3.11.35 \(\int \frac {(-1+x^3)^{2/3} (1-2 x^3+x^6)}{x^6 (1-x^3+x^6)} \, dx\) [1035]

3.11.35.1 Optimal result

Integrand size = 35, antiderivative size = 78 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1-2 x^3+x^6\right )}{x^6 \left (1-x^3+x^6\right )} \, dx=\frac {\left (-1+x^3\right )^{2/3} \left (-2+7 x^3\right )}{10 x^5}-\frac {1}{3} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right )}{-\text {$\#$1}+2 \text {$\#$1}^4}\&\right ] \]

output

Unintegrable
 

3.11.35.2 Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1-2 x^3+x^6\right )}{x^6 \left (1-x^3+x^6\right )} \, dx=\frac {\left (-1+x^3\right )^{2/3} \left (-2+7 x^3\right )}{10 x^5}-\frac {1}{3} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right )}{-\text {$\#$1}+2 \text {$\#$1}^4}\&\right ] \]

input

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

output

((-1 + x^3)^(2/3)*(-2 + 7*x^3))/(10*x^5) - RootSum[1 - #1^3 + #1^6 & , (-L 
og[x] + Log[(-1 + x^3)^(1/3) - x*#1])/(-#1 + 2*#1^4) & ]/3
 

3.11.35.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.73 (sec) , antiderivative size = 703, normalized size of antiderivative = 9.01, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.371, Rules used = {1380, 1844, 25, 809, 809, 769, 1758, 25, 933, 25, 1026, 769, 901}

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.

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

\(\Big \downarrow \) 1380

\(\displaystyle \int \frac {\left (x^3-1\right )^{8/3}}{x^6 \left (x^6-x^3+1\right )}dx\)

\(\Big \downarrow \) 1844

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 809

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

\(\Big \downarrow \) 809

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

\(\Big \downarrow \) 769

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

\(\Big \downarrow \) 1758

\(\displaystyle \frac {2 i \int -\frac {\left (x^3-1\right )^{5/3}}{-2 x^3-i \sqrt {3}+1}dx}{\sqrt {3}}-\frac {2 i \int -\frac {\left (x^3-1\right )^{5/3}}{-2 x^3+i \sqrt {3}+1}dx}{\sqrt {3}}-\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {\left (x^3-1\right )^{5/3}}{5 x^5}+\frac {\left (x^3-1\right )^{2/3}}{2 x^2}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {2 i \int \frac {\left (x^3-1\right )^{5/3}}{-2 x^3-i \sqrt {3}+1}dx}{\sqrt {3}}+\frac {2 i \int \frac {\left (x^3-1\right )^{5/3}}{-2 x^3+i \sqrt {3}+1}dx}{\sqrt {3}}-\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {\left (x^3-1\right )^{5/3}}{5 x^5}+\frac {\left (x^3-1\right )^{2/3}}{2 x^2}\)

\(\Big \downarrow \) 933

\(\displaystyle \frac {2 i \left (-\frac {1}{6} \left (x^3-1\right )^{2/3} x-\frac {1}{6} \int -\frac {-\left (\left (7-3 i \sqrt {3}\right ) x^3\right )-i \sqrt {3}+5}{\left (-2 x^3+i \sqrt {3}+1\right ) \sqrt [3]{x^3-1}}dx\right )}{\sqrt {3}}-\frac {2 i \left (-\frac {1}{6} \left (x^3-1\right )^{2/3} x-\frac {1}{6} \int -\frac {-\left (\left (7+3 i \sqrt {3}\right ) x^3\right )+i \sqrt {3}+5}{\left (-2 x^3-i \sqrt {3}+1\right ) \sqrt [3]{x^3-1}}dx\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {\left (x^3-1\right )^{5/3}}{5 x^5}+\frac {\left (x^3-1\right )^{2/3}}{2 x^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 i \left (-\frac {1}{6} x \left (x^3-1\right )^{2/3}+\frac {1}{6} \int \frac {-\left (\left (7-3 i \sqrt {3}\right ) x^3\right )-i \sqrt {3}+5}{\left (-2 x^3+i \sqrt {3}+1\right ) \sqrt [3]{x^3-1}}dx\right )}{\sqrt {3}}-\frac {2 i \left (-\frac {1}{6} x \left (x^3-1\right )^{2/3}+\frac {1}{6} \int \frac {-\left (\left (7+3 i \sqrt {3}\right ) x^3\right )+i \sqrt {3}+5}{\left (-2 x^3-i \sqrt {3}+1\right ) \sqrt [3]{x^3-1}}dx\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {\left (x^3-1\right )^{5/3}}{5 x^5}+\frac {\left (x^3-1\right )^{2/3}}{2 x^2}\)

\(\Big \downarrow \) 1026

\(\displaystyle -\frac {2 i \left (-\frac {1}{6} x \left (x^3-1\right )^{2/3}+\frac {1}{6} \left (\frac {1}{2} \left (7+3 i \sqrt {3}\right ) \int \frac {1}{\sqrt [3]{x^3-1}}dx-3 \left (1-i \sqrt {3}\right ) \int \frac {1}{\left (-2 x^3-i \sqrt {3}+1\right ) \sqrt [3]{x^3-1}}dx\right )\right )}{\sqrt {3}}+\frac {2 i \left (-\frac {1}{6} x \left (x^3-1\right )^{2/3}+\frac {1}{6} \left (\frac {1}{2} \left (7-3 i \sqrt {3}\right ) \int \frac {1}{\sqrt [3]{x^3-1}}dx-3 \left (1+i \sqrt {3}\right ) \int \frac {1}{\left (-2 x^3+i \sqrt {3}+1\right ) \sqrt [3]{x^3-1}}dx\right )\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {\left (x^3-1\right )^{5/3}}{5 x^5}+\frac {\left (x^3-1\right )^{2/3}}{2 x^2}\)

\(\Big \downarrow \) 769

\(\displaystyle -\frac {2 i \left (-\frac {1}{6} x \left (x^3-1\right )^{2/3}+\frac {1}{6} \left (\frac {1}{2} \left (7+3 i \sqrt {3}\right ) \left (\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )\right )-3 \left (1-i \sqrt {3}\right ) \int \frac {1}{\left (-2 x^3-i \sqrt {3}+1\right ) \sqrt [3]{x^3-1}}dx\right )\right )}{\sqrt {3}}+\frac {2 i \left (-\frac {1}{6} x \left (x^3-1\right )^{2/3}+\frac {1}{6} \left (\frac {1}{2} \left (7-3 i \sqrt {3}\right ) \left (\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )\right )-3 \left (1+i \sqrt {3}\right ) \int \frac {1}{\left (-2 x^3+i \sqrt {3}+1\right ) \sqrt [3]{x^3-1}}dx\right )\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {\left (x^3-1\right )^{5/3}}{5 x^5}+\frac {\left (x^3-1\right )^{2/3}}{2 x^2}\)

\(\Big \downarrow \) 901

\(\displaystyle -\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {2 i \left (-\frac {1}{6} x \left (x^3-1\right )^{2/3}+\frac {1}{6} \left (\frac {1}{2} \left (7-3 i \sqrt {3}\right ) \left (\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )\right )-3 \left (1+i \sqrt {3}\right ) \left (-\frac {i \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \sqrt [3]{x^3-1}}}{\sqrt {3}}\right )}{\sqrt {3} \left (\sqrt {3}-i\right )^{2/3} \sqrt [3]{\sqrt {3}+i}}-\frac {i \log \left (-2 x^3+i \sqrt {3}+1\right )}{6 \left (\sqrt {3}-i\right )^{2/3} \sqrt [3]{\sqrt {3}+i}}+\frac {i \log \left (-\sqrt [3]{x^3-1}+\frac {x}{\sqrt [3]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}}}\right )}{2 \left (\sqrt {3}-i\right )^{2/3} \sqrt [3]{\sqrt {3}+i}}\right )\right )\right )}{\sqrt {3}}-\frac {2 i \left (-\frac {1}{6} x \left (x^3-1\right )^{2/3}+\frac {1}{6} \left (\frac {1}{2} \left (7+3 i \sqrt {3}\right ) \left (\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )\right )-3 \left (1-i \sqrt {3}\right ) \left (\frac {i \arctan \left (\frac {1+\frac {2 \sqrt [3]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} x}{\sqrt [3]{x^3-1}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{\sqrt {3}-i} \left (\sqrt {3}+i\right )^{2/3}}+\frac {i \log \left (-2 x^3-i \sqrt {3}+1\right )}{6 \sqrt [3]{\sqrt {3}-i} \left (\sqrt {3}+i\right )^{2/3}}-\frac {i \log \left (-\sqrt [3]{x^3-1}+\sqrt [3]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} x\right )}{2 \sqrt [3]{\sqrt {3}-i} \left (\sqrt {3}+i\right )^{2/3}}\right )\right )\right )}{\sqrt {3}}+\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {\left (x^3-1\right )^{5/3}}{5 x^5}+\frac {\left (x^3-1\right )^{2/3}}{2 x^2}\)

input

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

output

(-1 + x^3)^(2/3)/(2*x^2) + (-1 + x^3)^(5/3)/(5*x^5) - ArcTan[(1 + (2*x)/(- 
1 + x^3)^(1/3))/Sqrt[3]]/Sqrt[3] + ((2*I)*(-1/6*(x*(-1 + x^3)^(2/3)) + (-3 
*(1 + I*Sqrt[3])*(((-I)*ArcTan[(1 + (2*x)/((-((I - Sqrt[3])/(I + Sqrt[3])) 
)^(1/3)*(-1 + x^3)^(1/3)))/Sqrt[3]])/(Sqrt[3]*(-I + Sqrt[3])^(2/3)*(I + Sq 
rt[3])^(1/3)) - ((I/6)*Log[1 + I*Sqrt[3] - 2*x^3])/((-I + Sqrt[3])^(2/3)*( 
I + Sqrt[3])^(1/3)) + ((I/2)*Log[x/(-((I - Sqrt[3])/(I + Sqrt[3])))^(1/3) 
- (-1 + x^3)^(1/3)])/((-I + Sqrt[3])^(2/3)*(I + Sqrt[3])^(1/3))) + ((7 - ( 
3*I)*Sqrt[3])*(ArcTan[(1 + (2*x)/(-1 + x^3)^(1/3))/Sqrt[3]]/Sqrt[3] - Log[ 
-x + (-1 + x^3)^(1/3)]/2))/2)/6))/Sqrt[3] - ((2*I)*(-1/6*(x*(-1 + x^3)^(2/ 
3)) + (-3*(1 - I*Sqrt[3])*((I*ArcTan[(1 + (2*(-((I - Sqrt[3])/(I + Sqrt[3] 
)))^(1/3)*x)/(-1 + x^3)^(1/3))/Sqrt[3]])/(Sqrt[3]*(-I + Sqrt[3])^(1/3)*(I 
+ Sqrt[3])^(2/3)) + ((I/6)*Log[1 - I*Sqrt[3] - 2*x^3])/((-I + Sqrt[3])^(1/ 
3)*(I + Sqrt[3])^(2/3)) - ((I/2)*Log[(-((I - Sqrt[3])/(I + Sqrt[3])))^(1/3 
)*x - (-1 + x^3)^(1/3)])/((-I + Sqrt[3])^(1/3)*(I + Sqrt[3])^(2/3))) + ((7 
 + (3*I)*Sqrt[3])*(ArcTan[(1 + (2*x)/(-1 + x^3)^(1/3))/Sqrt[3]]/Sqrt[3] - 
Log[-x + (-1 + x^3)^(1/3)]/2))/2)/6))/Sqrt[3] + Log[-x + (-1 + x^3)^(1/3)] 
/2
 

3.11.35.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 769

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + 2*Rt[b, 3]* 
(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^ 
3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]
 

rule 809

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* 
x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1))), x] - Simp[b*n*(p/(c^n*(m + 1)))   I 
nt[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && IGtQ 
[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntB 
inomialQ[a, b, c, n, m, p, x]
 

rule 901

Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> Wit 
h[{q = Rt[(b*c - a*d)/c, 3]}, Simp[ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/S 
qrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q), x] 
 + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0]
 

rule 933

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] 
:> Simp[d*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(b*(n*(p + q) + 1))), 
x] + Simp[1/(b*(n*(p + q) + 1))   Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Sim 
p[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q 
- 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d 
, 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntBinomialQ[ 
a, b, c, d, n, p, q, x]
 

rule 1026

Int[(((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)* 
(x_)^(n_)), x_Symbol] :> Simp[f/d   Int[(a + b*x^n)^p, x], x] + Simp[(d*e - 
 c*f)/d   Int[(a + b*x^n)^p/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, 
 p, n}, x]
 

rule 1380

Int[(u_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S 
imp[1/c^p   Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] 
&& EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
 

rule 1758

Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_ 
)), x_Symbol] :> With[{r = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/r)   Int[(d + e*x 
^n)^q/(b - r + 2*c*x^n), x], x] - Simp[2*(c/r)   Int[(d + e*x^n)^q/(b + r + 
 2*c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[n2, 2*n] && Ne 
Q[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[q]
 

rule 1844

Int[(((f_.)*(x_))^(m_)*((d_.) + (e_.)*(x_)^(n_))^(q_))/((a_) + (c_.)*(x_)^( 
n2_.) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[d/a   Int[(f*x)^m*(d + e*x^n)^( 
q - 1), x], x] - Simp[1/(a*f^n)   Int[(f*x)^(m + n)*(d + e*x^n)^(q - 1)*(Si 
mp[b*d - a*e + c*d*x^n, x]/(a + b*x^n + c*x^(2*n))), x], x] /; FreeQ[{a, b, 
 c, d, e, f}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] &&  ! 
IntegerQ[q] && GtQ[q, 0] && LtQ[m, 0]
 

3.11.35.4 Maple [N/A] (verified)

Time = 21.26 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.97

method	result	size
pseudoelliptic	\(\frac {-10 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right )}{2 \textit {\_R}^{4}-\textit {\_R}}\right ) x^{5}+21 x^{3} \left (x^{3}-1\right )^{\frac {2}{3}}-6 \left (x^{3}-1\right )^{\frac {2}{3}}}{30 x^{5}}\)	\(76\)
risch	\(\text {Expression too large to display}\)	\(1460\)
trager	\(\text {Expression too large to display}\)	\(4954\)

input

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

output

1/30*(-10*sum(ln((-_R*x+(x^3-1)^(1/3))/x)/(2*_R^4-_R),_R=RootOf(_Z^6-_Z^3+ 
1))*x^5+21*x^3*(x^3-1)^(2/3)-6*(x^3-1)^(2/3))/x^5
 

3.11.35.5 Fricas [F(-2)]

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

input

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

output

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

3.11.35.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.11.35.7 Maxima [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.45 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1-2 x^3+x^6\right )}{x^6 \left (1-x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{6} - 2 \, x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - x^{3} + 1\right )} x^{6}} \,d x } \]

input

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

output

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

3.11.35.8 Giac [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.45 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1-2 x^3+x^6\right )}{x^6 \left (1-x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{6} - 2 \, x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - x^{3} + 1\right )} x^{6}} \,d x } \]

input

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

output

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

3.11.35.9 Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.45 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1-2 x^3+x^6\right )}{x^6 \left (1-x^3+x^6\right )} \, dx=\int \frac {{\left (x^3-1\right )}^{2/3}\,\left (x^6-2\,x^3+1\right )}{x^6\,\left (x^6-x^3+1\right )} \,d x \]

input

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

output

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