Integrand size = 24, antiderivative size = 211 \[ \int \frac {1}{x^3 \left (1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\frac {3 \left (5+6 x+9 x^2\right ) \left (-x^2+x^3\right )^{2/3}}{40 x^4}-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-x^2+x^3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{-x^2+x^3}\right )}{3 \sqrt [3]{2}}-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-x^2+x^3}+\sqrt [3]{2} \left (-x^2+x^3\right )^{2/3}\right )}{6 \sqrt [3]{2}}+\frac {1}{3} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]
Time = 0.39 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.19 \[ \int \frac {1}{x^3 \left (1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\frac {-45-9 x-27 x^2+81 x^3-20\ 2^{2/3} \sqrt {3} \sqrt [3]{-1+x} x^{8/3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{2^{2/3} \sqrt [3]{-1+x}+\sqrt [3]{x}}\right )+20\ 2^{2/3} \sqrt [3]{-1+x} x^{8/3} \log \left (2^{2/3} \sqrt [3]{-1+x}-2 \sqrt [3]{x}\right )-10\ 2^{2/3} \sqrt [3]{-1+x} x^{8/3} \log \left (\sqrt [3]{2} (-1+x)^{2/3}+2^{2/3} \sqrt [3]{-1+x} \sqrt [3]{x}+2 x^{2/3}\right )+40 \sqrt [3]{-1+x} x^{8/3} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{-1+x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{120 x^2 \sqrt [3]{(-1+x) x^2}} \]
(-45 - 9*x - 27*x^2 + 81*x^3 - 20*2^(2/3)*Sqrt[3]*(-1 + x)^(1/3)*x^(8/3)*A rcTan[(Sqrt[3]*x^(1/3))/(2^(2/3)*(-1 + x)^(1/3) + x^(1/3))] + 20*2^(2/3)*( -1 + x)^(1/3)*x^(8/3)*Log[2^(2/3)*(-1 + x)^(1/3) - 2*x^(1/3)] - 10*2^(2/3) *(-1 + x)^(1/3)*x^(8/3)*Log[2^(1/3)*(-1 + x)^(2/3) + 2^(2/3)*(-1 + x)^(1/3 )*x^(1/3) + 2*x^(2/3)] + 40*(-1 + x)^(1/3)*x^(8/3)*RootSum[1 - #1^3 + #1^6 & , (-Log[x^(1/3)] + Log[(-1 + x)^(1/3) - x^(1/3)*#1])/#1 & ])/(120*x^2*( (-1 + x)*x^2)^(1/3))
Result contains complex when optimal does not.
Time = 1.46 (sec) , antiderivative size = 892, normalized size of antiderivative = 4.23, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, 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.
| \(\displaystyle \int \frac {1}{x^3 \left (x^3+1\right ) \sqrt [3]{x^3-x^2}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{x-1} x^{2/3} \int \frac {1}{\sqrt [3]{x-1} x^{11/3} \left (x^3+1\right )}dx}{\sqrt [3]{x^3-x^2}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {3 \sqrt [3]{x-1} x^{2/3} \int \frac {1}{\sqrt [3]{x-1} x^3 \left (x^3+1\right )}d\sqrt [3]{x}}{\sqrt [3]{x^3-x^2}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {3 \sqrt [3]{x-1} x^{2/3} \int \left (\frac {\sqrt [3]{x}-2}{9 \left (x^{2/3}-\sqrt [3]{x}+1\right ) \sqrt [3]{x-1}}-\frac {1}{9 \left (\sqrt [3]{x}+1\right ) \sqrt [3]{x-1}}+\frac {x-2}{3 \sqrt [3]{x-1} \left (x^2-x+1\right )}+\frac {1}{\sqrt [3]{x-1} x^3}\right )d\sqrt [3]{x}}{\sqrt [3]{x^3-x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 \sqrt [3]{x-1} x^{2/3} \left (-\frac {\arctan \left (\frac {1-\frac {\sqrt [3]{2} \left (1-\sqrt [3]{x}\right )}{\sqrt [3]{x-1}}}{\sqrt {3}}\right )}{9 \sqrt [3]{2} \sqrt {3}}+\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{2} \left (1-\sqrt [3]{x}\right )}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right )}{9 \sqrt [3]{2} \sqrt {3}}-\frac {\arctan \left (\frac {1-2^{2/3} \sqrt [3]{x-1}}{\sqrt {3}}\right )}{9 \sqrt [3]{2} \sqrt {3}}-\frac {2^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \arctan \left (\frac {\frac {2 \sqrt [3]{x}}{\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x-1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}}}-\frac {\log \left (1-\frac {\sqrt [3]{2} \left (1-\sqrt [3]{x}\right )}{\sqrt [3]{x-1}}\right )}{27 \sqrt [3]{2}}+\frac {\log \left (\frac {2^{2/3} \left (1-\sqrt [3]{x}\right )^2}{(x-1)^{2/3}}+\frac {\sqrt [3]{2} \left (1-\sqrt [3]{x}\right )}{\sqrt [3]{x-1}}+1\right )}{54 \sqrt [3]{2}}-\frac {\log \left (\sqrt [3]{x-1}+\sqrt [3]{2}\right )}{18 \sqrt [3]{2}}+\frac {\log \left (2^{2/3} \sqrt [3]{x-1}-\sqrt [3]{x}+1\right )}{18 \sqrt [3]{2}}-\frac {\log \left (\left (1-\sqrt [3]{x}\right ) \left (\sqrt [3]{x}+1\right )^2\right )}{54 \sqrt [3]{2}}+\frac {\log \left (\sqrt [3]{2} \sqrt [3]{x}-\sqrt [3]{x-1}\right )}{9 \sqrt [3]{2}}+\frac {1}{6} \sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}}}-\sqrt [3]{x-1}\right )+\frac {\log \left (\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x}-\sqrt [3]{x-1}\right )}{6 \sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}}}-\frac {\log (x+1)}{54 \sqrt [3]{2}}-\frac {1}{18} \sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \log \left (2 x-i \sqrt {3}-1\right )-\frac {\log \left (2 x+i \sqrt {3}-1\right )}{18 \sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}}}+\frac {9 (x-1)^{2/3}}{40 x^{2/3}}+\frac {3 (x-1)^{2/3}}{20 x^{5/3}}+\frac {(x-1)^{2/3}}{8 x^{8/3}}\right )}{\sqrt [3]{x^3-x^2}}\) |
(3*(-1 + x)^(1/3)*x^(2/3)*((-1 + x)^(2/3)/(8*x^(8/3)) + (3*(-1 + x)^(2/3)) /(20*x^(5/3)) + (9*(-1 + x)^(2/3))/(40*x^(2/3)) - ArcTan[(1 - (2^(1/3)*(1 - x^(1/3)))/(-1 + x)^(1/3))/Sqrt[3]]/(9*2^(1/3)*Sqrt[3]) + ArcTan[(1 + (2* 2^(1/3)*(1 - x^(1/3)))/(-1 + x)^(1/3))/Sqrt[3]]/(9*2^(1/3)*Sqrt[3]) - ArcT an[(1 - 2^(2/3)*(-1 + x)^(1/3))/Sqrt[3]]/(9*2^(1/3)*Sqrt[3]) - (2^(2/3)*Ar cTan[(1 + (2*2^(1/3)*x^(1/3))/(-1 + x)^(1/3))/Sqrt[3]])/(9*Sqrt[3]) - ((-( (I - Sqrt[3])/(I + Sqrt[3])))^(1/3)*ArcTan[(1 + (2*x^(1/3))/((-((I - Sqrt[ 3])/(I + Sqrt[3])))^(1/3)*(-1 + x)^(1/3)))/Sqrt[3]])/(3*Sqrt[3]) - ArcTan[ (1 + (2*(-((I - Sqrt[3])/(I + Sqrt[3])))^(1/3)*x^(1/3))/(-1 + x)^(1/3))/Sq rt[3]]/(3*Sqrt[3]*(-((I - Sqrt[3])/(I + Sqrt[3])))^(1/3)) - Log[1 - (2^(1/ 3)*(1 - x^(1/3)))/(-1 + x)^(1/3)]/(27*2^(1/3)) + Log[1 + (2^(2/3)*(1 - x^( 1/3))^2)/(-1 + x)^(2/3) + (2^(1/3)*(1 - x^(1/3)))/(-1 + x)^(1/3)]/(54*2^(1 /3)) - Log[2^(1/3) + (-1 + x)^(1/3)]/(18*2^(1/3)) + Log[1 + 2^(2/3)*(-1 + x)^(1/3) - x^(1/3)]/(18*2^(1/3)) - Log[(1 - x^(1/3))*(1 + x^(1/3))^2]/(54* 2^(1/3)) + Log[-(-1 + x)^(1/3) + 2^(1/3)*x^(1/3)]/(9*2^(1/3)) + ((-((I - S qrt[3])/(I + Sqrt[3])))^(1/3)*Log[-(-1 + x)^(1/3) + x^(1/3)/(-((I - Sqrt[3 ])/(I + Sqrt[3])))^(1/3)])/6 + Log[-(-1 + x)^(1/3) + (-((I - Sqrt[3])/(I + Sqrt[3])))^(1/3)*x^(1/3)]/(6*(-((I - Sqrt[3])/(I + Sqrt[3])))^(1/3)) - Lo g[1 + x]/(54*2^(1/3)) - ((-((I - Sqrt[3])/(I + Sqrt[3])))^(1/3)*Log[-1 - I *Sqrt[3] + 2*x])/18 - Log[-1 + I*Sqrt[3] + 2*x]/(18*(-((I - Sqrt[3])/(I...
3.26.22.3.1 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 = 14.29 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.83
| method | result | size |
| pseudoelliptic | \(\frac {-10 \,2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}} x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{4}+40 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right ) x^{4}+20 \sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (2^{\frac {2}{3}} \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}+x \right )}{3 x}\right ) x^{4}+20 \,2^{\frac {2}{3}} \ln \left (\frac {-2^{\frac {1}{3}} x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}{x}\right ) x^{4}+81 \left (\left (-1+x \right ) x^{2}\right )^{\frac {2}{3}} \left (x^{2}+\frac {2}{3} x +\frac {5}{9}\right )}{120 x^{4}}\) | \(176\) |
| trager | \(\text {Expression too large to display}\) | \(1758\) |
| risch | \(\text {Expression too large to display}\) | \(4518\) |
1/120*(-10*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)*x^4+40*sum(ln((-_R*x+((-1+x)*x^2)^(1/3))/x)/_R,_R=RootOf(_Z ^6-_Z^3+1))*x^4+20*3^(1/2)*2^(2/3)*arctan(1/3*3^(1/2)*(2^(2/3)*((-1+x)*x^2 )^(1/3)+x)/x)*x^4+20*2^(2/3)*ln((-2^(1/3)*x+((-1+x)*x^2)^(1/3))/x)*x^4+81* ((-1+x)*x^2)^(2/3)*(x^2+2/3*x+5/9))/x^4
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.28 (sec) , antiderivative size = 621, normalized size of antiderivative = 2.94 \[ \int \frac {1}{x^3 \left (1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\frac {20 \cdot 2^{\frac {2}{3}} x^{4} {\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 ) + 20 \cdot 2^{\frac {2}{3}} x^{4} {\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 ) + 20 \, \sqrt {6} 2^{\frac {1}{6}} x^{4} \arctan \left (\frac {2^{\frac {1}{6}} {\left (\sqrt {6} 2^{\frac {1}{3}} x + 2 \, \sqrt {6} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}\right )}}{6 \, x}\right ) + 20 \cdot 2^{\frac {2}{3}} x^{4} \log \left (-\frac {2^{\frac {1}{3}} x - {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - 10 \cdot 2^{\frac {2}{3}} x^{4} \log \left (\frac {2^{\frac {2}{3}} x^{2} + 2^{\frac {1}{3}} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} x + {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) - 10 \cdot 2^{\frac {2}{3}} {\left (\sqrt {-3} x^{4} + x^{4}\right )} {\left (i \, \sqrt {3} + 1\right )}^{\frac {1}{3}} \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 ) + 10 \cdot 2^{\frac {2}{3}} {\left (\sqrt {-3} x^{4} - x^{4}\right )} {\left (i \, \sqrt {3} + 1\right )}^{\frac {1}{3}} \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 ) + 10 \cdot 2^{\frac {2}{3}} {\left (\sqrt {-3} x^{4} - x^{4}\right )} {\left (-i \, \sqrt {3} + 1\right )}^{\frac {1}{3}} \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 ) - 10 \cdot 2^{\frac {2}{3}} {\left (\sqrt {-3} x^{4} + x^{4}\right )} {\left (-i \, \sqrt {3} + 1\right )}^{\frac {1}{3}} \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 ) + 9 \, {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}} {\left (9 \, x^{2} + 6 \, x + 5\right )}}{120 \, x^{4}} \]
1/120*(20*2^(2/3)*x^4*(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) + 20*2^(2/3)*x^4* (-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) + 20*sqrt(6)*2^(1/6)*x^4*arctan(1/6* 2^(1/6)*(sqrt(6)*2^(1/3)*x + 2*sqrt(6)*(x^3 - x^2)^(1/3))/x) + 20*2^(2/3)* x^4*log(-(2^(1/3)*x - (x^3 - x^2)^(1/3))/x) - 10*2^(2/3)*x^4*log((2^(2/3)* x^2 + 2^(1/3)*(x^3 - x^2)^(1/3)*x + (x^3 - x^2)^(2/3))/x^2) - 10*2^(2/3)*( sqrt(-3)*x^4 + x^4)*(I*sqrt(3) + 1)^(1/3)*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) + 10*2^(2/3)*(sqrt(-3)*x^4 - x^4)*(I*sqrt(3) + 1)^(1/3)*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) + 10*2^(2/3)*(sqrt(-3)*x^4 - x^4) *(-I*sqrt(3) + 1)^(1/3)*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) - 10 *2^(2/3)*(sqrt(-3)*x^4 + x^4)*(-I*sqrt(3) + 1)^(1/3)*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) + 9*(x^3 - x^2)^(2/3)*(9*x^2 + 6*x + 5))/x^4
Not integrable
Time = 0.80 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.12 \[ \int \frac {1}{x^3 \left (1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\int \frac {1}{x^{3} \sqrt [3]{x^{2} \left (x - 1\right )} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]
Not integrable
Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.11 \[ \int \frac {1}{x^3 \left (1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\int { \frac {1}{{\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} + 1\right )} x^{3}} \,d x } \]
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 38.66 (sec) , antiderivative size = 1007, normalized size of antiderivative = 4.77 \[ \int \frac {1}{x^3 \left (1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\text {Too large to display} \]
3/8*(1/x - 1)^2*(-1/x + 1)^(2/3) + 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*p i)^5 + sqrt(3)*cos(4/9*pi)^2 - sqrt(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*co s(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)*sin(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*s qrt(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))*arct an(-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*sqrt(3)*cos(4/9*pi)^4*sin(4/9*pi) - 10*sqr t(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...
Not integrable
Time = 7.32 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.11 \[ \int \frac {1}{x^3 \left (1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\int \frac {1}{x^3\,\left (x^3+1\right )\,{\left (x^3-x^2\right )}^{1/3}} \,d x \]