3.29.23 \(\int \frac {1-\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}}{x^2+\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}} \, dx\)
Optimal. Leaf size=282 \[ 3 \text {RootSum}\left [\text {$\#$1}^{10}-3 \text {$\#$1}^7-3 \text {$\#$1}^6+3 \text {$\#$1}^4+6 \text {$\#$1}^3-28 \text {$\#$1}-3\& ,\frac {\text {$\#$1}^8 \left (-\log \left (x^3+1\right )\right )+\text {$\#$1}^8 \log \left (-\text {$\#$1} x^3-\text {$\#$1}+\sqrt [3]{-3 x^{10}+x^9-9 x^7+3 x^6-9 x^4+3 x^3-3 x+1}\right )+2 \text {$\#$1}^5 \log \left (x^3+1\right )-2 \text {$\#$1}^5 \log \left (-\text {$\#$1} x^3-\text {$\#$1}+\sqrt [3]{-3 x^{10}+x^9-9 x^7+3 x^6-9 x^4+3 x^3-3 x+1}\right )-10 \text {$\#$1}^2 \log \left (x^3+1\right )+10 \text {$\#$1}^2 \log \left (-\text {$\#$1} x^3-\text {$\#$1}+\sqrt [3]{-3 x^{10}+x^9-9 x^7+3 x^6-9 x^4+3 x^3-3 x+1}\right )}{10 \text {$\#$1}^9-21 \text {$\#$1}^6-18 \text {$\#$1}^5+12 \text {$\#$1}^3+18 \text {$\#$1}^2-28}\& \right ]-x \]
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Rubi [F] time = 54.38, 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 {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 \end {gather*}
Verification is not applicable to the result.
[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*} \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]{-\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\\ &=\text {rest of steps removed due to Latex formating problem} \end {align*}
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Mathematica [B] time = 8.80, size = 1433, normalized size = 5.08
result too large to display
Antiderivative was successfully verified.
[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]
-x - ((-((-1 + 3*x)*(1 + x^3)^3))^(2/3)*RootSum[-3 + 28*#1 - 6*#1^3 + 3*#1^4 - 3*#1^6 + 3*#1^7 + #1^10 & , (10
*Log[(-1 + 3*x)^(1/3) - #1]*#1^2 + 2*Log[(-1 + 3*x)^(1/3) - #1]*#1^5 + Log[(-1 + 3*x)^(1/3) - #1]*#1^8)/(28 -
18*#1^2 + 12*#1^3 - 18*#1^5 + 21*#1^6 + 10*#1^9) & ])/((-1 + 3*x)^(2/3)*(1 + x^3)^2) + ((-((-1 + 3*x)*(1 + x^3
)^3))^(1/3)*(x^2 + x^4)*RootSum[-3 + 28*#1 - 6*#1^3 + 3*#1^4 - 3*#1^6 + 3*#1^7 + #1^10 & , (10*Log[(-1 + 3*x)^
(1/3) - #1]*#1^2 + 2*Log[(-1 + 3*x)^(1/3) - #1]*#1^5 + Log[(-1 + 3*x)^(1/3) - #1]*#1^8)/(28 - 18*#1^2 + 12*#1^
3 - 18*#1^5 + 21*#1^6 + 10*#1^9) & ])/(x^2*(-1 + 3*x)^(1/3)*(1 + x^2)*(1 + x^3)) - RootSum[-1 + 3*#1 - 3*#1^3
+ 9*#1^4 - 4*#1^6 + 9*#1^7 - #1^9 + 3*#1^10 & , (Log[x - #1]*#1^4 + Log[x - #1]*#1^6)/(1 - 3*#1^2 + 12*#1^3 -
8*#1^5 + 21*#1^6 - 3*#1^8 + 10*#1^9) & ]/3 - ((-((-1 + 3*x)*(1 + x^3)^3))^(1/3)*(x^2 + x^4)*RootSum[9 + 84*#1
+ 784*#1^2 + 36*#1^3 + 177*#1^4 + 168*#1^5 + 54*#1^6 + 111*#1^7 + 177*#1^8 + 36*#1^9 + 30*#1^10 + 74*#1^11 + 9
*#1^12 + 15*#1^13 + 15*#1^14 + 3*#1^16 + 6*#1^17 + #1^20 & , (-30*Log[(-1 + 3*x)^(1/3) - #1]*#1^2 + 280*Log[(-
1 + 3*x)^(1/3) - #1]*#1^3 - 66*Log[(-1 + 3*x)^(1/3) - #1]*#1^5 + 86*Log[(-1 + 3*x)^(1/3) - #1]*#1^6 - 45*Log[(
-1 + 3*x)^(1/3) - #1]*#1^8 + 64*Log[(-1 + 3*x)^(1/3) - #1]*#1^9 - 12*Log[(-1 + 3*x)^(1/3) - #1]*#1^11 + 19*Log
[(-1 + 3*x)^(1/3) - #1]*#1^12 - 3*Log[(-1 + 3*x)^(1/3) - #1]*#1^14 + 5*Log[(-1 + 3*x)^(1/3) - #1]*#1^15 + Log[
(-1 + 3*x)^(1/3) - #1]*#1^18)/(84 + 1568*#1 + 108*#1^2 + 708*#1^3 + 840*#1^4 + 324*#1^5 + 777*#1^6 + 1416*#1^7
+ 324*#1^8 + 300*#1^9 + 814*#1^10 + 108*#1^11 + 195*#1^12 + 210*#1^13 + 48*#1^15 + 102*#1^16 + 20*#1^19) & ])
/(x^2*(-1 + 3*x)^(1/3)*(1 + x^2)*(1 + x^3)) - ((-((-1 + 3*x)*(1 + x^3)^3))^(2/3)*RootSum[9 + 84*#1 + 784*#1^2
+ 36*#1^3 + 177*#1^4 + 168*#1^5 + 54*#1^6 + 111*#1^7 + 177*#1^8 + 36*#1^9 + 30*#1^10 + 74*#1^11 + 9*#1^12 + 15
*#1^13 + 15*#1^14 + 3*#1^16 + 6*#1^17 + #1^20 & , (30*Log[(-1 + 3*x)^(1/3) - #1]*#1^2 + 560*Log[(-1 + 3*x)^(1/
3) - #1]*#1^3 + 66*Log[(-1 + 3*x)^(1/3) - #1]*#1^5 + 172*Log[(-1 + 3*x)^(1/3) - #1]*#1^6 + 45*Log[(-1 + 3*x)^(
1/3) - #1]*#1^8 + 128*Log[(-1 + 3*x)^(1/3) - #1]*#1^9 + 12*Log[(-1 + 3*x)^(1/3) - #1]*#1^11 + 38*Log[(-1 + 3*x
)^(1/3) - #1]*#1^12 + 3*Log[(-1 + 3*x)^(1/3) - #1]*#1^14 + 10*Log[(-1 + 3*x)^(1/3) - #1]*#1^15 + 2*Log[(-1 + 3
*x)^(1/3) - #1]*#1^18)/(84 + 1568*#1 + 108*#1^2 + 708*#1^3 + 840*#1^4 + 324*#1^5 + 777*#1^6 + 1416*#1^7 + 324*
#1^8 + 300*#1^9 + 814*#1^10 + 108*#1^11 + 195*#1^12 + 210*#1^13 + 48*#1^15 + 102*#1^16 + 20*#1^19) & ])/((-1 +
3*x)^(2/3)*(1 + x^3)^2)
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IntegrateAlgebraic [A] time = 0.32, size = 288, normalized size = 1.02 \begin {gather*} \frac {1}{3} (1-3 x)+3 \text {RootSum}\left [-3-28 \text {$\#$1}+6 \text {$\#$1}^3+3 \text {$\#$1}^4-3 \text {$\#$1}^6-3 \text {$\#$1}^7+\text {$\#$1}^{10}\&,\frac {-10 \log \left (1+x^3\right ) \text {$\#$1}^2+10 \log \left (\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}-\text {$\#$1}-x^3 \text {$\#$1}\right ) \text {$\#$1}^2+2 \log \left (1+x^3\right ) \text {$\#$1}^5-2 \log \left (\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}-\text {$\#$1}-x^3 \text {$\#$1}\right ) \text {$\#$1}^5-\log \left (1+x^3\right ) \text {$\#$1}^8+\log \left (\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}-\text {$\#$1}-x^3 \text {$\#$1}\right ) \text {$\#$1}^8}{-28+18 \text {$\#$1}^2+12 \text {$\#$1}^3-18 \text {$\#$1}^5-21 \text {$\#$1}^6+10 \text {$\#$1}^9}\&\right ] \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(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 + 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 + 3*x^3 - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10)^(1/3) - #1 - x^3*#1]*#1^2 + 2*Log[1 + x^3]*#1^5
- 2*Log[(1 - 3*x + 3*x^3 - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10)^(1/3) - #1 - x^3*#1]*#1^5 - Log[1 + x^3]*#1^8
+ Log[(1 - 3*x + 3*x^3 - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10)^(1/3) - #1 - x^3*#1]*#1^8)/(-28 + 18*#1^2 + 12
*#1^3 - 18*#1^5 - 21*#1^6 + 10*#1^9) & ]
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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maple [F] time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {1-\left (-3 x^{10}+x^{9}-9 x^{7}+3 x^{6}-9 x^{4}+3 x^{3}-3 x +1\right )^{\frac {1}{3}}}{x^{2}+\left (-3 x^{10}+x^{9}-9 x^{7}+3 x^{6}-9 x^{4}+3 x^{3}-3 x +1\right )^{\frac {1}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[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)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -x - \int -\frac {x^{2} + 1}{x^{2} - {\left (x^{3} + 1\right )} {\left (3 \, x - 1\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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