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

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

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Rubi [F]  time = 8.91, 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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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} \operatorname {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} \operatorname {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} \operatorname {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} \operatorname {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} \operatorname {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 ) \operatorname {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} \operatorname {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} \operatorname {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} \operatorname {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} \operatorname {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} \operatorname {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 ) \operatorname {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} \operatorname {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} \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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} \operatorname {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} \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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} \operatorname {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} \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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} \operatorname {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} \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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} \operatorname {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} \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 [F]  time = 2.67, size = 0, normalized size = 0.00 \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]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.21, size = 126, normalized size = 1.00 \begin {gather*} -\frac {x \sqrt [3]{x+x^3-x^7}}{2 \left (-1+x^6\right )}-\frac {\tan ^{-1}\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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.14, 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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (-x^{7} + x^{3} + x\right )}^{\frac {1}{3}} {\left (2 \, x^{6} + 1\right )}}{{\left (x^{6} - 1\right )}^{2}}\,{d x} \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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maple [C]  time = 48.87, size = 476, normalized size = 3.78

method	result	size
trager	\(-\frac {x \left (-x^{7}+x^{3}+x \right )^{\frac {1}{3}}}{2 \left (x^{6}-1\right )}-\frac {\ln \left (\frac {23084281021492945478344725970280579664 \RootOf \left (142884 \textit {\_Z}^{2}-378 \textit {\_Z} +1\right )^{2} x^{6}+779765626827203872050827736565976550 \RootOf \left (142884 \textit {\_Z}^{2}-378 \textit {\_Z} +1\right ) x^{6}-2180370336718022168439941441690525 x^{6}-363577426088513891283929434031919129708 \RootOf \left (142884 \textit {\_Z}^{2}-378 \textit {\_Z} +1\right )^{2} x^{2}+1726084667130388450019245844753992776 \RootOf \left (142884 \textit {\_Z}^{2}-378 \textit {\_Z} +1\right ) \left (-x^{7}+x^{3}+x \right )^{\frac {2}{3}}-929663874243506986438828114966559238 \RootOf \left (142884 \textit {\_Z}^{2}-378 \textit {\_Z} +1\right ) \left (-x^{7}+x^{3}+x \right )^{\frac {1}{3}} x +165424249675324598017279185640923948 \RootOf \left (142884 \textit {\_Z}^{2}-378 \textit {\_Z} +1\right ) x^{2}-2459428238739436472060391838535871 \left (-x^{7}+x^{3}+x \right )^{\frac {2}{3}}-2106933314515559427461422565575221 x \left (-x^{7}+x^{3}+x \right )^{\frac {1}{3}}-23084281021492945478344725970280579664 \RootOf \left (142884 \textit {\_Z}^{2}-378 \textit {\_Z} +1\right )^{2}+2021797948593075101644309336840305 x^{2}-779765626827203872050827736565976550 \RootOf \left (142884 \textit {\_Z}^{2}-378 \textit {\_Z} +1\right )+2180370336718022168439941441690525}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )}\right )}{6}+63 \RootOf \left (142884 \textit {\_Z}^{2}-378 \textit {\_Z} +1\right ) \ln \left (-\frac {426823916648008786317628300861745184 \RootOf \left (142884 \textit {\_Z}^{2}-378 \textit {\_Z} +1\right )^{2} x^{6}+43285196651552399719158942411926172 \RootOf \left (142884 \textit {\_Z}^{2}-378 \textit {\_Z} +1\right ) x^{6}-2338942724842969235235573546540745 x^{6}-6722476687206138384502645738572486648 \RootOf \left (142884 \textit {\_Z}^{2}-378 \textit {\_Z} +1\right )^{2} x^{2}+1726084667130388450019245844753992776 \RootOf \left (142884 \textit {\_Z}^{2}-378 \textit {\_Z} +1\right ) \left (-x^{7}+x^{3}+x \right )^{\frac {2}{3}}-796420792886881463580417729787433538 \RootOf \left (142884 \textit {\_Z}^{2}-378 \textit {\_Z} +1\right ) \left (-x^{7}+x^{3}+x \right )^{\frac {1}{3}} x -911879544383173287008926935763986522 \RootOf \left (142884 \textit {\_Z}^{2}-378 \textit {\_Z} +1\right ) x^{2}-2106933314515559427461422565575221 \left (-x^{7}+x^{3}+x \right )^{\frac {2}{3}}-2459428238739436472060391838535871 x \left (-x^{7}+x^{3}+x \right )^{\frac {1}{3}}-426823916648008786317628300861745184 \RootOf \left (142884 \textit {\_Z}^{2}-378 \textit {\_Z} +1\right )^{2}+4519313061560991403675514988231270 x^{2}-43285196651552399719158942411926172 \RootOf \left (142884 \textit {\_Z}^{2}-378 \textit {\_Z} +1\right )+2338942724842969235235573546540745}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )}\right )\)	\(476\)
risch	\(\text {Expression too large to display}\)	\(1670\)

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((23084281021492945478344725970280579664*RootOf(142884*_Z^2-378*_Z+1)^
2*x^6+779765626827203872050827736565976550*RootOf(142884*_Z^2-378*_Z+1)*x^6-2180370336718022168439941441690525
*x^6-363577426088513891283929434031919129708*RootOf(142884*_Z^2-378*_Z+1)^2*x^2+172608466713038845001924584475
3992776*RootOf(142884*_Z^2-378*_Z+1)*(-x^7+x^3+x)^(2/3)-929663874243506986438828114966559238*RootOf(142884*_Z^
2-378*_Z+1)*(-x^7+x^3+x)^(1/3)*x+165424249675324598017279185640923948*RootOf(142884*_Z^2-378*_Z+1)*x^2-2459428
238739436472060391838535871*(-x^7+x^3+x)^(2/3)-2106933314515559427461422565575221*x*(-x^7+x^3+x)^(1/3)-2308428
1021492945478344725970280579664*RootOf(142884*_Z^2-378*_Z+1)^2+2021797948593075101644309336840305*x^2-77976562
6827203872050827736565976550*RootOf(142884*_Z^2-378*_Z+1)+2180370336718022168439941441690525)/(-1+x)/(1+x)/(x^
2+x+1)/(x^2-x+1))+63*RootOf(142884*_Z^2-378*_Z+1)*ln(-(426823916648008786317628300861745184*RootOf(142884*_Z^2
-378*_Z+1)^2*x^6+43285196651552399719158942411926172*RootOf(142884*_Z^2-378*_Z+1)*x^6-233894272484296923523557
3546540745*x^6-6722476687206138384502645738572486648*RootOf(142884*_Z^2-378*_Z+1)^2*x^2+1726084667130388450019
245844753992776*RootOf(142884*_Z^2-378*_Z+1)*(-x^7+x^3+x)^(2/3)-796420792886881463580417729787433538*RootOf(14
2884*_Z^2-378*_Z+1)*(-x^7+x^3+x)^(1/3)*x-911879544383173287008926935763986522*RootOf(142884*_Z^2-378*_Z+1)*x^2
-2106933314515559427461422565575221*(-x^7+x^3+x)^(2/3)-2459428238739436472060391838535871*x*(-x^7+x^3+x)^(1/3)
-426823916648008786317628300861745184*RootOf(142884*_Z^2-378*_Z+1)^2+4519313061560991403675514988231270*x^2-43
285196651552399719158942411926172*RootOf(142884*_Z^2-378*_Z+1)+2338942724842969235235573546540745)/(-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*} \int \frac {{\left (-x^{7} + x^{3} + x\right )}^{\frac {1}{3}} {\left (2 \, x^{6} + 1\right )}}{{\left (x^{6} - 1\right )}^{2}}\,{d x} \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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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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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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