3.19.49 \(\int \frac {(1+2 x^6) \sqrt [3]{x+x^3-x^7}}{(-1+x^6)^2} \, dx\) [1849]

Optimal. Leaf size=126 \[ -\frac {x \sqrt [3]{x+x^3-x^7}}{2 \left (-1+x^6\right )}-\frac {\text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+x^3-x^7}}\right )}{2 \sqrt {3}}-\frac {1}{6} \log \left (-x+\sqrt [3]{x+x^3-x^7}\right )+\frac {1}{12} \log \left (x^2+x \sqrt [3]{x+x^3-x^7}+\left (x+x^3-x^7\right )^{2/3}\right ) \]

[Out]

-x*(-x^7+x^3+x)^(1/3)/(2*x^6-2)-1/6*arctan(3^(1/2)*x/(x+2*(-x^7+x^3+x)^(1/3)))*3^(1/2)-1/6*ln(-x+(-x^7+x^3+x)^
(1/3))+1/12*ln(x^2+x*(-x^7+x^3+x)^(1/3)+(-x^7+x^3+x)^(2/3))

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Rubi [F]
time = 5.86, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (1+2 x^6\right ) \sqrt [3]{x+x^3-x^7}}{\left (-1+x^6\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((1 + 2*x^6)*(x + x^3 - x^7)^(1/3))/(-1 + x^6)^2,x]

[Out]

(2*(x + x^3 - x^7)^(1/3)*Defer[Subst][Defer[Int][(1 + x^3 - x^9)^(1/3)/(-1 + I*Sqrt[3] - 2*x)^2, x], x, x^(2/3
)])/(9*x^(1/3)*(1 + x^2 - x^6)^(1/3)) - (((2*I)/9)*(x + x^3 - x^7)^(1/3)*Defer[Subst][Defer[Int][(1 + x^3 - x^
9)^(1/3)/(-1 + I*Sqrt[3] - 2*x), x], x, x^(2/3)])/(Sqrt[3]*x^(1/3)*(1 + x^2 - x^6)^(1/3)) + ((x + x^3 - x^7)^(
1/3)*Defer[Subst][Defer[Int][(1 + x^3 - x^9)^(1/3)/(-1 + x)^2, x], x, x^(2/3)])/(18*x^(1/3)*(1 + x^2 - x^6)^(1
/3)) - ((x + x^3 - x^7)^(1/3)*Defer[Subst][Defer[Int][(1 + x^3 - x^9)^(1/3)/(-1 + x), x], x, x^(2/3)])/(18*x^(
1/3)*(1 + x^2 - x^6)^(1/3)) + ((3 - I*Sqrt[3])*(x + x^3 - x^7)^(1/3)*Defer[Subst][Defer[Int][(1 + x^3 - x^9)^(
1/3)/(1 - I*Sqrt[3] + 2*x), x], x, x^(2/3)])/(54*x^(1/3)*(1 + x^2 - x^6)^(1/3)) + (2*(x + x^3 - x^7)^(1/3)*Def
er[Subst][Defer[Int][(1 + x^3 - x^9)^(1/3)/(1 + I*Sqrt[3] + 2*x)^2, x], x, x^(2/3)])/(9*x^(1/3)*(1 + x^2 - x^6
)^(1/3)) - (((2*I)/9)*(x + x^3 - x^7)^(1/3)*Defer[Subst][Defer[Int][(1 + x^3 - x^9)^(1/3)/(1 + I*Sqrt[3] + 2*x
), x], x, x^(2/3)])/(Sqrt[3]*x^(1/3)*(1 + x^2 - x^6)^(1/3)) + ((3 + I*Sqrt[3])*(x + x^3 - x^7)^(1/3)*Defer[Sub
st][Defer[Int][(1 + x^3 - x^9)^(1/3)/(1 + I*Sqrt[3] + 2*x), x], x, x^(2/3)])/(54*x^(1/3)*(1 + x^2 - x^6)^(1/3)
) - (2*(x + x^3 - x^7)^(1/3)*Defer[Subst][Defer[Int][(x*(1 + x^3 - x^9)^(1/3))/(-1 + I*Sqrt[3] - 2*x^3)^2, x],
x, x^(2/3)])/(x^(1/3)*(1 + x^2 - x^6)^(1/3)) + ((1 - I*Sqrt[3])*(x + x^3 - x^7)^(1/3)*Defer[Subst][Defer[Int]
[(x*(1 + x^3 - x^9)^(1/3))/(-1 + I*Sqrt[3] - 2*x^3)^2, x], x, x^(2/3)])/(x^(1/3)*(1 + x^2 - x^6)^(1/3)) - (2*(
x + x^3 - x^7)^(1/3)*Defer[Subst][Defer[Int][(x*(1 + x^3 - x^9)^(1/3))/(1 + I*Sqrt[3] + 2*x^3)^2, x], x, x^(2/
3)])/(x^(1/3)*(1 + x^2 - x^6)^(1/3)) + ((1 + I*Sqrt[3])*(x + x^3 - x^7)^(1/3)*Defer[Subst][Defer[Int][(x*(1 +
x^3 - x^9)^(1/3))/(1 + I*Sqrt[3] + 2*x^3)^2, x], x, x^(2/3)])/(x^(1/3)*(1 + x^2 - x^6)^(1/3))

Rubi steps

\begin {align*} \int \frac {\left (1+2 x^6\right ) \sqrt [3]{x+x^3-x^7}}{\left (-1+x^6\right )^2} \, dx &=\frac {\sqrt [3]{x+x^3-x^7} \int \frac {\sqrt [3]{x} \sqrt [3]{1+x^2-x^6} \left (1+2 x^6\right )}{\left (-1+x^6\right )^2} \, dx}{\sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}\\ &=\frac {\left (3 \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {x^3 \sqrt [3]{1+x^6-x^{18}} \left (1+2 x^{18}\right )}{\left (-1+x^{18}\right )^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}\\ &=\frac {\left (3 \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{1+x^3-x^9} \left (1+2 x^9\right )}{\left (-1+x^9\right )^2} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}\\ &=\frac {\left (3 \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \left (\frac {\sqrt [3]{1+x^3-x^9}}{27 (-1+x)^2}-\frac {\sqrt [3]{1+x^3-x^9}}{27 (-1+x)}-\frac {\sqrt [3]{1+x^3-x^9}}{9 \left (1+x+x^2\right )^2}+\frac {(1+x) \sqrt [3]{1+x^3-x^9}}{27 \left (1+x+x^2\right )}+\frac {x \left (1+x^3\right ) \sqrt [3]{1+x^3-x^9}}{\left (1+x^3+x^6\right )^2}-\frac {x \sqrt [3]{1+x^3-x^9}}{3 \left (1+x^3+x^6\right )}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}\\ &=\frac {\sqrt [3]{x+x^3-x^7} \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{(-1+x)^2} \, dx,x,x^{2/3}\right )}{18 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\sqrt [3]{x+x^3-x^7} \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{-1+x} \, dx,x,x^{2/3}\right )}{18 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\sqrt [3]{x+x^3-x^7} \text {Subst}\left (\int \frac {(1+x) \sqrt [3]{1+x^3-x^9}}{1+x+x^2} \, dx,x,x^{2/3}\right )}{18 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\sqrt [3]{x+x^3-x^7} \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{\left (1+x+x^2\right )^2} \, dx,x,x^{2/3}\right )}{6 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\sqrt [3]{x+x^3-x^7} \text {Subst}\left (\int \frac {x \sqrt [3]{1+x^3-x^9}}{1+x^3+x^6} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (3 \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {x \left (1+x^3\right ) \sqrt [3]{1+x^3-x^9}}{\left (1+x^3+x^6\right )^2} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}\\ &=\frac {\sqrt [3]{x+x^3-x^7} \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{(-1+x)^2} \, dx,x,x^{2/3}\right )}{18 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\sqrt [3]{x+x^3-x^7} \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{-1+x} \, dx,x,x^{2/3}\right )}{18 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\sqrt [3]{x+x^3-x^7} \text {Subst}\left (\int \left (\frac {\left (1-\frac {i}{\sqrt {3}}\right ) \sqrt [3]{1+x^3-x^9}}{1-i \sqrt {3}+2 x}+\frac {\left (1+\frac {i}{\sqrt {3}}\right ) \sqrt [3]{1+x^3-x^9}}{1+i \sqrt {3}+2 x}\right ) \, dx,x,x^{2/3}\right )}{18 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\sqrt [3]{x+x^3-x^7} \text {Subst}\left (\int \left (-\frac {4 \sqrt [3]{1+x^3-x^9}}{3 \left (-1+i \sqrt {3}-2 x\right )^2}+\frac {4 i \sqrt [3]{1+x^3-x^9}}{3 \sqrt {3} \left (-1+i \sqrt {3}-2 x\right )}-\frac {4 \sqrt [3]{1+x^3-x^9}}{3 \left (1+i \sqrt {3}+2 x\right )^2}+\frac {4 i \sqrt [3]{1+x^3-x^9}}{3 \sqrt {3} \left (1+i \sqrt {3}+2 x\right )}\right ) \, dx,x,x^{2/3}\right )}{6 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\sqrt [3]{x+x^3-x^7} \text {Subst}\left (\int \left (\frac {2 i x \sqrt [3]{1+x^3-x^9}}{\sqrt {3} \left (-1+i \sqrt {3}-2 x^3\right )}+\frac {2 i x \sqrt [3]{1+x^3-x^9}}{\sqrt {3} \left (1+i \sqrt {3}+2 x^3\right )}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (3 \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \left (\frac {x \sqrt [3]{1+x^3-x^9}}{\left (1+x^3+x^6\right )^2}+\frac {x^4 \sqrt [3]{1+x^3-x^9}}{\left (1+x^3+x^6\right )^2}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}\\ &=\frac {\sqrt [3]{x+x^3-x^7} \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{(-1+x)^2} \, dx,x,x^{2/3}\right )}{18 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\sqrt [3]{x+x^3-x^7} \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{-1+x} \, dx,x,x^{2/3}\right )}{18 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (2 \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{\left (-1+i \sqrt {3}-2 x\right )^2} \, dx,x,x^{2/3}\right )}{9 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (2 \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{\left (1+i \sqrt {3}+2 x\right )^2} \, dx,x,x^{2/3}\right )}{9 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (3 \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{1+x^3-x^9}}{\left (1+x^3+x^6\right )^2} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (3 \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {x^4 \sqrt [3]{1+x^3-x^9}}{\left (1+x^3+x^6\right )^2} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\left (2 i \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{-1+i \sqrt {3}-2 x} \, dx,x,x^{2/3}\right )}{9 \sqrt {3} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\left (2 i \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{1+i \sqrt {3}+2 x} \, dx,x,x^{2/3}\right )}{9 \sqrt {3} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\left (i \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{1+x^3-x^9}}{-1+i \sqrt {3}-2 x^3} \, dx,x,x^{2/3}\right )}{\sqrt {3} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\left (i \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{1+x^3-x^9}}{1+i \sqrt {3}+2 x^3} \, dx,x,x^{2/3}\right )}{\sqrt {3} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (\left (3-i \sqrt {3}\right ) \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{1-i \sqrt {3}+2 x} \, dx,x,x^{2/3}\right )}{54 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (\left (3+i \sqrt {3}\right ) \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{1+i \sqrt {3}+2 x} \, dx,x,x^{2/3}\right )}{54 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}\\ &=\frac {\sqrt [3]{x+x^3-x^7} \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{(-1+x)^2} \, dx,x,x^{2/3}\right )}{18 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\sqrt [3]{x+x^3-x^7} \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{-1+x} \, dx,x,x^{2/3}\right )}{18 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (2 \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{\left (-1+i \sqrt {3}-2 x\right )^2} \, dx,x,x^{2/3}\right )}{9 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (2 \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{\left (1+i \sqrt {3}+2 x\right )^2} \, dx,x,x^{2/3}\right )}{9 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (3 \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \left (-\frac {2 \left (-1+i \sqrt {3}\right ) x \sqrt [3]{1+x^3-x^9}}{3 \left (-1+i \sqrt {3}-2 x^3\right )^2}-\frac {2 i x \sqrt [3]{1+x^3-x^9}}{3 \sqrt {3} \left (-1+i \sqrt {3}-2 x^3\right )}+\frac {2 \left (1+i \sqrt {3}\right ) x \sqrt [3]{1+x^3-x^9}}{3 \left (1+i \sqrt {3}+2 x^3\right )^2}-\frac {2 i x \sqrt [3]{1+x^3-x^9}}{3 \sqrt {3} \left (1+i \sqrt {3}+2 x^3\right )}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (3 \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \left (-\frac {4 x \sqrt [3]{1+x^3-x^9}}{3 \left (-1+i \sqrt {3}-2 x^3\right )^2}+\frac {4 i x \sqrt [3]{1+x^3-x^9}}{3 \sqrt {3} \left (-1+i \sqrt {3}-2 x^3\right )}-\frac {4 x \sqrt [3]{1+x^3-x^9}}{3 \left (1+i \sqrt {3}+2 x^3\right )^2}+\frac {4 i x \sqrt [3]{1+x^3-x^9}}{3 \sqrt {3} \left (1+i \sqrt {3}+2 x^3\right )}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\left (2 i \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{-1+i \sqrt {3}-2 x} \, dx,x,x^{2/3}\right )}{9 \sqrt {3} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\left (2 i \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{1+i \sqrt {3}+2 x} \, dx,x,x^{2/3}\right )}{9 \sqrt {3} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\left (i \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \left (-\frac {(-1)^{2/3} \left (1-i \sqrt {3}\right )^{2/3} \sqrt [3]{1+x^3-x^9}}{3 \sqrt [3]{2} \left (-1+i \sqrt {3}\right ) \left (\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{-2} x\right )}-\frac {\left (1-i \sqrt {3}\right )^{2/3} \sqrt [3]{1+x^3-x^9}}{3 \sqrt [3]{2} \left (-1+i \sqrt {3}\right ) \left (\sqrt [3]{1-i \sqrt {3}}+\sqrt [3]{2} x\right )}+\frac {\sqrt [3]{-\frac {1}{2}} \left (1-i \sqrt {3}\right )^{2/3} \sqrt [3]{1+x^3-x^9}}{3 \left (-1+i \sqrt {3}\right ) \left (\sqrt [3]{1-i \sqrt {3}}+(-1)^{2/3} \sqrt [3]{2} x\right )}\right ) \, dx,x,x^{2/3}\right )}{\sqrt {3} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\left (i \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \left (-\frac {(-1)^{2/3} \sqrt [3]{1+x^3-x^9}}{3 \sqrt [3]{2} \sqrt [3]{1+i \sqrt {3}} \left (\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{-2} x\right )}-\frac {\sqrt [3]{1+x^3-x^9}}{3 \sqrt [3]{2} \sqrt [3]{1+i \sqrt {3}} \left (\sqrt [3]{1+i \sqrt {3}}+\sqrt [3]{2} x\right )}+\frac {\sqrt [3]{-\frac {1}{2}} \sqrt [3]{1+x^3-x^9}}{3 \sqrt [3]{1+i \sqrt {3}} \left (\sqrt [3]{1+i \sqrt {3}}+(-1)^{2/3} \sqrt [3]{2} x\right )}\right ) \, dx,x,x^{2/3}\right )}{\sqrt {3} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (\left (3-i \sqrt {3}\right ) \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{1-i \sqrt {3}+2 x} \, dx,x,x^{2/3}\right )}{54 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (\left (3+i \sqrt {3}\right ) \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{1+i \sqrt {3}+2 x} \, dx,x,x^{2/3}\right )}{54 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}\\ &=\frac {\sqrt [3]{x+x^3-x^7} \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{(-1+x)^2} \, dx,x,x^{2/3}\right )}{18 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\sqrt [3]{x+x^3-x^7} \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{-1+x} \, dx,x,x^{2/3}\right )}{18 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (2 \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{\left (-1+i \sqrt {3}-2 x\right )^2} \, dx,x,x^{2/3}\right )}{9 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (2 \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{\left (1+i \sqrt {3}+2 x\right )^2} \, dx,x,x^{2/3}\right )}{9 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\left (2 \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{1+x^3-x^9}}{\left (-1+i \sqrt {3}-2 x^3\right )^2} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\left (2 \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{1+x^3-x^9}}{\left (1+i \sqrt {3}+2 x^3\right )^2} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\left (2 i \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{-1+i \sqrt {3}-2 x} \, dx,x,x^{2/3}\right )}{9 \sqrt {3} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\left (2 i \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{1+i \sqrt {3}+2 x} \, dx,x,x^{2/3}\right )}{9 \sqrt {3} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\left (i \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{1+x^3-x^9}}{-1+i \sqrt {3}-2 x^3} \, dx,x,x^{2/3}\right )}{\sqrt {3} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\left (i \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{1+x^3-x^9}}{1+i \sqrt {3}+2 x^3} \, dx,x,x^{2/3}\right )}{\sqrt {3} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (2 i \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{1+x^3-x^9}}{-1+i \sqrt {3}-2 x^3} \, dx,x,x^{2/3}\right )}{\sqrt {3} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (2 i \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{1+x^3-x^9}}{1+i \sqrt {3}+2 x^3} \, dx,x,x^{2/3}\right )}{\sqrt {3} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\left (i \sqrt [3]{-\frac {i}{2 \left (i-\sqrt {3}\right )}} \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{\sqrt [3]{1+i \sqrt {3}}+(-1)^{2/3} \sqrt [3]{2} x} \, dx,x,x^{2/3}\right )}{3 \sqrt {3} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\left (i \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{\sqrt [3]{1-i \sqrt {3}}+\sqrt [3]{2} x} \, dx,x,x^{2/3}\right )}{3 \sqrt [3]{2} \sqrt {3} \sqrt [3]{1-i \sqrt {3}} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{1+x^3-x^9}}{\left (-1+i \sqrt {3}-2 x^3\right )^2} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (\left (3-i \sqrt {3}\right ) \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{1-i \sqrt {3}+2 x} \, dx,x,x^{2/3}\right )}{54 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\left ((-1)^{5/6} \left (1-i \sqrt {3}\right )^{2/3} \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{\sqrt [3]{1-i \sqrt {3}}+(-1)^{2/3} \sqrt [3]{2} x} \, dx,x,x^{2/3}\right )}{3 \sqrt [3]{2} \sqrt {3} \left (-1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (i \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{-2} x} \, dx,x,x^{2/3}\right )}{3 \sqrt [3]{2} \sqrt {3} \sqrt [3]{-1+i \sqrt {3}} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (i \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{\sqrt [3]{1+i \sqrt {3}}+\sqrt [3]{2} x} \, dx,x,x^{2/3}\right )}{3 \sqrt [3]{2} \sqrt {3} \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\left (\sqrt [6]{-1} \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{-2} x} \, dx,x,x^{2/3}\right )}{3 \sqrt [3]{2} \sqrt {3} \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{1+x^3-x^9}}{\left (1+i \sqrt {3}+2 x^3\right )^2} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (\left (3+i \sqrt {3}\right ) \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{1+i \sqrt {3}+2 x} \, dx,x,x^{2/3}\right )}{54 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}\\ &=\frac {\sqrt [3]{x+x^3-x^7} \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{(-1+x)^2} \, dx,x,x^{2/3}\right )}{18 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\sqrt [3]{x+x^3-x^7} \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{-1+x} \, dx,x,x^{2/3}\right )}{18 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (2 \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{\left (-1+i \sqrt {3}-2 x\right )^2} \, dx,x,x^{2/3}\right )}{9 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (2 \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{\left (1+i \sqrt {3}+2 x\right )^2} \, dx,x,x^{2/3}\right )}{9 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\left (2 \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{1+x^3-x^9}}{\left (-1+i \sqrt {3}-2 x^3\right )^2} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\left (2 \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{1+x^3-x^9}}{\left (1+i \sqrt {3}+2 x^3\right )^2} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\left (2 i \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{-1+i \sqrt {3}-2 x} \, dx,x,x^{2/3}\right )}{9 \sqrt {3} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\left (2 i \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{1+i \sqrt {3}+2 x} \, dx,x,x^{2/3}\right )}{9 \sqrt {3} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\left (i \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \left (-\frac {(-1)^{2/3} \left (1-i \sqrt {3}\right )^{2/3} \sqrt [3]{1+x^3-x^9}}{3 \sqrt [3]{2} \left (-1+i \sqrt {3}\right ) \left (\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{-2} x\right )}-\frac {\left (1-i \sqrt {3}\right )^{2/3} \sqrt [3]{1+x^3-x^9}}{3 \sqrt [3]{2} \left (-1+i \sqrt {3}\right ) \left (\sqrt [3]{1-i \sqrt {3}}+\sqrt [3]{2} x\right )}+\frac {\sqrt [3]{-\frac {1}{2}} \left (1-i \sqrt {3}\right )^{2/3} \sqrt [3]{1+x^3-x^9}}{3 \left (-1+i \sqrt {3}\right ) \left (\sqrt [3]{1-i \sqrt {3}}+(-1)^{2/3} \sqrt [3]{2} x\right )}\right ) \, dx,x,x^{2/3}\right )}{\sqrt {3} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\left (i \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \left (-\frac {(-1)^{2/3} \sqrt [3]{1+x^3-x^9}}{3 \sqrt [3]{2} \sqrt [3]{1+i \sqrt {3}} \left (\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{-2} x\right )}-\frac {\sqrt [3]{1+x^3-x^9}}{3 \sqrt [3]{2} \sqrt [3]{1+i \sqrt {3}} \left (\sqrt [3]{1+i \sqrt {3}}+\sqrt [3]{2} x\right )}+\frac {\sqrt [3]{-\frac {1}{2}} \sqrt [3]{1+x^3-x^9}}{3 \sqrt [3]{1+i \sqrt {3}} \left (\sqrt [3]{1+i \sqrt {3}}+(-1)^{2/3} \sqrt [3]{2} x\right )}\right ) \, dx,x,x^{2/3}\right )}{\sqrt {3} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (2 i \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \left (-\frac {(-1)^{2/3} \left (1-i \sqrt {3}\right )^{2/3} \sqrt [3]{1+x^3-x^9}}{3 \sqrt [3]{2} \left (-1+i \sqrt {3}\right ) \left (\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{-2} x\right )}-\frac {\left (1-i \sqrt {3}\right )^{2/3} \sqrt [3]{1+x^3-x^9}}{3 \sqrt [3]{2} \left (-1+i \sqrt {3}\right ) \left (\sqrt [3]{1-i \sqrt {3}}+\sqrt [3]{2} x\right )}+\frac {\sqrt [3]{-\frac {1}{2}} \left (1-i \sqrt {3}\right )^{2/3} \sqrt [3]{1+x^3-x^9}}{3 \left (-1+i \sqrt {3}\right ) \left (\sqrt [3]{1-i \sqrt {3}}+(-1)^{2/3} \sqrt [3]{2} x\right )}\right ) \, dx,x,x^{2/3}\right )}{\sqrt {3} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (2 i \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \left (-\frac {(-1)^{2/3} \sqrt [3]{1+x^3-x^9}}{3 \sqrt [3]{2} \sqrt [3]{1+i \sqrt {3}} \left (\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{-2} x\right )}-\frac {\sqrt [3]{1+x^3-x^9}}{3 \sqrt [3]{2} \sqrt [3]{1+i \sqrt {3}} \left (\sqrt [3]{1+i \sqrt {3}}+\sqrt [3]{2} x\right )}+\frac {\sqrt [3]{-\frac {1}{2}} \sqrt [3]{1+x^3-x^9}}{3 \sqrt [3]{1+i \sqrt {3}} \left (\sqrt [3]{1+i \sqrt {3}}+(-1)^{2/3} \sqrt [3]{2} x\right )}\right ) \, dx,x,x^{2/3}\right )}{\sqrt {3} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\left (i \sqrt [3]{-\frac {i}{2 \left (i-\sqrt {3}\right )}} \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{\sqrt [3]{1+i \sqrt {3}}+(-1)^{2/3} \sqrt [3]{2} x} \, dx,x,x^{2/3}\right )}{3 \sqrt {3} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\left (i \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{\sqrt [3]{1-i \sqrt {3}}+\sqrt [3]{2} x} \, dx,x,x^{2/3}\right )}{3 \sqrt [3]{2} \sqrt {3} \sqrt [3]{1-i \sqrt {3}} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{1+x^3-x^9}}{\left (-1+i \sqrt {3}-2 x^3\right )^2} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (\left (3-i \sqrt {3}\right ) \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{1-i \sqrt {3}+2 x} \, dx,x,x^{2/3}\right )}{54 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\left ((-1)^{5/6} \left (1-i \sqrt {3}\right )^{2/3} \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{\sqrt [3]{1-i \sqrt {3}}+(-1)^{2/3} \sqrt [3]{2} x} \, dx,x,x^{2/3}\right )}{3 \sqrt [3]{2} \sqrt {3} \left (-1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (i \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{-2} x} \, dx,x,x^{2/3}\right )}{3 \sqrt [3]{2} \sqrt {3} \sqrt [3]{-1+i \sqrt {3}} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (i \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{\sqrt [3]{1+i \sqrt {3}}+\sqrt [3]{2} x} \, dx,x,x^{2/3}\right )}{3 \sqrt [3]{2} \sqrt {3} \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\left (\sqrt [6]{-1} \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{-2} x} \, dx,x,x^{2/3}\right )}{3 \sqrt [3]{2} \sqrt {3} \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{1+x^3-x^9}}{\left (1+i \sqrt {3}+2 x^3\right )^2} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (\left (3+i \sqrt {3}\right ) \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{1+i \sqrt {3}+2 x} \, dx,x,x^{2/3}\right )}{54 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}\\ &=\frac {\sqrt [3]{x+x^3-x^7} \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{(-1+x)^2} \, dx,x,x^{2/3}\right )}{18 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\sqrt [3]{x+x^3-x^7} \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{-1+x} \, dx,x,x^{2/3}\right )}{18 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (2 \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{\left (-1+i \sqrt {3}-2 x\right )^2} \, dx,x,x^{2/3}\right )}{9 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (2 \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{\left (1+i \sqrt {3}+2 x\right )^2} \, dx,x,x^{2/3}\right )}{9 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\left (2 \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{1+x^3-x^9}}{\left (-1+i \sqrt {3}-2 x^3\right )^2} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\left (2 \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{1+x^3-x^9}}{\left (1+i \sqrt {3}+2 x^3\right )^2} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\left (2 i \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{-1+i \sqrt {3}-2 x} \, dx,x,x^{2/3}\right )}{9 \sqrt {3} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\left (2 i \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{1+i \sqrt {3}+2 x} \, dx,x,x^{2/3}\right )}{9 \sqrt {3} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-2 \frac {\left (i \sqrt [3]{-\frac {i}{2 \left (i-\sqrt {3}\right )}} \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{\sqrt [3]{1+i \sqrt {3}}+(-1)^{2/3} \sqrt [3]{2} x} \, dx,x,x^{2/3}\right )}{3 \sqrt {3} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (i 2^{2/3} \sqrt [3]{-\frac {i}{i-\sqrt {3}}} \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{\sqrt [3]{1+i \sqrt {3}}+(-1)^{2/3} \sqrt [3]{2} x} \, dx,x,x^{2/3}\right )}{3 \sqrt {3} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-2 \frac {\left (i \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{\sqrt [3]{1-i \sqrt {3}}+\sqrt [3]{2} x} \, dx,x,x^{2/3}\right )}{3 \sqrt [3]{2} \sqrt {3} \sqrt [3]{1-i \sqrt {3}} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (i 2^{2/3} \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{\sqrt [3]{1-i \sqrt {3}}+\sqrt [3]{2} x} \, dx,x,x^{2/3}\right )}{3 \sqrt {3} \sqrt [3]{1-i \sqrt {3}} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{1+x^3-x^9}}{\left (-1+i \sqrt {3}-2 x^3\right )^2} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (\left (3-i \sqrt {3}\right ) \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{1-i \sqrt {3}+2 x} \, dx,x,x^{2/3}\right )}{54 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-2 \frac {\left ((-1)^{5/6} \left (1-i \sqrt {3}\right )^{2/3} \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{\sqrt [3]{1-i \sqrt {3}}+(-1)^{2/3} \sqrt [3]{2} x} \, dx,x,x^{2/3}\right )}{3 \sqrt [3]{2} \sqrt {3} \left (-1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left ((-1)^{5/6} \left (2 \left (1-i \sqrt {3}\right )\right )^{2/3} \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{\sqrt [3]{1-i \sqrt {3}}+(-1)^{2/3} \sqrt [3]{2} x} \, dx,x,x^{2/3}\right )}{3 \sqrt {3} \left (-1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+2 \frac {\left (i \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{-2} x} \, dx,x,x^{2/3}\right )}{3 \sqrt [3]{2} \sqrt {3} \sqrt [3]{-1+i \sqrt {3}} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\left (i 2^{2/3} \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{-2} x} \, dx,x,x^{2/3}\right )}{3 \sqrt {3} \sqrt [3]{-1+i \sqrt {3}} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+2 \frac {\left (i \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{\sqrt [3]{1+i \sqrt {3}}+\sqrt [3]{2} x} \, dx,x,x^{2/3}\right )}{3 \sqrt [3]{2} \sqrt {3} \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-2 \frac {\left (\sqrt [6]{-1} \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{-2} x} \, dx,x,x^{2/3}\right )}{3 \sqrt [3]{2} \sqrt {3} \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}-\frac {\left (i 2^{2/3} \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{\sqrt [3]{1+i \sqrt {3}}+\sqrt [3]{2} x} \, dx,x,x^{2/3}\right )}{3 \sqrt {3} \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (\sqrt [6]{-1} 2^{2/3} \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{-2} x} \, dx,x,x^{2/3}\right )}{3 \sqrt {3} \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{1+x^3-x^9}}{\left (1+i \sqrt {3}+2 x^3\right )^2} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}+\frac {\left (\left (3+i \sqrt {3}\right ) \sqrt [3]{x+x^3-x^7}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3-x^9}}{1+i \sqrt {3}+2 x} \, dx,x,x^{2/3}\right )}{54 \sqrt [3]{x} \sqrt [3]{1+x^2-x^6}}\\ \end {align*}

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Mathematica [A]
time = 2.92, size = 231, normalized size = 1.83 \begin {gather*} \frac {\sqrt [3]{x+x^3-x^7} \left (-6 x^{4/3} \sqrt [3]{-1-x^2+x^6}+2 \sqrt {3} \left (-1+x^6\right ) \text {ArcTan}\left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}-2 \sqrt [3]{-1-x^2+x^6}}\right )+2 \left (-1+x^6\right ) \log \left (x^{2/3}+\sqrt [3]{-1-x^2+x^6}\right )+\log \left (x^{4/3}-x^{2/3} \sqrt [3]{-1-x^2+x^6}+\left (-1-x^2+x^6\right )^{2/3}\right )-x^6 \log \left (x^{4/3}-x^{2/3} \sqrt [3]{-1-x^2+x^6}+\left (-1-x^2+x^6\right )^{2/3}\right )\right )}{12 \sqrt [3]{x} \left (-1+x^6\right ) \sqrt [3]{-1-x^2+x^6}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 + 2*x^6)*(x + x^3 - x^7)^(1/3))/(-1 + x^6)^2,x]

[Out]

((x + x^3 - x^7)^(1/3)*(-6*x^(4/3)*(-1 - x^2 + x^6)^(1/3) + 2*Sqrt[3]*(-1 + x^6)*ArcTan[(Sqrt[3]*x^(2/3))/(x^(
2/3) - 2*(-1 - x^2 + x^6)^(1/3))] + 2*(-1 + x^6)*Log[x^(2/3) + (-1 - x^2 + x^6)^(1/3)] + Log[x^(4/3) - x^(2/3)
*(-1 - x^2 + x^6)^(1/3) + (-1 - x^2 + x^6)^(2/3)] - x^6*Log[x^(4/3) - x^(2/3)*(-1 - x^2 + x^6)^(1/3) + (-1 - x
^2 + x^6)^(2/3)]))/(12*x^(1/3)*(-1 + x^6)*(-1 - x^2 + x^6)^(1/3))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 28.18, size = 709, normalized size = 5.63

method	result	size
trager	\(\text {Expression too large to display}\)	\(709\)
risch	\(\text {Expression too large to display}\)	\(1138\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^6+1)*(-x^7+x^3+x)^(1/3)/(x^6-1)^2,x,method=_RETURNVERBOSE)

[Out]

-1/2*x/(x^6-1)*(-x^7+x^3+x)^(1/3)+1/6*ln(-(22657457104844936692027097669418834480*RootOf(142884*_Z^2-378*_Z+1)
^2*x^6-945189876502528470068106922206900498*RootOf(142884*_Z^2-378*_Z+1)*x^6+238300401071513217579004442807899
1*x^6-356854949401307752899426788293346643060*RootOf(142884*_Z^2-378*_Z+1)^2*x^2+17260846671303884500192458447
53992776*RootOf(142884*_Z^2-378*_Z+1)*(-x^7+x^3+x)^(2/3)+1726084667130388450019245844753992776*RootOf(142884*_
Z^2-378*_Z+1)*(-x^7+x^3+x)^(1/3)*x+2670145379832260812187041580979777546*RootOf(142884*_Z^2-378*_Z+1)*x^2-2265
7457104844936692027097669418834480*RootOf(142884*_Z^2-378*_Z+1)^2-2106933314515559427461422565575221*(-x^7+x^3
+x)^(2/3)-2106933314515559427461422565575221*x*(-x^7+x^3+x)^(1/3)-4604448427483475729492628216966186*x^2+94518
9876502528470068106922206900498*RootOf(142884*_Z^2-378*_Z+1)-2383004010715132175790044428078991)/(-1+x)/(1+x)/
(x^2+x+1)/(x^2-x+1))-63*ln(-(22657457104844936692027097669418834480*RootOf(142884*_Z^2-378*_Z+1)^2*x^6-9451898
76502528470068106922206900498*RootOf(142884*_Z^2-378*_Z+1)*x^6+2383004010715132175790044428078991*x^6-35685494
9401307752899426788293346643060*RootOf(142884*_Z^2-378*_Z+1)^2*x^2+1726084667130388450019245844753992776*RootO
f(142884*_Z^2-378*_Z+1)*(-x^7+x^3+x)^(2/3)+1726084667130388450019245844753992776*RootOf(142884*_Z^2-378*_Z+1)*
(-x^7+x^3+x)^(1/3)*x+2670145379832260812187041580979777546*RootOf(142884*_Z^2-378*_Z+1)*x^2-226574571048449366
92027097669418834480*RootOf(142884*_Z^2-378*_Z+1)^2-2106933314515559427461422565575221*(-x^7+x^3+x)^(2/3)-2106
933314515559427461422565575221*x*(-x^7+x^3+x)^(1/3)-4604448427483475729492628216966186*x^2+9451898765025284700
68106922206900498*RootOf(142884*_Z^2-378*_Z+1)-2383004010715132175790044428078991)/(-1+x)/(1+x)/(x^2+x+1)/(x^2
-x+1))*RootOf(142884*_Z^2-378*_Z+1)+63*RootOf(142884*_Z^2-378*_Z+1)*ln(-(2265745710484493669202709766941883448
0*RootOf(142884*_Z^2-378*_Z+1)^2*x^6+825309151080068487570609050940134178*RootOf(142884*_Z^2-378*_Z+1)*x^6+410
74080050321385262642347196670*x^6-356854949401307752899426788293346643060*RootOf(142884*_Z^2-378*_Z+1)^2*x^2-1
726084667130388450019245844753992776*RootOf(142884*_Z^2-378*_Z+1)*(-x^7+x^3+x)^(2/3)-1726084667130388450019245
844753992776*RootOf(142884*_Z^2-378*_Z+1)*(-x^7+x^3+x)^(1/3)*x-782023954428516087851450108528208006*RootOf(142
884*_Z^2-378*_Z+1)*x^2-22657457104844936692027097669418834480*RootOf(142884*_Z^2-378*_Z+1)^2+24594282387394364
72060391838535871*(-x^7+x^3+x)^(2/3)+2459428238739436472060391838535871*x*(-x^7+x^3+x)^(1/3)-38086874228479829
970813812855094*x^2-825309151080068487570609050940134178*RootOf(142884*_Z^2-378*_Z+1)-410740800503213852626423
47196670)/(-1+x)/(1+x)/(x^2+x+1)/(x^2-x+1))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^6+1)*(-x^7+x^3+x)^(1/3)/(x^6-1)^2,x, algorithm="maxima")

[Out]

integrate((-x^7 + x^3 + x)^(1/3)*(2*x^6 + 1)/(x^6 - 1)^2, x)

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Fricas [A]
time = 0.83, size = 151, normalized size = 1.20 \begin {gather*} \frac {2 \, \sqrt {3} {\left (x^{6} - 1\right )} \arctan \left (-\frac {4 \, \sqrt {3} {\left (-x^{7} + x^{3} + x\right )}^{\frac {1}{3}} x - \sqrt {3} {\left (x^{6} - x^{2} - 1\right )} - 2 \, \sqrt {3} {\left (-x^{7} + x^{3} + x\right )}^{\frac {2}{3}}}{x^{6} - 9 \, x^{2} - 1}\right ) - {\left (x^{6} - 1\right )} \log \left (\frac {x^{6} - 3 \, {\left (-x^{7} + x^{3} + x\right )}^{\frac {1}{3}} x + 3 \, {\left (-x^{7} + x^{3} + x\right )}^{\frac {2}{3}} - 1}{x^{6} - 1}\right ) - 6 \, {\left (-x^{7} + x^{3} + x\right )}^{\frac {1}{3}} x}{12 \, {\left (x^{6} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^6+1)*(-x^7+x^3+x)^(1/3)/(x^6-1)^2,x, algorithm="fricas")

[Out]

1/12*(2*sqrt(3)*(x^6 - 1)*arctan(-(4*sqrt(3)*(-x^7 + x^3 + x)^(1/3)*x - sqrt(3)*(x^6 - x^2 - 1) - 2*sqrt(3)*(-
x^7 + x^3 + x)^(2/3))/(x^6 - 9*x^2 - 1)) - (x^6 - 1)*log((x^6 - 3*(-x^7 + x^3 + x)^(1/3)*x + 3*(-x^7 + x^3 + x
)^(2/3) - 1)/(x^6 - 1)) - 6*(-x^7 + x^3 + x)^(1/3)*x)/(x^6 - 1)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{- x \left (x^{6} - x^{2} - 1\right )} \left (2 x^{6} + 1\right )}{\left (x - 1\right )^{2} \left (x + 1\right )^{2} \left (x^{2} - x + 1\right )^{2} \left (x^{2} + x + 1\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**6+1)*(-x**7+x**3+x)**(1/3)/(x**6-1)**2,x)

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^6+1)*(-x^7+x^3+x)^(1/3)/(x^6-1)^2,x, algorithm="giac")

[Out]

integrate((-x^7 + x^3 + x)^(1/3)*(2*x^6 + 1)/(x^6 - 1)^2, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (2\,x^6+1\right )\,{\left (-x^7+x^3+x\right )}^{1/3}}{{\left (x^6-1\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^6 + 1)*(x + x^3 - x^7)^(1/3))/(x^6 - 1)^2,x)

[Out]

int(((2*x^6 + 1)*(x + x^3 - x^7)^(1/3))/(x^6 - 1)^2, x)

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