\(\int \frac {(-1+2 x^6) \sqrt [3]{x+x^7}}{(1-2 x^2+x^6) (1-x^2+x^6)} \, dx\) [2388]

Optimal result

Integrand size = 41, antiderivative size = 191 \[ \int \frac {\left (-1+2 x^6\right ) \sqrt [3]{x+x^7}}{\left (1-2 x^2+x^6\right ) \left (1-x^2+x^6\right )} \, dx=-\frac {1}{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+x^7}}\right )+\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{x+x^7}}\right )}{2^{2/3}}-\frac {1}{2} \log \left (-x+\sqrt [3]{x+x^7}\right )+\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{x+x^7}\right )}{2^{2/3}}+\frac {1}{4} \log \left (x^2+x \sqrt [3]{x+x^7}+\left (x+x^7\right )^{2/3}\right )-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{x+x^7}+\sqrt [3]{2} \left (x+x^7\right )^{2/3}\right )}{2\ 2^{2/3}} \]

[Out]

-1/2*3^(1/2)*arctan(3^(1/2)*x/(x+2*(x^7+x)^(1/3)))+1/2*3^(1/2)*arctan(3^(1/2)*x/(x+2^(2/3)*(x^7+x)^(1/3)))*2^(
1/3)-1/2*ln(-x+(x^7+x)^(1/3))+1/2*ln(-2*x+2^(2/3)*(x^7+x)^(1/3))*2^(1/3)+1/4*ln(x^2+x*(x^7+x)^(1/3)+(x^7+x)^(2
/3))-1/4*ln(2*x^2+2^(2/3)*x*(x^7+x)^(1/3)+2^(1/3)*(x^7+x)^(2/3))*2^(1/3)

Rubi [F]

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

[In]

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

[Out]

((x + x^7)^(1/3)*Defer[Subst][Defer[Int][(1 + x^9)^(1/3)/(-1 + x), x], x, x^(2/3)])/(2*x^(1/3)*(1 + x^6)^(1/3)
) - ((1 + I*Sqrt[3])*(x + x^7)^(1/3)*Defer[Subst][Defer[Int][(1 + x^9)^(1/3)/(1 - I*Sqrt[3] + 2*x), x], x, x^(
2/3)])/(2*x^(1/3)*(1 + x^6)^(1/3)) - ((1 - I*Sqrt[3])*(x + x^7)^(1/3)*Defer[Subst][Defer[Int][(1 + x^9)^(1/3)/
(1 + I*Sqrt[3] + 2*x), x], x, x^(2/3)])/(2*x^(1/3)*(1 + x^6)^(1/3)) - ((-1 - Sqrt[5])^(2/3)*(x + x^7)^(1/3)*De
fer[Subst][Defer[Int][(1 + x^9)^(1/3)/((1 + Sqrt[5])^(1/3) - (-2)^(1/3)*x), x], x, x^(2/3)])/(2^(1/3)*Sqrt[5]*
x^(1/3)*(1 + x^6)^(1/3)) + ((-1)^(2/3)*(x + x^7)^(1/3)*Defer[Subst][Defer[Int][(1 + x^9)^(1/3)/((1 + Sqrt[5])^
(1/3) - (-2)^(1/3)*x), x], x, x^(2/3)])/(Sqrt[5]*(2*(1 + Sqrt[5]))^(1/3)*x^(1/3)*(1 + x^6)^(1/3)) - ((1 - Sqrt
[5])^(2/3)*(x + x^7)^(1/3)*Defer[Subst][Defer[Int][(1 + x^9)^(1/3)/((-1 + Sqrt[5])^(1/3) + (-2)^(1/3)*x), x],
x, x^(2/3)])/(2^(1/3)*Sqrt[5]*x^(1/3)*(1 + x^6)^(1/3)) - ((-1)^(2/3)*(x + x^7)^(1/3)*Defer[Subst][Defer[Int][(
1 + x^9)^(1/3)/((-1 + Sqrt[5])^(1/3) + (-2)^(1/3)*x), x], x, x^(2/3)])/(Sqrt[5]*(2*(-1 + Sqrt[5]))^(1/3)*x^(1/
3)*(1 + x^6)^(1/3)) - ((-1 + Sqrt[5])^(2/3)*(x + x^7)^(1/3)*Defer[Subst][Defer[Int][(1 + x^9)^(1/3)/((-1 + Sqr
t[5])^(1/3) - 2^(1/3)*x), x], x, x^(2/3)])/(2^(1/3)*Sqrt[5]*x^(1/3)*(1 + x^6)^(1/3)) - ((x + x^7)^(1/3)*Defer[
Subst][Defer[Int][(1 + x^9)^(1/3)/((-1 + Sqrt[5])^(1/3) - 2^(1/3)*x), x], x, x^(2/3)])/(Sqrt[5]*(2*(-1 + Sqrt[
5]))^(1/3)*x^(1/3)*(1 + x^6)^(1/3)) - ((1 + Sqrt[5])^(2/3)*(x + x^7)^(1/3)*Defer[Subst][Defer[Int][(1 + x^9)^(
1/3)/((1 + Sqrt[5])^(1/3) + 2^(1/3)*x), x], x, x^(2/3)])/(2^(1/3)*Sqrt[5]*x^(1/3)*(1 + x^6)^(1/3)) + ((x + x^7
)^(1/3)*Defer[Subst][Defer[Int][(1 + x^9)^(1/3)/((1 + Sqrt[5])^(1/3) + 2^(1/3)*x), x], x, x^(2/3)])/(Sqrt[5]*(
2*(1 + Sqrt[5]))^(1/3)*x^(1/3)*(1 + x^6)^(1/3)) + ((-1 - Sqrt[5])^(1/3)*(x + x^7)^(1/3)*Defer[Subst][Defer[Int
][(1 + x^9)^(1/3)/((-1 + Sqrt[5])^(1/3) - (-1)^(2/3)*2^(1/3)*x), x], x, x^(2/3)])/(2*Sqrt[5]*x^(1/3)*(1 + x^6)
^(1/3)) + ((-1 - Sqrt[5])^(1/3)*(5 - Sqrt[5])*(x + x^7)^(1/3)*Defer[Subst][Defer[Int][(1 + x^9)^(1/3)/((-1 + S
qrt[5])^(1/3) - (-1)^(2/3)*2^(1/3)*x), x], x, x^(2/3)])/(10*x^(1/3)*(1 + x^6)^(1/3)) - ((1 - Sqrt[5])^(1/3)*(x
+ x^7)^(1/3)*Defer[Subst][Defer[Int][(1 + x^9)^(1/3)/((1 + Sqrt[5])^(1/3) + (-1)^(2/3)*2^(1/3)*x), x], x, x^(
2/3)])/(2*Sqrt[5]*x^(1/3)*(1 + x^6)^(1/3)) + ((1 - Sqrt[5])^(1/3)*(5 + Sqrt[5])*(x + x^7)^(1/3)*Defer[Subst][D
efer[Int][(1 + x^9)^(1/3)/((1 + Sqrt[5])^(1/3) + (-1)^(2/3)*2^(1/3)*x), x], x, x^(2/3)])/(10*x^(1/3)*(1 + x^6)
^(1/3)) + (3*(x + x^7)^(1/3)*Defer[Subst][Defer[Int][(x*(1 + x^9)^(1/3))/(1 - x^3 + x^9), x], x, x^(2/3)])/(2*
x^(1/3)*(1 + x^6)^(1/3)) - (9*(x + x^7)^(1/3)*Defer[Subst][Defer[Int][(x^7*(1 + x^9)^(1/3))/(1 - x^3 + x^9), x
], x, x^(2/3)])/(2*x^(1/3)*(1 + x^6)^(1/3))

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

Mathematica [F]

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

[In]

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

[Out]

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

Maple [A] (verified)

Time = 6.36 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.84

method	result	size
pseudoelliptic	\(\frac {\left (-2 \arctan \left (\frac {\sqrt {3}\, \left (x +2^{\frac {2}{3}} \left (x^{7}+x \right )^{\frac {1}{3}}\right )}{3 x}\right ) \sqrt {3}+2 \ln \left (\frac {-2^{\frac {1}{3}} x +\left (x^{7}+x \right )^{\frac {1}{3}}}{x}\right )-\ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} \left (x^{7}+x \right )^{\frac {1}{3}} x +\left (x^{7}+x \right )^{\frac {2}{3}}}{x^{2}}\right )\right ) 2^{\frac {1}{3}}}{4}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{7}+x \right )^{\frac {1}{3}}\right )}{3 x}\right )}{2}+\frac {\ln \left (\frac {x^{2}+x \left (x^{7}+x \right )^{\frac {1}{3}}+\left (x^{7}+x \right )^{\frac {2}{3}}}{x^{2}}\right )}{4}-\frac {\ln \left (\frac {-x +\left (x^{7}+x \right )^{\frac {1}{3}}}{x}\right )}{2}\)	\(160\)

[In]

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

[Out]

1/4*(-2*arctan(1/3*3^(1/2)/x*(x+2^(2/3)*(x^7+x)^(1/3)))*3^(1/2)+2*ln((-2^(1/3)*x+(x^7+x)^(1/3))/x)-ln((2^(2/3)
*x^2+2^(1/3)*(x^7+x)^(1/3)*x+(x^7+x)^(2/3))/x^2))*2^(1/3)+1/2*3^(1/2)*arctan(1/3*3^(1/2)/x*(x+2*(x^7+x)^(1/3))
)+1/4*ln((x^2+x*(x^7+x)^(1/3)+(x^7+x)^(2/3))/x^2)-1/2*ln((-x+(x^7+x)^(1/3))/x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {\left (-1+2 x^6\right ) \sqrt [3]{x+x^7}}{\left (1-2 x^2+x^6\right ) \left (1-x^2+x^6\right )} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (re
sidue poly has multiple non-linear factors)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (-1+2 x^6\right ) \sqrt [3]{x+x^7}}{\left (1-2 x^2+x^6\right ) \left (1-x^2+x^6\right )} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-1+2 x^6\right ) \sqrt [3]{x+x^7}}{\left (1-2 x^2+x^6\right ) \left (1-x^2+x^6\right )} \, dx=\int \frac {\left (2\,x^6-1\right )\,{\left (x^7+x\right )}^{1/3}}{\left (x^6-x^2+1\right )\,\left (x^6-2\,x^2+1\right )} \,d x \]

[In]

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

[Out]

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