\(\int \frac {(1+x^3)^{2/3} (1-2 x^3+2 x^6)}{x^6 (-1-x^3+2 x^6)} \, dx\) [2576]

Optimal result

Integrand size = 39, antiderivative size = 221 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (1-2 x^3+2 x^6\right )}{x^6 \left (-1-x^3+2 x^6\right )} \, dx=\frac {\left (2-13 x^3\right ) \left (1+x^3\right )^{2/3}}{10 x^5}-\frac {10 \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1+x^3}}\right )}{3 \sqrt {3}}-\frac {2^{2/3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{1+x^3}}\right )}{3 \sqrt {3}}-\frac {10}{9} \log \left (x+\sqrt [3]{1+x^3}\right )+\frac {1}{9} 2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{1+x^3}\right )+\frac {5}{9} \log \left (x^2-x \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right )-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{1+x^3}+\sqrt [3]{2} \left (1+x^3\right )^{2/3}\right )}{9 \sqrt [3]{2}} \] Output:

1/10*(-13*x^3+2)*(x^3+1)^(2/3)/x^5-10/9*3^(1/2)*arctan(3^(1/2)*x/(-x+2*(x^ 
3+1)^(1/3)))-1/9*2^(2/3)*arctan(3^(1/2)*x/(x+2^(2/3)*(x^3+1)^(1/3)))*3^(1/ 
2)-10/9*ln(x+(x^3+1)^(1/3))+1/9*2^(2/3)*ln(-2*x+2^(2/3)*(x^3+1)^(1/3))+5/9 
*ln(x^2-x*(x^3+1)^(1/3)+(x^3+1)^(2/3))-1/18*ln(2*x^2+2^(2/3)*x*(x^3+1)^(1/ 
3)+2^(1/3)*(x^3+1)^(2/3))*2^(2/3)
 

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.95 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (1-2 x^3+2 x^6\right )}{x^6 \left (-1-x^3+2 x^6\right )} \, dx=\frac {1}{90} \left (\frac {9 \left (2-13 x^3\right ) \left (1+x^3\right )^{2/3}}{x^5}+100 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x-2 \sqrt [3]{1+x^3}}\right )-10\ 2^{2/3} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{1+x^3}}\right )-100 \log \left (x+\sqrt [3]{1+x^3}\right )+10\ 2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{1+x^3}\right )+50 \log \left (x^2-x \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right )-5\ 2^{2/3} \log \left (2 x^2+2^{2/3} x \sqrt [3]{1+x^3}+\sqrt [3]{2} \left (1+x^3\right )^{2/3}\right )\right ) \] Input:

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

Output:

((9*(2 - 13*x^3)*(1 + x^3)^(2/3))/x^5 + 100*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x 
- 2*(1 + x^3)^(1/3))] - 10*2^(2/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3) 
*(1 + x^3)^(1/3))] - 100*Log[x + (1 + x^3)^(1/3)] + 10*2^(2/3)*Log[-2*x + 
2^(2/3)*(1 + x^3)^(1/3)] + 50*Log[x^2 - x*(1 + x^3)^(1/3) + (1 + x^3)^(2/3 
)] - 5*2^(2/3)*Log[2*x^2 + 2^(2/3)*x*(1 + x^3)^(1/3) + 2^(1/3)*(1 + x^3)^( 
2/3)])/90
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(529\) vs. \(2(221)=442\).

Time = 1.28 (sec) , antiderivative size = 529, normalized size of antiderivative = 2.39, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {7279, 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.

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

\(\Big \downarrow \) 7279

\(\displaystyle \int \left (\frac {\left (x^3+1\right )^{2/3}}{9 (x-1)}+\frac {3 \left (x^3+1\right )^{2/3}}{x^3}-\frac {20 \left (x^3+1\right )^{2/3}}{3 \left (2 x^3+1\right )}-\frac {\left (x^3+1\right )^{2/3}}{x^6}+\frac {\left (x^3+1\right )^{2/3} (-x-2)}{9 \left (x^2+x+1\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {10 \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{x^3+1}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {2\ 2^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {2^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (x+1)}{\sqrt [3]{x^3+1}}}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {2^{2/3} \arctan \left (\frac {\frac {\sqrt [3]{2} (x+1)}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {2^{2/3} \arctan \left (\frac {2^{2/3} \sqrt [3]{x^3+1}+1}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {1}{27} 2^{2/3} \log \left (1-x^3\right )+\frac {\log \left (1-x^3\right )}{27 \sqrt [3]{2}}+\frac {5}{9} \log \left (2 x^3+1\right )+\frac {\log \left (\frac {2^{2/3} (x+1)^2}{\left (x^3+1\right )^{2/3}}-\frac {\sqrt [3]{2} (x+1)}{\sqrt [3]{x^3+1}}+1\right )}{27 \sqrt [3]{2}}-\frac {1}{27} 2^{2/3} \log \left (\frac {\sqrt [3]{2} (x+1)}{\sqrt [3]{x^3+1}}+1\right )-\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{x^3+1}\right )}{9 \sqrt [3]{2}}-\frac {5}{3} \log \left (-\sqrt [3]{x^3+1}-x\right )+\frac {1}{9} 2^{2/3} \log \left (\sqrt [3]{2} x-\sqrt [3]{x^3+1}\right )-\frac {\log \left (-2^{2/3} \sqrt [3]{x^3+1}+x+1\right )}{18 \sqrt [3]{2}}+\frac {\log \left (2^{2/3} \sqrt [3]{x^3+1}-x-1\right )}{6 \sqrt [3]{2}}+\frac {\left (x^3+1\right )^{5/3}}{5 x^5}-\frac {3 \left (x^3+1\right )^{2/3}}{2 x^2}-\frac {\log \left (-(1-x)^2 (x+1)\right )}{18 \sqrt [3]{2}}+\frac {\log \left ((1-x)^2 (x+1)\right )}{54 \sqrt [3]{2}}\)

Input:

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

Output:

(-3*(1 + x^3)^(2/3))/(2*x^2) + (1 + x^3)^(5/3)/(5*x^5) + (10*ArcTan[(1 - ( 
2*x)/(1 + x^3)^(1/3))/Sqrt[3]])/(3*Sqrt[3]) - (2*2^(2/3)*ArcTan[(1 + (2*2^ 
(1/3)*x)/(1 + x^3)^(1/3))/Sqrt[3]])/(9*Sqrt[3]) + (2^(2/3)*ArcTan[(1 - (2* 
2^(1/3)*(1 + x))/(1 + x^3)^(1/3))/Sqrt[3]])/(9*Sqrt[3]) - (2^(2/3)*ArcTan[ 
(1 + (2^(1/3)*(1 + x))/(1 + x^3)^(1/3))/Sqrt[3]])/(9*Sqrt[3]) - (2^(2/3)*A 
rcTan[(1 + 2^(2/3)*(1 + x^3)^(1/3))/Sqrt[3]])/(9*Sqrt[3]) - Log[-((1 - x)^ 
2*(1 + x))]/(18*2^(1/3)) + Log[(1 - x)^2*(1 + x)]/(54*2^(1/3)) + Log[1 - x 
^3]/(27*2^(1/3)) - (2^(2/3)*Log[1 - x^3])/27 + (5*Log[1 + 2*x^3])/9 + Log[ 
1 + (2^(2/3)*(1 + x)^2)/(1 + x^3)^(2/3) - (2^(1/3)*(1 + x))/(1 + x^3)^(1/3 
)]/(27*2^(1/3)) - (2^(2/3)*Log[1 + (2^(1/3)*(1 + x))/(1 + x^3)^(1/3)])/27 
- Log[2^(1/3) - (1 + x^3)^(1/3)]/(9*2^(1/3)) - (5*Log[-x - (1 + x^3)^(1/3) 
])/3 + (2^(2/3)*Log[2^(1/3)*x - (1 + x^3)^(1/3)])/9 - Log[1 + x - 2^(2/3)* 
(1 + x^3)^(1/3)]/(18*2^(1/3)) + Log[-1 - x + 2^(2/3)*(1 + x^3)^(1/3)]/(6*2 
^(1/3))
 

Defintions of rubi rules used

rule 2009

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

rule 7279

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ 
{v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su 
mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
 

Maple [A] (verified)

Time = 4.50 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.87

method	result	size
pseudoelliptic	\(\frac {10 \,2^{\frac {2}{3}} \ln \left (\frac {-2^{\frac {1}{3}} x +\left (x^{3}+1\right )^{\frac {1}{3}}}{x}\right ) x^{5}-100 \ln \left (\frac {x +\left (x^{3}+1\right )^{\frac {1}{3}}}{x}\right ) x^{5}+\left (-117 x^{3}+18\right ) \left (x^{3}+1\right )^{\frac {2}{3}}+5 x^{5} \left (\left (2 \arctan \left (\frac {\sqrt {3}\, \left (x +2^{\frac {2}{3}} \left (x^{3}+1\right )^{\frac {1}{3}}\right )}{3 x}\right ) \sqrt {3}-\ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} x \left (x^{3}+1\right )^{\frac {1}{3}}+\left (x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right )\right ) 2^{\frac {2}{3}}-20 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x -2 \left (x^{3}+1\right )^{\frac {1}{3}}\right )}{3 x}\right )+10 \ln \left (\frac {x^{2}-x \left (x^{3}+1\right )^{\frac {1}{3}}+\left (x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right )\right )}{90 x^{5}}\)	\(193\)

Input:

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

Output:

1/90*(10*2^(2/3)*ln((-2^(1/3)*x+(x^3+1)^(1/3))/x)*x^5-100*ln((x+(x^3+1)^(1 
/3))/x)*x^5+(-117*x^3+18)*(x^3+1)^(2/3)+5*x^5*((2*arctan(1/3*3^(1/2)/x*(x+ 
2^(2/3)*(x^3+1)^(1/3)))*3^(1/2)-ln((2^(2/3)*x^2+2^(1/3)*x*(x^3+1)^(1/3)+(x 
^3+1)^(2/3))/x^2))*2^(2/3)-20*3^(1/2)*arctan(1/3*3^(1/2)*(x-2*(x^3+1)^(1/3 
))/x)+10*ln((x^2-x*(x^3+1)^(1/3)+(x^3+1)^(2/3))/x^2)))/x^5
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 367 vs. \(2 (166) = 332\).

Time = 1.78 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.66 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (1-2 x^3+2 x^6\right )}{x^6 \left (-1-x^3+2 x^6\right )} \, dx=-\frac {10 \cdot 4^{\frac {1}{3}} \sqrt {3} x^{5} \arctan \left (\frac {3 \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (5 \, x^{7} - 4 \, x^{4} - x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} - 6 \cdot 4^{\frac {1}{3}} \sqrt {3} {\left (19 \, x^{8} + 16 \, x^{5} + x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}} - \sqrt {3} {\left (71 \, x^{9} + 111 \, x^{6} + 33 \, x^{3} + 1\right )}}{3 \, {\left (109 \, x^{9} + 105 \, x^{6} + 3 \, x^{3} - 1\right )}}\right ) - 300 \, \sqrt {3} x^{5} \arctan \left (\frac {4 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 2 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (x^{3} + 1\right )}}{7 \, x^{3} - 1}\right ) - 10 \cdot 4^{\frac {1}{3}} x^{5} \log \left (\frac {3 \cdot 4^{\frac {2}{3}} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x - 4^{\frac {1}{3}} {\left (x^{3} - 1\right )}}{x^{3} - 1}\right ) + 5 \cdot 4^{\frac {1}{3}} x^{5} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} {\left (5 \, x^{4} + x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 4^{\frac {2}{3}} {\left (19 \, x^{6} + 16 \, x^{3} + 1\right )} + 24 \, {\left (2 \, x^{5} + x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x^{6} - 2 \, x^{3} + 1}\right ) + 150 \, x^{5} \log \left (\frac {2 \, x^{3} + 3 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + 1}{2 \, x^{3} + 1}\right ) + 27 \, {\left (13 \, x^{3} - 2\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{270 \, x^{5}} \] Input:

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

Output:

-1/270*(10*4^(1/3)*sqrt(3)*x^5*arctan(1/3*(3*4^(2/3)*sqrt(3)*(5*x^7 - 4*x^ 
4 - x)*(x^3 + 1)^(2/3) - 6*4^(1/3)*sqrt(3)*(19*x^8 + 16*x^5 + x^2)*(x^3 + 
1)^(1/3) - sqrt(3)*(71*x^9 + 111*x^6 + 33*x^3 + 1))/(109*x^9 + 105*x^6 + 3 
*x^3 - 1)) - 300*sqrt(3)*x^5*arctan((4*sqrt(3)*(x^3 + 1)^(1/3)*x^2 + 2*sqr 
t(3)*(x^3 + 1)^(2/3)*x + sqrt(3)*(x^3 + 1))/(7*x^3 - 1)) - 10*4^(1/3)*x^5* 
log((3*4^(2/3)*(x^3 + 1)^(1/3)*x^2 - 6*(x^3 + 1)^(2/3)*x - 4^(1/3)*(x^3 - 
1))/(x^3 - 1)) + 5*4^(1/3)*x^5*log((6*4^(1/3)*(5*x^4 + x)*(x^3 + 1)^(2/3) 
+ 4^(2/3)*(19*x^6 + 16*x^3 + 1) + 24*(2*x^5 + x^2)*(x^3 + 1)^(1/3))/(x^6 - 
 2*x^3 + 1)) + 150*x^5*log((2*x^3 + 3*(x^3 + 1)^(1/3)*x^2 + 3*(x^3 + 1)^(2 
/3)*x + 1)/(2*x^3 + 1)) + 27*(13*x^3 - 2)*(x^3 + 1)^(2/3))/x^5
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (1+x^3\right )^{2/3} \left (1-2 x^3+2 x^6\right )}{x^6 \left (-1-x^3+2 x^6\right )} \, dx=\frac {\left (x^{3}+1\right )^{\frac {2}{3}} x^{3}+\left (x^{3}+1\right )^{\frac {2}{3}}-15 \left (\int \frac {\left (x^{3}+1\right )^{\frac {2}{3}}}{2 x^{12}+x^{9}-2 x^{6}-x^{3}}d x \right ) x^{5}+5 \left (\int \frac {\left (x^{3}+1\right )^{\frac {2}{3}}}{2 x^{9}+x^{6}-2 x^{3}-1}d x \right ) x^{5}+20 \left (\int \frac {\left (x^{3}+1\right )^{\frac {2}{3}} x^{3}}{2 x^{9}+x^{6}-2 x^{3}-1}d x \right ) x^{5}}{5 x^{5}} \] Input:

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

Output:

((x**3 + 1)**(2/3)*x**3 + (x**3 + 1)**(2/3) - 15*int((x**3 + 1)**(2/3)/(2* 
x**12 + x**9 - 2*x**6 - x**3),x)*x**5 + 5*int((x**3 + 1)**(2/3)/(2*x**9 + 
x**6 - 2*x**3 - 1),x)*x**5 + 20*int(((x**3 + 1)**(2/3)*x**3)/(2*x**9 + x** 
6 - 2*x**3 - 1),x)*x**5)/(5*x**5)