Integrand size = 24, antiderivative size = 237 \[ \int \frac {1+2 x^6}{\sqrt [3]{x+x^3} \left (-1+x^6\right )} \, dx=\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+x^3}}\right )-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{x+x^3}}\right )}{2 \sqrt [3]{2}}-\log \left (-x+\sqrt [3]{x+x^3}\right )+\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{x+x^3}\right )}{2 \sqrt [3]{2}}+\frac {1}{2} \log \left (x^2+x \sqrt [3]{x+x^3}+\left (x+x^3\right )^{2/3}\right )-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{x+x^3}+\sqrt [3]{2} \left (x+x^3\right )^{2/3}\right )}{4 \sqrt [3]{2}}+\frac {1}{2} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{x+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]
Time = 3.49 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.26 \[ \int \frac {1+2 x^6}{\sqrt [3]{x+x^3} \left (-1+x^6\right )} \, dx=\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \left (8 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{1+x^2}}\right )-2\ 2^{2/3} \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2^{2/3} \sqrt [3]{1+x^2}}\right )-8 \log \left (-x^{2/3}+\sqrt [3]{1+x^2}\right )+2\ 2^{2/3} \log \left (-2 x^{2/3}+2^{2/3} \sqrt [3]{1+x^2}\right )+4 \log \left (x^{4/3}+x^{2/3} \sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right )-2^{2/3} \log \left (2 x^{4/3}+2^{2/3} x^{2/3} \sqrt [3]{1+x^2}+\sqrt [3]{2} \left (1+x^2\right )^{2/3}\right )+4 \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-2 \log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{1+x^2}-x^{2/3} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right )}{8 \sqrt [3]{x+x^3}} \]
(x^(1/3)*(1 + x^2)^(1/3)*(8*Sqrt[3]*ArcTan[(Sqrt[3]*x^(2/3))/(x^(2/3) + 2* (1 + x^2)^(1/3))] - 2*2^(2/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x^(2/3))/(x^(2/3) + 2^(2/3)*(1 + x^2)^(1/3))] - 8*Log[-x^(2/3) + (1 + x^2)^(1/3)] + 2*2^(2/3)* Log[-2*x^(2/3) + 2^(2/3)*(1 + x^2)^(1/3)] + 4*Log[x^(4/3) + x^(2/3)*(1 + x ^2)^(1/3) + (1 + x^2)^(2/3)] - 2^(2/3)*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 & ]))/(8*(x + x^3)^ (1/3))
Leaf count is larger than twice the leaf count of optimal. \(1683\) vs. \(2(237)=474\).
Time = 2.90 (sec) , antiderivative size = 1683, normalized size of antiderivative = 7.10, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2467, 25, 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.
| \(\displaystyle \int \frac {2 x^6+1}{\sqrt [3]{x^3+x} \left (x^6-1\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{x} \sqrt [3]{x^2+1} \int -\frac {2 x^6+1}{\sqrt [3]{x} \sqrt [3]{x^2+1} \left (1-x^6\right )}dx}{\sqrt [3]{x^3+x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [3]{x} \sqrt [3]{x^2+1} \int \frac {2 x^6+1}{\sqrt [3]{x} \sqrt [3]{x^2+1} \left (1-x^6\right )}dx}{\sqrt [3]{x^3+x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{x^2+1} \int \frac {\sqrt [3]{x} \left (2 x^6+1\right )}{\sqrt [3]{x^2+1} \left (1-x^6\right )}d\sqrt [3]{x}}{\sqrt [3]{x^3+x}}\) |
| \(\Big \downarrow \) 7266 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{x^2+1} \int \frac {2 x^3+1}{\sqrt [3]{x+1} \left (1-x^3\right )}dx^{2/3}}{2 \sqrt [3]{x^3+x}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{x^2+1} \int \left (\frac {3}{\sqrt [3]{x+1} \left (1-x^3\right )}-\frac {2}{\sqrt [3]{x+1}}\right )dx^{2/3}}{2 \sqrt [3]{x^3+x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{x^2+1} \left (-\frac {1}{6} (-1)^{7/9} x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x,-\sqrt [3]{-1} x\right )+\frac {1}{6} (-1)^{4/9} x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x,-\sqrt [3]{-1} x\right )-\frac {1}{6} \sqrt [9]{-1} x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x,-\sqrt [3]{-1} x\right )+\frac {1}{6} (-1)^{8/9} x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x,(-1)^{2/3} x\right )-\frac {1}{6} (-1)^{5/9} x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x,(-1)^{2/3} x\right )+\frac {1}{6} (-1)^{2/9} x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x,(-1)^{2/3} x\right )+\frac {\arctan \left (\frac {\frac {\sqrt [3]{2} \left (x^{2/3}+1\right )}{\sqrt [3]{x+1}}+1}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {\arctan \left (\frac {\frac {(-1)^{2/3} \sqrt [3]{2} \left (1-\sqrt [3]{-1} x^{2/3}\right )}{\sqrt [3]{x+1}}+1}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {\arctan \left (\frac {1-\frac {\sqrt [3]{-2} \left ((-1)^{2/3} x^{2/3}+1\right )}{\sqrt [3]{x+1}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}-\frac {2 \arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x+1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{1-\sqrt [3]{-1}} x^{2/3}}{\sqrt [3]{x+1}}+1}{\sqrt {3}}\right )}{\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 )}{\sqrt {3} \sqrt [3]{1+(-1)^{2/3}}}+\frac {(-1)^{7/9} \arctan \left (\frac {\frac {2 \sqrt [3]{x+1}}{\sqrt [3]{1-\sqrt [3]{-1}}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}}}-\frac {(-1)^{4/9} \arctan \left (\frac {\frac {2 \sqrt [3]{x+1}}{\sqrt [3]{1-\sqrt [3]{-1}}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}}}+\frac {\sqrt [9]{-1} \arctan \left (\frac {\frac {2 \sqrt [3]{x+1}}{\sqrt [3]{1-\sqrt [3]{-1}}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}}}-\frac {(-1)^{8/9} \arctan \left (\frac {\frac {2 \sqrt [3]{x+1}}{\sqrt [3]{1+(-1)^{2/3}}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{1+(-1)^{2/3}}}+\frac {(-1)^{5/9} \arctan \left (\frac {\frac {2 \sqrt [3]{x+1}}{\sqrt [3]{1+(-1)^{2/3}}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{1+(-1)^{2/3}}}-\frac {(-1)^{2/9} \arctan \left (\frac {\frac {2 \sqrt [3]{x+1}}{\sqrt [3]{1+(-1)^{2/3}}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{1+(-1)^{2/3}}}+\frac {\log \left (\left (1-x^{2/3}\right )^2 \left (x^{2/3}+1\right )\right )}{12 \sqrt [3]{2}}+\frac {\log \left (\left (1-\sqrt [3]{-1} x^{2/3}\right ) \left (\sqrt [3]{-1} x^{2/3}+1\right )^2\right )}{12 \sqrt [3]{2}}+\frac {\log \left (\left (1-(-1)^{2/3} x^{2/3}\right )^2 \left ((-1)^{2/3} x^{2/3}+1\right )\right )}{12 \sqrt [3]{2}}-\frac {(-1)^{7/9} \log \left (x+\sqrt [3]{-1}\right )}{18 \sqrt [3]{1-\sqrt [3]{-1}}}+\frac {(-1)^{4/9} \log \left (x+\sqrt [3]{-1}\right )}{18 \sqrt [3]{1-\sqrt [3]{-1}}}-\frac {\sqrt [9]{-1} \log \left (x+\sqrt [3]{-1}\right )}{18 \sqrt [3]{1-\sqrt [3]{-1}}}+\frac {(-1)^{8/9} \log \left (x-(-1)^{2/3}\right )}{18 \sqrt [3]{1+(-1)^{2/3}}}-\frac {(-1)^{5/9} \log \left (x-(-1)^{2/3}\right )}{18 \sqrt [3]{1+(-1)^{2/3}}}+\frac {(-1)^{2/9} \log \left (x-(-1)^{2/3}\right )}{18 \sqrt [3]{1+(-1)^{2/3}}}+\frac {\log \left (\sqrt [3]{-1} x+1\right )}{6 \sqrt [3]{1-\sqrt [3]{-1}}}+\frac {\log \left (1-(-1)^{2/3} x\right )}{6 \sqrt [3]{1+(-1)^{2/3}}}+\frac {(-1)^{7/9} \log \left (\sqrt [3]{1-\sqrt [3]{-1}}-\sqrt [3]{x+1}\right )}{6 \sqrt [3]{1-\sqrt [3]{-1}}}-\frac {(-1)^{4/9} \log \left (\sqrt [3]{1-\sqrt [3]{-1}}-\sqrt [3]{x+1}\right )}{6 \sqrt [3]{1-\sqrt [3]{-1}}}+\frac {\sqrt [9]{-1} \log \left (\sqrt [3]{1-\sqrt [3]{-1}}-\sqrt [3]{x+1}\right )}{6 \sqrt [3]{1-\sqrt [3]{-1}}}-\frac {(-1)^{8/9} \log \left (\sqrt [3]{1+(-1)^{2/3}}-\sqrt [3]{x+1}\right )}{6 \sqrt [3]{1+(-1)^{2/3}}}+\frac {(-1)^{5/9} \log \left (\sqrt [3]{1+(-1)^{2/3}}-\sqrt [3]{x+1}\right )}{6 \sqrt [3]{1+(-1)^{2/3}}}-\frac {(-1)^{2/9} \log \left (\sqrt [3]{1+(-1)^{2/3}}-\sqrt [3]{x+1}\right )}{6 \sqrt [3]{1+(-1)^{2/3}}}-\frac {\log \left (\sqrt [3]{1-\sqrt [3]{-1}} x^{2/3}-\sqrt [3]{x+1}\right )}{2 \sqrt [3]{1-\sqrt [3]{-1}}}-\frac {\log \left (\sqrt [3]{1+(-1)^{2/3}} x^{2/3}-\sqrt [3]{x+1}\right )}{2 \sqrt [3]{1+(-1)^{2/3}}}+\log \left (\sqrt [3]{x+1}-x^{2/3}\right )-\frac {\log \left ((-1)^{2/3} x^{2/3}-(-2)^{2/3} \sqrt [3]{x+1}+1\right )}{4 \sqrt [3]{2}}-\frac {\log \left (x^{2/3}-2^{2/3} \sqrt [3]{x+1}+1\right )}{4 \sqrt [3]{2}}-\frac {\log \left (-\sqrt [3]{-1} x^{2/3}+\sqrt [3]{-1} 2^{2/3} \sqrt [3]{x+1}+1\right )}{4 \sqrt [3]{2}}\right )}{2 \sqrt [3]{x^3+x}}\) |
(-3*x^(1/3)*(1 + x^2)^(1/3)*(-1/6*((-1)^(1/9)*x^(2/3)*AppellF1[2/3, 1/3, 1 , 5/3, -x, -((-1)^(1/3)*x)]) + ((-1)^(4/9)*x^(2/3)*AppellF1[2/3, 1/3, 1, 5 /3, -x, -((-1)^(1/3)*x)])/6 - ((-1)^(7/9)*x^(2/3)*AppellF1[2/3, 1/3, 1, 5/ 3, -x, -((-1)^(1/3)*x)])/6 + ((-1)^(2/9)*x^(2/3)*AppellF1[2/3, 1/3, 1, 5/3 , -x, (-1)^(2/3)*x])/6 - ((-1)^(5/9)*x^(2/3)*AppellF1[2/3, 1/3, 1, 5/3, -x , (-1)^(2/3)*x])/6 + ((-1)^(8/9)*x^(2/3)*AppellF1[2/3, 1/3, 1, 5/3, -x, (- 1)^(2/3)*x])/6 + ArcTan[(1 + (2^(1/3)*(1 + x^(2/3)))/(1 + x)^(1/3))/Sqrt[3 ]]/(2*2^(1/3)*Sqrt[3]) + ArcTan[(1 + ((-1)^(2/3)*2^(1/3)*(1 - (-1)^(1/3)*x ^(2/3)))/(1 + x)^(1/3))/Sqrt[3]]/(2*2^(1/3)*Sqrt[3]) + ArcTan[(1 - ((-2)^( 1/3)*(1 + (-1)^(2/3)*x^(2/3)))/(1 + x)^(1/3))/Sqrt[3]]/(2*2^(1/3)*Sqrt[3]) - (2*ArcTan[(1 + (2*x^(2/3))/(1 + x)^(1/3))/Sqrt[3]])/Sqrt[3] + ArcTan[(1 + (2*(1 - (-1)^(1/3))^(1/3)*x^(2/3))/(1 + x)^(1/3))/Sqrt[3]]/(Sqrt[3]*(1 - (-1)^(1/3))^(1/3)) + ArcTan[(1 + (2*(1 + (-1)^(2/3))^(1/3)*x^(2/3))/(1 + x)^(1/3))/Sqrt[3]]/(Sqrt[3]*(1 + (-1)^(2/3))^(1/3)) + ((-1)^(1/9)*ArcTan[ (1 + (2*(1 + x)^(1/3))/(1 - (-1)^(1/3))^(1/3))/Sqrt[3]])/(3*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]])/(3*Sqrt[3]*(1 - (-1)^(1/3))^(1/3)) + ((-1)^(7/9)*Arc Tan[(1 + (2*(1 + x)^(1/3))/(1 - (-1)^(1/3))^(1/3))/Sqrt[3]])/(3*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]])/(3*Sqrt[3]*(1 + (-1)^(2/3))^(1/3)) + ((-1)^(5...
3.27.58.3.1 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 = 5.26 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.92
| method | result | size |
| pseudoelliptic | \(\frac {\ln \left (\frac {{\left (x \left (x^{2}+1\right )\right )}^{\frac {2}{3}}+{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\left (2 {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right )}{2}-\ln \left (\frac {{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}-x}{x}\right )+\frac {2^{\frac {2}{3}} \ln \left (\frac {-2^{\frac {1}{3}} x +{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}}{x}\right )}{4}-\frac {2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}} x +{\left (x \left (x^{2}+1\right )\right )}^{\frac {2}{3}}}{x^{2}}\right )}{8}+\frac {\sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (2^{\frac {2}{3}} {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}+x \right )}{3 x}\right )}{4}\) | \(218\) |
1/2*ln(((x*(x^2+1))^(2/3)+(x*(x^2+1))^(1/3)*x+x^2)/x^2)-3^(1/2)*arctan(1/3 *(2*(x*(x^2+1))^(1/3)+x)*3^(1/2)/x)+1/2*sum(ln((-_R*x+(x*(x^2+1))^(1/3))/x )/_R,_R=RootOf(_Z^6-_Z^3+1))-ln(((x*(x^2+1))^(1/3)-x)/x)+1/4*2^(2/3)*ln((- 2^(1/3)*x+(x*(x^2+1))^(1/3))/x)-1/8*2^(2/3)*ln((2^(2/3)*x^2+2^(1/3)*(x*(x^ 2+1))^(1/3)*x+(x*(x^2+1))^(2/3))/x^2)+1/4*3^(1/2)*2^(2/3)*arctan(1/3*3^(1/ 2)*(2^(2/3)*(x*(x^2+1))^(1/3)+x)/x)
Exception generated. \[ \int \frac {1+2 x^6}{\sqrt [3]{x+x^3} \left (-1+x^6\right )} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (residue poly has multiple non-linear fac tors)
Not integrable
Time = 1.84 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.16 \[ \int \frac {1+2 x^6}{\sqrt [3]{x+x^3} \left (-1+x^6\right )} \, dx=\int \frac {2 x^{6} + 1}{\sqrt [3]{x \left (x^{2} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}\, dx \]
Integral((2*x**6 + 1)/((x*(x**2 + 1))**(1/3)*(x - 1)*(x + 1)*(x**2 - x + 1 )*(x**2 + x + 1)), x)
Not integrable
Time = 0.30 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.10 \[ \int \frac {1+2 x^6}{\sqrt [3]{x+x^3} \left (-1+x^6\right )} \, dx=\int { \frac {2 \, x^{6} + 1}{{\left (x^{6} - 1\right )} {\left (x^{3} + x\right )}^{\frac {1}{3}}} \,d x } \]
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.39 (sec) , antiderivative size = 995, normalized size of antiderivative = 4.20 \[ \int \frac {1+2 x^6}{\sqrt [3]{x+x^3} \left (-1+x^6\right )} \, dx=\text {Too large to display} \]
1/4*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3) + 2*(1/x^2 + 1)^(1 /3))) - 1/2*(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^2 + 1)^(1/3))/((1/2*I*sqrt(3) + 1/2)*sin(4/9*pi))) - 1/2*(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)*co s(2/9*pi) + 2*(1/x^2 + 1)^(1/3))/((1/2*I*sqrt(3) + 1/2)*sin(2/9*pi))) + 1/ 2*(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*p i)^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^2 + 1)^(1/3))/((1/2*I*sqrt(3) + 1/2)*sin(1/9*pi))) - 1/4*(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*pi))*(1...
Not integrable
Time = 6.39 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.10 \[ \int \frac {1+2 x^6}{\sqrt [3]{x+x^3} \left (-1+x^6\right )} \, dx=\int \frac {2\,x^6+1}{\left (x^6-1\right )\,{\left (x^3+x\right )}^{1/3}} \,d x \]