\(\int \frac {1-\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}}{x^2+\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}} \, dx\) [2823]

Optimal result

Integrand size = 85, antiderivative size = 282 \[ \int \frac {1-\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}}{x^2+\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}} \, dx=-x+3 \text {RootSum}\left [-3-28 \text {$\#$1}+6 \text {$\#$1}^3+3 \text {$\#$1}^4-3 \text {$\#$1}^6-3 \text {$\#$1}^7+\text {$\#$1}^{10}\&,\frac {-10 \log \left (1+x^3\right ) \text {$\#$1}^2+10 \log \left (\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}-\text {$\#$1}-x^3 \text {$\#$1}\right ) \text {$\#$1}^2+2 \log \left (1+x^3\right ) \text {$\#$1}^5-2 \log \left (\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}-\text {$\#$1}-x^3 \text {$\#$1}\right ) \text {$\#$1}^5-\log \left (1+x^3\right ) \text {$\#$1}^8+\log \left (\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}-\text {$\#$1}-x^3 \text {$\#$1}\right ) \text {$\#$1}^8}{-28+18 \text {$\#$1}^2+12 \text {$\#$1}^3-18 \text {$\#$1}^5-21 \text {$\#$1}^6+10 \text {$\#$1}^9}\&\right ] \]

[Out]

Unintegrable

Rubi [F]

\[ \int \frac {1-\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}}{x^2+\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}} \, dx=\int \frac {1-\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}}{x^2+\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}} \, dx \]

[In]

Int[(1 - (1 - 3*x + 3*x^3 - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10)^(1/3))/(x^2 + (1 - 3*x + 3*x^3 - 9*x^4 + 3*x
^6 - 9*x^7 + x^9 - 3*x^10)^(1/3)),x]

[Out]

-x - Defer[Int][(1 - 3*x + 3*x^3 - 9*x^4 + 4*x^6 - 9*x^7 + x^9 - 3*x^10)^(-1), x] - Defer[Int][(-1 + 3*x - 3*x
^3 + 9*x^4 - 4*x^6 + 9*x^7 - x^9 + 3*x^10)^(-1), x] - Defer[Int][x^4/(-1 + 3*x - 3*x^3 + 9*x^4 - 4*x^6 + 9*x^7
- x^9 + 3*x^10), x] - Defer[Int][x^6/(-1 + 3*x - 3*x^3 + 9*x^4 - 4*x^6 + 9*x^7 - x^9 + 3*x^10), x] + (10*((1
- 3*x)*(1 + x^3)^3)^(1/3)*Defer[Subst][Defer[Int][x^2/(-3 + 28*x - 6*x^3 + 3*x^4 - 3*x^6 + 3*x^7 + x^10), x],
x, (-1 + 3*x)^(1/3)])/((-1 + 3*x)^(1/3)*(1 + x^3)) - (10*((1 - 3*x)*(1 + x^3)^3)^(2/3)*Defer[Subst][Defer[Int]
[x^2/(-3 + 28*x - 6*x^3 + 3*x^4 - 3*x^6 + 3*x^7 + x^10), x], x, (-1 + 3*x)^(1/3)])/((-1 + 3*x)^(2/3)*(1 + x^3)
^2) + (2*((1 - 3*x)*(1 + x^3)^3)^(1/3)*Defer[Subst][Defer[Int][x^5/(-3 + 28*x - 6*x^3 + 3*x^4 - 3*x^6 + 3*x^7
+ x^10), x], x, (-1 + 3*x)^(1/3)])/((-1 + 3*x)^(1/3)*(1 + x^3)) - (2*((1 - 3*x)*(1 + x^3)^3)^(2/3)*Defer[Subst
][Defer[Int][x^5/(-3 + 28*x - 6*x^3 + 3*x^4 - 3*x^6 + 3*x^7 + x^10), x], x, (-1 + 3*x)^(1/3)])/((-1 + 3*x)^(2/
3)*(1 + x^3)^2) + (((1 - 3*x)*(1 + x^3)^3)^(1/3)*Defer[Subst][Defer[Int][x^8/(-3 + 28*x - 6*x^3 + 3*x^4 - 3*x^
6 + 3*x^7 + x^10), x], x, (-1 + 3*x)^(1/3)])/((-1 + 3*x)^(1/3)*(1 + x^3)) - (((1 - 3*x)*(1 + x^3)^3)^(2/3)*Def
er[Subst][Defer[Int][x^8/(-3 + 28*x - 6*x^3 + 3*x^4 - 3*x^6 + 3*x^7 + x^10), x], x, (-1 + 3*x)^(1/3)])/((-1 +
3*x)^(2/3)*(1 + x^3)^2) + (30*((1 - 3*x)*(1 + x^3)^3)^(1/3)*Defer[Subst][Defer[Int][x^2/(9 + 84*x + 784*x^2 +
36*x^3 + 177*x^4 + 168*x^5 + 54*x^6 + 111*x^7 + 177*x^8 + 36*x^9 + 30*x^10 + 74*x^11 + 9*x^12 + 15*x^13 + 15*x
^14 + 3*x^16 + 6*x^17 + x^20), x], x, (-1 + 3*x)^(1/3)])/((-1 + 3*x)^(1/3)*(1 + x^3)) - (30*((1 - 3*x)*(1 + x^
3)^3)^(2/3)*Defer[Subst][Defer[Int][x^2/(9 + 84*x + 784*x^2 + 36*x^3 + 177*x^4 + 168*x^5 + 54*x^6 + 111*x^7 +
177*x^8 + 36*x^9 + 30*x^10 + 74*x^11 + 9*x^12 + 15*x^13 + 15*x^14 + 3*x^16 + 6*x^17 + x^20), x], x, (-1 + 3*x)
^(1/3)])/((-1 + 3*x)^(2/3)*(1 + x^3)^2) - (280*((1 - 3*x)*(1 + x^3)^3)^(1/3)*Defer[Subst][Defer[Int][x^3/(9 +
84*x + 784*x^2 + 36*x^3 + 177*x^4 + 168*x^5 + 54*x^6 + 111*x^7 + 177*x^8 + 36*x^9 + 30*x^10 + 74*x^11 + 9*x^12
+ 15*x^13 + 15*x^14 + 3*x^16 + 6*x^17 + x^20), x], x, (-1 + 3*x)^(1/3)])/((-1 + 3*x)^(1/3)*(1 + x^3)) - (560*
((1 - 3*x)*(1 + x^3)^3)^(2/3)*Defer[Subst][Defer[Int][x^3/(9 + 84*x + 784*x^2 + 36*x^3 + 177*x^4 + 168*x^5 + 5
4*x^6 + 111*x^7 + 177*x^8 + 36*x^9 + 30*x^10 + 74*x^11 + 9*x^12 + 15*x^13 + 15*x^14 + 3*x^16 + 6*x^17 + x^20),
x], x, (-1 + 3*x)^(1/3)])/((-1 + 3*x)^(2/3)*(1 + x^3)^2) + (66*((1 - 3*x)*(1 + x^3)^3)^(1/3)*Defer[Subst][Def
er[Int][x^5/(9 + 84*x + 784*x^2 + 36*x^3 + 177*x^4 + 168*x^5 + 54*x^6 + 111*x^7 + 177*x^8 + 36*x^9 + 30*x^10 +
74*x^11 + 9*x^12 + 15*x^13 + 15*x^14 + 3*x^16 + 6*x^17 + x^20), x], x, (-1 + 3*x)^(1/3)])/((-1 + 3*x)^(1/3)*(
1 + x^3)) - (66*((1 - 3*x)*(1 + x^3)^3)^(2/3)*Defer[Subst][Defer[Int][x^5/(9 + 84*x + 784*x^2 + 36*x^3 + 177*x
^4 + 168*x^5 + 54*x^6 + 111*x^7 + 177*x^8 + 36*x^9 + 30*x^10 + 74*x^11 + 9*x^12 + 15*x^13 + 15*x^14 + 3*x^16 +
6*x^17 + x^20), x], x, (-1 + 3*x)^(1/3)])/((-1 + 3*x)^(2/3)*(1 + x^3)^2) - (86*((1 - 3*x)*(1 + x^3)^3)^(1/3)*
Defer[Subst][Defer[Int][x^6/(9 + 84*x + 784*x^2 + 36*x^3 + 177*x^4 + 168*x^5 + 54*x^6 + 111*x^7 + 177*x^8 + 36
*x^9 + 30*x^10 + 74*x^11 + 9*x^12 + 15*x^13 + 15*x^14 + 3*x^16 + 6*x^17 + x^20), x], x, (-1 + 3*x)^(1/3)])/((-
1 + 3*x)^(1/3)*(1 + x^3)) - (172*((1 - 3*x)*(1 + x^3)^3)^(2/3)*Defer[Subst][Defer[Int][x^6/(9 + 84*x + 784*x^2
+ 36*x^3 + 177*x^4 + 168*x^5 + 54*x^6 + 111*x^7 + 177*x^8 + 36*x^9 + 30*x^10 + 74*x^11 + 9*x^12 + 15*x^13 + 1
5*x^14 + 3*x^16 + 6*x^17 + x^20), x], x, (-1 + 3*x)^(1/3)])/((-1 + 3*x)^(2/3)*(1 + x^3)^2) + (45*((1 - 3*x)*(1
+ x^3)^3)^(1/3)*Defer[Subst][Defer[Int][x^8/(9 + 84*x + 784*x^2 + 36*x^3 + 177*x^4 + 168*x^5 + 54*x^6 + 111*x
^7 + 177*x^8 + 36*x^9 + 30*x^10 + 74*x^11 + 9*x^12 + 15*x^13 + 15*x^14 + 3*x^16 + 6*x^17 + x^20), x], x, (-1 +
3*x)^(1/3)])/((-1 + 3*x)^(1/3)*(1 + x^3)) - (45*((1 - 3*x)*(1 + x^3)^3)^(2/3)*Defer[Subst][Defer[Int][x^8/(9
+ 84*x + 784*x^2 + 36*x^3 + 177*x^4 + 168*x^5 + 54*x^6 + 111*x^7 + 177*x^8 + 36*x^9 + 30*x^10 + 74*x^11 + 9*x^
12 + 15*x^13 + 15*x^14 + 3*x^16 + 6*x^17 + x^20), x], x, (-1 + 3*x)^(1/3)])/((-1 + 3*x)^(2/3)*(1 + x^3)^2) - (
64*((1 - 3*x)*(1 + x^3)^3)^(1/3)*Defer[Subst][Defer[Int][x^9/(9 + 84*x + 784*x^2 + 36*x^3 + 177*x^4 + 168*x^5
+ 54*x^6 + 111*x^7 + 177*x^8 + 36*x^9 + 30*x^10 + 74*x^11 + 9*x^12 + 15*x^13 + 15*x^14 + 3*x^16 + 6*x^17 + x^2
0), x], x, (-1 + 3*x)^(1/3)])/((-1 + 3*x)^(1/3)*(1 + x^3)) - (128*((1 - 3*x)*(1 + x^3)^3)^(2/3)*Defer[Subst][D
efer[Int][x^9/(9 + 84*x + 784*x^2 + 36*x^3 + 177*x^4 + 168*x^5 + 54*x^6 + 111*x^7 + 177*x^8 + 36*x^9 + 30*x^10
+ 74*x^11 + 9*x^12 + 15*x^13 + 15*x^14 + 3*x^16 + 6*x^17 + x^20), x], x, (-1 + 3*x)^(1/3)])/((-1 + 3*x)^(2/3)
*(1 + x^3)^2) + (12*((1 - 3*x)*(1 + x^3)^3)^(1/3)*Defer[Subst][Defer[Int][x^11/(9 + 84*x + 784*x^2 + 36*x^3 +
177*x^4 + 168*x^5 + 54*x^6 + 111*x^7 + 177*x^8 + 36*x^9 + 30*x^10 + 74*x^11 + 9*x^12 + 15*x^13 + 15*x^14 + 3*x
^16 + 6*x^17 + x^20), x], x, (-1 + 3*x)^(1/3)])/((-1 + 3*x)^(1/3)*(1 + x^3)) - (12*((1 - 3*x)*(1 + x^3)^3)^(2/
3)*Defer[Subst][Defer[Int][x^11/(9 + 84*x + 784*x^2 + 36*x^3 + 177*x^4 + 168*x^5 + 54*x^6 + 111*x^7 + 177*x^8
+ 36*x^9 + 30*x^10 + 74*x^11 + 9*x^12 + 15*x^13 + 15*x^14 + 3*x^16 + 6*x^17 + x^20), x], x, (-1 + 3*x)^(1/3)])
/((-1 + 3*x)^(2/3)*(1 + x^3)^2) - (19*((1 - 3*x)*(1 + x^3)^3)^(1/3)*Defer[Subst][Defer[Int][x^12/(9 + 84*x + 7
84*x^2 + 36*x^3 + 177*x^4 + 168*x^5 + 54*x^6 + 111*x^7 + 177*x^8 + 36*x^9 + 30*x^10 + 74*x^11 + 9*x^12 + 15*x^
13 + 15*x^14 + 3*x^16 + 6*x^17 + x^20), x], x, (-1 + 3*x)^(1/3)])/((-1 + 3*x)^(1/3)*(1 + x^3)) - (38*((1 - 3*x
)*(1 + x^3)^3)^(2/3)*Defer[Subst][Defer[Int][x^12/(9 + 84*x + 784*x^2 + 36*x^3 + 177*x^4 + 168*x^5 + 54*x^6 +
111*x^7 + 177*x^8 + 36*x^9 + 30*x^10 + 74*x^11 + 9*x^12 + 15*x^13 + 15*x^14 + 3*x^16 + 6*x^17 + x^20), x], x,
(-1 + 3*x)^(1/3)])/((-1 + 3*x)^(2/3)*(1 + x^3)^2) + (3*((1 - 3*x)*(1 + x^3)^3)^(1/3)*Defer[Subst][Defer[Int][x
^14/(9 + 84*x + 784*x^2 + 36*x^3 + 177*x^4 + 168*x^5 + 54*x^6 + 111*x^7 + 177*x^8 + 36*x^9 + 30*x^10 + 74*x^11
+ 9*x^12 + 15*x^13 + 15*x^14 + 3*x^16 + 6*x^17 + x^20), x], x, (-1 + 3*x)^(1/3)])/((-1 + 3*x)^(1/3)*(1 + x^3)
) - (3*((1 - 3*x)*(1 + x^3)^3)^(2/3)*Defer[Subst][Defer[Int][x^14/(9 + 84*x + 784*x^2 + 36*x^3 + 177*x^4 + 168
*x^5 + 54*x^6 + 111*x^7 + 177*x^8 + 36*x^9 + 30*x^10 + 74*x^11 + 9*x^12 + 15*x^13 + 15*x^14 + 3*x^16 + 6*x^17
+ x^20), x], x, (-1 + 3*x)^(1/3)])/((-1 + 3*x)^(2/3)*(1 + x^3)^2) - (5*((1 - 3*x)*(1 + x^3)^3)^(1/3)*Defer[Sub
st][Defer[Int][x^15/(9 + 84*x + 784*x^2 + 36*x^3 + 177*x^4 + 168*x^5 + 54*x^6 + 111*x^7 + 177*x^8 + 36*x^9 + 3
0*x^10 + 74*x^11 + 9*x^12 + 15*x^13 + 15*x^14 + 3*x^16 + 6*x^17 + x^20), x], x, (-1 + 3*x)^(1/3)])/((-1 + 3*x)
^(1/3)*(1 + x^3)) - (10*((1 - 3*x)*(1 + x^3)^3)^(2/3)*Defer[Subst][Defer[Int][x^15/(9 + 84*x + 784*x^2 + 36*x^
3 + 177*x^4 + 168*x^5 + 54*x^6 + 111*x^7 + 177*x^8 + 36*x^9 + 30*x^10 + 74*x^11 + 9*x^12 + 15*x^13 + 15*x^14 +
3*x^16 + 6*x^17 + x^20), x], x, (-1 + 3*x)^(1/3)])/((-1 + 3*x)^(2/3)*(1 + x^3)^2) - (((1 - 3*x)*(1 + x^3)^3)^
(1/3)*Defer[Subst][Defer[Int][x^18/(9 + 84*x + 784*x^2 + 36*x^3 + 177*x^4 + 168*x^5 + 54*x^6 + 111*x^7 + 177*x
^8 + 36*x^9 + 30*x^10 + 74*x^11 + 9*x^12 + 15*x^13 + 15*x^14 + 3*x^16 + 6*x^17 + x^20), x], x, (-1 + 3*x)^(1/3
)])/((-1 + 3*x)^(1/3)*(1 + x^3)) - (2*((1 - 3*x)*(1 + x^3)^3)^(2/3)*Defer[Subst][Defer[Int][x^18/(9 + 84*x + 7
84*x^2 + 36*x^3 + 177*x^4 + 168*x^5 + 54*x^6 + 111*x^7 + 177*x^8 + 36*x^9 + 30*x^10 + 74*x^11 + 9*x^12 + 15*x^
13 + 15*x^14 + 3*x^16 + 6*x^17 + x^20), x], x, (-1 + 3*x)^(1/3)])/((-1 + 3*x)^(2/3)*(1 + x^3)^2)

Rubi steps \begin{align*} \text {integral}& = \int \frac {1-\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}{x^2+\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}} \, dx \\ & = \int \left (\frac {1}{x^2+\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}-\frac {\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}{x^2+\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}\right ) \, dx \\ & = \int \frac {1}{x^2+\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}} \, dx-\int \frac {\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}{x^2+\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}} \, dx \\ & = -\frac {\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )} \int \frac {\sqrt [3]{-1+3 x} \left (1+x^3\right )}{x^2+\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}} \, dx}{\sqrt [3]{-1+3 x} \left (1+x^3\right )}+\int \left (-\frac {x^4}{-1+3 x-3 x^3+9 x^4-4 x^6+9 x^7-x^9+3 x^{10}}+\frac {x^2 \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}{-1+3 x-3 x^3+9 x^4-4 x^6+9 x^7-x^9+3 x^{10}}-\frac {\left (-\left ((-1+3 x) \left (1+x^3\right )^3\right )\right )^{2/3}}{-1+3 x-3 x^3+9 x^4-4 x^6+9 x^7-x^9+3 x^{10}}\right ) \, dx \\ & = -\frac {\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )} \int \left (\frac {\sqrt [3]{-1+3 x}}{x^2+\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}+\frac {x^3 \sqrt [3]{-1+3 x}}{x^2+\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}\right ) \, dx}{\sqrt [3]{-1+3 x} \left (1+x^3\right )}-\int \frac {x^4}{-1+3 x-3 x^3+9 x^4-4 x^6+9 x^7-x^9+3 x^{10}} \, dx+\int \frac {x^2 \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}{-1+3 x-3 x^3+9 x^4-4 x^6+9 x^7-x^9+3 x^{10}} \, dx-\int \frac {\left (-\left ((-1+3 x) \left (1+x^3\right )^3\right )\right )^{2/3}}{-1+3 x-3 x^3+9 x^4-4 x^6+9 x^7-x^9+3 x^{10}} \, dx \\ & = \frac {\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )} \int \frac {x^2 \sqrt [3]{-1+3 x} \left (1+x^3\right )}{-1+3 x-3 x^3+9 x^4-4 x^6+9 x^7-x^9+3 x^{10}} \, dx}{\sqrt [3]{-1+3 x} \left (1+x^3\right )}-\frac {\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )} \int \frac {\sqrt [3]{-1+3 x}}{x^2+\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}} \, dx}{\sqrt [3]{-1+3 x} \left (1+x^3\right )}-\frac {\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )} \int \frac {x^3 \sqrt [3]{-1+3 x}}{x^2+\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}} \, dx}{\sqrt [3]{-1+3 x} \left (1+x^3\right )}-\frac {\left (-\left ((-1+3 x) \left (1+x^3\right )^3\right )\right )^{2/3} \int \frac {(-1+3 x)^{2/3} \left (1+x^3\right )^2}{-1+3 x-3 x^3+9 x^4-4 x^6+9 x^7-x^9+3 x^{10}} \, dx}{(-1+3 x)^{2/3} \left (1+x^3\right )^2}-\int \frac {x^4}{-1+3 x-3 x^3+9 x^4-4 x^6+9 x^7-x^9+3 x^{10}} \, dx \\ & = -\frac {\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )} \int \left (-\frac {x^4 \sqrt [3]{-1+3 x}}{-1+3 x-3 x^3+9 x^4-4 x^6+9 x^7-x^9+3 x^{10}}+\frac {x^2 \sqrt [3]{-1+3 x} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}{-1+3 x-3 x^3+9 x^4-4 x^6+9 x^7-x^9+3 x^{10}}-\frac {\sqrt [3]{-1+3 x} \left (-\left ((-1+3 x) \left (1+x^3\right )^3\right )\right )^{2/3}}{-1+3 x-3 x^3+9 x^4-4 x^6+9 x^7-x^9+3 x^{10}}\right ) \, dx}{\sqrt [3]{-1+3 x} \left (1+x^3\right )}-\frac {\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )} \int \left (-\frac {x^7 \sqrt [3]{-1+3 x}}{-1+3 x-3 x^3+9 x^4-4 x^6+9 x^7-x^9+3 x^{10}}+\frac {x^5 \sqrt [3]{-1+3 x} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}{-1+3 x-3 x^3+9 x^4-4 x^6+9 x^7-x^9+3 x^{10}}-\frac {x^3 \sqrt [3]{-1+3 x} \left (-\left ((-1+3 x) \left (1+x^3\right )^3\right )\right )^{2/3}}{-1+3 x-3 x^3+9 x^4-4 x^6+9 x^7-x^9+3 x^{10}}\right ) \, dx}{\sqrt [3]{-1+3 x} \left (1+x^3\right )}+\frac {\left (81 \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}\right ) \text {Subst}\left (\int \frac {x^3 \left (1+x^3\right )^2 \left (28+3 x^3+3 x^6+x^9\right )}{-27+21790 x^3+6651 x^6+7272 x^9+3486 x^{12}+1179 x^{15}+585 x^{18}+165 x^{21}+36 x^{24}+9 x^{27}+x^{30}} \, dx,x,\sqrt [3]{-1+3 x}\right )}{\sqrt [3]{-1+3 x} \left (1+x^3\right )}-\frac {\left (27 \left (-\left ((-1+3 x) \left (1+x^3\right )^3\right )\right )^{2/3}\right ) \text {Subst}\left (\int \frac {x^4 \left (28+3 x^3+3 x^6+x^9\right )^2}{-27+21790 x^3+6651 x^6+7272 x^9+3486 x^{12}+1179 x^{15}+585 x^{18}+165 x^{21}+36 x^{24}+9 x^{27}+x^{30}} \, dx,x,\sqrt [3]{-1+3 x}\right )}{(-1+3 x)^{2/3} \left (1+x^3\right )^2}-\int \frac {x^4}{-1+3 x-3 x^3+9 x^4-4 x^6+9 x^7-x^9+3 x^{10}} \, dx \\ & = \frac {\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )} \int \frac {x^4 \sqrt [3]{-1+3 x}}{-1+3 x-3 x^3+9 x^4-4 x^6+9 x^7-x^9+3 x^{10}} \, dx}{\sqrt [3]{-1+3 x} \left (1+x^3\right )}+\frac {\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )} \int \frac {x^7 \sqrt [3]{-1+3 x}}{-1+3 x-3 x^3+9 x^4-4 x^6+9 x^7-x^9+3 x^{10}} \, dx}{\sqrt [3]{-1+3 x} \left (1+x^3\right )}-\frac {\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )} \int \frac {x^2 \sqrt [3]{-1+3 x} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}{-1+3 x-3 x^3+9 x^4-4 x^6+9 x^7-x^9+3 x^{10}} \, dx}{\sqrt [3]{-1+3 x} \left (1+x^3\right )}-\frac {\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )} \int \frac {x^5 \sqrt [3]{-1+3 x} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}{-1+3 x-3 x^3+9 x^4-4 x^6+9 x^7-x^9+3 x^{10}} \, dx}{\sqrt [3]{-1+3 x} \left (1+x^3\right )}+\frac {\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )} \int \frac {\sqrt [3]{-1+3 x} \left (-\left ((-1+3 x) \left (1+x^3\right )^3\right )\right )^{2/3}}{-1+3 x-3 x^3+9 x^4-4 x^6+9 x^7-x^9+3 x^{10}} \, dx}{\sqrt [3]{-1+3 x} \left (1+x^3\right )}+\frac {\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )} \int \frac {x^3 \sqrt [3]{-1+3 x} \left (-\left ((-1+3 x) \left (1+x^3\right )^3\right )\right )^{2/3}}{-1+3 x-3 x^3+9 x^4-4 x^6+9 x^7-x^9+3 x^{10}} \, dx}{\sqrt [3]{-1+3 x} \left (1+x^3\right )}+\frac {\left (81 \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}\right ) \text {Subst}\left (\int \left (\frac {x^2}{9 \left (-3+28 x-6 x^3+3 x^4-3 x^6+3 x^7+x^{10}\right )}-\frac {x^2 \left (-3+28 x-6 x^3+3 x^4-3 x^6+3 x^7+x^{10}\right )}{9 \left (9+84 x+784 x^2+36 x^3+177 x^4+168 x^5+54 x^6+111 x^7+177 x^8+36 x^9+30 x^{10}+74 x^{11}+9 x^{12}+15 x^{13}+15 x^{14}+3 x^{16}+6 x^{17}+x^{20}\right )}\right ) \, dx,x,\sqrt [3]{-1+3 x}\right )}{\sqrt [3]{-1+3 x} \left (1+x^3\right )}-\frac {\left (27 \left (-\left ((-1+3 x) \left (1+x^3\right )^3\right )\right )^{2/3}\right ) \text {Subst}\left (\int \left (\frac {x^2}{3 \left (-3+28 x-6 x^3+3 x^4-3 x^6+3 x^7+x^{10}\right )}+\frac {x^2 \left (3+56 x+6 x^3+6 x^4+3 x^6+6 x^7+2 x^{10}\right )}{3 \left (9+84 x+784 x^2+36 x^3+177 x^4+168 x^5+54 x^6+111 x^7+177 x^8+36 x^9+30 x^{10}+74 x^{11}+9 x^{12}+15 x^{13}+15 x^{14}+3 x^{16}+6 x^{17}+x^{20}\right )}\right ) \, dx,x,\sqrt [3]{-1+3 x}\right )}{(-1+3 x)^{2/3} \left (1+x^3\right )^2}-\int \frac {x^4}{-1+3 x-3 x^3+9 x^4-4 x^6+9 x^7-x^9+3 x^{10}} \, dx \\ & = \frac {\left (9 \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}\right ) \text {Subst}\left (\int \frac {x^2}{-3+28 x-6 x^3+3 x^4-3 x^6+3 x^7+x^{10}} \, dx,x,\sqrt [3]{-1+3 x}\right )}{\sqrt [3]{-1+3 x} \left (1+x^3\right )}-\frac {\left (9 \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}\right ) \text {Subst}\left (\int \frac {x^2 \left (-3+28 x-6 x^3+3 x^4-3 x^6+3 x^7+x^{10}\right )}{9+84 x+784 x^2+36 x^3+177 x^4+168 x^5+54 x^6+111 x^7+177 x^8+36 x^9+30 x^{10}+74 x^{11}+9 x^{12}+15 x^{13}+15 x^{14}+3 x^{16}+6 x^{17}+x^{20}} \, dx,x,\sqrt [3]{-1+3 x}\right )}{\sqrt [3]{-1+3 x} \left (1+x^3\right )}+\frac {\left (9 \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}\right ) \text {Subst}\left (\int \frac {x^3 \left (1+x^3\right )^7}{-27+21790 x^3+6651 x^6+7272 x^9+3486 x^{12}+1179 x^{15}+585 x^{18}+165 x^{21}+36 x^{24}+9 x^{27}+x^{30}} \, dx,x,\sqrt [3]{-1+3 x}\right )}{\sqrt [3]{-1+3 x} \left (1+x^3\right )}+\frac {\left (243 \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}\right ) \text {Subst}\left (\int \frac {x^3 \left (1+x^3\right )^4}{-27+21790 x^3+6651 x^6+7272 x^9+3486 x^{12}+1179 x^{15}+585 x^{18}+165 x^{21}+36 x^{24}+9 x^{27}+x^{30}} \, dx,x,\sqrt [3]{-1+3 x}\right )}{\sqrt [3]{-1+3 x} \left (1+x^3\right )}-\frac {\left (-\left ((-1+3 x) \left (1+x^3\right )^3\right )\right )^{2/3} \int \frac {x^2 (-1+3 x)^{2/3} \left (1+x^3\right )}{-1+3 x-3 x^3+9 x^4-4 x^6+9 x^7-x^9+3 x^{10}} \, dx}{(-1+3 x)^{2/3} \left (1+x^3\right )^2}-\frac {\left (-\left ((-1+3 x) \left (1+x^3\right )^3\right )\right )^{2/3} \int \frac {x^5 (-1+3 x)^{2/3} \left (1+x^3\right )}{-1+3 x-3 x^3+9 x^4-4 x^6+9 x^7-x^9+3 x^{10}} \, dx}{(-1+3 x)^{2/3} \left (1+x^3\right )^2}-\frac {\left (9 \left (-\left ((-1+3 x) \left (1+x^3\right )^3\right )\right )^{2/3}\right ) \text {Subst}\left (\int \frac {x^2}{-3+28 x-6 x^3+3 x^4-3 x^6+3 x^7+x^{10}} \, dx,x,\sqrt [3]{-1+3 x}\right )}{(-1+3 x)^{2/3} \left (1+x^3\right )^2}-\frac {\left (9 \left (-\left ((-1+3 x) \left (1+x^3\right )^3\right )\right )^{2/3}\right ) \text {Subst}\left (\int \frac {x^2 \left (3+56 x+6 x^3+6 x^4+3 x^6+6 x^7+2 x^{10}\right )}{9+84 x+784 x^2+36 x^3+177 x^4+168 x^5+54 x^6+111 x^7+177 x^8+36 x^9+30 x^{10}+74 x^{11}+9 x^{12}+15 x^{13}+15 x^{14}+3 x^{16}+6 x^{17}+x^{20}} \, dx,x,\sqrt [3]{-1+3 x}\right )}{(-1+3 x)^{2/3} \left (1+x^3\right )^2}-\int \frac {x^4}{-1+3 x-3 x^3+9 x^4-4 x^6+9 x^7-x^9+3 x^{10}} \, dx-\int \frac {(-1+3 x) \left (1+x^3\right )^2}{-1+3 x-3 x^3+9 x^4-4 x^6+9 x^7-x^9+3 x^{10}} \, dx-\int \frac {x^3 (-1+3 x) \left (1+x^3\right )^2}{-1+3 x-3 x^3+9 x^4-4 x^6+9 x^7-x^9+3 x^{10}} \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.81 \[ \int \frac {1-\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}}{x^2+\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}} \, dx=\frac {1}{3}-x-3 \text {RootSum}\left [-3-28 \text {$\#$1}+6 \text {$\#$1}^3+3 \text {$\#$1}^4-3 \text {$\#$1}^6-3 \text {$\#$1}^7+\text {$\#$1}^{10}\&,\frac {10 \log \left (1+x^3\right ) \text {$\#$1}^2-10 \log \left (\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}-\text {$\#$1}-x^3 \text {$\#$1}\right ) \text {$\#$1}^2-2 \log \left (1+x^3\right ) \text {$\#$1}^5+2 \log \left (\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}-\text {$\#$1}-x^3 \text {$\#$1}\right ) \text {$\#$1}^5+\log \left (1+x^3\right ) \text {$\#$1}^8-\log \left (\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}-\text {$\#$1}-x^3 \text {$\#$1}\right ) \text {$\#$1}^8}{-28+18 \text {$\#$1}^2+12 \text {$\#$1}^3-18 \text {$\#$1}^5-21 \text {$\#$1}^6+10 \text {$\#$1}^9}\&\right ] \]

[In]

Integrate[(1 - (1 - 3*x + 3*x^3 - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10)^(1/3))/(x^2 + (1 - 3*x + 3*x^3 - 9*x^4
+ 3*x^6 - 9*x^7 + x^9 - 3*x^10)^(1/3)),x]

[Out]

1/3 - x - 3*RootSum[-3 - 28*#1 + 6*#1^3 + 3*#1^4 - 3*#1^6 - 3*#1^7 + #1^10 & , (10*Log[1 + x^3]*#1^2 - 10*Log[
(-((-1 + 3*x)*(1 + x^3)^3))^(1/3) - #1 - x^3*#1]*#1^2 - 2*Log[1 + x^3]*#1^5 + 2*Log[(-((-1 + 3*x)*(1 + x^3)^3)
)^(1/3) - #1 - x^3*#1]*#1^5 + Log[1 + x^3]*#1^8 - Log[(-((-1 + 3*x)*(1 + x^3)^3))^(1/3) - #1 - x^3*#1]*#1^8)/(
-28 + 18*#1^2 + 12*#1^3 - 18*#1^5 - 21*#1^6 + 10*#1^9) & ]

Maple [F(-1)]

Timed out.

\[\int \frac {1-\left (-3 x^{10}+x^{9}-9 x^{7}+3 x^{6}-9 x^{4}+3 x^{3}-3 x +1\right )^{\frac {1}{3}}}{x^{2}+\left (-3 x^{10}+x^{9}-9 x^{7}+3 x^{6}-9 x^{4}+3 x^{3}-3 x +1\right )^{\frac {1}{3}}}d x\]

[In]

int((1-(-3*x^10+x^9-9*x^7+3*x^6-9*x^4+3*x^3-3*x+1)^(1/3))/(x^2+(-3*x^10+x^9-9*x^7+3*x^6-9*x^4+3*x^3-3*x+1)^(1/
3)),x)

[Out]

int((1-(-3*x^10+x^9-9*x^7+3*x^6-9*x^4+3*x^3-3*x+1)^(1/3))/(x^2+(-3*x^10+x^9-9*x^7+3*x^6-9*x^4+3*x^3-3*x+1)^(1/
3)),x)

Fricas [F(-1)]

Timed out. \[ \int \frac {1-\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}}{x^2+\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}} \, dx=\text {Timed out} \]

[In]

integrate((1-(-3*x^10+x^9-9*x^7+3*x^6-9*x^4+3*x^3-3*x+1)^(1/3))/(x^2+(-3*x^10+x^9-9*x^7+3*x^6-9*x^4+3*x^3-3*x+
1)^(1/3)),x, algorithm="fricas")

[Out]

Timed out

Sympy [N/A]

Not integrable

Time = 2.31 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.45 \[ \int \frac {1-\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}}{x^2+\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}} \, dx=- \int \frac {\sqrt [3]{- 3 x^{10} + x^{9} - 9 x^{7} + 3 x^{6} - 9 x^{4} + 3 x^{3} - 3 x + 1}}{x^{2} + \sqrt [3]{- 3 x^{10} + x^{9} - 9 x^{7} + 3 x^{6} - 9 x^{4} + 3 x^{3} - 3 x + 1}}\, dx - \int \left (- \frac {1}{x^{2} + \sqrt [3]{- 3 x^{10} + x^{9} - 9 x^{7} + 3 x^{6} - 9 x^{4} + 3 x^{3} - 3 x + 1}}\right )\, dx \]

[In]

integrate((1-(-3*x**10+x**9-9*x**7+3*x**6-9*x**4+3*x**3-3*x+1)**(1/3))/(x**2+(-3*x**10+x**9-9*x**7+3*x**6-9*x*
*4+3*x**3-3*x+1)**(1/3)),x)

[Out]

-Integral((-3*x**10 + x**9 - 9*x**7 + 3*x**6 - 9*x**4 + 3*x**3 - 3*x + 1)**(1/3)/(x**2 + (-3*x**10 + x**9 - 9*
x**7 + 3*x**6 - 9*x**4 + 3*x**3 - 3*x + 1)**(1/3)), x) - Integral(-1/(x**2 + (-3*x**10 + x**9 - 9*x**7 + 3*x**
6 - 9*x**4 + 3*x**3 - 3*x + 1)**(1/3)), x)

Maxima [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.12 \[ \int \frac {1-\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}}{x^2+\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}} \, dx=\int { -\frac {{\left (-3 \, x^{10} + x^{9} - 9 \, x^{7} + 3 \, x^{6} - 9 \, x^{4} + 3 \, x^{3} - 3 \, x + 1\right )}^{\frac {1}{3}} - 1}{x^{2} + {\left (-3 \, x^{10} + x^{9} - 9 \, x^{7} + 3 \, x^{6} - 9 \, x^{4} + 3 \, x^{3} - 3 \, x + 1\right )}^{\frac {1}{3}}} \,d x } \]

[In]

integrate((1-(-3*x^10+x^9-9*x^7+3*x^6-9*x^4+3*x^3-3*x+1)^(1/3))/(x^2+(-3*x^10+x^9-9*x^7+3*x^6-9*x^4+3*x^3-3*x+
1)^(1/3)),x, algorithm="maxima")

[Out]

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

Giac [N/A]

Not integrable

Time = 0.41 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.29 \[ \int \frac {1-\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}}{x^2+\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}} \, dx=\int { -\frac {{\left (-3 \, x^{10} + x^{9} - 9 \, x^{7} + 3 \, x^{6} - 9 \, x^{4} + 3 \, x^{3} - 3 \, x + 1\right )}^{\frac {1}{3}} - 1}{x^{2} + {\left (-3 \, x^{10} + x^{9} - 9 \, x^{7} + 3 \, x^{6} - 9 \, x^{4} + 3 \, x^{3} - 3 \, x + 1\right )}^{\frac {1}{3}}} \,d x } \]

[In]

integrate((1-(-3*x^10+x^9-9*x^7+3*x^6-9*x^4+3*x^3-3*x+1)^(1/3))/(x^2+(-3*x^10+x^9-9*x^7+3*x^6-9*x^4+3*x^3-3*x+
1)^(1/3)),x, algorithm="giac")

[Out]

integrate(-((-3*x^10 + x^9 - 9*x^7 + 3*x^6 - 9*x^4 + 3*x^3 - 3*x + 1)^(1/3) - 1)/(x^2 + (-3*x^10 + x^9 - 9*x^7
+ 3*x^6 - 9*x^4 + 3*x^3 - 3*x + 1)^(1/3)), x)

Mupad [N/A]

Not integrable

Time = 7.26 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.29 \[ \int \frac {1-\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}}{x^2+\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}} \, dx=-\int \frac {{\left (-3\,x^{10}+x^9-9\,x^7+3\,x^6-9\,x^4+3\,x^3-3\,x+1\right )}^{1/3}-1}{{\left (-3\,x^{10}+x^9-9\,x^7+3\,x^6-9\,x^4+3\,x^3-3\,x+1\right )}^{1/3}+x^2} \,d x \]

[In]

int(-((3*x^3 - 3*x - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10 + 1)^(1/3) - 1)/((3*x^3 - 3*x - 9*x^4 + 3*x^6 - 9*x^
7 + x^9 - 3*x^10 + 1)^(1/3) + x^2),x)

[Out]

-int(((3*x^3 - 3*x - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10 + 1)^(1/3) - 1)/((3*x^3 - 3*x - 9*x^4 + 3*x^6 - 9*x^
7 + x^9 - 3*x^10 + 1)^(1/3) + x^2), x)