 x^3+x^6}+\left (1+a^3 x^3+x^6\right )^{2/3}\right )}{6 a} \] Input:

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

Output:

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

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.

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

\(\displaystyle -i \int \frac {1}{\left (i-x^3\right ) \sqrt [3]{x^6+a^3 x^3+1}}dx-i \int \frac {1}{\left (x^3+i\right ) \sqrt [3]{x^6+a^3 x^3+1}}dx+\frac {x \sqrt [3]{\frac {2 x^3}{a^3-\sqrt {a^6-4}}+1} \sqrt [3]{\frac {2 x^3}{\sqrt {a^6-4}+a^3}+1} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{3},\frac {1}{3},\frac {4}{3},-\frac {2 x^3}{a^3-\sqrt {a^6-4}},-\frac {2 x^3}{a^3+\sqrt {a^6-4}}\right )}{\sqrt [3]{a^3 x^3+x^6+1}}\)

Input:

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

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.82 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.94

method	result	size
pseudoelliptic	\(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (a x +2 \left (a^{3} x^{3}+x^{6}+1\right )^{\frac {1}{3}}\right )}{3 a x}\right )+2 \ln \left (\frac {-a x +\left (a^{3} x^{3}+x^{6}+1\right )^{\frac {1}{3}}}{x}\right )-\ln \left (\frac {a^{2} x^{2}+a x \left (a^{3} x^{3}+x^{6}+1\right )^{\frac {1}{3}}+\left (a^{3} x^{3}+x^{6}+1\right )^{\frac {2}{3}}}{x^{2}}\right )}{6 a}\)	\(116\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 2.73 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.19 \[ \int \frac {-1+x^6}{\left (1+x^6\right ) \sqrt [3]{1+a^3 x^3+x^6}} \, dx=-\frac {2 \, \sqrt {3} \arctan \left (-\frac {4 \, \sqrt {3} {\left (a^{3} x^{3} + x^{6} + 1\right )}^{\frac {1}{3}} a^{2} x^{2} - 2 \, \sqrt {3} {\left (a^{3} x^{3} + x^{6} + 1\right )}^{\frac {2}{3}} a x + \sqrt {3} {\left (a^{3} x^{3} + x^{6} + 1\right )}}{9 \, a^{3} x^{3} + x^{6} + 1}\right ) - \log \left (\frac {x^{6} + 3 \, {\left (a^{3} x^{3} + x^{6} + 1\right )}^{\frac {1}{3}} a^{2} x^{2} - 3 \, {\left (a^{3} x^{3} + x^{6} + 1\right )}^{\frac {2}{3}} a x + 1}{x^{6} + 1}\right )}{6 \, a} \] Input:

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

Output:

-1/6*(2*sqrt(3)*arctan(-(4*sqrt(3)*(a^3*x^3 + x^6 + 1)^(1/3)*a^2*x^2 - 2*s 
qrt(3)*(a^3*x^3 + x^6 + 1)^(2/3)*a*x + sqrt(3)*(a^3*x^3 + x^6 + 1))/(9*a^3 
*x^3 + x^6 + 1)) - log((x^6 + 3*(a^3*x^3 + x^6 + 1)^(1/3)*a^2*x^2 - 3*(a^3 
*x^3 + x^6 + 1)^(2/3)*a*x + 1)/(x^6 + 1)))/a
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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