\(\int \frac {1+x^6}{\sqrt [3]{-x^2+x^4} (-1+x^6)} \, dx\) [2242]

Optimal result

Integrand size = 26, antiderivative size = 167 \[ \int \frac {1+x^6}{\sqrt [3]{-x^2+x^4} \left (-1+x^6\right )} \, dx=-\frac {\left (-x^2+x^4\right )^{2/3}}{x \left (-1+x^2\right )}-\frac {2 \arctan \left (\frac {\sqrt [6]{3} x}{\sqrt [3]{-x^2+x^4}}\right )}{3 \sqrt [6]{3}}-\frac {\arctan \left (\frac {3^{5/6} x \sqrt [3]{-x^2+x^4}}{-3 x^2+3^{2/3} \left (-x^2+x^4\right )^{2/3}}\right )}{3 \sqrt [6]{3}}-\frac {\text {arctanh}\left (\frac {\frac {x^2}{\sqrt [3]{3}}+\frac {\left (-x^2+x^4\right )^{2/3}}{3^{2/3}}}{x \sqrt [3]{-x^2+x^4}}\right )}{3^{2/3}} \]

[Out]

-(x^4-x^2)^(2/3)/x/(x^2-1)-2/9*arctan(3^(1/6)*x/(x^4-x^2)^(1/3))*3^(5/6)-1/9*arctan(3^(5/6)*x*(x^4-x^2)^(1/3)/
(-3*x^2+3^(2/3)*(x^4-x^2)^(2/3)))*3^(5/6)-1/3*arctanh((1/3*3^(2/3)*x^2+1/3*(x^4-x^2)^(2/3)*3^(1/3))/x/(x^4-x^2
)^(1/3))*3^(1/3)

Rubi [F]

\[ \int \frac {1+x^6}{\sqrt [3]{-x^2+x^4} \left (-1+x^6\right )} \, dx=\int \frac {1+x^6}{\sqrt [3]{-x^2+x^4} \left (-1+x^6\right )} \, dx \]

[In]

Int[(1 + x^6)/((-x^2 + x^4)^(1/3)*(-1 + x^6)),x]

[Out]

(-2*x*(1 - x^2)^(1/3)*AppellF1[1/6, 1/3, 1, 7/6, x^2, (-2*x^2)/(1 - I*Sqrt[3])])/(-x^2 + x^4)^(1/3) - (2*x*(1
- x^2)^(1/3)*AppellF1[1/6, 1/3, 1, 7/6, x^2, (-2*x^2)/(1 + I*Sqrt[3])])/(-x^2 + x^4)^(1/3) + (3*x*(1 - x^2)^(1
/3)*Hypergeometric2F1[1/6, 1/3, 7/6, x^2])/(-x^2 + x^4)^(1/3) + (x^(2/3)*(-1 + x^2)^(1/3)*Defer[Subst][Defer[I
nt][1/((-1 + x)*(-1 + x^6)^(1/3)), x], x, x^(1/3)])/(3*(-x^2 + x^4)^(1/3)) - (x^(2/3)*(-1 + x^2)^(1/3)*Defer[S
ubst][Defer[Int][1/((1 + x)*(-1 + x^6)^(1/3)), x], x, x^(1/3)])/(3*(-x^2 + x^4)^(1/3)) + ((1 + I*Sqrt[3])*x^(2
/3)*(-1 + x^2)^(1/3)*Defer[Subst][Defer[Int][1/((-1 - I*Sqrt[3] + 2*x)*(-1 + x^6)^(1/3)), x], x, x^(1/3)])/(3*
(-x^2 + x^4)^(1/3)) - ((1 - I*Sqrt[3])*x^(2/3)*(-1 + x^2)^(1/3)*Defer[Subst][Defer[Int][1/((1 - I*Sqrt[3] + 2*
x)*(-1 + x^6)^(1/3)), x], x, x^(1/3)])/(3*(-x^2 + x^4)^(1/3)) + ((1 - I*Sqrt[3])*x^(2/3)*(-1 + x^2)^(1/3)*Defe
r[Subst][Defer[Int][1/((-1 + I*Sqrt[3] + 2*x)*(-1 + x^6)^(1/3)), x], x, x^(1/3)])/(3*(-x^2 + x^4)^(1/3)) - ((1
+ I*Sqrt[3])*x^(2/3)*(-1 + x^2)^(1/3)*Defer[Subst][Defer[Int][1/((1 + I*Sqrt[3] + 2*x)*(-1 + x^6)^(1/3)), x],
x, x^(1/3)])/(3*(-x^2 + x^4)^(1/3))

Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {1+x^6}{x^{2/3} \sqrt [3]{-1+x^2} \left (-1+x^6\right )} \, dx}{\sqrt [3]{-x^2+x^4}} \\ & = \frac {\left (3 x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1+x^{18}}{\sqrt [3]{-1+x^6} \left (-1+x^{18}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^4}} \\ & = \frac {\left (3 x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{\sqrt [3]{-1+x^6}}+\frac {2}{\sqrt [3]{-1+x^6} \left (-1+x^{18}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^4}} \\ & = \frac {\left (3 x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^4}}+\frac {\left (6 x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^6} \left (-1+x^{18}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^4}} \\ & = \frac {\left (3 x^{2/3} \sqrt [3]{1-x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^4}}+\frac {\left (6 x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{9 \left (-1+x^2\right ) \sqrt [3]{-1+x^6}}+\frac {-2+x}{18 \left (1-x+x^2\right ) \sqrt [3]{-1+x^6}}+\frac {-2-x}{18 \left (1+x+x^2\right ) \sqrt [3]{-1+x^6}}+\frac {-2+x^3}{6 \sqrt [3]{-1+x^6} \left (1-x^3+x^6\right )}+\frac {-2-x^3}{6 \sqrt [3]{-1+x^6} \left (1+x^3+x^6\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^4}} \\ & = \frac {3 x \sqrt [3]{1-x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},x^2\right )}{\sqrt [3]{-x^2+x^4}}+\frac {\left (x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {-2+x}{\left (1-x+x^2\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}}+\frac {\left (x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {-2-x}{\left (1+x+x^2\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}}+\frac {\left (2 x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+x^2\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}}+\frac {\left (x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {-2+x^3}{\sqrt [3]{-1+x^6} \left (1-x^3+x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^4}}+\frac {\left (x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {-2-x^3}{\sqrt [3]{-1+x^6} \left (1+x^3+x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^4}} \\ & = \frac {3 x \sqrt [3]{1-x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},x^2\right )}{\sqrt [3]{-x^2+x^4}}+\frac {\left (x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \left (\frac {1+i \sqrt {3}}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^6}}+\frac {1-i \sqrt {3}}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^6}}\right ) \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}}+\frac {\left (x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \left (\frac {-1+i \sqrt {3}}{\left (1-i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^6}}+\frac {-1-i \sqrt {3}}{\left (1+i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^6}}\right ) \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}}+\frac {\left (2 x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{2 (-1+x) \sqrt [3]{-1+x^6}}-\frac {1}{2 (1+x) \sqrt [3]{-1+x^6}}\right ) \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}}+\frac {\left (x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \left (\frac {1+i \sqrt {3}}{\left (-1-i \sqrt {3}+2 x^3\right ) \sqrt [3]{-1+x^6}}+\frac {1-i \sqrt {3}}{\left (-1+i \sqrt {3}+2 x^3\right ) \sqrt [3]{-1+x^6}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^4}}+\frac {\left (x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \left (\frac {-1+i \sqrt {3}}{\left (1-i \sqrt {3}+2 x^3\right ) \sqrt [3]{-1+x^6}}+\frac {-1-i \sqrt {3}}{\left (1+i \sqrt {3}+2 x^3\right ) \sqrt [3]{-1+x^6}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^4}} \\ & = \frac {3 x \sqrt [3]{1-x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},x^2\right )}{\sqrt [3]{-x^2+x^4}}+\frac {\left (x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}}-\frac {\left (x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}}+\frac {\left (\left (-1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}}+\frac {\left (\left (-1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1+i \sqrt {3}+2 x^3\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^4}}+\frac {\left (\left (1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}}+\frac {\left (\left (1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+i \sqrt {3}+2 x^3\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^4}}+\frac {\left (\left (-1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}}+\frac {\left (\left (-1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-i \sqrt {3}+2 x^3\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^4}}+\frac {\left (\left (1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}}+\frac {\left (\left (1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-1-i \sqrt {3}+2 x^3\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^4}} \\ & = \frac {3 x \sqrt [3]{1-x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},x^2\right )}{\sqrt [3]{-x^2+x^4}}+\frac {\left (x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}}-\frac {\left (x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}}+\frac {\left (\left (-1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}}+\frac {\left (\left (-1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \left (\frac {-i+\sqrt {3}}{2 \left (i+\sqrt {3}+2 i x^6\right ) \sqrt [3]{-1+x^6}}+\frac {x^3}{\sqrt [3]{-1+x^6} \left (1-i \sqrt {3}+2 x^6\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^4}}+\frac {\left (\left (1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}}+\frac {\left (\left (1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \left (\frac {-i-\sqrt {3}}{2 \left (-i+\sqrt {3}-2 i x^6\right ) \sqrt [3]{-1+x^6}}+\frac {x^3}{\sqrt [3]{-1+x^6} \left (1+i \sqrt {3}+2 x^6\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^4}}+\frac {\left (\left (-1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}}+\frac {\left (\left (-1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \left (\frac {i+\sqrt {3}}{2 \left (-i+\sqrt {3}-2 i x^6\right ) \sqrt [3]{-1+x^6}}+\frac {x^3}{\sqrt [3]{-1+x^6} \left (1+i \sqrt {3}+2 x^6\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^4}}+\frac {\left (\left (1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}}+\frac {\left (\left (1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \left (\frac {i-\sqrt {3}}{2 \left (i+\sqrt {3}+2 i x^6\right ) \sqrt [3]{-1+x^6}}+\frac {x^3}{\sqrt [3]{-1+x^6} \left (1-i \sqrt {3}+2 x^6\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^4}} \\ & = \frac {3 x \sqrt [3]{1-x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},x^2\right )}{\sqrt [3]{-x^2+x^4}}+\frac {\left (x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}}-\frac {\left (x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}}+\frac {\left (\left (-1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}}+\frac {\left (\left (-1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt [3]{-1+x^6} \left (1-i \sqrt {3}+2 x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^4}}+\frac {\left (\left (1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}}+\frac {\left (\left (1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt [3]{-1+x^6} \left (1+i \sqrt {3}+2 x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^4}}+\frac {\left (\left (-i-\sqrt {3}\right ) \left (1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-i+\sqrt {3}-2 i x^6\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{-x^2+x^4}}+\frac {\left (\left (-1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}}+\frac {\left (\left (-1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt [3]{-1+x^6} \left (1+i \sqrt {3}+2 x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^4}}+\frac {\left (\left (1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}}+\frac {\left (\left (1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt [3]{-1+x^6} \left (1-i \sqrt {3}+2 x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^4}}+\frac {\left (\left (i-\sqrt {3}\right ) \left (1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (i+\sqrt {3}+2 i x^6\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{-x^2+x^4}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \left (-i+\sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (i+\sqrt {3}+2 i x^6\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{-x^2+x^4}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \left (i+\sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-i+\sqrt {3}-2 i x^6\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{-x^2+x^4}} \\ & = \frac {3 x \sqrt [3]{1-x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},x^2\right )}{\sqrt [3]{-x^2+x^4}}+\frac {\left (\left (-i-\sqrt {3}\right ) \left (1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1-x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x^6} \left (-i+\sqrt {3}-2 i x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{-x^2+x^4}}+\frac {\left (\left (i-\sqrt {3}\right ) \left (1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1-x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x^6} \left (i+\sqrt {3}+2 i x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{-x^2+x^4}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \left (-i+\sqrt {3}\right ) x^{2/3} \sqrt [3]{1-x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x^6} \left (i+\sqrt {3}+2 i x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{-x^2+x^4}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \left (i+\sqrt {3}\right ) x^{2/3} \sqrt [3]{1-x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x^6} \left (-i+\sqrt {3}-2 i x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{-x^2+x^4}}+\frac {\left (x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}}-\frac {\left (x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}}+\frac {\left (\left (-1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}}+\frac {\left (\left (-1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt [3]{-1+x^3} \left (1-i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x^2+x^4}}+\frac {\left (\left (1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}}+\frac {\left (\left (1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt [3]{-1+x^3} \left (1+i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x^2+x^4}}+\frac {\left (\left (-1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}}+\frac {\left (\left (-1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt [3]{-1+x^3} \left (1+i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x^2+x^4}}+\frac {\left (\left (1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}}+\frac {\left (\left (1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt [3]{-1+x^3} \left (1-i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x^2+x^4}} \\ & = -\frac {2 x \sqrt [3]{1-x^2} \operatorname {AppellF1}\left (\frac {1}{6},\frac {1}{3},1,\frac {7}{6},x^2,-\frac {2 x^2}{1-i \sqrt {3}}\right )}{\sqrt [3]{-x^2+x^4}}-\frac {2 x \sqrt [3]{1-x^2} \operatorname {AppellF1}\left (\frac {1}{6},\frac {1}{3},1,\frac {7}{6},x^2,-\frac {2 x^2}{1+i \sqrt {3}}\right )}{\sqrt [3]{-x^2+x^4}}+\frac {3 x \sqrt [3]{1-x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},x^2\right )}{\sqrt [3]{-x^2+x^4}}+\frac {\left (\left (-1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1-x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt [3]{1-x^3} \left (1-i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x^2+x^4}}+\frac {\left (\left (1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1-x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt [3]{1-x^3} \left (1+i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x^2+x^4}}+\frac {\left (\left (-1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1-x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt [3]{1-x^3} \left (1+i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x^2+x^4}}+\frac {\left (\left (1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1-x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt [3]{1-x^3} \left (1-i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x^2+x^4}}+\frac {\left (x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}}-\frac {\left (x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}}+\frac {\left (\left (-1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}}+\frac {\left (\left (1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}}+\frac {\left (\left (-1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}}+\frac {\left (\left (1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}} \\ & = -\frac {2 x \sqrt [3]{1-x^2} \operatorname {AppellF1}\left (\frac {1}{6},\frac {1}{3},1,\frac {7}{6},x^2,-\frac {2 x^2}{1-i \sqrt {3}}\right )}{\sqrt [3]{-x^2+x^4}}-\frac {2 x \sqrt [3]{1-x^2} \operatorname {AppellF1}\left (\frac {1}{6},\frac {1}{3},1,\frac {7}{6},x^2,-\frac {2 x^2}{1+i \sqrt {3}}\right )}{\sqrt [3]{-x^2+x^4}}+\frac {3 x \sqrt [3]{1-x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},x^2\right )}{\sqrt [3]{-x^2+x^4}}+\frac {\left (x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}}-\frac {\left (x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}}+\frac {\left (\left (-1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}}+\frac {\left (\left (1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}}+\frac {\left (\left (-1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}}+\frac {\left (\left (1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{-x^2+x^4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.79 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.14 \[ \int \frac {1+x^6}{\sqrt [3]{-x^2+x^4} \left (-1+x^6\right )} \, dx=-\frac {x^{2/3} \left (9 \sqrt [3]{x}+2\ 3^{5/6} \sqrt [3]{-1+x^2} \arctan \left (\frac {\sqrt [6]{3} \sqrt [3]{x}}{\sqrt [3]{-1+x^2}}\right )+3^{5/6} \sqrt [3]{-1+x^2} \arctan \left (\frac {3^{5/6} \sqrt [3]{x} \sqrt [3]{-1+x^2}}{-3 x^{2/3}+3^{2/3} \left (-1+x^2\right )^{2/3}}\right )+3 \sqrt [3]{3} \sqrt [3]{-1+x^2} \text {arctanh}\left (\frac {3 \sqrt [3]{3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}{3 x^{2/3}+3^{2/3} \left (-1+x^2\right )^{2/3}}\right )\right )}{9 \sqrt [3]{x^2 \left (-1+x^2\right )}} \]

[In]

Integrate[(1 + x^6)/((-x^2 + x^4)^(1/3)*(-1 + x^6)),x]

[Out]

-1/9*(x^(2/3)*(9*x^(1/3) + 2*3^(5/6)*(-1 + x^2)^(1/3)*ArcTan[(3^(1/6)*x^(1/3))/(-1 + x^2)^(1/3)] + 3^(5/6)*(-1
+ x^2)^(1/3)*ArcTan[(3^(5/6)*x^(1/3)*(-1 + x^2)^(1/3))/(-3*x^(2/3) + 3^(2/3)*(-1 + x^2)^(2/3))] + 3*3^(1/3)*(
-1 + x^2)^(1/3)*ArcTanh[(3*3^(1/3)*x^(1/3)*(-1 + x^2)^(1/3))/(3*x^(2/3) + 3^(2/3)*(-1 + x^2)^(2/3))]))/(x^2*(-
1 + x^2))^(1/3)

Maple [A] (verified)

Time = 61.49 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.23

method	result	size
pseudoelliptic	\(\frac {\left (\left (\arctan \left (\frac {2 \left (x^{4}-x^{2}\right )^{\frac {1}{3}} 3^{\frac {5}{6}}-3 x \sqrt {3}}{3 x}\right )+2 \arctan \left (\frac {\left (x^{4}-x^{2}\right )^{\frac {1}{3}} 3^{\frac {5}{6}}}{3 x}\right )+\arctan \left (\frac {2 \left (x^{4}-x^{2}\right )^{\frac {1}{3}} 3^{\frac {5}{6}}+3 x \sqrt {3}}{3 x}\right )\right ) 3^{\frac {5}{6}}+\frac {3 \,3^{\frac {1}{3}} \left (\ln \left (\frac {-3^{\frac {2}{3}} \left (x^{4}-x^{2}\right )^{\frac {1}{3}} x +3^{\frac {1}{3}} x^{2}+\left (x^{4}-x^{2}\right )^{\frac {2}{3}}}{x^{2}}\right )-\ln \left (\frac {3^{\frac {2}{3}} \left (x^{4}-x^{2}\right )^{\frac {1}{3}} x +3^{\frac {1}{3}} x^{2}+\left (x^{4}-x^{2}\right )^{\frac {2}{3}}}{x^{2}}\right )\right )}{2}\right ) \left (x^{4}-x^{2}\right )^{\frac {1}{3}}-9 x}{9 \left (x^{4}-x^{2}\right )^{\frac {1}{3}}}\)	\(205\)
risch	\(\text {Expression too large to display}\)	\(758\)
trager	\(\text {Expression too large to display}\)	\(2605\)

[In]

int((x^6+1)/(x^4-x^2)^(1/3)/(x^6-1),x,method=_RETURNVERBOSE)

[Out]

1/9*(((arctan(1/3*(2*(x^4-x^2)^(1/3)*3^(5/6)-3*x*3^(1/2))/x)+2*arctan(1/3*(x^4-x^2)^(1/3)/x*3^(5/6))+arctan(1/
3*(2*(x^4-x^2)^(1/3)*3^(5/6)+3*x*3^(1/2))/x))*3^(5/6)+3/2*3^(1/3)*(ln((-3^(2/3)*(x^4-x^2)^(1/3)*x+3^(1/3)*x^2+
(x^4-x^2)^(2/3))/x^2)-ln((3^(2/3)*(x^4-x^2)^(1/3)*x+3^(1/3)*x^2+(x^4-x^2)^(2/3))/x^2)))*(x^4-x^2)^(1/3)-9*x)/(
x^4-x^2)^(1/3)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 4.02 (sec) , antiderivative size = 1561, normalized size of antiderivative = 9.35 \[ \int \frac {1+x^6}{\sqrt [3]{-x^2+x^4} \left (-1+x^6\right )} \, dx=\text {Too large to display} \]

[In]

integrate((x^6+1)/(x^4-x^2)^(1/3)/(x^6-1),x, algorithm="fricas")

[Out]

1/36*(3^(5/6)*(-1)^(1/6)*(x^3 + sqrt(-3)*(x^3 - x) - x)*log(-(3^(5/6)*(-1)^(1/6)*(419*x^5 + 2880*x^4 - 2095*x^
3 - 2880*x^2 + sqrt(-3)*(419*x^5 + 2880*x^4 - 2095*x^3 - 2880*x^2 + 419*x) + 419*x) + 6*3^(1/3)*(-1)^(2/3)*(24
0*x^5 - 419*x^4 - 1200*x^3 + 419*x^2 + sqrt(-3)*(240*x^5 - 419*x^4 - 1200*x^3 + 419*x^2 + 240*x) + 240*x) - 12
*(x^4 - x^2)^(2/3)*(419*x^2 - sqrt(3)*(-480*I*x^2 + 419*I*x + 480*I) + 1440*x - 419) + 6*(x^4 - x^2)^(1/3)*(3^
(2/3)*(-1)^(1/3)*(480*x^3 - 419*x^2 - sqrt(-3)*(480*x^3 - 419*x^2 - 480*x) - 480*x) - 3^(1/6)*(-1)^(5/6)*(419*
x^3 + 1440*x^2 - sqrt(-3)*(419*x^3 + 1440*x^2 - 419*x) - 419*x)))/(x^5 + x^3 + x)) - 3^(5/6)*(-1)^(1/6)*(x^3 +
sqrt(-3)*(x^3 - x) - x)*log((3^(5/6)*(-1)^(1/6)*(419*x^5 + 2880*x^4 - 2095*x^3 - 2880*x^2 + sqrt(-3)*(419*x^5
+ 2880*x^4 - 2095*x^3 - 2880*x^2 + 419*x) + 419*x) - 6*3^(1/3)*(-1)^(2/3)*(240*x^5 - 419*x^4 - 1200*x^3 + 419
*x^2 + sqrt(-3)*(240*x^5 - 419*x^4 - 1200*x^3 + 419*x^2 + 240*x) + 240*x) + 12*(x^4 - x^2)^(2/3)*(419*x^2 - sq
rt(3)*(480*I*x^2 - 419*I*x - 480*I) + 1440*x - 419) - 6*(x^4 - x^2)^(1/3)*(3^(2/3)*(-1)^(1/3)*(480*x^3 - 419*x
^2 - sqrt(-3)*(480*x^3 - 419*x^2 - 480*x) - 480*x) + 3^(1/6)*(-1)^(5/6)*(419*x^3 + 1440*x^2 - sqrt(-3)*(419*x^
3 + 1440*x^2 - 419*x) - 419*x)))/(x^5 + x^3 + x)) + 3^(5/6)*(-1)^(1/6)*(x^3 - sqrt(-3)*(x^3 - x) - x)*log(-(3^
(5/6)*(-1)^(1/6)*(419*x^5 + 2880*x^4 - 2095*x^3 - 2880*x^2 - sqrt(-3)*(419*x^5 + 2880*x^4 - 2095*x^3 - 2880*x^
2 + 419*x) + 419*x) + 6*3^(1/3)*(-1)^(2/3)*(240*x^5 - 419*x^4 - 1200*x^3 + 419*x^2 - sqrt(-3)*(240*x^5 - 419*x
^4 - 1200*x^3 + 419*x^2 + 240*x) + 240*x) - 12*(x^4 - x^2)^(2/3)*(419*x^2 - sqrt(3)*(-480*I*x^2 + 419*I*x + 48
0*I) + 1440*x - 419) + 6*(x^4 - x^2)^(1/3)*(3^(2/3)*(-1)^(1/3)*(480*x^3 - 419*x^2 + sqrt(-3)*(480*x^3 - 419*x^
2 - 480*x) - 480*x) - 3^(1/6)*(-1)^(5/6)*(419*x^3 + 1440*x^2 + sqrt(-3)*(419*x^3 + 1440*x^2 - 419*x) - 419*x))
)/(x^5 + x^3 + x)) - 3^(5/6)*(-1)^(1/6)*(x^3 - sqrt(-3)*(x^3 - x) - x)*log((3^(5/6)*(-1)^(1/6)*(419*x^5 + 2880
*x^4 - 2095*x^3 - 2880*x^2 - sqrt(-3)*(419*x^5 + 2880*x^4 - 2095*x^3 - 2880*x^2 + 419*x) + 419*x) - 6*3^(1/3)*
(-1)^(2/3)*(240*x^5 - 419*x^4 - 1200*x^3 + 419*x^2 - sqrt(-3)*(240*x^5 - 419*x^4 - 1200*x^3 + 419*x^2 + 240*x)
+ 240*x) + 12*(x^4 - x^2)^(2/3)*(419*x^2 - sqrt(3)*(480*I*x^2 - 419*I*x - 480*I) + 1440*x - 419) - 6*(x^4 - x
^2)^(1/3)*(3^(2/3)*(-1)^(1/3)*(480*x^3 - 419*x^2 + sqrt(-3)*(480*x^3 - 419*x^2 - 480*x) - 480*x) + 3^(1/6)*(-1
)^(5/6)*(419*x^3 + 1440*x^2 + sqrt(-3)*(419*x^3 + 1440*x^2 - 419*x) - 419*x)))/(x^5 + x^3 + x)) - 2*3^(5/6)*(-
1)^(1/6)*(x^3 - x)*log((3^(5/6)*(-1)^(1/6)*(419*x^5 + 2880*x^4 - 2095*x^3 - 2880*x^2 + 419*x) + 6*3^(1/3)*(-1)
^(2/3)*(240*x^5 - 419*x^4 - 1200*x^3 + 419*x^2 + 240*x) + 6*(x^4 - x^2)^(2/3)*(419*x^2 - sqrt(3)*(-480*I*x^2 +
419*I*x + 480*I) + 1440*x - 419) + 6*(x^4 - x^2)^(1/3)*(3^(2/3)*(-1)^(1/3)*(480*x^3 - 419*x^2 - 480*x) - 3^(1
/6)*(-1)^(5/6)*(419*x^3 + 1440*x^2 - 419*x)))/(x^5 + x^3 + x)) + 2*3^(5/6)*(-1)^(1/6)*(x^3 - x)*log(-(3^(5/6)*
(-1)^(1/6)*(419*x^5 + 2880*x^4 - 2095*x^3 - 2880*x^2 + 419*x) - 6*3^(1/3)*(-1)^(2/3)*(240*x^5 - 419*x^4 - 1200
*x^3 + 419*x^2 + 240*x) - 6*(x^4 - x^2)^(2/3)*(419*x^2 - sqrt(3)*(480*I*x^2 - 419*I*x - 480*I) + 1440*x - 419)
- 6*(x^4 - x^2)^(1/3)*(3^(2/3)*(-1)^(1/3)*(480*x^3 - 419*x^2 - 480*x) + 3^(1/6)*(-1)^(5/6)*(419*x^3 + 1440*x^
2 - 419*x)))/(x^5 + x^3 + x)) - 36*(x^4 - x^2)^(2/3))/(x^3 - x)

Sympy [F]

\[ \int \frac {1+x^6}{\sqrt [3]{-x^2+x^4} \left (-1+x^6\right )} \, dx=\int \frac {\left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 1\right )}{\sqrt [3]{x^{2} \left (x - 1\right ) \left (x + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}\, dx \]

[In]

integrate((x**6+1)/(x**4-x**2)**(1/3)/(x**6-1),x)

[Out]

Integral((x**2 + 1)*(x**4 - x**2 + 1)/((x**2*(x - 1)*(x + 1))**(1/3)*(x - 1)*(x + 1)*(x**2 - x + 1)*(x**2 + x
+ 1)), x)

Maxima [F]

\[ \int \frac {1+x^6}{\sqrt [3]{-x^2+x^4} \left (-1+x^6\right )} \, dx=\int { \frac {x^{6} + 1}{{\left (x^{6} - 1\right )} {\left (x^{4} - x^{2}\right )}^{\frac {1}{3}}} \,d x } \]

[In]

integrate((x^6+1)/(x^4-x^2)^(1/3)/(x^6-1),x, algorithm="maxima")

[Out]

integrate((x^6 + 1)/((x^6 - 1)*(x^4 - x^2)^(1/3)), x)

Giac [F]

\[ \int \frac {1+x^6}{\sqrt [3]{-x^2+x^4} \left (-1+x^6\right )} \, dx=\int { \frac {x^{6} + 1}{{\left (x^{6} - 1\right )} {\left (x^{4} - x^{2}\right )}^{\frac {1}{3}}} \,d x } \]

[In]

integrate((x^6+1)/(x^4-x^2)^(1/3)/(x^6-1),x, algorithm="giac")

[Out]

integrate((x^6 + 1)/((x^6 - 1)*(x^4 - x^2)^(1/3)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1+x^6}{\sqrt [3]{-x^2+x^4} \left (-1+x^6\right )} \, dx=\int \frac {x^6+1}{\left (x^6-1\right )\,{\left (x^4-x^2\right )}^{1/3}} \,d x \]

[In]

int((x^6 + 1)/((x^6 - 1)*(x^4 - x^2)^(1/3)),x)

[Out]

int((x^6 + 1)/((x^6 - 1)*(x^4 - x^2)^(1/3)), x)