Integrand size = 19, antiderivative size = 173 \[ \int \frac {1}{\sqrt [3]{-x+x^3} \left (1+x^6\right )} \, dx=\frac {\arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-x+x^3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}-\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{-x+x^3}\right )}{6 \sqrt [3]{2}}+\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-x+x^3}+\sqrt [3]{2} \left (-x+x^3\right )^{2/3}\right )}{12 \sqrt [3]{2}}-\frac {1}{6} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{-x+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]
Time = 1.72 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.16 \[ \int \frac {1}{\sqrt [3]{-x+x^3} \left (1+x^6\right )} \, dx=\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \left (2^{2/3} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2^{2/3} \sqrt [3]{-1+x^2}}\right )-2 \log \left (-2 x^{2/3}+2^{2/3} \sqrt [3]{-1+x^2}\right )+\log \left (2 x^{4/3}+2^{2/3} x^{2/3} \sqrt [3]{-1+x^2}+\sqrt [3]{2} \left (-1+x^2\right )^{2/3}\right )\right )-4 \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-2 \log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{-1+x^2}-x^{2/3} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right )}{24 \sqrt [3]{x \left (-1+x^2\right )}} \]
(x^(1/3)*(-1 + x^2)^(1/3)*(2^(2/3)*(2*Sqrt[3]*ArcTan[(Sqrt[3]*x^(2/3))/(x^ (2/3) + 2^(2/3)*(-1 + x^2)^(1/3))] - 2*Log[-2*x^(2/3) + 2^(2/3)*(-1 + x^2) ^(1/3)] + Log[2*x^(4/3) + 2^(2/3)*x^(2/3)*(-1 + x^2)^(1/3) + 2^(1/3)*(-1 + x^2)^(2/3)]) - 4*RootSum[1 - #1^3 + #1^6 & , (-2*Log[x^(1/3)] + Log[(-1 + x^2)^(1/3) - x^(2/3)*#1])/#1 & ]))/(24*(x*(-1 + x^2))^(1/3))
Leaf count is larger than twice the leaf count of optimal. \(1733\) vs. \(2(173)=346\).
Time = 2.73 (sec) , antiderivative size = 1733, normalized size of antiderivative = 10.02, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2467, 2035, 7266, 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.
| \(\displaystyle \int \frac {1}{\sqrt [3]{x^3-x} \left (x^6+1\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{x} \sqrt [3]{x^2-1} \int \frac {1}{\sqrt [3]{x} \sqrt [3]{x^2-1} \left (x^6+1\right )}dx}{\sqrt [3]{x^3-x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{x^2-1} \int \frac {\sqrt [3]{x}}{\sqrt [3]{x^2-1} \left (x^6+1\right )}d\sqrt [3]{x}}{\sqrt [3]{x^3-x}}\) |
| \(\Big \downarrow \) 7266 |
| \(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{x^2-1} \int \frac {1}{\sqrt [3]{x-1} \left (x^3+1\right )}dx^{2/3}}{2 \sqrt [3]{x^3-x}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{x^2-1} \int \left (-\frac {1}{9 \left (-x^{2/3}-1\right ) \sqrt [3]{x-1}}-\frac {1}{9 \left (\sqrt [9]{-1} x^{2/3}-1\right ) \sqrt [3]{x-1}}-\frac {1}{9 \left (-(-1)^{2/9} x^{2/3}-1\right ) \sqrt [3]{x-1}}-\frac {1}{9 \left (\sqrt [3]{-1} x^{2/3}-1\right ) \sqrt [3]{x-1}}-\frac {1}{9 \left (-(-1)^{4/9} x^{2/3}-1\right ) \sqrt [3]{x-1}}-\frac {1}{9 \left ((-1)^{5/9} x^{2/3}-1\right ) \sqrt [3]{x-1}}-\frac {1}{9 \left (-(-1)^{2/3} x^{2/3}-1\right ) \sqrt [3]{x-1}}-\frac {1}{9 \left ((-1)^{7/9} x^{2/3}-1\right ) \sqrt [3]{x-1}}-\frac {1}{9 \left (-(-1)^{8/9} x^{2/3}-1\right ) \sqrt [3]{x-1}}\right )dx^{2/3}}{2 \sqrt [3]{x^3-x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{x^2-1} \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-x^{2/3}\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}-x^{2/3}\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} x^{2/3}\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}} x^{2/3}}{\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}} x^{2/3}}{\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-x^{2/3}\right ) \left (x^{2/3}+1\right )^2\right )\right )}{36 \sqrt [3]{2}}-\frac {\log \left (x^{2/3}-2^{2/3} \sqrt [3]{x-1}-1\right )}{12 \sqrt [3]{2}}-\frac {\log \left (-\sqrt [3]{-1} x^{2/3}+\sqrt [3]{-1} 2^{2/3} \sqrt [3]{x-1}-1\right )}{12 \sqrt [3]{2}}+\frac {\log \left (-(-1)^{2/3} \left (x^{2/3}+(-1)^{2/3}\right )^2 \left (\sqrt [3]{-1} x^{2/3}+1\right )\right )}{36 \sqrt [3]{2}}+\frac {\log \left ((-1)^{2/3} \left (x^{2/3}+\sqrt [3]{-1}\right ) \left ((-1)^{2/3} x^{2/3}+1\right )^2\right )}{36 \sqrt [3]{2}}-\frac {\log \left ((-1)^{2/3} x^{2/3}-(-2)^{2/3} \sqrt [3]{x-1}-1\right )}{12 \sqrt [3]{2}}-\frac {\log \left (\sqrt [3]{1-\sqrt [3]{-1}} x^{2/3}-\sqrt [3]{x-1}\right )}{6 \sqrt [3]{1-\sqrt [3]{-1}}}-\frac {\log \left (\sqrt [3]{1+(-1)^{2/3}} x^{2/3}-\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 )}{2 \sqrt [3]{x^3-x}}\) |
(3*x^(1/3)*(-1 + x^2)^(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)*Appe llF1[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)])/ (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)) + ArcTan[(1 - (2^(1/3)*( 1 - x^(2/3)))/(-1 + x)^(1/3))/Sqrt[3]]/(6*2^(1/3)*Sqrt[3]) + ArcTan[(1 - ( 2^(1/3)*((-1)^(2/3) - x^(2/3)))/(-1 + x)^(1/3))/Sqrt[3]]/(6*2^(1/3)*Sqrt[3 ]) + ArcTan[(1 + ((-2)^(1/3)*(1 - (-1)^(2/3)*x^(2/3)))/(-1 + x)^(1/3))/Sqr t[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* Sqrt[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))^...
3.23.76.3.1 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_)*(x_)^(m_.), x_Symbol] :> Simp[1/(m + 1) Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /; FreeQ[m, x] && NeQ[m, -1] && Function OfQ[x^(m + 1), u, 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 = 59.18 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.78
| method | result | size |
| pseudoelliptic | \(\frac {\left (-2 \arctan \left (\frac {\sqrt {3}\, \left (x +2^{\frac {2}{3}} \left (x^{3}-x \right )^{\frac {1}{3}}\right )}{3 x}\right ) \sqrt {3}-2 \ln \left (\frac {-2^{\frac {1}{3}} x +\left (x^{3}-x \right )^{\frac {1}{3}}}{x}\right )+\ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} \left (x^{3}-x \right )^{\frac {1}{3}} x +\left (x^{3}-x \right )^{\frac {2}{3}}}{x^{2}}\right )\right ) 2^{\frac {2}{3}}}{24}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{3}-x \right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right )}{6}\) | \(135\) |
| trager | \(\text {Expression too large to display}\) | \(4998\) |
1/24*(-2*arctan(1/3*3^(1/2)*(x+2^(2/3)*(x^3-x)^(1/3))/x)*3^(1/2)-2*ln((-2^ (1/3)*x+(x^3-x)^(1/3))/x)+ln((2^(2/3)*x^2+2^(1/3)*(x^3-x)^(1/3)*x+(x^3-x)^ (2/3))/x^2))*2^(2/3)-1/6*sum(ln((-_R*x+(x^3-x)^(1/3))/x)/_R,_R=RootOf(_Z^6 -_Z^3+1))
Exception generated. \[ \int \frac {1}{\sqrt [3]{-x+x^3} \left (1+x^6\right )} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (trace 0)
Not integrable
Time = 1.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.16 \[ \int \frac {1}{\sqrt [3]{-x+x^3} \left (1+x^6\right )} \, dx=\int \frac {1}{\sqrt [3]{x \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 1\right )}\, dx \]
Not integrable
Time = 0.31 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.11 \[ \int \frac {1}{\sqrt [3]{-x+x^3} \left (1+x^6\right )} \, dx=\int { \frac {1}{{\left (x^{6} + 1\right )} {\left (x^{3} - x\right )}^{\frac {1}{3}}} \,d x } \]
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.37 (sec) , antiderivative size = 967, normalized size of antiderivative = 5.59 \[ \int \frac {1}{\sqrt [3]{-x+x^3} \left (1+x^6\right )} \, dx=\text {Too large to display} \]
-1/12*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3) + 2*(-1/x^2 + 1) ^(1/3))) + 1/6*(sqrt(3)*cos(4/9*pi)^5 - 10*sqrt(3)*cos(4/9*pi)^3*sin(4/9*p i)^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^2 + 1)^(1/3))/((1/2*I*sqrt(3) + 1/2)*sin(4/9*p i))) + 1/6*(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^2 + 1)^(1/3))/((1/2*I*sqrt(3) + 1/2)*sin(2/9*pi))) - 1/6*(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)*c os(1/9*pi) - 2*(-1/x^2 + 1)^(1/3))/((1/2*I*sqrt(3) + 1/2)*sin(1/9*pi))) + 1/12*(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) - cos(4/9*...
Not integrable
Time = 5.95 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.11 \[ \int \frac {1}{\sqrt [3]{-x+x^3} \left (1+x^6\right )} \, dx=\int \frac {1}{{\left (x^3-x\right )}^{1/3}\,\left (x^6+1\right )} \,d x \]