Integrand size = 24, antiderivative size = 288 \[ \int \frac {1+x^6}{\sqrt [3]{x^2+x^4} \left (-1+x^6\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{x^2+x^4}}\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x^2+x^4}}\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {\sqrt {3} x}{-x+2^{2/3} \sqrt [3]{x^2+x^4}}\right )}{2 \sqrt [3]{2} \sqrt {3}}-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{x^2+x^4}}\right )}{2 \sqrt [3]{2} \sqrt {3}}-\frac {2}{3} \text {arctanh}\left (\frac {x}{\sqrt [3]{x^2+x^4}}\right )-\frac {\text {arctanh}\left (\frac {\sqrt [3]{2} x}{\sqrt [3]{x^2+x^4}}\right )}{3 \sqrt [3]{2}}-\frac {1}{3} \text {arctanh}\left (\frac {x^2+\left (x^2+x^4\right )^{2/3}}{x \sqrt [3]{x^2+x^4}}\right )-\frac {\text {arctanh}\left (\frac {\sqrt [3]{2} x^2+\frac {\left (x^2+x^4\right )^{2/3}}{\sqrt [3]{2}}}{x \sqrt [3]{x^2+x^4}}\right )}{6 \sqrt [3]{2}} \]
-1/3*3^(1/2)*arctan(3^(1/2)*x/(-x+2*(x^4+x^2)^(1/3)))-1/3*arctan(3^(1/2)*x /(x+2*(x^4+x^2)^(1/3)))*3^(1/2)-1/12*3^(1/2)*arctan(3^(1/2)*x/(-x+2^(2/3)* (x^4+x^2)^(1/3)))*2^(2/3)-1/12*3^(1/2)*arctan(3^(1/2)*x/(x+2^(2/3)*(x^4+x^ 2)^(1/3)))*2^(2/3)-2/3*arctanh(x/(x^4+x^2)^(1/3))-1/6*arctanh(2^(1/3)*x/(x ^4+x^2)^(1/3))*2^(2/3)-1/3*arctanh((x^2+(x^4+x^2)^(2/3))/x/(x^4+x^2)^(1/3) )-1/12*arctanh((2^(1/3)*x^2+1/2*(x^4+x^2)^(2/3)*2^(2/3))/x/(x^4+x^2)^(1/3) )*2^(2/3)
Time = 0.95 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.16 \[ \int \frac {1+x^6}{\sqrt [3]{x^2+x^4} \left (-1+x^6\right )} \, dx=-\frac {x^{2/3} \sqrt [3]{1+x^2} \left (-4 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}-2 \sqrt [3]{1+x^2}}\right )+4 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}+2 \sqrt [3]{1+x^2}}\right )+2^{2/3} \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{-\sqrt [3]{x}+2^{2/3} \sqrt [3]{1+x^2}}\right )+2^{2/3} \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}+2^{2/3} \sqrt [3]{1+x^2}}\right )+8 \text {arctanh}\left (\frac {\sqrt [3]{x}}{\sqrt [3]{1+x^2}}\right )+2\ 2^{2/3} \text {arctanh}\left (\frac {\sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{1+x^2}}\right )+4 \text {arctanh}\left (\frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{x^{2/3}+\left (1+x^2\right )^{2/3}}\right )+2^{2/3} \text {arctanh}\left (\frac {2^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}}{2 x^{2/3}+\sqrt [3]{2} \left (1+x^2\right )^{2/3}}\right )\right )}{12 \sqrt [3]{x^2+x^4}} \]
-1/12*(x^(2/3)*(1 + x^2)^(1/3)*(-4*Sqrt[3]*ArcTan[(Sqrt[3]*x^(1/3))/(x^(1/ 3) - 2*(1 + x^2)^(1/3))] + 4*Sqrt[3]*ArcTan[(Sqrt[3]*x^(1/3))/(x^(1/3) + 2 *(1 + x^2)^(1/3))] + 2^(2/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x^(1/3))/(-x^(1/3) + 2^(2/3)*(1 + x^2)^(1/3))] + 2^(2/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x^(1/3))/(x^(1 /3) + 2^(2/3)*(1 + x^2)^(1/3))] + 8*ArcTanh[x^(1/3)/(1 + x^2)^(1/3)] + 2*2 ^(2/3)*ArcTanh[(2^(1/3)*x^(1/3))/(1 + x^2)^(1/3)] + 4*ArcTanh[(x^(1/3)*(1 + x^2)^(1/3))/(x^(2/3) + (1 + x^2)^(2/3))] + 2^(2/3)*ArcTanh[(2^(2/3)*x^(1 /3)*(1 + x^2)^(1/3))/(2*x^(2/3) + 2^(1/3)*(1 + x^2)^(2/3))]))/(x^2 + x^4)^ (1/3)
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.
| \(\displaystyle \int \frac {x^6+1}{\sqrt [3]{x^4+x^2} \left (x^6-1\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{2/3} \sqrt [3]{x^2+1} \int -\frac {x^6+1}{x^{2/3} \sqrt [3]{x^2+1} \left (1-x^6\right )}dx}{\sqrt [3]{x^4+x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x^{2/3} \sqrt [3]{x^2+1} \int \frac {x^6+1}{x^{2/3} \sqrt [3]{x^2+1} \left (1-x^6\right )}dx}{\sqrt [3]{x^4+x^2}}\) |
| \(\Big \downarrow \) 2019 |
| \(\displaystyle -\frac {x^{2/3} \sqrt [3]{x^2+1} \int \frac {\left (x^2+1\right )^{2/3} \left (x^4-x^2+1\right )}{x^{2/3} \left (1-x^6\right )}dx}{\sqrt [3]{x^4+x^2}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{x^2+1} \int \frac {\left (x^2+1\right )^{2/3} \left (x^4-x^2+1\right )}{1-x^6}d\sqrt [3]{x}}{\sqrt [3]{x^4+x^2}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{x^2+1} \int \left (\frac {\left (x^2+1\right )^{2/3}}{18 \left (1-\sqrt [3]{x}\right )}+\frac {\left (x^2+1\right )^{2/3}}{18 \left (\sqrt [3]{x}+1\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{2/3}}{18 \left (1-\sqrt [9]{-1} \sqrt [3]{x}\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{2/3}}{18 \left (\sqrt [9]{-1} \sqrt [3]{x}+1\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{2/3}}{18 \left (1-(-1)^{2/9} \sqrt [3]{x}\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{2/3}}{18 \left ((-1)^{2/9} \sqrt [3]{x}+1\right )}+\frac {\left (x^2+1\right )^{2/3}}{18 \left (1-\sqrt [3]{-1} \sqrt [3]{x}\right )}+\frac {\left (x^2+1\right )^{2/3}}{18 \left (\sqrt [3]{-1} \sqrt [3]{x}+1\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{2/3}}{18 \left (1-(-1)^{4/9} \sqrt [3]{x}\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{2/3}}{18 \left ((-1)^{4/9} \sqrt [3]{x}+1\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{2/3}}{18 \left (1-(-1)^{5/9} \sqrt [3]{x}\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{2/3}}{18 \left ((-1)^{5/9} \sqrt [3]{x}+1\right )}+\frac {\left (x^2+1\right )^{2/3}}{18 \left (1-(-1)^{2/3} \sqrt [3]{x}\right )}+\frac {\left (x^2+1\right )^{2/3}}{18 \left ((-1)^{2/3} \sqrt [3]{x}+1\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{2/3}}{18 \left (1-(-1)^{7/9} \sqrt [3]{x}\right )}+\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (x^2+1\right )^{2/3}}{18 \left ((-1)^{7/9} \sqrt [3]{x}+1\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{2/3}}{18 \left (1-(-1)^{8/9} \sqrt [3]{x}\right )}+\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (x^2+1\right )^{2/3}}{18 \left ((-1)^{8/9} \sqrt [3]{x}+1\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x^4+x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{x^2+1} \left (\frac {1}{18} \int \frac {\left (x^2+1\right )^{2/3}}{1-\sqrt [3]{x}}d\sqrt [3]{x}+\frac {1}{18} \int \frac {\left (x^2+1\right )^{2/3}}{\sqrt [3]{x}+1}d\sqrt [3]{x}+\frac {1}{18} \left (1+i \sqrt {3}\right ) \int \frac {\left (x^2+1\right )^{2/3}}{1-\sqrt [9]{-1} \sqrt [3]{x}}d\sqrt [3]{x}+\frac {1}{18} \left (1+i \sqrt {3}\right ) \int \frac {\left (x^2+1\right )^{2/3}}{\sqrt [9]{-1} \sqrt [3]{x}+1}d\sqrt [3]{x}+\frac {1}{18} \left (1-i \sqrt {3}\right ) \int \frac {\left (x^2+1\right )^{2/3}}{1-(-1)^{2/9} \sqrt [3]{x}}d\sqrt [3]{x}+\frac {1}{18} \left (1-i \sqrt {3}\right ) \int \frac {\left (x^2+1\right )^{2/3}}{(-1)^{2/9} \sqrt [3]{x}+1}d\sqrt [3]{x}+\frac {1}{18} \int \frac {\left (x^2+1\right )^{2/3}}{1-\sqrt [3]{-1} \sqrt [3]{x}}d\sqrt [3]{x}+\frac {1}{18} \int \frac {\left (x^2+1\right )^{2/3}}{\sqrt [3]{-1} \sqrt [3]{x}+1}d\sqrt [3]{x}+\frac {1}{18} \left (1+i \sqrt {3}\right ) \int \frac {\left (x^2+1\right )^{2/3}}{1-(-1)^{4/9} \sqrt [3]{x}}d\sqrt [3]{x}+\frac {1}{18} \left (1+i \sqrt {3}\right ) \int \frac {\left (x^2+1\right )^{2/3}}{(-1)^{4/9} \sqrt [3]{x}+1}d\sqrt [3]{x}+\frac {1}{18} \left (1-i \sqrt {3}\right ) \int \frac {\left (x^2+1\right )^{2/3}}{1-(-1)^{5/9} \sqrt [3]{x}}d\sqrt [3]{x}+\frac {1}{18} \left (1-i \sqrt {3}\right ) \int \frac {\left (x^2+1\right )^{2/3}}{(-1)^{5/9} \sqrt [3]{x}+1}d\sqrt [3]{x}+\frac {1}{18} \int \frac {\left (x^2+1\right )^{2/3}}{1-(-1)^{2/3} \sqrt [3]{x}}d\sqrt [3]{x}+\frac {1}{18} \int \frac {\left (x^2+1\right )^{2/3}}{(-1)^{2/3} \sqrt [3]{x}+1}d\sqrt [3]{x}+\frac {1}{18} \left (1+i \sqrt {3}\right ) \int \frac {\left (x^2+1\right )^{2/3}}{1-(-1)^{7/9} \sqrt [3]{x}}d\sqrt [3]{x}+\frac {1}{18} \left (1+i \sqrt {3}\right ) \int \frac {\left (x^2+1\right )^{2/3}}{(-1)^{7/9} \sqrt [3]{x}+1}d\sqrt [3]{x}+\frac {1}{18} \left (1-i \sqrt {3}\right ) \int \frac {\left (x^2+1\right )^{2/3}}{1-(-1)^{8/9} \sqrt [3]{x}}d\sqrt [3]{x}+\frac {1}{18} \left (1-i \sqrt {3}\right ) \int \frac {\left (x^2+1\right )^{2/3}}{(-1)^{8/9} \sqrt [3]{x}+1}d\sqrt [3]{x}\right )}{\sqrt [3]{x^4+x^2}}\) |
3.29.33.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px , Qx, x]^p*Qx^(p + q), x] /; FreeQ[q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]
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 = 35.54 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.35
| method | result | size |
| pseudoelliptic | \(\frac {\ln \left (\frac {\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {2}{3}}-\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )}{6}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (-2 \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )}{3}-\frac {\ln \left (\frac {\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {2}{3}}+\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )}{6}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )}{3}+\frac {2^{\frac {2}{3}} \ln \left (\frac {-2^{\frac {1}{3}} x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}}{x}\right )}{12}-\frac {2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}} x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {2}{3}}}{x^{2}}\right )}{24}+\frac {\sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}} 2^{\frac {2}{3}}+x \right )}{3 x}\right )}{12}-\frac {\ln \left (\frac {\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}+x}{x}\right )}{3}+\frac {\ln \left (\frac {\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}-x}{x}\right )}{3}-\frac {2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {1}{3}} x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}}{x}\right )}{12}+\frac {2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}-2^{\frac {1}{3}} \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}} x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {2}{3}}}{x^{2}}\right )}{24}-\frac {\sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (-\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}} 2^{\frac {2}{3}}+x \right )}{3 x}\right )}{12}\) | \(388\) |
| trager | \(\text {Expression too large to display}\) | \(5994\) |
1/6*ln(((x^2*(x^2+1))^(2/3)-(x^2*(x^2+1))^(1/3)*x+x^2)/x^2)-1/3*3^(1/2)*ar ctan(1/3*(-2*(x^2*(x^2+1))^(1/3)+x)*3^(1/2)/x)-1/6*ln(((x^2*(x^2+1))^(2/3) +(x^2*(x^2+1))^(1/3)*x+x^2)/x^2)+1/3*3^(1/2)*arctan(1/3*(2*(x^2*(x^2+1))^( 1/3)+x)*3^(1/2)/x)+1/12*2^(2/3)*ln((-2^(1/3)*x+(x^2*(x^2+1))^(1/3))/x)-1/2 4*2^(2/3)*ln((2^(2/3)*x^2+2^(1/3)*(x^2*(x^2+1))^(1/3)*x+(x^2*(x^2+1))^(2/3 ))/x^2)+1/12*3^(1/2)*2^(2/3)*arctan(1/3*3^(1/2)*((x^2*(x^2+1))^(1/3)*2^(2/ 3)+x)/x)-1/3*ln(((x^2*(x^2+1))^(1/3)+x)/x)+1/3*ln(((x^2*(x^2+1))^(1/3)-x)/ x)-1/12*2^(2/3)*ln((2^(1/3)*x+(x^2*(x^2+1))^(1/3))/x)+1/24*2^(2/3)*ln((2^( 2/3)*x^2-2^(1/3)*(x^2*(x^2+1))^(1/3)*x+(x^2*(x^2+1))^(2/3))/x^2)-1/12*3^(1 /2)*2^(2/3)*arctan(1/3*3^(1/2)*(-(x^2*(x^2+1))^(1/3)*2^(2/3)+x)/x)
Leaf count of result is larger than twice the leaf count of optimal. 456 vs. \(2 (223) = 446\).
Time = 1.75 (sec) , antiderivative size = 456, normalized size of antiderivative = 1.58 \[ \int \frac {1+x^6}{\sqrt [3]{x^2+x^4} \left (-1+x^6\right )} \, dx=\frac {1}{12} \, \sqrt {6} 2^{\frac {1}{6}} \left (-1\right )^{\frac {1}{3}} \arctan \left (\frac {2^{\frac {1}{6}} {\left (4 \, \sqrt {6} 2^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}} {\left (x^{2} + 2 \, x + 1\right )} + 8 \, \sqrt {6} \left (-1\right )^{\frac {1}{3}} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} - 2 \, x^{2} + x\right )} - \sqrt {6} 2^{\frac {1}{3}} {\left (x^{5} - 8 \, x^{4} - 2 \, x^{3} - 8 \, x^{2} + x\right )}\right )}}{6 \, {\left (x^{5} + 8 \, x^{4} - 2 \, x^{3} + 8 \, x^{2} + x\right )}}\right ) + \frac {1}{12} \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-\frac {4 \cdot 2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} x - 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{3} + 2 \, x^{2} + x\right )} + 4 \, {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}}}{x^{3} - 2 \, x^{2} + x}\right ) - \frac {1}{24} \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (\frac {2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{3} - 2 \, x^{2} + x\right )} + 2 \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}} + 4 \, {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} x}{x^{3} - 2 \, x^{2} + x}\right ) - \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}} {\left (x^{2} + x + 1\right )} - 4 \, \sqrt {3} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} - x^{2} + x\right )} - \sqrt {3} {\left (x^{5} - 4 \, x^{4} + x^{3} - 4 \, x^{2} + x\right )}}{3 \, {\left (x^{5} + 4 \, x^{4} + x^{3} + 4 \, x^{2} + x\right )}}\right ) + \frac {1}{3} \, \log \left (\frac {x^{3} - x^{2} + 2 \, {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} x + x - 2 \, {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}}}{x^{3} + x^{2} + x}\right ) + \frac {1}{6} \, \log \left (\frac {x^{3} - x^{2} + 2 \, {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} x + x - 2 \, {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}}}{x^{3} - x^{2} + x}\right ) \]
1/12*sqrt(6)*2^(1/6)*(-1)^(1/3)*arctan(1/6*2^(1/6)*(4*sqrt(6)*2^(2/3)*(-1) ^(2/3)*(x^4 + x^2)^(2/3)*(x^2 + 2*x + 1) + 8*sqrt(6)*(-1)^(1/3)*(x^4 + x^2 )^(1/3)*(x^3 - 2*x^2 + x) - sqrt(6)*2^(1/3)*(x^5 - 8*x^4 - 2*x^3 - 8*x^2 + x))/(x^5 + 8*x^4 - 2*x^3 + 8*x^2 + x)) + 1/12*2^(2/3)*(-1)^(1/3)*log(-(4* 2^(1/3)*(-1)^(2/3)*(x^4 + x^2)^(1/3)*x - 2^(2/3)*(-1)^(1/3)*(x^3 + 2*x^2 + x) + 4*(x^4 + x^2)^(2/3))/(x^3 - 2*x^2 + x)) - 1/24*2^(2/3)*(-1)^(1/3)*lo g((2^(1/3)*(-1)^(2/3)*(x^3 - 2*x^2 + x) + 2*2^(2/3)*(-1)^(1/3)*(x^4 + x^2) ^(2/3) + 4*(x^4 + x^2)^(1/3)*x)/(x^3 - 2*x^2 + x)) - 1/3*sqrt(3)*arctan(1/ 3*(4*sqrt(3)*(x^4 + x^2)^(2/3)*(x^2 + x + 1) - 4*sqrt(3)*(x^4 + x^2)^(1/3) *(x^3 - x^2 + x) - sqrt(3)*(x^5 - 4*x^4 + x^3 - 4*x^2 + x))/(x^5 + 4*x^4 + x^3 + 4*x^2 + x)) + 1/3*log((x^3 - x^2 + 2*(x^4 + x^2)^(1/3)*x + x - 2*(x ^4 + x^2)^(2/3))/(x^3 + x^2 + x)) + 1/6*log((x^3 - x^2 + 2*(x^4 + x^2)^(1/ 3)*x + x - 2*(x^4 + x^2)^(2/3))/(x^3 - x^2 + x))
\[ \int \frac {1+x^6}{\sqrt [3]{x^2+x^4} \left (-1+x^6\right )} \, dx=\int \frac {\left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 1\right )}{\sqrt [3]{x^{2} \left (x^{2} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}\, dx \]
Integral((x**2 + 1)*(x**4 - x**2 + 1)/((x**2*(x**2 + 1))**(1/3)*(x - 1)*(x + 1)*(x**2 - x + 1)*(x**2 + x + 1)), x)
\[ \int \frac {1+x^6}{\sqrt [3]{x^2+x^4} \left (-1+x^6\right )} \, dx=\int { \frac {x^{6} + 1}{{\left (x^{6} - 1\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}}} \,d x } \]
\[ \int \frac {1+x^6}{\sqrt [3]{x^2+x^4} \left (-1+x^6\right )} \, dx=\int { \frac {x^{6} + 1}{{\left (x^{6} - 1\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}}} \,d x } \]
Timed out. \[ \int \frac {1+x^6}{\sqrt [3]{x^2+x^4} \left (-1+x^6\right )} \, dx=\int \frac {x^6+1}{{\left (x^4+x^2\right )}^{1/3}\,\left (x^6-1\right )} \,d x \]