Integrand size = 21, antiderivative size = 180 \[ \int \frac {1}{\left (1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\frac {\arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-x^2+x^3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{-x^2+x^3}\right )}{3 \sqrt [3]{2}}+\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-x^2+x^3}+\sqrt [3]{2} \left (-x^2+x^3\right )^{2/3}\right )}{6 \sqrt [3]{2}}-\frac {1}{3} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]
Time = 0.00 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.04 \[ \int \frac {1}{\left (1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\frac {\sqrt [3]{-1+x} x^{2/3} \left (2^{2/3} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{2^{2/3} \sqrt [3]{-1+x}+\sqrt [3]{x}}\right )-2 \log \left (2^{2/3} \sqrt [3]{-1+x}-2 \sqrt [3]{x}\right )+\log \left (\sqrt [3]{2} (-1+x)^{2/3}+2^{2/3} \sqrt [3]{-1+x} \sqrt [3]{x}+2 x^{2/3}\right )\right )-4 \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{-1+x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right )}{12 \sqrt [3]{(-1+x) x^2}} \]
((-1 + x)^(1/3)*x^(2/3)*(2^(2/3)*(2*Sqrt[3]*ArcTan[(Sqrt[3]*x^(1/3))/(2^(2 /3)*(-1 + x)^(1/3) + x^(1/3))] - 2*Log[2^(2/3)*(-1 + x)^(1/3) - 2*x^(1/3)] + Log[2^(1/3)*(-1 + x)^(2/3) + 2^(2/3)*(-1 + x)^(1/3)*x^(1/3) + 2*x^(2/3) ]) - 4*RootSum[1 - #1^3 + #1^6 & , (-Log[x^(1/3)] + Log[(-1 + x)^(1/3) - x ^(1/3)*#1])/#1 & ]))/(12*((-1 + x)*x^2)^(1/3))
Leaf count is larger than twice the leaf count of optimal. \(1731\) vs. \(2(180)=360\).
Time = 2.40 (sec) , antiderivative size = 1731, normalized size of antiderivative = 9.62, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2467, 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 |
| \(\displaystyle \frac {\sqrt [3]{x-1} x^{2/3} \int \frac {1}{\sqrt [3]{x-1} x^{2/3} \left (x^3+1\right )}dx}{\sqrt [3]{x^3-x^2}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {3 \sqrt [3]{x-1} x^{2/3} \int \frac {1}{\sqrt [3]{x-1} \left (x^3+1\right )}d\sqrt [3]{x}}{\sqrt [3]{x^3-x^2}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {3 \sqrt [3]{x-1} x^{2/3} \int \left (-\frac {1}{9 \left (-\sqrt [3]{x}-1\right ) \sqrt [3]{x-1}}-\frac {1}{9 \left (\sqrt [9]{-1} \sqrt [3]{x}-1\right ) \sqrt [3]{x-1}}-\frac {1}{9 \left (-(-1)^{2/9} \sqrt [3]{x}-1\right ) \sqrt [3]{x-1}}-\frac {1}{9 \left (\sqrt [3]{-1} \sqrt [3]{x}-1\right ) \sqrt [3]{x-1}}-\frac {1}{9 \left (-(-1)^{4/9} \sqrt [3]{x}-1\right ) \sqrt [3]{x-1}}-\frac {1}{9 \left ((-1)^{5/9} \sqrt [3]{x}-1\right ) \sqrt [3]{x-1}}-\frac {1}{9 \left (-(-1)^{2/3} \sqrt [3]{x}-1\right ) \sqrt [3]{x-1}}-\frac {1}{9 \left ((-1)^{7/9} \sqrt [3]{x}-1\right ) \sqrt [3]{x-1}}-\frac {1}{9 \left (-(-1)^{8/9} \sqrt [3]{x}-1\right ) \sqrt [3]{x-1}}\right )d\sqrt [3]{x}}{\sqrt [3]{x^3-x^2}}\) |
| \(\Big \downarrow \) 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 {\arctan \left (\frac {1-\frac {\sqrt [3]{2} \left (1-\sqrt [3]{x}\right )}{\sqrt [3]{x-1}}}{\sqrt {3}}\right )}{6 \sqrt [3]{2} \sqrt {3}}+\frac {\arctan \left (\frac {1-\frac {\sqrt [3]{2} \left ((-1)^{2/3}-\sqrt [3]{x}\right )}{\sqrt [3]{x-1}}}{\sqrt {3}}\right )}{6 \sqrt [3]{2} \sqrt {3}}+\frac {\arctan \left (\frac {\frac {\sqrt [3]{-2} \left (1-(-1)^{2/3} \sqrt [3]{x}\right )}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right )}{6 \sqrt [3]{2} \sqrt {3}}+\frac {(-1)^{7/9} \arctan \left (\frac {1-\frac {2 \sqrt [3]{x-1}}{\sqrt [3]{1-\sqrt [3]{-1}}}}{\sqrt {3}}\right )}{9 \sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}}}-\frac {(-1)^{4/9} \arctan \left (\frac {1-\frac {2 \sqrt [3]{x-1}}{\sqrt [3]{1-\sqrt [3]{-1}}}}{\sqrt {3}}\right )}{9 \sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}}}+\frac {\sqrt [9]{-1} \arctan \left (\frac {1-\frac {2 \sqrt [3]{x-1}}{\sqrt [3]{1-\sqrt [3]{-1}}}}{\sqrt {3}}\right )}{9 \sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}}}-\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)^{7/9} \log \left (\sqrt [3]{x-1}+\sqrt [3]{1-\sqrt [3]{-1}}\right )}{18 \sqrt [3]{1-\sqrt [3]{-1}}}-\frac {(-1)^{4/9} \log \left (\sqrt [3]{x-1}+\sqrt [3]{1-\sqrt [3]{-1}}\right )}{18 \sqrt [3]{1-\sqrt [3]{-1}}}+\frac {\sqrt [9]{-1} \log \left (\sqrt [3]{x-1}+\sqrt [3]{1-\sqrt [3]{-1}}\right )}{18 \sqrt [3]{1-\sqrt [3]{-1}}}-\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 (-\left (\left (1-\sqrt [3]{x}\right ) \left (\sqrt [3]{x}+1\right )^2\right )\right )}{36 \sqrt [3]{2}}-\frac {\log \left (-2^{2/3} \sqrt [3]{x-1}+\sqrt [3]{x}-1\right )}{12 \sqrt [3]{2}}-\frac {\log \left (\sqrt [3]{-1} 2^{2/3} \sqrt [3]{x-1}-\sqrt [3]{-1} \sqrt [3]{x}-1\right )}{12 \sqrt [3]{2}}+\frac {\log \left (-(-1)^{2/3} \left (\sqrt [3]{x}+(-1)^{2/3}\right )^2 \left (\sqrt [3]{-1} \sqrt [3]{x}+1\right )\right )}{36 \sqrt [3]{2}}+\frac {\log \left ((-1)^{2/3} \left (\sqrt [3]{x}+\sqrt [3]{-1}\right ) \left ((-1)^{2/3} \sqrt [3]{x}+1\right )^2\right )}{36 \sqrt [3]{2}}-\frac {\log \left (-(-2)^{2/3} \sqrt [3]{x-1}+(-1)^{2/3} \sqrt [3]{x}-1\right )}{12 \sqrt [3]{2}}-\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)^{7/9} \log \left (\sqrt [3]{-1}-x\right )}{54 \sqrt [3]{1-\sqrt [3]{-1}}}+\frac {(-1)^{4/9} \log \left (\sqrt [3]{-1}-x\right )}{54 \sqrt [3]{1-\sqrt [3]{-1}}}-\frac {\sqrt [9]{-1} \log \left (\sqrt [3]{-1}-x\right )}{54 \sqrt [3]{1-\sqrt [3]{-1}}}+\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)^{2/3} x-1\right )}{18 \sqrt [3]{1+(-1)^{2/3}}}\right )}{\sqrt [3]{x^3-x^2}}\) |
(3*(-1 + x)^(1/3)*x^(2/3)*(((-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)*Appell F1[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)*AppellF1[2/3, 1/3, 1, 5/3, x, -((-1)^(2/3)*x)])/(1 8*(-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)) + ArcTan[(1 - (2^(1/3)*(1 - x^(1/3)))/(-1 + x)^(1/3))/Sqrt[3]]/(6*2^(1/3)*Sqrt[3]) + ArcTan[(1 - (2^ (1/3)*((-1)^(2/3) - x^(1/3)))/(-1 + x)^(1/3))/Sqrt[3]]/(6*2^(1/3)*Sqrt[3]) + ArcTan[(1 + ((-2)^(1/3)*(1 - (-1)^(2/3)*x^(1/3)))/(-1 + x)^(1/3))/Sqrt[ 3]]/(6*2^(1/3)*Sqrt[3]) + ((-1)^(1/9)*ArcTan[(1 - (2*(-1 + x)^(1/3))/(1 - (-1)^(1/3))^(1/3))/Sqrt[3]])/(9*Sqrt[3]*(1 - (-1)^(1/3))^(1/3)) - ((-1)^(4 /9)*ArcTan[(1 - (2*(-1 + x)^(1/3))/(1 - (-1)^(1/3))^(1/3))/Sqrt[3]])/(9*Sq rt[3]*(1 - (-1)^(1/3))^(1/3)) + ((-1)^(7/9)*ArcTan[(1 - (2*(-1 + x)^(1/3)) /(1 - (-1)^(1/3))^(1/3))/Sqrt[3]])/(9*Sqrt[3]*(1 - (-1)^(1/3))^(1/3)) - (( -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...
3.24.22.3.1 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 = 5.20 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.78
| method | result | size |
| pseudoelliptic | \(-\frac {2^{\frac {2}{3}} \ln \left (\frac {-2^{\frac {1}{3}} x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}{x}\right )}{6}+\frac {2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}} x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {2}{3}}}{x^{2}}\right )}{12}-\frac {\sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (2^{\frac {2}{3}} \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}+x \right )}{3 x}\right )}{6}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right )}{3}\) | \(140\) |
| trager | \(\text {Expression too large to display}\) | \(2155\) |
-1/6*2^(2/3)*ln((-2^(1/3)*x+((-1+x)*x^2)^(1/3))/x)+1/12*2^(2/3)*ln((2^(2/3 )*x^2+2^(1/3)*((-1+x)*x^2)^(1/3)*x+((-1+x)*x^2)^(2/3))/x^2)-1/6*3^(1/2)*2^ (2/3)*arctan(1/3*3^(1/2)*(2^(2/3)*((-1+x)*x^2)^(1/3)+x)/x)-1/3*sum(ln((-_R *x+((-1+x)*x^2)^(1/3))/x)/_R,_R=RootOf(_Z^6-_Z^3+1))
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.32 (sec) , antiderivative size = 575, normalized size of antiderivative = 3.19 \[ \int \frac {1}{\left (1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\frac {1}{12} \cdot 2^{\frac {2}{3}} {\left (i \, \sqrt {3} - 1\right )}^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} \log \left (\frac {{\left (\sqrt {3} 2^{\frac {1}{3}} {\left (i \, \sqrt {-3} x + i \, x\right )} + 2^{\frac {1}{3}} {\left (\sqrt {-3} x + x\right )}\right )} {\left (i \, \sqrt {3} - 1\right )}^{\frac {2}{3}} + 8 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{12} \cdot 2^{\frac {2}{3}} {\left (i \, \sqrt {3} - 1\right )}^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} \log \left (\frac {{\left (\sqrt {3} 2^{\frac {1}{3}} {\left (-i \, \sqrt {-3} x + i \, x\right )} - 2^{\frac {1}{3}} {\left (\sqrt {-3} x - x\right )}\right )} {\left (i \, \sqrt {3} - 1\right )}^{\frac {2}{3}} + 8 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{12} \cdot 2^{\frac {2}{3}} {\left (-i \, \sqrt {3} - 1\right )}^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} \log \left (\frac {{\left (\sqrt {3} 2^{\frac {1}{3}} {\left (i \, \sqrt {-3} x - i \, x\right )} - 2^{\frac {1}{3}} {\left (\sqrt {-3} x - x\right )}\right )} {\left (-i \, \sqrt {3} - 1\right )}^{\frac {2}{3}} + 8 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{12} \cdot 2^{\frac {2}{3}} {\left (-i \, \sqrt {3} - 1\right )}^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} \log \left (\frac {{\left (\sqrt {3} 2^{\frac {1}{3}} {\left (-i \, \sqrt {-3} x - i \, x\right )} + 2^{\frac {1}{3}} {\left (\sqrt {-3} x + x\right )}\right )} {\left (-i \, \sqrt {3} - 1\right )}^{\frac {2}{3}} + 8 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{6} \, \sqrt {6} 2^{\frac {1}{6}} \left (-1\right )^{\frac {1}{3}} \arctan \left (-\frac {2^{\frac {1}{6}} {\left (\sqrt {6} 2^{\frac {1}{3}} x - 2 \, \sqrt {6} \left (-1\right )^{\frac {1}{3}} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}\right )}}{6 \, x}\right ) + \frac {1}{6} \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-\frac {2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} x - {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{6} \cdot 2^{\frac {2}{3}} {\left (i \, \sqrt {3} - 1\right )}^{\frac {1}{3}} \log \left (\frac {{\left (-i \, \sqrt {3} 2^{\frac {1}{3}} x - 2^{\frac {1}{3}} x\right )} {\left (i \, \sqrt {3} - 1\right )}^{\frac {2}{3}} + 4 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{6} \cdot 2^{\frac {2}{3}} {\left (-i \, \sqrt {3} - 1\right )}^{\frac {1}{3}} \log \left (\frac {{\left (i \, \sqrt {3} 2^{\frac {1}{3}} x - 2^{\frac {1}{3}} x\right )} {\left (-i \, \sqrt {3} - 1\right )}^{\frac {2}{3}} + 4 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{12} \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-\frac {2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{2} - 2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} x - {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) \]
1/12*2^(2/3)*(I*sqrt(3) - 1)^(1/3)*(sqrt(-3) - 1)*log(((sqrt(3)*2^(1/3)*(I *sqrt(-3)*x + I*x) + 2^(1/3)*(sqrt(-3)*x + x))*(I*sqrt(3) - 1)^(2/3) + 8*( x^3 - x^2)^(1/3))/x) - 1/12*2^(2/3)*(I*sqrt(3) - 1)^(1/3)*(sqrt(-3) + 1)*l og(((sqrt(3)*2^(1/3)*(-I*sqrt(-3)*x + I*x) - 2^(1/3)*(sqrt(-3)*x - x))*(I* sqrt(3) - 1)^(2/3) + 8*(x^3 - x^2)^(1/3))/x) - 1/12*2^(2/3)*(-I*sqrt(3) - 1)^(1/3)*(sqrt(-3) + 1)*log(((sqrt(3)*2^(1/3)*(I*sqrt(-3)*x - I*x) - 2^(1/ 3)*(sqrt(-3)*x - x))*(-I*sqrt(3) - 1)^(2/3) + 8*(x^3 - x^2)^(1/3))/x) + 1/ 12*2^(2/3)*(-I*sqrt(3) - 1)^(1/3)*(sqrt(-3) - 1)*log(((sqrt(3)*2^(1/3)*(-I *sqrt(-3)*x - I*x) + 2^(1/3)*(sqrt(-3)*x + x))*(-I*sqrt(3) - 1)^(2/3) + 8* (x^3 - x^2)^(1/3))/x) - 1/6*sqrt(6)*2^(1/6)*(-1)^(1/3)*arctan(-1/6*2^(1/6) *(sqrt(6)*2^(1/3)*x - 2*sqrt(6)*(-1)^(1/3)*(x^3 - x^2)^(1/3))/x) + 1/6*2^( 2/3)*(-1)^(1/3)*log(-(2^(1/3)*(-1)^(2/3)*x - (x^3 - x^2)^(1/3))/x) + 1/6*2 ^(2/3)*(I*sqrt(3) - 1)^(1/3)*log(((-I*sqrt(3)*2^(1/3)*x - 2^(1/3)*x)*(I*sq rt(3) - 1)^(2/3) + 4*(x^3 - x^2)^(1/3))/x) + 1/6*2^(2/3)*(-I*sqrt(3) - 1)^ (1/3)*log(((I*sqrt(3)*2^(1/3)*x - 2^(1/3)*x)*(-I*sqrt(3) - 1)^(2/3) + 4*(x ^3 - x^2)^(1/3))/x) - 1/12*2^(2/3)*(-1)^(1/3)*log(-(2^(2/3)*(-1)^(1/3)*x^2 - 2^(1/3)*(-1)^(2/3)*(x^3 - x^2)^(1/3)*x - (x^3 - x^2)^(2/3))/x^2)
Not integrable
Time = 0.74 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.12 \[ \int \frac {1}{\left (1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\int \frac {1}{\sqrt [3]{x^{2} \left (x - 1\right )} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]
Not integrable
Time = 0.31 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.12 \[ \int \frac {1}{\left (1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\int { \frac {1}{{\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} + 1\right )}} \,d x } \]
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 34.57 (sec) , antiderivative size = 967, normalized size of antiderivative = 5.37 \[ \int \frac {1}{\left (1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\text {Too large to display} \]
-1/6*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3) + 2*(-1/x + 1)^(1 /3))) + 1/3*(sqrt(3)*cos(4/9*pi)^5 - 10*sqrt(3)*cos(4/9*pi)^3*sin(4/9*pi)^ 2 + 5*sqrt(3)*cos(4/9*pi)*sin(4/9*pi)^4 - 5*cos(4/9*pi)^4*sin(4/9*pi) + 10 *cos(4/9*pi)^2*sin(4/9*pi)^3 - sin(4/9*pi)^5 + sqrt(3)*cos(4/9*pi)^2 - sqr t(3)*sin(4/9*pi)^2 - 2*cos(4/9*pi)*sin(4/9*pi))*arctan(1/2*((-I*sqrt(3) - 1)*cos(4/9*pi) + 2*(-1/x + 1)^(1/3))/((1/2*I*sqrt(3) + 1/2)*sin(4/9*pi))) + 1/3*(sqrt(3)*cos(2/9*pi)^5 - 10*sqrt(3)*cos(2/9*pi)^3*sin(2/9*pi)^2 + 5* sqrt(3)*cos(2/9*pi)*sin(2/9*pi)^4 - 5*cos(2/9*pi)^4*sin(2/9*pi) + 10*cos(2 /9*pi)^2*sin(2/9*pi)^3 - sin(2/9*pi)^5 + sqrt(3)*cos(2/9*pi)^2 - sqrt(3)*s in(2/9*pi)^2 - 2*cos(2/9*pi)*sin(2/9*pi))*arctan(1/2*((-I*sqrt(3) - 1)*cos (2/9*pi) + 2*(-1/x + 1)^(1/3))/((1/2*I*sqrt(3) + 1/2)*sin(2/9*pi))) - 1/3* (sqrt(3)*cos(1/9*pi)^5 - 10*sqrt(3)*cos(1/9*pi)^3*sin(1/9*pi)^2 + 5*sqrt(3 )*cos(1/9*pi)*sin(1/9*pi)^4 + 5*cos(1/9*pi)^4*sin(1/9*pi) - 10*cos(1/9*pi) ^2*sin(1/9*pi)^3 + sin(1/9*pi)^5 - sqrt(3)*cos(1/9*pi)^2 + sqrt(3)*sin(1/9 *pi)^2 - 2*cos(1/9*pi)*sin(1/9*pi))*arctan(-1/2*((-I*sqrt(3) - 1)*cos(1/9* pi) - 2*(-1/x + 1)^(1/3))/((1/2*I*sqrt(3) + 1/2)*sin(1/9*pi))) + 1/6*(5*sq rt(3)*cos(4/9*pi)^4*sin(4/9*pi) - 10*sqrt(3)*cos(4/9*pi)^2*sin(4/9*pi)^3 + sqrt(3)*sin(4/9*pi)^5 + cos(4/9*pi)^5 - 10*cos(4/9*pi)^3*sin(4/9*pi)^2 + 5*cos(4/9*pi)*sin(4/9*pi)^4 + 2*sqrt(3)*cos(4/9*pi)*sin(4/9*pi) + cos(4/9* pi)^2 - sin(4/9*pi)^2)*log((-I*sqrt(3)*cos(4/9*pi) - cos(4/9*pi))*(-1/x...
Not integrable
Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.12 \[ \int \frac {1}{\left (1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\int \frac {1}{\left (x^3+1\right )\,{\left (x^3-x^2\right )}^{1/3}} \,d x \]