Integrand size = 22, antiderivative size = 199 \[ \int \frac {1}{x^3 \left (-1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\frac {3 \left (5-6 x+9 x^2\right ) \left (x^2+x^3\right )^{2/3}}{40 x^4}-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{x^2+x^3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{x^2+x^3}\right )}{3 \sqrt [3]{2}}-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{x^2+x^3}+\sqrt [3]{2} \left (x^2+x^3\right )^{2/3}\right )}{6 \sqrt [3]{2}}+\frac {1}{3} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{x^2+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]
Time = 0.00 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.26 \[ \int \frac {1}{x^3 \left (-1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\frac {(1+x) \left (45-9 x+27 x^2+81 x^3-20\ 2^{2/3} \sqrt {3} x^{8/3} \sqrt [3]{1+x} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}+2^{2/3} \sqrt [3]{1+x}}\right )+20\ 2^{2/3} x^{8/3} \sqrt [3]{1+x} \log \left (-2 \sqrt [3]{x}+2^{2/3} \sqrt [3]{1+x}\right )-10\ 2^{2/3} x^{8/3} \sqrt [3]{1+x} \log \left (2 x^{2/3}+2^{2/3} \sqrt [3]{x} \sqrt [3]{1+x}+\sqrt [3]{2} (1+x)^{2/3}\right )+40 x^{8/3} \sqrt [3]{1+x} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{1+x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right )}{120 \left (x^2 (1+x)\right )^{4/3}} \]
((1 + x)*(45 - 9*x + 27*x^2 + 81*x^3 - 20*2^(2/3)*Sqrt[3]*x^(8/3)*(1 + x)^ (1/3)*ArcTan[(Sqrt[3]*x^(1/3))/(x^(1/3) + 2^(2/3)*(1 + x)^(1/3))] + 20*2^( 2/3)*x^(8/3)*(1 + x)^(1/3)*Log[-2*x^(1/3) + 2^(2/3)*(1 + x)^(1/3)] - 10*2^ (2/3)*x^(8/3)*(1 + x)^(1/3)*Log[2*x^(2/3) + 2^(2/3)*x^(1/3)*(1 + x)^(1/3) + 2^(1/3)*(1 + x)^(2/3)] + 40*x^(8/3)*(1 + x)^(1/3)*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))^(4/3))
Result contains complex when optimal does not.
Time = 1.31 (sec) , antiderivative size = 943, normalized size of antiderivative = 4.74, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, 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.
| \(\displaystyle \int \frac {1}{x^3 \left (x^3-1\right ) \sqrt [3]{x^3+x^2}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{2/3} \sqrt [3]{x+1} \int -\frac {1}{x^{11/3} \sqrt [3]{x+1} \left (1-x^3\right )}dx}{\sqrt [3]{x^3+x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x^{2/3} \sqrt [3]{x+1} \int \frac {1}{x^{11/3} \sqrt [3]{x+1} \left (1-x^3\right )}dx}{\sqrt [3]{x^3+x^2}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{x+1} \int \frac {1}{x^3 \sqrt [3]{x+1} \left (1-x^3\right )}d\sqrt [3]{x}}{\sqrt [3]{x^3+x^2}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{x+1} \int \left (\frac {\sqrt [3]{x}+2}{9 \left (x^{2/3}+\sqrt [3]{x}+1\right ) \sqrt [3]{x+1}}-\frac {1}{9 \left (\sqrt [3]{x}-1\right ) \sqrt [3]{x+1}}+\frac {1}{x^3 \sqrt [3]{x+1}}+\frac {x+2}{3 \sqrt [3]{x+1} \left (x^2+x+1\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x^3+x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{x+1} \left (-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{x}+1\right )}{\sqrt [3]{x+1}}}{\sqrt {3}}\right )}{9 \sqrt [3]{2} \sqrt {3}}+\frac {\arctan \left (\frac {\frac {\sqrt [3]{2} \left (\sqrt [3]{x}+1\right )}{\sqrt [3]{x+1}}+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 {\arctan \left (\frac {2^{2/3} \sqrt [3]{x+1}+1}{\sqrt {3}}\right )}{9 \sqrt [3]{2} \sqrt {3}}+\frac {\log \left (-\left (1-\sqrt [3]{x}\right )^2 \left (\sqrt [3]{x}+1\right )\right )}{36 \sqrt [3]{2}}-\frac {\log \left (\left (1-\sqrt [3]{x}\right )^2 \left (\sqrt [3]{x}+1\right )\right )}{108 \sqrt [3]{2}}+\frac {\log (1-x)}{54 \sqrt [3]{2}}+\frac {\log \left (2 x-i \sqrt {3}+1\right )}{18 \sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}}}+\frac {1}{18} \sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \log \left (2 x+i \sqrt {3}+1\right )-\frac {\log \left (\frac {2^{2/3} \left (\sqrt [3]{x}+1\right )^2}{(x+1)^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{x}+1\right )}{\sqrt [3]{x+1}}+1\right )}{54 \sqrt [3]{2}}+\frac {\log \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{x}+1\right )}{\sqrt [3]{x+1}}+1\right )}{27 \sqrt [3]{2}}+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{x+1}\right )}{18 \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 \left (\sqrt [3]{x}-2^{2/3} \sqrt [3]{x+1}+1\right )}{36 \sqrt [3]{2}}-\frac {\log \left (-\sqrt [3]{x}+2^{2/3} \sqrt [3]{x+1}-1\right )}{12 \sqrt [3]{2}}-\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*x^(2/3)*(1 + x)^(1/3)*(-1/8*(1 + x)^(2/3)/x^(8/3) + (3*(1 + x)^(2/3))/ (20*x^(5/3)) - (9*(1 + x)^(2/3))/(40*x^(2/3)) - ArcTan[(1 - (2*2^(1/3)*(1 + x^(1/3)))/(1 + x)^(1/3))/Sqrt[3]]/(9*2^(1/3)*Sqrt[3]) + ArcTan[(1 + (2^( 1/3)*(1 + x^(1/3)))/(1 + x)^(1/3))/Sqrt[3]]/(9*2^(1/3)*Sqrt[3]) + (2^(2/3) *ArcTan[(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 - Sqr t[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)) + ArcTan[(1 + 2^ (2/3)*(1 + x)^(1/3))/Sqrt[3]]/(9*2^(1/3)*Sqrt[3]) + Log[-((1 - x^(1/3))^2* (1 + x^(1/3)))]/(36*2^(1/3)) - Log[(1 - x^(1/3))^2*(1 + x^(1/3))]/(108*2^( 1/3)) + Log[1 - x]/(54*2^(1/3)) + Log[1 - I*Sqrt[3] + 2*x]/(18*(-((I - Sqr t[3])/(I + Sqrt[3])))^(1/3)) + ((-((I - Sqrt[3])/(I + Sqrt[3])))^(1/3)*Log [1 + I*Sqrt[3] + 2*x])/18 - 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[1 + (2^(1/3) *(1 + x^(1/3)))/(1 + x)^(1/3)]/(27*2^(1/3)) + Log[2^(1/3) - (1 + x)^(1/3)] /(18*2^(1/3)) - Log[2^(1/3)*x^(1/3) - (1 + x)^(1/3)]/(9*2^(1/3)) - ((-((I - Sqrt[3])/(I + Sqrt[3])))^(1/3)*Log[x^(1/3)/(-((I - Sqrt[3])/(I + Sqrt[3] )))^(1/3) - (1 + x)^(1/3)])/6 - Log[(-((I - Sqrt[3])/(I + Sqrt[3])))^(1/3) *x^(1/3) - (1 + x)^(1/3)]/(6*(-((I - Sqrt[3])/(I + Sqrt[3])))^(1/3)) + ...
3.25.51.3.1 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 = 30.82 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.88
| method | result | size |
| pseudoelliptic | \(\frac {-10 \,2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}} x +\left (x^{2} \left (1+x \right )\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{4}+40 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right ) x^{4}+20 \sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (2^{\frac {2}{3}} \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}+x \right )}{3 x}\right ) x^{4}+20 \,2^{\frac {2}{3}} \ln \left (\frac {-2^{\frac {1}{3}} x +\left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}}{x}\right ) x^{4}+81 \left (x^{2} \left (1+x \right )\right )^{\frac {2}{3}} \left (x^{2}-\frac {2}{3} x +\frac {5}{9}\right )}{120 x^{4}}\) | \(176\) |
| risch | \(\text {Expression too large to display}\) | \(2282\) |
| trager | \(\text {Expression too large to display}\) | \(4873\) |
1/120*(-10*2^(2/3)*ln((2^(2/3)*x^2+2^(1/3)*(x^2*(1+x))^(1/3)*x+(x^2*(1+x)) ^(2/3))/x^2)*x^4+40*sum(ln((-_R*x+(x^2*(1+x))^(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)*(x^2*(1+x))^(1 /3)+x)/x)*x^4+20*2^(2/3)*ln((-2^(1/3)*x+(x^2*(1+x))^(1/3))/x)*x^4+81*(x^2* (1+x))^(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 = 599, normalized size of antiderivative = 3.01 \[ \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 = 0.92 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.13 \[ \int \frac {1}{x^3 \left (-1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\int \frac {1}{x^{3} \sqrt [3]{x^{2} \left (x + 1\right )} \left (x - 1\right ) \left (x^{2} + x + 1\right )}\, dx \]
Not integrable
Time = 0.29 (sec) , antiderivative size = 22, 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 = 37.91 (sec) , antiderivative size = 968, normalized size of antiderivative = 4.86 \[ \int \frac {1}{x^3 \left (-1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\text {Too large to display} \]
3/8*(1/x + 1)^(8/3) + 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*pi)^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*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)*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*sqrt(3) + 1/2)*si n(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*sqr t(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*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) - c...
Not integrable
Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.13 \[ \int \frac {1}{x^3 \left (-1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=-\int \frac {1}{{\left (x^3+x^2\right )}^{1/3}\,\left (x^3-x^6\right )} \,d x \]