Integrand size = 24, antiderivative size = 210 \[ \int \frac {x^3 \sqrt [3]{-x^2+x^3}}{1+x^6} \, dx=\frac {1}{6} \text {RootSum}\left [2-2 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^3}\&\right ]-\frac {1}{6} \text {RootSum}\left [1-2 \text {$\#$1}^3+5 \text {$\#$1}^6-4 \text {$\#$1}^9+\text {$\#$1}^{12}\&,\frac {\log (x) \text {$\#$1}-\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}+2 \log (x) \text {$\#$1}^4-2 \log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^4-\log (x) \text {$\#$1}^7+\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^7}{-1+5 \text {$\#$1}^3-6 \text {$\#$1}^6+2 \text {$\#$1}^9}\&\right ] \]
Time = 0.29 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.09 \[ \int \frac {x^3 \sqrt [3]{-x^2+x^3}}{1+x^6} \, dx=\frac {(-1+x)^{2/3} x^{4/3} \left (3 \text {RootSum}\left [2-2 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log \left (\sqrt [3]{x}\right ) \text {$\#$1}+\log \left (\sqrt [3]{-1+x}-\sqrt [3]{x} \text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^3}\&\right ]+\text {RootSum}\left [1-2 \text {$\#$1}^3+5 \text {$\#$1}^6-4 \text {$\#$1}^9+\text {$\#$1}^{12}\&,\frac {-\log (x) \text {$\#$1}+3 \log \left (\sqrt [3]{-1+x}-\sqrt [3]{x} \text {$\#$1}\right ) \text {$\#$1}-2 \log (x) \text {$\#$1}^4+6 \log \left (\sqrt [3]{-1+x}-\sqrt [3]{x} \text {$\#$1}\right ) \text {$\#$1}^4+\log (x) \text {$\#$1}^7-3 \log \left (\sqrt [3]{-1+x}-\sqrt [3]{x} \text {$\#$1}\right ) \text {$\#$1}^7}{-1+5 \text {$\#$1}^3-6 \text {$\#$1}^6+2 \text {$\#$1}^9}\&\right ]\right )}{18 \left ((-1+x) x^2\right )^{2/3}} \]
((-1 + x)^(2/3)*x^(4/3)*(3*RootSum[2 - 2*#1^3 + #1^6 & , (-(Log[x^(1/3)]*# 1) + Log[(-1 + x)^(1/3) - x^(1/3)*#1]*#1)/(-1 + #1^3) & ] + RootSum[1 - 2* #1^3 + 5*#1^6 - 4*#1^9 + #1^12 & , (-(Log[x]*#1) + 3*Log[(-1 + x)^(1/3) - x^(1/3)*#1]*#1 - 2*Log[x]*#1^4 + 6*Log[(-1 + x)^(1/3) - x^(1/3)*#1]*#1^4 + Log[x]*#1^7 - 3*Log[(-1 + x)^(1/3) - x^(1/3)*#1]*#1^7)/(-1 + 5*#1^3 - 6*# 1^6 + 2*#1^9) & ]))/(18*((-1 + x)*x^2)^(2/3))
Result contains complex when optimal does not.
Time = 5.33 (sec) , antiderivative size = 3046, normalized size of antiderivative = 14.50, 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.
| \(\displaystyle \int \frac {x^3 \sqrt [3]{x^3-x^2}}{x^6+1} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{x^3-x^2} \int \frac {\sqrt [3]{x-1} x^{11/3}}{x^6+1}dx}{\sqrt [3]{x-1} x^{2/3}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {3 \sqrt [3]{x^3-x^2} \int \frac {\sqrt [3]{x-1} x^{13/3}}{x^6+1}d\sqrt [3]{x}}{\sqrt [3]{x-1} x^{2/3}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {3 \sqrt [3]{x^3-x^2} \int \left (\frac {\sqrt [3]{x-1} x^{4/3}}{2 \left (x^3-i\right )}+\frac {\sqrt [3]{x-1} x^{4/3}}{2 \left (x^3+i\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x-1} x^{2/3}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 \sqrt [3]{x^3-x^2} \left (\frac {i \sqrt [3]{1-i} \arctan \left (\frac {\frac {2 \sqrt [3]{1-i} \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right )}{18 \sqrt {3}}+\frac {2 \arctan \left (\frac {\frac {2 \sqrt [3]{1-i} \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right )}{9 (1-i)^{5/3} \sqrt {3}}-\frac {i \sqrt [3]{1+i} \arctan \left (\frac {\frac {2 \sqrt [3]{1+i} \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right )}{18 \sqrt {3}}+\frac {2 \arctan \left (\frac {\frac {2 \sqrt [3]{1+i} \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right )}{9 (1+i)^{5/3} \sqrt {3}}+\frac {i \arctan \left (\frac {1-\frac {2 \sqrt [3]{-\frac {(2+i)-\sqrt {3}}{i-\sqrt {3}}} \sqrt [3]{x}}{\sqrt [3]{x-1}}}{\sqrt {3}}\right )}{9 \left ((2+i)-\sqrt {3}\right )^{2/3} \sqrt [3]{-i+\sqrt {3}}}-\frac {1}{18} \sqrt [3]{-\frac {(2+i)-\sqrt {3}}{i-\sqrt {3}}} \arctan \left (\frac {1-\frac {2 \sqrt [3]{-\frac {(2+i)-\sqrt {3}}{i-\sqrt {3}}} \sqrt [3]{x}}{\sqrt [3]{x-1}}}{\sqrt {3}}\right )+\frac {\left (\frac {1}{18}-\frac {i}{18}\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{-\frac {(2+i)-\sqrt {3}}{i-\sqrt {3}}} \sqrt [3]{x}}{\sqrt [3]{x-1}}}{\sqrt {3}}\right )}{\left (-\frac {(2+i)-\sqrt {3}}{i-\sqrt {3}}\right )^{2/3}}-\frac {i \arctan \left (\frac {1-\frac {2 \sqrt [3]{-\frac {(-2+i)+\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x}}{\sqrt [3]{x-1}}}{\sqrt {3}}\right )}{9 \left ((2-i)-\sqrt {3}\right )^{2/3} \sqrt [3]{i+\sqrt {3}}}-\frac {1}{18} \sqrt [3]{-\frac {(-2+i)+\sqrt {3}}{i+\sqrt {3}}} \arctan \left (\frac {1-\frac {2 \sqrt [3]{-\frac {(-2+i)+\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x}}{\sqrt [3]{x-1}}}{\sqrt {3}}\right )+\frac {\left (\frac {1}{18}+\frac {i}{18}\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{-\frac {(-2+i)+\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x}}{\sqrt [3]{x-1}}}{\sqrt {3}}\right )}{\left (-\frac {(-2+i)+\sqrt {3}}{i+\sqrt {3}}\right )^{2/3}}-\frac {i \arctan \left (\frac {\frac {2 \sqrt [3]{x}}{\sqrt [3]{-\frac {i-\sqrt {3}}{(2-i)+\sqrt {3}}} \sqrt [3]{x-1}}+1}{\sqrt {3}}\right )}{9 \sqrt [3]{-i+\sqrt {3}} \left ((2-i)+\sqrt {3}\right )^{2/3}}-\left (\frac {1}{18}+\frac {i}{18}\right ) \left (-\frac {i-\sqrt {3}}{(2-i)+\sqrt {3}}\right )^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{x}}{\sqrt [3]{-\frac {i-\sqrt {3}}{(2-i)+\sqrt {3}}} \sqrt [3]{x-1}}+1}{\sqrt {3}}\right )-\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{x}}{\sqrt [3]{-\frac {i-\sqrt {3}}{(2-i)+\sqrt {3}}} \sqrt [3]{x-1}}+1}{\sqrt {3}}\right )}{18 \sqrt [3]{-\frac {i-\sqrt {3}}{(2-i)+\sqrt {3}}}}+\frac {i \arctan \left (\frac {\frac {2 \sqrt [3]{x}}{\sqrt [3]{\frac {i+\sqrt {3}}{(2+i)+\sqrt {3}}} \sqrt [3]{x-1}}+1}{\sqrt {3}}\right )}{9 \sqrt [3]{i+\sqrt {3}} \left ((2+i)+\sqrt {3}\right )^{2/3}}-\left (\frac {1}{18}-\frac {i}{18}\right ) \left (\frac {i+\sqrt {3}}{(2+i)+\sqrt {3}}\right )^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{x}}{\sqrt [3]{\frac {i+\sqrt {3}}{(2+i)+\sqrt {3}}} \sqrt [3]{x-1}}+1}{\sqrt {3}}\right )-\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{x}}{\sqrt [3]{\frac {i+\sqrt {3}}{(2+i)+\sqrt {3}}} \sqrt [3]{x-1}}+1}{\sqrt {3}}\right )}{18 \sqrt [3]{\frac {i+\sqrt {3}}{(2+i)+\sqrt {3}}}}+\frac {1}{36} i \sqrt [3]{1-i} \log \left (\sqrt [3]{1-i} \sqrt [3]{x}-\sqrt [3]{x-1}\right )+\frac {\log \left (\sqrt [3]{1-i} \sqrt [3]{x}-\sqrt [3]{x-1}\right )}{9 (1-i)^{5/3}}-\frac {1}{36} i \sqrt [3]{1+i} \log \left (\sqrt [3]{1+i} \sqrt [3]{x}-\sqrt [3]{x-1}\right )+\frac {\log \left (\sqrt [3]{1+i} \sqrt [3]{x}-\sqrt [3]{x-1}\right )}{9 (1+i)^{5/3}}+\frac {i \log \left (-\sqrt [3]{x-1}-\sqrt [3]{-\frac {(2+i)-\sqrt {3}}{i-\sqrt {3}}} \sqrt [3]{x}\right )}{6 \sqrt {3} \left ((2+i)-\sqrt {3}\right )^{2/3} \sqrt [3]{-i+\sqrt {3}}}-\frac {\sqrt [3]{-\frac {(2+i)-\sqrt {3}}{i-\sqrt {3}}} \log \left (-\sqrt [3]{x-1}-\sqrt [3]{-\frac {(2+i)-\sqrt {3}}{i-\sqrt {3}}} \sqrt [3]{x}\right )}{12 \sqrt {3}}+\frac {\left (\frac {1}{12}-\frac {i}{12}\right ) \log \left (-\sqrt [3]{x-1}-\sqrt [3]{-\frac {(2+i)-\sqrt {3}}{i-\sqrt {3}}} \sqrt [3]{x}\right )}{\sqrt {3} \left (-\frac {(2+i)-\sqrt {3}}{i-\sqrt {3}}\right )^{2/3}}-\frac {i \log \left (-\sqrt [3]{x-1}-\sqrt [3]{-\frac {(-2+i)+\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x}\right )}{6 \sqrt {3} \left ((2-i)-\sqrt {3}\right )^{2/3} \sqrt [3]{i+\sqrt {3}}}-\frac {\sqrt [3]{-\frac {(-2+i)+\sqrt {3}}{i+\sqrt {3}}} \log \left (-\sqrt [3]{x-1}-\sqrt [3]{-\frac {(-2+i)+\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x}\right )}{12 \sqrt {3}}+\frac {\left (\frac {1}{12}+\frac {i}{12}\right ) \log \left (-\sqrt [3]{x-1}-\sqrt [3]{-\frac {(-2+i)+\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x}\right )}{\sqrt {3} \left (-\frac {(-2+i)+\sqrt {3}}{i+\sqrt {3}}\right )^{2/3}}-\frac {i \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{-\frac {i-\sqrt {3}}{(2-i)+\sqrt {3}}}}-\sqrt [3]{x-1}\right )}{6 \sqrt {3} \sqrt [3]{-i+\sqrt {3}} \left ((2-i)+\sqrt {3}\right )^{2/3}}-\frac {\left (\frac {1}{12}+\frac {i}{12}\right ) \left (-\frac {i-\sqrt {3}}{(2-i)+\sqrt {3}}\right )^{2/3} \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{-\frac {i-\sqrt {3}}{(2-i)+\sqrt {3}}}}-\sqrt [3]{x-1}\right )}{\sqrt {3}}-\frac {\log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{-\frac {i-\sqrt {3}}{(2-i)+\sqrt {3}}}}-\sqrt [3]{x-1}\right )}{12 \sqrt {3} \sqrt [3]{-\frac {i-\sqrt {3}}{(2-i)+\sqrt {3}}}}+\frac {i \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{\frac {i+\sqrt {3}}{(2+i)+\sqrt {3}}}}-\sqrt [3]{x-1}\right )}{6 \sqrt {3} \sqrt [3]{i+\sqrt {3}} \left ((2+i)+\sqrt {3}\right )^{2/3}}-\frac {\left (\frac {1}{12}-\frac {i}{12}\right ) \left (\frac {i+\sqrt {3}}{(2+i)+\sqrt {3}}\right )^{2/3} \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{\frac {i+\sqrt {3}}{(2+i)+\sqrt {3}}}}-\sqrt [3]{x-1}\right )}{\sqrt {3}}-\frac {\log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{\frac {i+\sqrt {3}}{(2+i)+\sqrt {3}}}}-\sqrt [3]{x-1}\right )}{12 \sqrt {3} \sqrt [3]{\frac {i+\sqrt {3}}{(2+i)+\sqrt {3}}}}-\frac {i \log \left (-2 x+\sqrt {3}-i\right )}{18 \sqrt {3} \left ((2+i)-\sqrt {3}\right )^{2/3} \sqrt [3]{-i+\sqrt {3}}}+\frac {\sqrt [3]{-\frac {(2+i)-\sqrt {3}}{i-\sqrt {3}}} \log \left (-2 x+\sqrt {3}-i\right )}{36 \sqrt {3}}-\frac {\left (\frac {1}{36}-\frac {i}{36}\right ) \log \left (-2 x+\sqrt {3}-i\right )}{\sqrt {3} \left (-\frac {(2+i)-\sqrt {3}}{i-\sqrt {3}}\right )^{2/3}}+\frac {i \log \left (-2 x+\sqrt {3}+i\right )}{18 \sqrt {3} \left ((2-i)-\sqrt {3}\right )^{2/3} \sqrt [3]{i+\sqrt {3}}}+\frac {\sqrt [3]{-\frac {(-2+i)+\sqrt {3}}{i+\sqrt {3}}} \log \left (-2 x+\sqrt {3}+i\right )}{36 \sqrt {3}}-\frac {\left (\frac {1}{36}+\frac {i}{36}\right ) \log \left (-2 x+\sqrt {3}+i\right )}{\sqrt {3} \left (-\frac {(-2+i)+\sqrt {3}}{i+\sqrt {3}}\right )^{2/3}}-\frac {\log (-i x-1)}{27 (1+i)^{5/3}}-\frac {\log (i x-1)}{27 (1-i)^{5/3}}+\frac {1}{108} i \sqrt [3]{1+i} \log (x-i)-\frac {1}{108} i \sqrt [3]{1-i} \log (x+i)+\frac {i \log \left (2 x+\sqrt {3}-i\right )}{18 \sqrt {3} \sqrt [3]{-i+\sqrt {3}} \left ((2-i)+\sqrt {3}\right )^{2/3}}+\frac {\left (\frac {1}{36}+\frac {i}{36}\right ) \left (-\frac {i-\sqrt {3}}{(2-i)+\sqrt {3}}\right )^{2/3} \log \left (2 x+\sqrt {3}-i\right )}{\sqrt {3}}+\frac {\log \left (2 x+\sqrt {3}-i\right )}{36 \sqrt {3} \sqrt [3]{-\frac {i-\sqrt {3}}{(2-i)+\sqrt {3}}}}-\frac {i \log \left (2 x+\sqrt {3}+i\right )}{18 \sqrt {3} \sqrt [3]{i+\sqrt {3}} \left ((2+i)+\sqrt {3}\right )^{2/3}}+\frac {\left (\frac {1}{36}-\frac {i}{36}\right ) \left (\frac {i+\sqrt {3}}{(2+i)+\sqrt {3}}\right )^{2/3} \log \left (2 x+\sqrt {3}+i\right )}{\sqrt {3}}+\frac {\log \left (2 x+\sqrt {3}+i\right )}{36 \sqrt {3} \sqrt [3]{\frac {i+\sqrt {3}}{(2+i)+\sqrt {3}}}}\right )}{\sqrt [3]{x-1} x^{2/3}}\) |
(3*(-x^2 + x^3)^(1/3)*((2*ArcTan[(1 + (2*(1 - I)^(1/3)*x^(1/3))/(-1 + x)^( 1/3))/Sqrt[3]])/(9*(1 - I)^(5/3)*Sqrt[3]) + ((I/18)*(1 - I)^(1/3)*ArcTan[( 1 + (2*(1 - I)^(1/3)*x^(1/3))/(-1 + x)^(1/3))/Sqrt[3]])/Sqrt[3] + (2*ArcTa n[(1 + (2*(1 + I)^(1/3)*x^(1/3))/(-1 + x)^(1/3))/Sqrt[3]])/(9*(1 + I)^(5/3 )*Sqrt[3]) - ((I/18)*(1 + I)^(1/3)*ArcTan[(1 + (2*(1 + I)^(1/3)*x^(1/3))/( -1 + x)^(1/3))/Sqrt[3]])/Sqrt[3] + ((1/18 - I/18)*ArcTan[(1 - (2*(-(((2 + I) - Sqrt[3])/(I - Sqrt[3])))^(1/3)*x^(1/3))/(-1 + x)^(1/3))/Sqrt[3]])/(-( ((2 + I) - Sqrt[3])/(I - Sqrt[3])))^(2/3) - ((-(((2 + I) - Sqrt[3])/(I - S qrt[3])))^(1/3)*ArcTan[(1 - (2*(-(((2 + I) - Sqrt[3])/(I - Sqrt[3])))^(1/3 )*x^(1/3))/(-1 + x)^(1/3))/Sqrt[3]])/18 + ((I/9)*ArcTan[(1 - (2*(-(((2 + I ) - Sqrt[3])/(I - Sqrt[3])))^(1/3)*x^(1/3))/(-1 + x)^(1/3))/Sqrt[3]])/(((2 + I) - Sqrt[3])^(2/3)*(-I + Sqrt[3])^(1/3)) + ((1/18 + I/18)*ArcTan[(1 - (2*(-(((-2 + I) + Sqrt[3])/(I + Sqrt[3])))^(1/3)*x^(1/3))/(-1 + x)^(1/3))/ Sqrt[3]])/(-(((-2 + I) + Sqrt[3])/(I + Sqrt[3])))^(2/3) - ((-(((-2 + I) + Sqrt[3])/(I + Sqrt[3])))^(1/3)*ArcTan[(1 - (2*(-(((-2 + I) + Sqrt[3])/(I + Sqrt[3])))^(1/3)*x^(1/3))/(-1 + x)^(1/3))/Sqrt[3]])/18 - ((I/9)*ArcTan[(1 - (2*(-(((-2 + I) + Sqrt[3])/(I + Sqrt[3])))^(1/3)*x^(1/3))/(-1 + x)^(1/3 ))/Sqrt[3]])/(((2 - I) - Sqrt[3])^(2/3)*(I + Sqrt[3])^(1/3)) - ArcTan[(1 + (2*x^(1/3))/((-((I - Sqrt[3])/((2 - I) + Sqrt[3])))^(1/3)*(-1 + x)^(1/3)) )/Sqrt[3]]/(18*(-((I - Sqrt[3])/((2 - I) + Sqrt[3])))^(1/3)) - (1/18 + ...
3.26.19.3.1 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 = 83.20 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.58
| method | result | size |
| pseudoelliptic | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-2 \textit {\_Z}^{3}+2\right )}{\sum }\frac {\textit {\_R} \ln \left (\frac {-\textit {\_R} x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}^{3}-1}\right )}{6}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{12}-4 \textit {\_Z}^{9}+5 \textit {\_Z}^{6}-2 \textit {\_Z}^{3}+1\right )}{\sum }\frac {\textit {\_R} \left (\textit {\_R}^{6}-2 \textit {\_R}^{3}-1\right ) \ln \left (\frac {-\textit {\_R} x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}{x}\right )}{2 \textit {\_R}^{9}-6 \textit {\_R}^{6}+5 \textit {\_R}^{3}-1}\right )}{6}\) | \(122\) |
| trager | \(\text {Expression too large to display}\) | \(78775\) |
1/6*sum(_R*ln((-_R*x+((-1+x)*x^2)^(1/3))/x)/(_R^3-1),_R=RootOf(_Z^6-2*_Z^3 +2))-1/6*sum(_R*(_R^6-2*_R^3-1)*ln((-_R*x+((-1+x)*x^2)^(1/3))/x)/(2*_R^9-6 *_R^6+5*_R^3-1),_R=RootOf(_Z^12-4*_Z^9+5*_Z^6-2*_Z^3+1))
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.30 (sec) , antiderivative size = 1493, normalized size of antiderivative = 7.11 \[ \int \frac {x^3 \sqrt [3]{-x^2+x^3}}{1+x^6} \, dx=\text {Too large to display} \]
-1/24*2^(2/3)*(sqrt(-3) + 1)*(sqrt(4*sqrt(3) - 7) + 1)^(1/3)*log(((sqrt(3) *2^(2/3)*(sqrt(-3)*x + x) - (sqrt(3)*2^(2/3)*(sqrt(-3)*x + x) + 2*2^(2/3)* (sqrt(-3)*x + x))*sqrt(4*sqrt(3) - 7))*(sqrt(4*sqrt(3) - 7) + 1)^(1/3) + 8 *(x^3 - x^2)^(1/3))/x) + 1/24*2^(2/3)*(sqrt(-3) - 1)*(sqrt(4*sqrt(3) - 7) + 1)^(1/3)*log(-((sqrt(3)*2^(2/3)*(sqrt(-3)*x - x) - (sqrt(3)*2^(2/3)*(sqr t(-3)*x - x) + 2*2^(2/3)*(sqrt(-3)*x - x))*sqrt(4*sqrt(3) - 7))*(sqrt(4*sq rt(3) - 7) + 1)^(1/3) - 8*(x^3 - x^2)^(1/3))/x) - 1/24*2^(2/3)*(sqrt(-3) + 1)*(-sqrt(4*sqrt(3) - 7) + 1)^(1/3)*log(((sqrt(3)*2^(2/3)*(sqrt(-3)*x + x ) + (sqrt(3)*2^(2/3)*(sqrt(-3)*x + x) + 2*2^(2/3)*(sqrt(-3)*x + x))*sqrt(4 *sqrt(3) - 7))*(-sqrt(4*sqrt(3) - 7) + 1)^(1/3) + 8*(x^3 - x^2)^(1/3))/x) + 1/24*2^(2/3)*(sqrt(-3) - 1)*(-sqrt(4*sqrt(3) - 7) + 1)^(1/3)*log(-((sqrt (3)*2^(2/3)*(sqrt(-3)*x - x) + (sqrt(3)*2^(2/3)*(sqrt(-3)*x - x) + 2*2^(2/ 3)*(sqrt(-3)*x - x))*sqrt(4*sqrt(3) - 7))*(-sqrt(4*sqrt(3) - 7) + 1)^(1/3) - 8*(x^3 - x^2)^(1/3))/x) - 1/24*2^(2/3)*(sqrt(-3) + 1)*(sqrt(-4*sqrt(3) - 7) + 1)^(1/3)*log(-((sqrt(3)*2^(2/3)*(sqrt(-3)*x + x) - (sqrt(3)*2^(2/3) *(sqrt(-3)*x + x) - 2*2^(2/3)*(sqrt(-3)*x + x))*sqrt(-4*sqrt(3) - 7))*(sqr t(-4*sqrt(3) - 7) + 1)^(1/3) - 8*(x^3 - x^2)^(1/3))/x) + 1/24*2^(2/3)*(sqr t(-3) - 1)*(sqrt(-4*sqrt(3) - 7) + 1)^(1/3)*log(((sqrt(3)*2^(2/3)*(sqrt(-3 )*x - x) - (sqrt(3)*2^(2/3)*(sqrt(-3)*x - x) - 2*2^(2/3)*(sqrt(-3)*x - x)) *sqrt(-4*sqrt(3) - 7))*(sqrt(-4*sqrt(3) - 7) + 1)^(1/3) + 8*(x^3 - x^2)...
Not integrable
Time = 0.58 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.13 \[ \int \frac {x^3 \sqrt [3]{-x^2+x^3}}{1+x^6} \, dx=\int \frac {x^{3} \sqrt [3]{x^{2} \left (x - 1\right )}}{\left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 1\right )}\, dx \]
Not integrable
Time = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.11 \[ \int \frac {x^3 \sqrt [3]{-x^2+x^3}}{1+x^6} \, dx=\int { \frac {{\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} x^{3}}{x^{6} + 1} \,d x } \]
Not integrable
Time = 0.64 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.11 \[ \int \frac {x^3 \sqrt [3]{-x^2+x^3}}{1+x^6} \, dx=\int { \frac {{\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} x^{3}}{x^{6} + 1} \,d x } \]
Not integrable
Time = 6.72 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.11 \[ \int \frac {x^3 \sqrt [3]{-x^2+x^3}}{1+x^6} \, dx=\int \frac {x^3\,{\left (x^3-x^2\right )}^{1/3}}{x^6+1} \,d x \]