Integrand size = 21, antiderivative size = 74 \[ \int \frac {1}{\left (-1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\frac {\left (-x^2+x^3\right )^{2/3}}{(1-x) x}+\frac {1}{3} \text {RootSum}\left [3-3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]
Time = 0.00 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\left (-1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\frac {-3 x+\sqrt [3]{-1+x} x^{2/3} \text {RootSum}\left [3-3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{-1+x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{3 \sqrt [3]{(-1+x) x^2}} \]
(-3*x + (-1 + x)^(1/3)*x^(2/3)*RootSum[3 - 3*#1^3 + #1^6 & , (-Log[x^(1/3) ] + Log[(-1 + x)^(1/3) - x^(1/3)*#1])/#1 & ])/(3*((-1 + x)*x^2)^(1/3))
Leaf count is larger than twice the leaf count of optimal. \(1561\) vs. \(2(74)=148\).
Time = 2.65 (sec) , antiderivative size = 1561, normalized size of antiderivative = 21.09, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2467, 25, 2035, 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.
| \(\displaystyle \int \frac {1}{\left (x^3-1\right ) \sqrt [3]{x^3-x^2}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{x-1} x^{2/3} \int -\frac {1}{\sqrt [3]{x-1} x^{2/3} \left (1-x^3\right )}dx}{\sqrt [3]{x^3-x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [3]{x-1} x^{2/3} \int \frac {1}{\sqrt [3]{x-1} x^{2/3} \left (1-x^3\right )}dx}{\sqrt [3]{x^3-x^2}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 \sqrt [3]{x-1} x^{2/3} \int \frac {1}{\sqrt [3]{x-1} \left (1-x^3\right )}d\sqrt [3]{x}}{\sqrt [3]{x^3-x^2}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {3 \sqrt [3]{x-1} x^{2/3} \int \left (\frac {1}{9 \left (1-\sqrt [3]{x}\right ) \sqrt [3]{x-1}}+\frac {1}{9 \left (\sqrt [9]{-1} \sqrt [3]{x}+1\right ) \sqrt [3]{x-1}}+\frac {1}{9 \left (1-(-1)^{2/9} \sqrt [3]{x}\right ) \sqrt [3]{x-1}}+\frac {1}{9 \left (\sqrt [3]{-1} \sqrt [3]{x}+1\right ) \sqrt [3]{x-1}}+\frac {1}{9 \left (1-(-1)^{4/9} \sqrt [3]{x}\right ) \sqrt [3]{x-1}}+\frac {1}{9 \left ((-1)^{5/9} \sqrt [3]{x}+1\right ) \sqrt [3]{x-1}}+\frac {1}{9 \left (1-(-1)^{2/3} \sqrt [3]{x}\right ) \sqrt [3]{x-1}}+\frac {1}{9 \left ((-1)^{7/9} \sqrt [3]{x}+1\right ) \sqrt [3]{x-1}}+\frac {1}{9 \left (1-(-1)^{8/9} \sqrt [3]{x}\right ) \sqrt [3]{x-1}}\right )d\sqrt [3]{x}}{\sqrt [3]{x^3-x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 \sqrt [3]{x-1} x^{2/3} \left (-\frac {(-1)^{7/9} \sqrt [3]{1-x} x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x,-\sqrt [3]{-1} x\right )}{18 \sqrt [3]{x-1}}+\frac {(-1)^{4/9} \sqrt [3]{1-x} x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x,-\sqrt [3]{-1} x\right )}{18 \sqrt [3]{x-1}}-\frac {\sqrt [9]{-1} \sqrt [3]{1-x} x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x,-\sqrt [3]{-1} x\right )}{18 \sqrt [3]{x-1}}+\frac {(-1)^{8/9} \sqrt [3]{1-x} x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x,(-1)^{2/3} x\right )}{18 \sqrt [3]{x-1}}-\frac {(-1)^{5/9} \sqrt [3]{1-x} x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x,(-1)^{2/3} x\right )}{18 \sqrt [3]{x-1}}+\frac {(-1)^{2/9} \sqrt [3]{1-x} x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x,(-1)^{2/3} x\right )}{18 \sqrt [3]{x-1}}+\frac {(-1)^{7/9} \sqrt [3]{-\frac {1}{1+\sqrt [3]{-1}}} \arctan \left (\frac {1-\frac {2 \sqrt [3]{x-1}}{\sqrt [3]{1+\sqrt [3]{-1}}}}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {(-1)^{4/9} \sqrt [3]{-\frac {1}{1+\sqrt [3]{-1}}} \arctan \left (\frac {1-\frac {2 \sqrt [3]{x-1}}{\sqrt [3]{1+\sqrt [3]{-1}}}}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {\sqrt [9]{-1} \sqrt [3]{-\frac {1}{1+\sqrt [3]{-1}}} \arctan \left (\frac {1-\frac {2 \sqrt [3]{x-1}}{\sqrt [3]{1+\sqrt [3]{-1}}}}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {(-1)^{8/9} \arctan \left (\frac {1-\frac {2 \sqrt [3]{x-1}}{\sqrt [3]{1-(-1)^{2/3}}}}{\sqrt {3}}\right )}{9 \sqrt {3} \sqrt [3]{1-(-1)^{2/3}}}-\frac {(-1)^{5/9} \arctan \left (\frac {1-\frac {2 \sqrt [3]{x-1}}{\sqrt [3]{1-(-1)^{2/3}}}}{\sqrt {3}}\right )}{9 \sqrt {3} \sqrt [3]{1-(-1)^{2/3}}}+\frac {(-1)^{2/9} \arctan \left (\frac {1-\frac {2 \sqrt [3]{x-1}}{\sqrt [3]{1-(-1)^{2/3}}}}{\sqrt {3}}\right )}{9 \sqrt {3} \sqrt [3]{1-(-1)^{2/3}}}+\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{1+\sqrt [3]{-1}} \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{1+\sqrt [3]{-1}}}+\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{1-(-1)^{2/3}} \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{1-(-1)^{2/3}}}+\frac {(-1)^{2/3} \sqrt [3]{1-x} x^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {4}{3},\frac {5}{3},x\right )}{18 \sqrt [3]{x-1}}-\frac {\sqrt [3]{-1} \sqrt [3]{1-x} x^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {4}{3},\frac {5}{3},x\right )}{18 \sqrt [3]{x-1}}+\frac {\sqrt [3]{1-x} x^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {4}{3},\frac {5}{3},x\right )}{18 \sqrt [3]{x-1}}+\frac {1}{18} (-1)^{7/9} \sqrt [3]{-\frac {1}{1+\sqrt [3]{-1}}} \log \left (\sqrt [3]{x-1}+\sqrt [3]{1+\sqrt [3]{-1}}\right )-\frac {1}{18} (-1)^{4/9} \sqrt [3]{-\frac {1}{1+\sqrt [3]{-1}}} \log \left (\sqrt [3]{x-1}+\sqrt [3]{1+\sqrt [3]{-1}}\right )+\frac {1}{18} \sqrt [9]{-1} \sqrt [3]{-\frac {1}{1+\sqrt [3]{-1}}} \log \left (\sqrt [3]{x-1}+\sqrt [3]{1+\sqrt [3]{-1}}\right )+\frac {(-1)^{8/9} \log \left (\sqrt [3]{x-1}+\sqrt [3]{1-(-1)^{2/3}}\right )}{18 \sqrt [3]{1-(-1)^{2/3}}}-\frac {(-1)^{5/9} \log \left (\sqrt [3]{x-1}+\sqrt [3]{1-(-1)^{2/3}}\right )}{18 \sqrt [3]{1-(-1)^{2/3}}}+\frac {(-1)^{2/9} \log \left (\sqrt [3]{x-1}+\sqrt [3]{1-(-1)^{2/3}}\right )}{18 \sqrt [3]{1-(-1)^{2/3}}}-\frac {\log \left (\sqrt [3]{1+\sqrt [3]{-1}} \sqrt [3]{x}-\sqrt [3]{x-1}\right )}{6 \sqrt [3]{1+\sqrt [3]{-1}}}-\frac {\log \left (\sqrt [3]{1-(-1)^{2/3}} \sqrt [3]{x}-\sqrt [3]{x-1}\right )}{6 \sqrt [3]{1-(-1)^{2/3}}}-\frac {1}{54} (-1)^{7/9} \sqrt [3]{-\frac {1}{1+\sqrt [3]{-1}}} \log \left (x+\sqrt [3]{-1}\right )+\frac {1}{54} (-1)^{4/9} \sqrt [3]{-\frac {1}{1+\sqrt [3]{-1}}} \log \left (x+\sqrt [3]{-1}\right )-\frac {1}{54} \sqrt [9]{-1} \sqrt [3]{-\frac {1}{1+\sqrt [3]{-1}}} \log \left (x+\sqrt [3]{-1}\right )-\frac {(-1)^{8/9} \log \left (x-(-1)^{2/3}\right )}{54 \sqrt [3]{1-(-1)^{2/3}}}+\frac {(-1)^{5/9} \log \left (x-(-1)^{2/3}\right )}{54 \sqrt [3]{1-(-1)^{2/3}}}-\frac {(-1)^{2/9} \log \left (x-(-1)^{2/3}\right )}{54 \sqrt [3]{1-(-1)^{2/3}}}+\frac {\log \left (\sqrt [3]{-1} x+1\right )}{18 \sqrt [3]{1+\sqrt [3]{-1}}}+\frac {\log \left (1-(-1)^{2/3} x\right )}{18 \sqrt [3]{1-(-1)^{2/3}}}+\frac {\sqrt [3]{x}}{3 \sqrt [3]{x-1}}+\frac {(-1)^{2/3}}{9 \sqrt [3]{x-1}}-\frac {\sqrt [3]{-1}}{9 \sqrt [3]{x-1}}+\frac {1}{9 \sqrt [3]{x-1}}\right )}{\sqrt [3]{x^3-x^2}}\) |
(-3*(-1 + x)^(1/3)*x^(2/3)*(1/(9*(-1 + x)^(1/3)) - (-1)^(1/3)/(9*(-1 + x)^ (1/3)) + (-1)^(2/3)/(9*(-1 + x)^(1/3)) + x^(1/3)/(3*(-1 + x)^(1/3)) - ((-1 )^(1/9)*(1 - x)^(1/3)*x^(2/3)*AppellF1[2/3, 1/3, 1, 5/3, x, -((-1)^(1/3)*x )])/(18*(-1 + x)^(1/3)) + ((-1)^(4/9)*(1 - x)^(1/3)*x^(2/3)*AppellF1[2/3, 1/3, 1, 5/3, x, -((-1)^(1/3)*x)])/(18*(-1 + x)^(1/3)) - ((-1)^(7/9)*(1 - x )^(1/3)*x^(2/3)*AppellF1[2/3, 1/3, 1, 5/3, x, -((-1)^(1/3)*x)])/(18*(-1 + x)^(1/3)) + ((-1)^(2/9)*(1 - x)^(1/3)*x^(2/3)*AppellF1[2/3, 1/3, 1, 5/3, x , (-1)^(2/3)*x])/(18*(-1 + x)^(1/3)) - ((-1)^(5/9)*(1 - x)^(1/3)*x^(2/3)*A ppellF1[2/3, 1/3, 1, 5/3, x, (-1)^(2/3)*x])/(18*(-1 + x)^(1/3)) + ((-1)^(8 /9)*(1 - x)^(1/3)*x^(2/3)*AppellF1[2/3, 1/3, 1, 5/3, x, (-1)^(2/3)*x])/(18 *(-1 + x)^(1/3)) + ((-1)^(1/9)*(-(1 + (-1)^(1/3))^(-1))^(1/3)*ArcTan[(1 - (2*(-1 + x)^(1/3))/(1 + (-1)^(1/3))^(1/3))/Sqrt[3]])/(9*Sqrt[3]) - ((-1)^( 4/9)*(-(1 + (-1)^(1/3))^(-1))^(1/3)*ArcTan[(1 - (2*(-1 + x)^(1/3))/(1 + (- 1)^(1/3))^(1/3))/Sqrt[3]])/(9*Sqrt[3]) + ((-1)^(7/9)*(-(1 + (-1)^(1/3))^(- 1))^(1/3)*ArcTan[(1 - (2*(-1 + x)^(1/3))/(1 + (-1)^(1/3))^(1/3))/Sqrt[3]]) /(9*Sqrt[3]) + ((-1)^(2/9)*ArcTan[(1 - (2*(-1 + x)^(1/3))/(1 - (-1)^(2/3)) ^(1/3))/Sqrt[3]])/(9*Sqrt[3]*(1 - (-1)^(2/3))^(1/3)) - ((-1)^(5/9)*ArcTan[ (1 - (2*(-1 + x)^(1/3))/(1 - (-1)^(2/3))^(1/3))/Sqrt[3]])/(9*Sqrt[3]*(1 - (-1)^(2/3))^(1/3)) + ((-1)^(8/9)*ArcTan[(1 - (2*(-1 + x)^(1/3))/(1 - (-1)^ (2/3))^(1/3))/Sqrt[3]])/(9*Sqrt[3]*(1 - (-1)^(2/3))^(1/3)) + ArcTan[(1 ...
3.10.70.3.1 Defintions of rubi rules used
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
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]
Time = 14.62 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.85
| method | result | size |
| pseudoelliptic | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-3 \textit {\_Z}^{3}+3\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (\left (x -1\right ) x^{2}\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right ) \left (\left (x -1\right ) x^{2}\right )^{\frac {1}{3}}-3 x}{3 \left (\left (x -1\right ) x^{2}\right )^{\frac {1}{3}}}\) | \(63\) |
| trager | \(\text {Expression too large to display}\) | \(2138\) |
| risch | \(\text {Expression too large to display}\) | \(2357\) |
1/3*(sum(ln((-_R*x+((x-1)*x^2)^(1/3))/x)/_R,_R=RootOf(_Z^6-3*_Z^3+3))*((x- 1)*x^2)^(1/3)-3*x)/((x-1)*x^2)^(1/3)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.27 (sec) , antiderivative size = 501, normalized size of antiderivative = 6.77 \[ \int \frac {1}{\left (-1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=-\frac {6^{\frac {2}{3}} {\left (x^{2} + \sqrt {-3} {\left (x^{2} - x\right )} - x\right )} {\left (i \, \sqrt {3} + 3\right )}^{\frac {1}{3}} \log \left (\frac {6^{\frac {1}{3}} {\left (\sqrt {3} {\left (i \, \sqrt {-3} x - i \, x\right )} - 3 \, \sqrt {-3} x + 3 \, x\right )} {\left (i \, \sqrt {3} + 3\right )}^{\frac {2}{3}} + 24 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + 6^{\frac {2}{3}} {\left (x^{2} - \sqrt {-3} {\left (x^{2} - x\right )} - x\right )} {\left (i \, \sqrt {3} + 3\right )}^{\frac {1}{3}} \log \left (\frac {6^{\frac {1}{3}} {\left (\sqrt {3} {\left (-i \, \sqrt {-3} x - i \, x\right )} + 3 \, \sqrt {-3} x + 3 \, x\right )} {\left (i \, \sqrt {3} + 3\right )}^{\frac {2}{3}} + 24 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - 2 \cdot 6^{\frac {2}{3}} {\left (x^{2} - x\right )} {\left (i \, \sqrt {3} + 3\right )}^{\frac {1}{3}} \log \left (\frac {6^{\frac {1}{3}} {\left (i \, \sqrt {3} x - 3 \, x\right )} {\left (i \, \sqrt {3} + 3\right )}^{\frac {2}{3}} + 12 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + 6^{\frac {2}{3}} {\left (x^{2} - \sqrt {-3} {\left (x^{2} - x\right )} - x\right )} {\left (-i \, \sqrt {3} + 3\right )}^{\frac {1}{3}} \log \left (\frac {6^{\frac {1}{3}} {\left (\sqrt {3} {\left (i \, \sqrt {-3} x + i \, x\right )} + 3 \, \sqrt {-3} x + 3 \, x\right )} {\left (-i \, \sqrt {3} + 3\right )}^{\frac {2}{3}} + 24 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + 6^{\frac {2}{3}} {\left (x^{2} + \sqrt {-3} {\left (x^{2} - x\right )} - x\right )} {\left (-i \, \sqrt {3} + 3\right )}^{\frac {1}{3}} \log \left (\frac {6^{\frac {1}{3}} {\left (\sqrt {3} {\left (-i \, \sqrt {-3} x + i \, x\right )} - 3 \, \sqrt {-3} x + 3 \, x\right )} {\left (-i \, \sqrt {3} + 3\right )}^{\frac {2}{3}} + 24 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - 2 \cdot 6^{\frac {2}{3}} {\left (x^{2} - x\right )} {\left (-i \, \sqrt {3} + 3\right )}^{\frac {1}{3}} \log \left (\frac {6^{\frac {1}{3}} {\left (-i \, \sqrt {3} x - 3 \, x\right )} {\left (-i \, \sqrt {3} + 3\right )}^{\frac {2}{3}} + 12 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + 36 \, {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{36 \, {\left (x^{2} - x\right )}} \]
-1/36*(6^(2/3)*(x^2 + sqrt(-3)*(x^2 - x) - x)*(I*sqrt(3) + 3)^(1/3)*log((6 ^(1/3)*(sqrt(3)*(I*sqrt(-3)*x - I*x) - 3*sqrt(-3)*x + 3*x)*(I*sqrt(3) + 3) ^(2/3) + 24*(x^3 - x^2)^(1/3))/x) + 6^(2/3)*(x^2 - sqrt(-3)*(x^2 - x) - x) *(I*sqrt(3) + 3)^(1/3)*log((6^(1/3)*(sqrt(3)*(-I*sqrt(-3)*x - I*x) + 3*sqr t(-3)*x + 3*x)*(I*sqrt(3) + 3)^(2/3) + 24*(x^3 - x^2)^(1/3))/x) - 2*6^(2/3 )*(x^2 - x)*(I*sqrt(3) + 3)^(1/3)*log((6^(1/3)*(I*sqrt(3)*x - 3*x)*(I*sqrt (3) + 3)^(2/3) + 12*(x^3 - x^2)^(1/3))/x) + 6^(2/3)*(x^2 - sqrt(-3)*(x^2 - x) - x)*(-I*sqrt(3) + 3)^(1/3)*log((6^(1/3)*(sqrt(3)*(I*sqrt(-3)*x + I*x) + 3*sqrt(-3)*x + 3*x)*(-I*sqrt(3) + 3)^(2/3) + 24*(x^3 - x^2)^(1/3))/x) + 6^(2/3)*(x^2 + sqrt(-3)*(x^2 - x) - x)*(-I*sqrt(3) + 3)^(1/3)*log((6^(1/3 )*(sqrt(3)*(-I*sqrt(-3)*x + I*x) - 3*sqrt(-3)*x + 3*x)*(-I*sqrt(3) + 3)^(2 /3) + 24*(x^3 - x^2)^(1/3))/x) - 2*6^(2/3)*(x^2 - x)*(-I*sqrt(3) + 3)^(1/3 )*log((6^(1/3)*(-I*sqrt(3)*x - 3*x)*(-I*sqrt(3) + 3)^(2/3) + 12*(x^3 - x^2 )^(1/3))/x) + 36*(x^3 - x^2)^(2/3))/(x^2 - x)
Not integrable
Time = 0.69 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.30 \[ \int \frac {1}{\left (-1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\int \frac {1}{\sqrt [3]{x^{2} \left (x - 1\right )} \left (x - 1\right ) \left (x^{2} + x + 1\right )}\, dx \]
Not integrable
Time = 0.32 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.28 \[ \int \frac {1}{\left (-1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\int { \frac {1}{{\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} - 1\right )}} \,d x } \]
Exception generated. \[ \int \frac {1}{\left (-1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Invalid _EXT in replace_ext Error: Bad Argument ValueDone
Not integrable
Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.28 \[ \int \frac {1}{\left (-1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\int \frac {1}{\left (x^3-1\right )\,{\left (x^3-x^2\right )}^{1/3}} \,d x \]