\(\int \frac {(2-2 x+2 x^2-3 x^3+3 x^4) \sqrt [3]{-x-x^3-x^4+x^6}}{(1+x) (-1+2 x-2 x^2+x^3) (-1-x^3+x^5)} \, dx\) [2896]

Optimal result

Integrand size = 74, antiderivative size = 318 \[ \int \frac {\left (2-2 x+2 x^2-3 x^3+3 x^4\right ) \sqrt [3]{-x-x^3-x^4+x^6}}{(1+x) \left (-1+2 x-2 x^2+x^3\right ) \left (-1-x^3+x^5\right )} \, dx=\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-x-x^3-x^4+x^6}}{-2 x+\sqrt [3]{-x-x^3-x^4+x^6}}\right )-\sqrt [3]{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-x-x^3-x^4+x^6}}{-2 \sqrt [3]{2} x+\sqrt [3]{-x-x^3-x^4+x^6}}\right )-\log \left (x+\sqrt [3]{-x-x^3-x^4+x^6}\right )+\sqrt [3]{2} \log \left (\sqrt [3]{2} x+\sqrt [3]{-x-x^3-x^4+x^6}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{-x-x^3-x^4+x^6}+\left (-x-x^3-x^4+x^6\right )^{2/3}\right )-\frac {\log \left (2^{2/3} x^2-\sqrt [3]{2} x \sqrt [3]{-x-x^3-x^4+x^6}+\left (-x-x^3-x^4+x^6\right )^{2/3}\right )}{2^{2/3}} \]

[Out]

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

Rubi [F]

\[ \int \frac {\left (2-2 x+2 x^2-3 x^3+3 x^4\right ) \sqrt [3]{-x-x^3-x^4+x^6}}{(1+x) \left (-1+2 x-2 x^2+x^3\right ) \left (-1-x^3+x^5\right )} \, dx=\int \frac {\left (2-2 x+2 x^2-3 x^3+3 x^4\right ) \sqrt [3]{-x-x^3-x^4+x^6}}{(1+x) \left (-1+2 x-2 x^2+x^3\right ) \left (-1-x^3+x^5\right )} \, dx \]

[In]

Int[((2 - 2*x + 2*x^2 - 3*x^3 + 3*x^4)*(-x - x^3 - x^4 + x^6)^(1/3))/((1 + x)*(-1 + 2*x - 2*x^2 + x^3)*(-1 - x
^3 + x^5)),x]

[Out]

-(((-x - x^3 - x^4 + x^6)^(1/3)*Defer[Subst][Defer[Int][(-1 - x^6 - x^9 + x^15)^(1/3)/(-1 + x), x], x, x^(1/3)
])/(x^(1/3)*(-1 - x^2 - x^3 + x^5)^(1/3))) - (2*(-x - x^3 - x^4 + x^6)^(1/3)*Defer[Subst][Defer[Int][(-1 - x^6
- x^9 + x^15)^(1/3)/(1 + x), x], x, x^(1/3)])/(x^(1/3)*(-1 - x^2 - x^3 + x^5)^(1/3)) + (2*(1 + I*Sqrt[3])*(-x
- x^3 - x^4 + x^6)^(1/3)*Defer[Subst][Defer[Int][(-1 - x^6 - x^9 + x^15)^(1/3)/(-1 - I*Sqrt[3] + 2*x), x], x,
x^(1/3)])/(x^(1/3)*(-1 - x^2 - x^3 + x^5)^(1/3)) + ((1 - I*Sqrt[3])*(-x - x^3 - x^4 + x^6)^(1/3)*Defer[Subst]
[Defer[Int][(-1 - x^6 - x^9 + x^15)^(1/3)/(1 - I*Sqrt[3] + 2*x), x], x, x^(1/3)])/(x^(1/3)*(-1 - x^2 - x^3 + x
^5)^(1/3)) + (2*(1 - I*Sqrt[3])*(-x - x^3 - x^4 + x^6)^(1/3)*Defer[Subst][Defer[Int][(-1 - x^6 - x^9 + x^15)^(
1/3)/(-1 + I*Sqrt[3] + 2*x), x], x, x^(1/3)])/(x^(1/3)*(-1 - x^2 - x^3 + x^5)^(1/3)) + ((1 + I*Sqrt[3])*(-x -
x^3 - x^4 + x^6)^(1/3)*Defer[Subst][Defer[Int][(-1 - x^6 - x^9 + x^15)^(1/3)/(1 + I*Sqrt[3] + 2*x), x], x, x^(
1/3)])/(x^(1/3)*(-1 - x^2 - x^3 + x^5)^(1/3)) + (2*(-x - x^3 - x^4 + x^6)^(1/3)*Defer[Subst][Defer[Int][(-1 -
x^6 - x^9 + x^15)^(1/3)/((1 - I*Sqrt[3])^(1/3) + (-2)^(1/3)*x), x], x, x^(1/3)])/((1 - I*Sqrt[3])^(2/3)*x^(1/3
)*(-1 - x^2 - x^3 + x^5)^(1/3)) + (2*(-x - x^3 - x^4 + x^6)^(1/3)*Defer[Subst][Defer[Int][(-1 - x^6 - x^9 + x^
15)^(1/3)/((1 + I*Sqrt[3])^(1/3) + (-2)^(1/3)*x), x], x, x^(1/3)])/((1 + I*Sqrt[3])^(2/3)*x^(1/3)*(-1 - x^2 -
x^3 + x^5)^(1/3)) + (2*(-x - x^3 - x^4 + x^6)^(1/3)*Defer[Subst][Defer[Int][(-1 - x^6 - x^9 + x^15)^(1/3)/((1
- I*Sqrt[3])^(1/3) - 2^(1/3)*x), x], x, x^(1/3)])/((1 - I*Sqrt[3])^(2/3)*x^(1/3)*(-1 - x^2 - x^3 + x^5)^(1/3))
+ (2*(-x - x^3 - x^4 + x^6)^(1/3)*Defer[Subst][Defer[Int][(-1 - x^6 - x^9 + x^15)^(1/3)/((1 + I*Sqrt[3])^(1/3
) - 2^(1/3)*x), x], x, x^(1/3)])/((1 + I*Sqrt[3])^(2/3)*x^(1/3)*(-1 - x^2 - x^3 + x^5)^(1/3)) + (2*(-x - x^3 -
x^4 + x^6)^(1/3)*Defer[Subst][Defer[Int][(-1 - x^6 - x^9 + x^15)^(1/3)/((1 - I*Sqrt[3])^(1/3) - (-1)^(2/3)*2^
(1/3)*x), x], x, x^(1/3)])/((1 - I*Sqrt[3])^(2/3)*x^(1/3)*(-1 - x^2 - x^3 + x^5)^(1/3)) + (2*(-x - x^3 - x^4 +
x^6)^(1/3)*Defer[Subst][Defer[Int][(-1 - x^6 - x^9 + x^15)^(1/3)/((1 + I*Sqrt[3])^(1/3) - (-1)^(2/3)*2^(1/3)*
x), x], x, x^(1/3)])/((1 + I*Sqrt[3])^(2/3)*x^(1/3)*(-1 - x^2 - x^3 + x^5)^(1/3)) - (9*(-x - x^3 - x^4 + x^6)^
(1/3)*Defer[Subst][Defer[Int][(x^6*(-1 - x^6 - x^9 + x^15)^(1/3))/(-1 - x^9 + x^15), x], x, x^(1/3)])/(x^(1/3)
*(-1 - x^2 - x^3 + x^5)^(1/3)) + (15*(-x - x^3 - x^4 + x^6)^(1/3)*Defer[Subst][Defer[Int][(x^12*(-1 - x^6 - x^
9 + x^15)^(1/3))/(-1 - x^9 + x^15), x], x, x^(1/3)])/(x^(1/3)*(-1 - x^2 - x^3 + x^5)^(1/3))

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [3]{-x-x^3-x^4+x^6} \int \frac {\sqrt [3]{x} \left (2-2 x+2 x^2-3 x^3+3 x^4\right ) \sqrt [3]{-1-x^2-x^3+x^5}}{(1+x) \left (-1+2 x-2 x^2+x^3\right ) \left (-1-x^3+x^5\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}} \\ & = \frac {\left (3 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \text {Subst}\left (\int \frac {x^3 \left (2-2 x^3+2 x^6-3 x^9+3 x^{12}\right ) \sqrt [3]{-1-x^6-x^9+x^{15}}}{\left (1+x^3\right ) \left (-1+2 x^3-2 x^6+x^9\right ) \left (-1-x^9+x^{15}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}} \\ & = \frac {\left (3 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \text {Subst}\left (\int \left (-\frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{3 (-1+x)}-\frac {2 \sqrt [3]{-1-x^6-x^9+x^{15}}}{3 (1+x)}+\frac {2 (-2+x) \sqrt [3]{-1-x^6-x^9+x^{15}}}{3 \left (1-x+x^2\right )}+\frac {(2+x) \sqrt [3]{-1-x^6-x^9+x^{15}}}{3 \left (1+x+x^2\right )}+\frac {\left (1-2 x^3\right ) \sqrt [3]{-1-x^6-x^9+x^{15}}}{1-x^3+x^6}+\frac {x^6 \left (-3+5 x^6\right ) \sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-x^9+x^{15}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}} \\ & = -\frac {\sqrt [3]{-x-x^3-x^4+x^6} \text {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{-1+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\sqrt [3]{-x-x^3-x^4+x^6} \text {Subst}\left (\int \frac {(2+x) \sqrt [3]{-1-x^6-x^9+x^{15}}}{1+x+x^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}-\frac {\left (2 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{1+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (2 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \text {Subst}\left (\int \frac {(-2+x) \sqrt [3]{-1-x^6-x^9+x^{15}}}{1-x+x^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (3 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \text {Subst}\left (\int \frac {\left (1-2 x^3\right ) \sqrt [3]{-1-x^6-x^9+x^{15}}}{1-x^3+x^6} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (3 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \text {Subst}\left (\int \frac {x^6 \left (-3+5 x^6\right ) \sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-x^9+x^{15}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}} \\ & = -\frac {\sqrt [3]{-x-x^3-x^4+x^6} \text {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{-1+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\sqrt [3]{-x-x^3-x^4+x^6} \text {Subst}\left (\int \left (\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-1-x^6-x^9+x^{15}}}{1-i \sqrt {3}+2 x}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-1-x^6-x^9+x^{15}}}{1+i \sqrt {3}+2 x}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}-\frac {\left (2 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{1+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (2 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \text {Subst}\left (\int \left (\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-i \sqrt {3}+2 x}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-1-x^6-x^9+x^{15}}}{-1+i \sqrt {3}+2 x}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (3 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \text {Subst}\left (\int \left (-\frac {2 \sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-i \sqrt {3}+2 x^3}-\frac {2 \sqrt [3]{-1-x^6-x^9+x^{15}}}{-1+i \sqrt {3}+2 x^3}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (3 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \text {Subst}\left (\int \left (-\frac {3 x^6 \sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-x^9+x^{15}}+\frac {5 x^{12} \sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-x^9+x^{15}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}} \\ & = -\frac {\sqrt [3]{-x-x^3-x^4+x^6} \text {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{-1+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}-\frac {\left (2 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{1+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}-\frac {\left (6 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-i \sqrt {3}+2 x^3} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}-\frac {\left (6 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{-1+i \sqrt {3}+2 x^3} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}-\frac {\left (9 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \text {Subst}\left (\int \frac {x^6 \sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-x^9+x^{15}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (15 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \text {Subst}\left (\int \frac {x^{12} \sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-x^9+x^{15}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [3]{-x-x^3-x^4+x^6}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{1-i \sqrt {3}+2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (2 \left (1-i \sqrt {3}\right ) \sqrt [3]{-x-x^3-x^4+x^6}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{-1+i \sqrt {3}+2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [3]{-x-x^3-x^4+x^6}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{1+i \sqrt {3}+2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (2 \left (1+i \sqrt {3}\right ) \sqrt [3]{-x-x^3-x^4+x^6}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-i \sqrt {3}+2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}} \\ & = -\frac {\sqrt [3]{-x-x^3-x^4+x^6} \text {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{-1+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}-\frac {\left (2 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{1+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}-\frac {\left (6 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \text {Subst}\left (\int \left (\frac {\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{-1-x^6-x^9+x^{15}}}{3 \left (-1+i \sqrt {3}\right ) \left (\sqrt [3]{1-i \sqrt {3}}+\sqrt [3]{-2} x\right )}+\frac {\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{-1-x^6-x^9+x^{15}}}{3 \left (-1+i \sqrt {3}\right ) \left (\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x\right )}+\frac {\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{-1-x^6-x^9+x^{15}}}{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,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}-\frac {\left (6 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \text {Subst}\left (\int \left (\frac {\sqrt [3]{1+i \sqrt {3}} \sqrt [3]{-1-x^6-x^9+x^{15}}}{3 \left (-1-i \sqrt {3}\right ) \left (\sqrt [3]{1+i \sqrt {3}}+\sqrt [3]{-2} x\right )}+\frac {\sqrt [3]{1+i \sqrt {3}} \sqrt [3]{-1-x^6-x^9+x^{15}}}{3 \left (-1-i \sqrt {3}\right ) \left (\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x\right )}+\frac {\sqrt [3]{1+i \sqrt {3}} \sqrt [3]{-1-x^6-x^9+x^{15}}}{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,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}-\frac {\left (9 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \text {Subst}\left (\int \frac {x^6 \sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-x^9+x^{15}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (15 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \text {Subst}\left (\int \frac {x^{12} \sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-x^9+x^{15}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [3]{-x-x^3-x^4+x^6}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{1-i \sqrt {3}+2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (2 \left (1-i \sqrt {3}\right ) \sqrt [3]{-x-x^3-x^4+x^6}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{-1+i \sqrt {3}+2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [3]{-x-x^3-x^4+x^6}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{1+i \sqrt {3}+2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (2 \left (1+i \sqrt {3}\right ) \sqrt [3]{-x-x^3-x^4+x^6}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-i \sqrt {3}+2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.16 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.10 \[ \int \frac {\left (2-2 x+2 x^2-3 x^3+3 x^4\right ) \sqrt [3]{-x-x^3-x^4+x^6}}{(1+x) \left (-1+2 x-2 x^2+x^3\right ) \left (-1-x^3+x^5\right )} \, dx=\frac {x^{2/3} \left (-1-x^2-x^3+x^5\right )^{2/3} \left (-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}-2 \sqrt [3]{-1-x^2-x^3+x^5}}\right )+2 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}-2^{2/3} \sqrt [3]{-1-x^2-x^3+x^5}}\right )-2 \log \left (x^{2/3}+\sqrt [3]{-1-x^2-x^3+x^5}\right )+2 \sqrt [3]{2} \log \left (2 x^{2/3}+2^{2/3} \sqrt [3]{-1-x^2-x^3+x^5}\right )+\log \left (x^{4/3}-x^{2/3} \sqrt [3]{-1-x^2-x^3+x^5}+\left (-1-x^2-x^3+x^5\right )^{2/3}\right )-\sqrt [3]{2} \log \left (-2 x^{4/3}+2^{2/3} x^{2/3} \sqrt [3]{-1-x^2-x^3+x^5}-\sqrt [3]{2} \left (-1-x^2-x^3+x^5\right )^{2/3}\right )\right )}{2 \left (x \left (-1-x^2-x^3+x^5\right )\right )^{2/3}} \]

[In]

Integrate[((2 - 2*x + 2*x^2 - 3*x^3 + 3*x^4)*(-x - x^3 - x^4 + x^6)^(1/3))/((1 + x)*(-1 + 2*x - 2*x^2 + x^3)*(
-1 - x^3 + x^5)),x]

[Out]

(x^(2/3)*(-1 - x^2 - x^3 + x^5)^(2/3)*(-2*Sqrt[3]*ArcTan[(Sqrt[3]*x^(2/3))/(x^(2/3) - 2*(-1 - x^2 - x^3 + x^5)
^(1/3))] + 2*2^(1/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x^(2/3))/(x^(2/3) - 2^(2/3)*(-1 - x^2 - x^3 + x^5)^(1/3))] - 2*Lo
g[x^(2/3) + (-1 - x^2 - x^3 + x^5)^(1/3)] + 2*2^(1/3)*Log[2*x^(2/3) + 2^(2/3)*(-1 - x^2 - x^3 + x^5)^(1/3)] +
Log[x^(4/3) - x^(2/3)*(-1 - x^2 - x^3 + x^5)^(1/3) + (-1 - x^2 - x^3 + x^5)^(2/3)] - 2^(1/3)*Log[-2*x^(4/3) +
2^(2/3)*x^(2/3)*(-1 - x^2 - x^3 + x^5)^(1/3) - 2^(1/3)*(-1 - x^2 - x^3 + x^5)^(2/3)]))/(2*(x*(-1 - x^2 - x^3 +
x^5))^(2/3))

Maple [A] (verified)

Time = 75.37 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.80

method	result	size
pseudoelliptic	\(-\frac {2^{\frac {1}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}-2^{\frac {1}{3}} {\left (x \left (x^{5}-x^{3}-x^{2}-1\right )\right )}^{\frac {1}{3}} x +{\left (x \left (x^{5}-x^{3}-x^{2}-1\right )\right )}^{\frac {2}{3}}}{x^{2}}\right )}{2}+\frac {\ln \left (\frac {{\left (x \left (x^{5}-x^{3}-x^{2}-1\right )\right )}^{\frac {2}{3}}-{\left (x \left (x^{5}-x^{3}-x^{2}-1\right )\right )}^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )}{2}+2^{\frac {1}{3}} \ln \left (\frac {2^{\frac {1}{3}} x +{\left (x \left (x^{5}-x^{3}-x^{2}-1\right )\right )}^{\frac {1}{3}}}{x}\right )-\ln \left (\frac {{\left (x \left (x^{5}-x^{3}-x^{2}-1\right )\right )}^{\frac {1}{3}}+x}{x}\right )+\left (-\arctan \left (\frac {\sqrt {3}\, \left (-2^{\frac {2}{3}} {\left (x \left (x^{5}-x^{3}-x^{2}-1\right )\right )}^{\frac {1}{3}}+x \right )}{3 x}\right ) 2^{\frac {1}{3}}+\arctan \left (\frac {\left (x -2 {\left (x \left (x^{5}-x^{3}-x^{2}-1\right )\right )}^{\frac {1}{3}}\right ) \sqrt {3}}{3 x}\right )\right ) \sqrt {3}\)	\(255\)
trager	\(\text {Expression too large to display}\)	\(3486\)

[In]

int((3*x^4-3*x^3+2*x^2-2*x+2)*(x^6-x^4-x^3-x)^(1/3)/(1+x)/(x^3-2*x^2+2*x-1)/(x^5-x^3-1),x,method=_RETURNVERBOS
E)

[Out]

-1/2*2^(1/3)*ln((2^(2/3)*x^2-2^(1/3)*(x*(x^5-x^3-x^2-1))^(1/3)*x+(x*(x^5-x^3-x^2-1))^(2/3))/x^2)+1/2*ln(((x*(x
^5-x^3-x^2-1))^(2/3)-(x*(x^5-x^3-x^2-1))^(1/3)*x+x^2)/x^2)+2^(1/3)*ln((2^(1/3)*x+(x*(x^5-x^3-x^2-1))^(1/3))/x)
-ln(((x*(x^5-x^3-x^2-1))^(1/3)+x)/x)+(-arctan(1/3*3^(1/2)*(-2^(2/3)*(x*(x^5-x^3-x^2-1))^(1/3)+x)/x)*2^(1/3)+ar
ctan(1/3*(x-2*(x*(x^5-x^3-x^2-1))^(1/3))*3^(1/2)/x))*3^(1/2)

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (2-2 x+2 x^2-3 x^3+3 x^4\right ) \sqrt [3]{-x-x^3-x^4+x^6}}{(1+x) \left (-1+2 x-2 x^2+x^3\right ) \left (-1-x^3+x^5\right )} \, dx=\text {Timed out} \]

[In]

integrate((3*x^4-3*x^3+2*x^2-2*x+2)*(x^6-x^4-x^3-x)^(1/3)/(1+x)/(x^3-2*x^2+2*x-1)/(x^5-x^3-1),x, algorithm="fr
icas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {\left (2-2 x+2 x^2-3 x^3+3 x^4\right ) \sqrt [3]{-x-x^3-x^4+x^6}}{(1+x) \left (-1+2 x-2 x^2+x^3\right ) \left (-1-x^3+x^5\right )} \, dx=\int \frac {\sqrt [3]{x \left (x^{5} - x^{3} - x^{2} - 1\right )} \left (3 x^{4} - 3 x^{3} + 2 x^{2} - 2 x + 2\right )}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{5} - x^{3} - 1\right )}\, dx \]

[In]

integrate((3*x**4-3*x**3+2*x**2-2*x+2)*(x**6-x**4-x**3-x)**(1/3)/(1+x)/(x**3-2*x**2+2*x-1)/(x**5-x**3-1),x)

[Out]

Integral((x*(x**5 - x**3 - x**2 - 1))**(1/3)*(3*x**4 - 3*x**3 + 2*x**2 - 2*x + 2)/((x - 1)*(x + 1)*(x**2 - x +
1)*(x**5 - x**3 - 1)), x)

Maxima [F]

\[ \int \frac {\left (2-2 x+2 x^2-3 x^3+3 x^4\right ) \sqrt [3]{-x-x^3-x^4+x^6}}{(1+x) \left (-1+2 x-2 x^2+x^3\right ) \left (-1-x^3+x^5\right )} \, dx=\int { \frac {{\left (x^{6} - x^{4} - x^{3} - x\right )}^{\frac {1}{3}} {\left (3 \, x^{4} - 3 \, x^{3} + 2 \, x^{2} - 2 \, x + 2\right )}}{{\left (x^{5} - x^{3} - 1\right )} {\left (x^{3} - 2 \, x^{2} + 2 \, x - 1\right )} {\left (x + 1\right )}} \,d x } \]

[In]

integrate((3*x^4-3*x^3+2*x^2-2*x+2)*(x^6-x^4-x^3-x)^(1/3)/(1+x)/(x^3-2*x^2+2*x-1)/(x^5-x^3-1),x, algorithm="ma
xima")

[Out]

integrate((x^6 - x^4 - x^3 - x)^(1/3)*(3*x^4 - 3*x^3 + 2*x^2 - 2*x + 2)/((x^5 - x^3 - 1)*(x^3 - 2*x^2 + 2*x -
1)*(x + 1)), x)

Giac [F]

\[ \int \frac {\left (2-2 x+2 x^2-3 x^3+3 x^4\right ) \sqrt [3]{-x-x^3-x^4+x^6}}{(1+x) \left (-1+2 x-2 x^2+x^3\right ) \left (-1-x^3+x^5\right )} \, dx=\int { \frac {{\left (x^{6} - x^{4} - x^{3} - x\right )}^{\frac {1}{3}} {\left (3 \, x^{4} - 3 \, x^{3} + 2 \, x^{2} - 2 \, x + 2\right )}}{{\left (x^{5} - x^{3} - 1\right )} {\left (x^{3} - 2 \, x^{2} + 2 \, x - 1\right )} {\left (x + 1\right )}} \,d x } \]

[In]

integrate((3*x^4-3*x^3+2*x^2-2*x+2)*(x^6-x^4-x^3-x)^(1/3)/(1+x)/(x^3-2*x^2+2*x-1)/(x^5-x^3-1),x, algorithm="gi
ac")

[Out]

integrate((x^6 - x^4 - x^3 - x)^(1/3)*(3*x^4 - 3*x^3 + 2*x^2 - 2*x + 2)/((x^5 - x^3 - 1)*(x^3 - 2*x^2 + 2*x -
1)*(x + 1)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (2-2 x+2 x^2-3 x^3+3 x^4\right ) \sqrt [3]{-x-x^3-x^4+x^6}}{(1+x) \left (-1+2 x-2 x^2+x^3\right ) \left (-1-x^3+x^5\right )} \, dx=\int -\frac {{\left (x^6-x^4-x^3-x\right )}^{1/3}\,\left (3\,x^4-3\,x^3+2\,x^2-2\,x+2\right )}{\left (x+1\right )\,\left (-x^5+x^3+1\right )\,\left (x^3-2\,x^2+2\,x-1\right )} \,d x \]

[In]

int(-((x^6 - x^3 - x^4 - x)^(1/3)*(2*x^2 - 2*x - 3*x^3 + 3*x^4 + 2))/((x + 1)*(x^3 - x^5 + 1)*(2*x - 2*x^2 + x
^3 - 1)),x)

[Out]

int(-((x^6 - x^3 - x^4 - x)^(1/3)*(2*x^2 - 2*x - 3*x^3 + 3*x^4 + 2))/((x + 1)*(x^3 - x^5 + 1)*(2*x - 2*x^2 + x
^3 - 1)), x)