\(\int \frac {(-1+x^3)^{2/3} (-1+x^6)}{x^6 (-2+x^3+2 x^6)} \, dx\) [1598]

Optimal result

Integrand size = 30, antiderivative size = 109 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-1+x^6\right )}{x^6 \left (-2+x^3+2 x^6\right )} \, dx=\frac {\left (-4-x^3\right ) \left (-1+x^3\right )^{2/3}}{40 x^5}+\frac {1}{12} \text {RootSum}\left [-1-3 \text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {\log (x)-\log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-3 \text {$\#$1}+4 \text {$\#$1}^4}\&\right ] \] Output:

Unintegrable
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-1+x^6\right )}{x^6 \left (-2+x^3+2 x^6\right )} \, dx=\frac {\left (-4-x^3\right ) \left (-1+x^3\right )^{2/3}}{40 x^5}+\frac {1}{12} \text {RootSum}\left [-1-3 \text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {\log (x)-\log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-3 \text {$\#$1}+4 \text {$\#$1}^4}\&\right ] \] Input:

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

Output:

((-4 - x^3)*(-1 + x^3)^(2/3))/(40*x^5) + RootSum[-1 - 3*#1^3 + 2*#1^6 & , 
(Log[x] - Log[(-1 + x^3)^(1/3) - x*#1] + Log[x]*#1^3 - Log[(-1 + x^3)^(1/3 
) - x*#1]*#1^3)/(-3*#1 + 4*#1^4) & ]/12
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(579\) vs. \(2(109)=218\).

Time = 1.24 (sec) , antiderivative size = 579, normalized size of antiderivative = 5.31, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1388, 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.

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

\(\Big \downarrow \) 1388

\(\displaystyle \int \frac {\left (x^3-1\right )^{5/3} \left (x^3+1\right )}{x^6 \left (2 x^6+x^3-2\right )}dx\)

\(\Big \downarrow \) 7279

\(\displaystyle \int \left (-\frac {3 \left (x^3-1\right )^{5/3}}{4 x^3}+\frac {\left (6 x^3+7\right ) \left (x^3-1\right )^{5/3}}{4 \left (2 x^6+x^3-2\right )}-\frac {\left (x^3-1\right )^{5/3}}{2 x^6}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\left (25+9 \sqrt {17}\right ) \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{24 \sqrt {51}}+\frac {\left (25-9 \sqrt {17}\right ) \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{24 \sqrt {51}}+\frac {1}{4} \sqrt {3} \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )+\frac {\sqrt [3]{13 \sqrt {17}-43} \arctan \left (\frac {1-\frac {\sqrt [3]{2 \left (\sqrt {17}-3\right )} x}{\sqrt [3]{x^3-1}}}{\sqrt {3}}\right )}{8 \sqrt {51}}+\frac {\sqrt [3]{43+13 \sqrt {17}} \arctan \left (\frac {\frac {\sqrt [3]{2 \left (3+\sqrt {17}\right )} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{8 \sqrt {51}}+\frac {1}{408} \left (51+11 \sqrt {17}\right ) x \left (x^3-1\right )^{2/3}+\frac {1}{408} \left (51-11 \sqrt {17}\right ) x \left (x^3-1\right )^{2/3}-\frac {5}{8} x \left (x^3-1\right )^{2/3}+\frac {\sqrt [3]{13 \sqrt {17}-43} \log \left (4 x^3-\sqrt {17}+1\right )}{48 \sqrt {17}}+\frac {\sqrt [3]{43+13 \sqrt {17}} \log \left (4 x^3+\sqrt {17}+1\right )}{48 \sqrt {17}}-\frac {\sqrt [3]{13 \sqrt {17}-43} \log \left (-\sqrt [3]{x^3-1}-\frac {\sqrt [3]{\sqrt {17}-3} x}{2^{2/3}}\right )}{16 \sqrt {17}}-\frac {\sqrt [3]{43+13 \sqrt {17}} \log \left (\frac {\sqrt [3]{3+\sqrt {17}} x}{2^{2/3}}-\sqrt [3]{x^3-1}\right )}{16 \sqrt {17}}+\frac {1}{816} \left (153+25 \sqrt {17}\right ) \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {1}{816} \left (153-25 \sqrt {17}\right ) \log \left (\sqrt [3]{x^3-1}-x\right )-\frac {3}{8} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {\left (x^3-1\right )^{5/3}}{10 x^5}+\frac {3 \left (x^3-1\right )^{5/3}}{8 x^2}+\frac {\left (x^3-1\right )^{2/3}}{4 x^2}\)

Input:

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

Output:

(-1 + x^3)^(2/3)/(4*x^2) - (5*x*(-1 + x^3)^(2/3))/8 + ((51 - 11*Sqrt[17])* 
x*(-1 + x^3)^(2/3))/408 + ((51 + 11*Sqrt[17])*x*(-1 + x^3)^(2/3))/408 + (- 
1 + x^3)^(5/3)/(10*x^5) + (3*(-1 + x^3)^(5/3))/(8*x^2) + (Sqrt[3]*ArcTan[( 
1 + (2*x)/(-1 + x^3)^(1/3))/Sqrt[3]])/4 + ((25 - 9*Sqrt[17])*ArcTan[(1 + ( 
2*x)/(-1 + x^3)^(1/3))/Sqrt[3]])/(24*Sqrt[51]) - ((25 + 9*Sqrt[17])*ArcTan 
[(1 + (2*x)/(-1 + x^3)^(1/3))/Sqrt[3]])/(24*Sqrt[51]) + ((-43 + 13*Sqrt[17 
])^(1/3)*ArcTan[(1 - ((2*(-3 + Sqrt[17]))^(1/3)*x)/(-1 + x^3)^(1/3))/Sqrt[ 
3]])/(8*Sqrt[51]) + ((43 + 13*Sqrt[17])^(1/3)*ArcTan[(1 + ((2*(3 + Sqrt[17 
]))^(1/3)*x)/(-1 + x^3)^(1/3))/Sqrt[3]])/(8*Sqrt[51]) + ((-43 + 13*Sqrt[17 
])^(1/3)*Log[1 - Sqrt[17] + 4*x^3])/(48*Sqrt[17]) + ((43 + 13*Sqrt[17])^(1 
/3)*Log[1 + Sqrt[17] + 4*x^3])/(48*Sqrt[17]) - ((-43 + 13*Sqrt[17])^(1/3)* 
Log[-(((-3 + Sqrt[17])^(1/3)*x)/2^(2/3)) - (-1 + x^3)^(1/3)])/(16*Sqrt[17] 
) - ((43 + 13*Sqrt[17])^(1/3)*Log[((3 + Sqrt[17])^(1/3)*x)/2^(2/3) - (-1 + 
 x^3)^(1/3)])/(16*Sqrt[17]) - (3*Log[-x + (-1 + x^3)^(1/3)])/8 + ((153 - 2 
5*Sqrt[17])*Log[-x + (-1 + x^3)^(1/3)])/816 + ((153 + 25*Sqrt[17])*Log[-x 
+ (-1 + x^3)^(1/3)])/816
 

Defintions of rubi rules used

rule 1388

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), 
x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, 
 c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer 
Q[p] || (GtQ[a, 0] && GtQ[d, 0]))
 

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 [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 1.

Time = 34.37 (sec) , antiderivative size = 9292, normalized size of antiderivative = 85.25

\[\text {output too large to display}\]

Input:

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

Output:

result too large to display
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-1+x^6\right )}{x^6 \left (-2+x^3+2 x^6\right )} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (trace 0)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [N/A]

Not integrable

Time = 0.16 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.28 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-1+x^6\right )}{x^6 \left (-2+x^3+2 x^6\right )} \, dx=\int { \frac {{\left (x^{6} - 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{6} + x^{3} - 2\right )} x^{6}} \,d x } \] Input:

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

Output:

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

Giac [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.28 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-1+x^6\right )}{x^6 \left (-2+x^3+2 x^6\right )} \, dx=\int { \frac {{\left (x^{6} - 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{6} + x^{3} - 2\right )} x^{6}} \,d x } \] Input:

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

Output:

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

Mupad [N/A]

Not integrable

Time = 8.91 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.28 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-1+x^6\right )}{x^6 \left (-2+x^3+2 x^6\right )} \, dx=\int \frac {{\left (x^3-1\right )}^{2/3}\,\left (x^6-1\right )}{x^6\,\left (2\,x^6+x^3-2\right )} \,d x \] Input:

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

Output:

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

Reduce [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.25 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-1+x^6\right )}{x^6 \left (-2+x^3+2 x^6\right )} \, dx=\frac {6 \left (x^{3}-1\right )^{\frac {2}{3}} x^{3}-\left (x^{3}-1\right )^{\frac {2}{3}}+25 \left (\int \frac {\left (x^{3}-1\right )^{\frac {2}{3}}}{2 x^{12}-x^{9}-3 x^{6}+2 x^{3}}d x \right ) x^{5}-15 \left (\int \frac {\left (x^{3}-1\right )^{\frac {2}{3}}}{2 x^{9}-x^{6}-3 x^{3}+2}d x \right ) x^{5}-20 \left (\int \frac {\left (x^{3}-1\right )^{\frac {2}{3}} x^{3}}{2 x^{9}-x^{6}-3 x^{3}+2}d x \right ) x^{5}}{10 x^{5}} \] Input:

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

Output:

(6*(x**3 - 1)**(2/3)*x**3 - (x**3 - 1)**(2/3) + 25*int((x**3 - 1)**(2/3)/( 
2*x**12 - x**9 - 3*x**6 + 2*x**3),x)*x**5 - 15*int((x**3 - 1)**(2/3)/(2*x* 
*9 - x**6 - 3*x**3 + 2),x)*x**5 - 20*int(((x**3 - 1)**(2/3)*x**3)/(2*x**9 
- x**6 - 3*x**3 + 2),x)*x**5)/(10*x**5)