\(\int \frac {(-1+x^3)^{2/3} (4+x^6)}{x^6 (-4+x^6)} \, dx\) [2838]

Optimal result

Integrand size = 25, antiderivative size = 289 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^6\right )}{x^6 \left (-4+x^6\right )} \, dx=\frac {\left (1-x^3\right ) \left (-1+x^3\right )^{2/3}}{5 x^5}-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\sqrt [6]{3} \arctan \left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )}{2\ 2^{2/3}}+\frac {\log \left (-x+\sqrt [3]{2} \sqrt [3]{-1+x^3}\right )}{6\ 2^{2/3}}-\frac {\log \left (-3 x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{-1+x^3}\right )}{2\ 2^{2/3} \sqrt [3]{3}}-\frac {\log \left (x^2+\sqrt [3]{2} x \sqrt [3]{-1+x^3}+2^{2/3} \left (-1+x^3\right )^{2/3}\right )}{12\ 2^{2/3}}+\frac {\log \left (3 x^2+\sqrt [3]{2} 3^{2/3} x \sqrt [3]{-1+x^3}+2^{2/3} \sqrt [3]{3} \left (-1+x^3\right )^{2/3}\right )}{4\ 2^{2/3} \sqrt [3]{3}} \] Output:

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

Mathematica [A] (verified)

Time = 0.66 (sec) , antiderivative size = 272, normalized size of antiderivative = 0.94 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^6\right )}{x^6 \left (-4+x^6\right )} \, dx=\frac {1}{120} \left (-\frac {24 \left (-1+x^3\right )^{5/3}}{x^5}-10 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )+30 \sqrt [3]{2} \sqrt [6]{3} \arctan \left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )+10 \sqrt [3]{2} \log \left (-x+\sqrt [3]{2} \sqrt [3]{-1+x^3}\right )-10 \sqrt [3]{2} 3^{2/3} \log \left (-3 x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{-1+x^3}\right )-5 \sqrt [3]{2} \log \left (x^2+\sqrt [3]{2} x \sqrt [3]{-1+x^3}+2^{2/3} \left (-1+x^3\right )^{2/3}\right )+5 \sqrt [3]{2} 3^{2/3} \log \left (3 x^2+\sqrt [3]{2} 3^{2/3} x \sqrt [3]{-1+x^3}+2^{2/3} \sqrt [3]{3} \left (-1+x^3\right )^{2/3}\right )\right ) \] Input:

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

Output:

((-24*(-1 + x^3)^(5/3))/x^5 - 10*2^(1/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2 
*2^(1/3)*(-1 + x^3)^(1/3))] + 30*2^(1/3)*3^(1/6)*ArcTan[(3^(5/6)*x)/(3^(1/ 
3)*x + 2*2^(1/3)*(-1 + x^3)^(1/3))] + 10*2^(1/3)*Log[-x + 2^(1/3)*(-1 + x^ 
3)^(1/3)] - 10*2^(1/3)*3^(2/3)*Log[-3*x + 2^(1/3)*3^(2/3)*(-1 + x^3)^(1/3) 
] - 5*2^(1/3)*Log[x^2 + 2^(1/3)*x*(-1 + x^3)^(1/3) + 2^(2/3)*(-1 + x^3)^(2 
/3)] + 5*2^(1/3)*3^(2/3)*Log[3*x^2 + 2^(1/3)*3^(2/3)*x*(-1 + x^3)^(1/3) + 
2^(2/3)*3^(1/3)*(-1 + x^3)^(2/3)])/120
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.68, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {7276, 2009}

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.

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\arctan \left (\frac {\frac {2^{2/3} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\sqrt [6]{3} \arctan \left (\frac {\frac {2^{2/3} \sqrt [3]{3} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{2\ 2^{2/3}}-\frac {\log \left (x^3-2\right )}{12\ 2^{2/3}}+\frac {\log \left (x^3+2\right )}{4\ 2^{2/3} \sqrt [3]{3}}-\frac {1}{4} \left (\frac {3}{2}\right )^{2/3} \log \left (\sqrt [3]{\frac {3}{2}} x-\sqrt [3]{x^3-1}\right )+\frac {\log \left (\frac {x}{\sqrt [3]{2}}-\sqrt [3]{x^3-1}\right )}{4\ 2^{2/3}}-\frac {\left (x^3-1\right )^{5/3}}{5 x^5}\)

Input:

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

Output:

-1/5*(-1 + x^3)^(5/3)/x^5 - ArcTan[(1 + (2^(2/3)*x)/(-1 + x^3)^(1/3))/Sqrt 
[3]]/(2*2^(2/3)*Sqrt[3]) + (3^(1/6)*ArcTan[(1 + (2^(2/3)*3^(1/3)*x)/(-1 + 
x^3)^(1/3))/Sqrt[3]])/(2*2^(2/3)) - Log[-2 + x^3]/(12*2^(2/3)) + Log[2 + x 
^3]/(4*2^(2/3)*3^(1/3)) - ((3/2)^(2/3)*Log[(3/2)^(1/3)*x - (-1 + x^3)^(1/3 
)])/4 + Log[x/2^(1/3) - (-1 + x^3)^(1/3)]/(4*2^(2/3))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

Maple [A] (verified)

Time = 3.41 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.90

method	result	size
pseudoelliptic	\(\frac {\left (-24 x^{3}+24\right ) \left (x^{3}-1\right )^{\frac {2}{3}}+10 \left (\left (\frac {\ln \left (\frac {{\left (\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {1}{3}} 2^{\frac {2}{3}} 3^{\frac {1}{3}} x +2^{\frac {1}{3}} 3^{\frac {2}{3}} x^{2}+2 {\left (\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {2}{3}}}{x^{2}}\right )}{2}-\ln \left (\frac {-2^{\frac {2}{3}} 3^{\frac {1}{3}} x +2 {\left (\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {1}{3}}}{x}\right )+\frac {\ln \left (2\right )}{2}\right ) 3^{\frac {2}{3}}+\arctan \left (\frac {\sqrt {3}\, \left (x +2 \,2^{\frac {1}{3}} \left (x^{3}-1\right )^{\frac {1}{3}}\right )}{3 x}\right ) \sqrt {3}-3 \arctan \left (\frac {\sqrt {3}\, \left (2 \,2^{\frac {1}{3}} 3^{\frac {2}{3}} \left (x^{3}-1\right )^{\frac {1}{3}}+3 x \right )}{9 x}\right ) 3^{\frac {1}{6}}-\frac {\ln \left (\frac {{\left (\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {1}{3}} 2^{\frac {2}{3}} x +2^{\frac {1}{3}} x^{2}+2 {\left (\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {2}{3}}}{x^{2}}\right )}{2}+\ln \left (\frac {-2^{\frac {2}{3}} x +2 {\left (\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {1}{3}}}{x}\right )-\frac {\ln \left (2\right )}{2}\right ) x^{5} 2^{\frac {1}{3}}}{120 x^{5}}\)	\(259\)

Input:

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

Output:

1/120*((-24*x^3+24)*(x^3-1)^(2/3)+10*((1/2*ln((((-1+x)*(x^2+x+1))^(1/3)*2^ 
(2/3)*3^(1/3)*x+2^(1/3)*3^(2/3)*x^2+2*((-1+x)*(x^2+x+1))^(2/3))/x^2)-ln((- 
2^(2/3)*3^(1/3)*x+2*((-1+x)*(x^2+x+1))^(1/3))/x)+1/2*ln(2))*3^(2/3)+arctan 
(1/3*3^(1/2)*(x+2*2^(1/3)*(x^3-1)^(1/3))/x)*3^(1/2)-3*arctan(1/9*3^(1/2)*( 
2*2^(1/3)*3^(2/3)*(x^3-1)^(1/3)+3*x)/x)*3^(1/6)-1/2*ln((((-1+x)*(x^2+x+1)) 
^(1/3)*2^(2/3)*x+2^(1/3)*x^2+2*((-1+x)*(x^2+x+1))^(2/3))/x^2)+ln((-2^(2/3) 
*x+2*((-1+x)*(x^2+x+1))^(1/3))/x)-1/2*ln(2))*x^5*2^(1/3))/x^5
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 500 vs. \(2 (202) = 404\).

Time = 3.25 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.73 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^6\right )}{x^6 \left (-4+x^6\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

1/720*(60*4^(1/6)*sqrt(1/3)*x^5*arctan(1/2*4^(1/6)*sqrt(1/3)*(12*4^(2/3)*( 
2*x^7 - 5*x^4 + 2*x)*(x^3 - 1)^(2/3) + 4^(1/3)*(91*x^9 - 168*x^6 + 84*x^3 
- 8) + 12*(19*x^8 - 22*x^5 + 4*x^2)*(x^3 - 1)^(1/3))/(53*x^9 - 48*x^6 - 12 
*x^3 + 8)) - 10*12^(2/3)*x^5*log(-(18*12^(1/3)*(x^3 - 1)^(1/3)*x^2 - 12^(2 
/3)*(x^3 + 2) - 36*(x^3 - 1)^(2/3)*x)/(x^3 + 2)) + 10*4^(2/3)*x^5*log((6*4 
^(1/3)*(x^3 - 1)^(1/3)*x^2 + 4^(2/3)*(x^3 - 2) - 12*(x^3 - 1)^(2/3)*x)/(x^ 
3 - 2)) + 5*12^(2/3)*x^5*log((6*12^(2/3)*(4*x^4 - x)*(x^3 - 1)^(2/3) + 12^ 
(1/3)*(55*x^6 - 50*x^3 + 4) + 18*(7*x^5 - 4*x^2)*(x^3 - 1)^(1/3))/(x^6 + 4 
*x^3 + 4)) - 5*4^(2/3)*x^5*log((6*4^(2/3)*(2*x^4 - x)*(x^3 - 1)^(2/3) + 4^ 
(1/3)*(19*x^6 - 22*x^3 + 4) + 6*(5*x^5 - 4*x^2)*(x^3 - 1)^(1/3))/(x^6 - 4* 
x^3 + 4)) + 60*12^(1/6)*x^5*arctan(1/6*12^(1/6)*(12*12^(2/3)*(4*x^7 + 7*x^ 
4 - 2*x)*(x^3 - 1)^(2/3) - 12^(1/3)*(377*x^9 - 600*x^6 + 204*x^3 - 8) - 36 
*(55*x^8 - 50*x^5 + 4*x^2)*(x^3 - 1)^(1/3))/(487*x^9 - 480*x^6 + 12*x^3 + 
8)) - 144*(x^3 - 1)^(5/3))/x^5
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^6\right )}{x^6 \left (-4+x^6\right )} \, dx=\frac {\left (x^{3}-1\right )^{\frac {2}{3}} x^{3}-\left (x^{3}-1\right )^{\frac {2}{3}}+40 \left (\int \frac {\left (x^{3}-1\right )^{\frac {2}{3}}}{x^{12}-4 x^{6}}d x \right ) x^{5}}{5 x^{5}} \] Input:

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

Output:

((x**3 - 1)**(2/3)*x**3 - (x**3 - 1)**(2/3) + 40*int((x**3 - 1)**(2/3)/(x* 
*12 - 4*x**6),x)*x**5)/(5*x**5)