3.26.19 \(\int \frac {x^3 \sqrt [3]{-x^2+x^3}}{1+x^6} \, dx\)
Optimal. Leaf size=210 \[ \frac {1}{6} \text {RootSum}\left [\text {$\#$1}^6-2 \text {$\#$1}^3+2\& ,\frac {\text {$\#$1} \log \left (\sqrt [3]{x^3-x^2}-\text {$\#$1} x\right )-\text {$\#$1} \log (x)}{\text {$\#$1}^3-1}\& \right ]-\frac {1}{6} \text {RootSum}\left [\text {$\#$1}^{12}-4 \text {$\#$1}^9+5 \text {$\#$1}^6-2 \text {$\#$1}^3+1\& ,\frac {\text {$\#$1}^7 \log \left (\sqrt [3]{x^3-x^2}-\text {$\#$1} x\right )+\text {$\#$1}^7 (-\log (x))-2 \text {$\#$1}^4 \log \left (\sqrt [3]{x^3-x^2}-\text {$\#$1} x\right )+2 \text {$\#$1}^4 \log (x)-\text {$\#$1} \log \left (\sqrt [3]{x^3-x^2}-\text {$\#$1} x\right )+\text {$\#$1} \log (x)}{2 \text {$\#$1}^9-6 \text {$\#$1}^6+5 \text {$\#$1}^3-1}\& \right ] \]
________________________________________________________________________________________
Rubi [C] time = 5.15, antiderivative size = 4727, normalized size of antiderivative = 22.51,
number of steps used = 103, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used
= {2056, 6725, 101, 157, 50, 59, 105, 91}
result too large to display
Warning: Unable to verify antiderivative.
[In]
Int[(x^3*(-x^2 + x^3)^(1/3))/(1 + x^6),x]
[Out]
(-59*(-x^2 + x^3)^(1/3))/162 + ((5*I)/36)*(1 - (-1)^(1/6))*(-x^2 + x^3)^(1/3) + (5*(-1)^(2/3)*(12 - (-1)^(1/6)
)*(-x^2 + x^3)^(1/3))/486 + (5*(-1)^(2/3)*(12 + (-1)^(1/6))*(-x^2 + x^3)^(1/3))/486 + (I/6)*(1 - (-1)^(5/6))*(
-x^2 + x^3)^(1/3) - (I/6)*(1 + (-1)^(5/6))*(-x^2 + x^3)^(1/3) + (5*(-1)^(1/6)*((1 - 6*I) - 6*Sqrt[3])*(-x^2 +
x^3)^(1/3))/486 + (((1 + 2*I) - I*Sqrt[3])*(-x^2 + x^3)^(1/3))/12 + ((5*I)/72)*((-2 - I) + Sqrt[3])*(-x^2 + x^
3)^(1/3) - (5/72 + (5*I)/72)*(-1)^(1/3)*(1 + Sqrt[3])*(-x^2 + x^3)^(1/3) - (I/12)*((2 + I) + Sqrt[3])*(-x^2 +
x^3)^(1/3) + (5*(-1)^(1/3)*((2 + I) + Sqrt[3])*(-x^2 + x^3)^(1/3))/72 - (5*(-1)^(1/6)*((1 + 6*I) + 6*Sqrt[3])*
(-x^2 + x^3)^(1/3))/486 - (x*(-x^2 + x^3)^(1/3))/54 + (I/12)*(1 - (-1)^(1/6))*x*(-x^2 + x^3)^(1/3) + ((-1)^(2/
3)*(12 - (-1)^(1/6))*x*(-x^2 + x^3)^(1/3))/162 + ((-1)^(2/3)*(12 + (-1)^(1/6))*x*(-x^2 + x^3)^(1/3))/162 + ((-
1)^(1/6)*((1 - 6*I) - 6*Sqrt[3])*x*(-x^2 + x^3)^(1/3))/162 + (I/24)*((-2 - I) + Sqrt[3])*x*(-x^2 + x^3)^(1/3)
- (I/24)*((2 + I) + Sqrt[3])*x*(-x^2 + x^3)^(1/3) + ((-1)^(1/3)*((2 + I) + Sqrt[3])*x*(-x^2 + x^3)^(1/3))/24 -
((-1)^(1/6)*((1 + 6*I) + 6*Sqrt[3])*x*(-x^2 + x^3)^(1/3))/162 + (x^2*(-x^2 + x^3)^(1/3))/9 + ((-1)^(2/3)*(12
- (-1)^(1/6))*x^2*(-x^2 + x^3)^(1/3))/216 + ((-1)^(2/3)*(12 + (-1)^(1/6))*x^2*(-x^2 + x^3)^(1/3))/216 + ((-1)^
(1/6)*((1 - 6*I) - 6*Sqrt[3])*x^2*(-x^2 + x^3)^(1/3))/216 - ((-1)^(1/6)*((1 + 6*I) + 6*Sqrt[3])*x^2*(-x^2 + x^
3)^(1/3))/216 - (22*(-x^2 + x^3)^(1/3)*ArcTan[1/Sqrt[3] + (2*x^(1/3))/(Sqrt[3]*(-1 + x)^(1/3))])/(81*Sqrt[3]*(
-1 + x)^(1/3)*x^(2/3)) - (((5*I)/18)*(1 - (-1)^(1/6))*(-x^2 + x^3)^(1/3)*ArcTan[1/Sqrt[3] + (2*x^(1/3))/(Sqrt[
3]*(-1 + x)^(1/3))])/(Sqrt[3]*(-1 + x)^(1/3)*x^(2/3)) + ((I/3)*(1 + (-1)^(5/6))*(-x^2 + x^3)^(1/3)*ArcTan[1/Sq
rt[3] + (2*x^(1/3))/(Sqrt[3]*(-1 + x)^(1/3))])/(Sqrt[3]*(-1 + x)^(1/3)*x^(2/3)) - (5*((-3 - 36*I) - (12 + I)*S
qrt[3])*(-x^2 + x^3)^(1/3)*ArcTan[1/Sqrt[3] + (2*x^(1/3))/(Sqrt[3]*(-1 + x)^(1/3))])/(1458*(-1 + x)^(1/3)*x^(2
/3)) - (5*((3 + 36*I) - (12 + I)*Sqrt[3])*(-x^2 + x^3)^(1/3)*ArcTan[1/Sqrt[3] + (2*x^(1/3))/(Sqrt[3]*(-1 + x)^
(1/3))])/(1458*(-1 + x)^(1/3)*x^(2/3)) + ((I/18)*(3 - (2 - I)*Sqrt[3])*(-x^2 + x^3)^(1/3)*ArcTan[1/Sqrt[3] + (
2*x^(1/3))/(Sqrt[3]*(-1 + x)^(1/3))])/((-1 + x)^(1/3)*x^(2/3)) - ((3 - Sqrt[3])*(-x^2 + x^3)^(1/3)*ArcTan[1/Sq
rt[3] + (2*x^(1/3))/(Sqrt[3]*(-1 + x)^(1/3))])/(6*(-1 + x)^(1/3)*x^(2/3)) + ((3 + Sqrt[3])*(-x^2 + x^3)^(1/3)*
ArcTan[1/Sqrt[3] + (2*x^(1/3))/(Sqrt[3]*(-1 + x)^(1/3))])/(6*(-1 + x)^(1/3)*x^(2/3)) - (5*(3*I + (1 - 2*I)*Sqr
t[3])*(-x^2 + x^3)^(1/3)*ArcTan[1/Sqrt[3] + (2*x^(1/3))/(Sqrt[3]*(-1 + x)^(1/3))])/(108*(-1 + x)^(1/3)*x^(2/3)
) - (11*(3*I + (1 + 2*I)*Sqrt[3])*(-x^2 + x^3)^(1/3)*ArcTan[1/Sqrt[3] + (2*x^(1/3))/(Sqrt[3]*(-1 + x)^(1/3))])
/(108*(-1 + x)^(1/3)*x^(2/3)) - (((5*I)/1458)*((-36 - 3*I) + (1 + 12*I)*Sqrt[3])*(-x^2 + x^3)^(1/3)*ArcTan[1/S
qrt[3] + (2*x^(1/3))/(Sqrt[3]*(-1 + x)^(1/3))])/((-1 + x)^(1/3)*x^(2/3)) - (((5*I)/1458)*((36 + 3*I) + (1 + 12
*I)*Sqrt[3])*(-x^2 + x^3)^(1/3)*ArcTan[1/Sqrt[3] + (2*x^(1/3))/(Sqrt[3]*(-1 + x)^(1/3))])/((-1 + x)^(1/3)*x^(2
/3)) + (((11*I)/108)*(3 + (2 + I)*Sqrt[3])*(-x^2 + x^3)^(1/3)*ArcTan[1/Sqrt[3] + (2*x^(1/3))/(Sqrt[3]*(-1 + x)
^(1/3))])/((-1 + x)^(1/3)*x^(2/3)) + ((-1)^(1/6)*(-1 + I)^(1/3)*(-x^2 + x^3)^(1/3)*ArcTan[1/Sqrt[3] + (2*(1 -
I)^(1/3)*x^(1/3))/(Sqrt[3]*(-1 + x)^(1/3))])/(2*Sqrt[3]*(-1 + x)^(1/3)*x^(2/3)) - ((I/2)*(1 + I)^(1/3)*(-x^2 +
x^3)^(1/3)*ArcTan[1/Sqrt[3] + (2*(1 + I)^(1/3)*x^(1/3))/(Sqrt[3]*(-1 + x)^(1/3))])/(Sqrt[3]*(-1 + x)^(1/3)*x^
(2/3)) + ((1/12 - I/12)*(3 - Sqrt[3])*(-x^2 + x^3)^(1/3)*ArcTan[1/Sqrt[3] + (2*(1 - (-1)^(1/6))^(1/3)*x^(1/3))
/(Sqrt[3]*(-1 + x)^(1/3))])/((1 - (-1)^(1/6))^(2/3)*(-1 + x)^(1/3)*x^(2/3)) + ((-1)^(1/6)*(-(1 + (-1)^(1/6))^(
-1))^(2/3)*(I + (-1)^(1/3))*(-x^2 + x^3)^(1/3)*ArcTan[1/Sqrt[3] + (2*(1 + (-1)^(1/6))^(1/3)*x^(1/3))/(Sqrt[3]*
(-1 + x)^(1/3))])/(2*Sqrt[3]*(-1 + x)^(1/3)*x^(2/3)) + ((I/2)*(((-2 + I) - Sqrt[3])/2)^(1/3)*(-x^2 + x^3)^(1/3
)*ArcTan[1/Sqrt[3] + (2*(1 - (-1)^(5/6))^(1/3)*x^(1/3))/(Sqrt[3]*(-1 + x)^(1/3))])/(Sqrt[3]*(-1 + x)^(1/3)*x^(
2/3)) + ((1/12 + I/12)*(3 - Sqrt[3])*(-x^2 + x^3)^(1/3)*ArcTan[1/Sqrt[3] + (2*(1 + (-1)^(5/6))^(1/3)*x^(1/3))/
(Sqrt[3]*(-1 + x)^(1/3))])/((1 + (-1)^(5/6))^(2/3)*(-1 + x)^(1/3)*x^(2/3)) + ((-1)^(1/6)*(-1 + I)^(1/3)*(-x^2
+ x^3)^(1/3)*Log[-(-1 + x)^(1/3) + (1 - I)^(1/3)*x^(1/3)])/(4*(-1 + x)^(1/3)*x^(2/3)) - ((I/4)*(1 + I)^(1/3)*(
-x^2 + x^3)^(1/3)*Log[-(-1 + x)^(1/3) + (1 + I)^(1/3)*x^(1/3)])/((-1 + x)^(1/3)*x^(2/3)) - ((-1)^(5/6)*(-1 + (
-1)^(1/6))^(1/3)*(-x^2 + x^3)^(1/3)*Log[-(-1 + x)^(1/3) + (1 - (-1)^(1/6))^(1/3)*x^(1/3)])/(4*(-1 + x)^(1/3)*x
^(2/3)) + ((-1)^(1/6)*(-(1 + (-1)^(1/6))^(-1))^(2/3)*(I + (-1)^(1/3))*(-x^2 + x^3)^(1/3)*Log[-(-1 + x)^(1/3) +
(1 + (-1)^(1/6))^(1/3)*x^(1/3)])/(4*(-1 + x)^(1/3)*x^(2/3)) - ((-1)^(1/3)*((2 + I) + Sqrt[3])*(-x^2 + x^3)^(1
/3)*Log[-(-1 + x)^(1/3) + (1 - (-1)^(5/6))^(1/3)*x^(1/3)])/(8*(-1 + (-1)^(5/6))^(2/3)*(-1 + x)^(1/3)*x^(2/3))
- ((-(1 + (-1)^(5/6))^(-1))^(2/3)*((-2 + I) + Sqrt[3])*(-x^2 + x^3)^(1/3)*Log[-(-1 + x)^(1/3) + (1 + (-1)^(5/6
))^(1/3)*x^(1/3)])/(8*(-1 + x)^(1/3)*x^(2/3)) - (11*(-x^2 + x^3)^(1/3)*Log[-1 + x^(1/3)/(-1 + x)^(1/3)])/(81*(
-1 + x)^(1/3)*x^(2/3)) - (((5*I)/36)*(1 - (-1)^(1/6))*(-x^2 + x^3)^(1/3)*Log[-1 + x^(1/3)/(-1 + x)^(1/3)])/((-
1 + x)^(1/3)*x^(2/3)) - (5*(-1)^(2/3)*(12 - (-1)^(1/6))*(-x^2 + x^3)^(1/3)*Log[-1 + x^(1/3)/(-1 + x)^(1/3)])/(
486*(-1 + x)^(1/3)*x^(2/3)) + ((-1)^(1/6)*(1 + (-1)^(1/6))*(-x^2 + x^3)^(1/3)*Log[-1 + x^(1/3)/(-1 + x)^(1/3)]
)/(4*(-1 + x)^(1/3)*x^(2/3)) - (5*(-1)^(2/3)*(12 + (-1)^(1/6))*(-x^2 + x^3)^(1/3)*Log[-1 + x^(1/3)/(-1 + x)^(1
/3)])/(486*(-1 + x)^(1/3)*x^(2/3)) - ((-1)^(5/6)*(I - (-1)^(1/3))*(-x^2 + x^3)^(1/3)*Log[-1 + x^(1/3)/(-1 + x)
^(1/3)])/(4*(-1 + x)^(1/3)*x^(2/3)) - ((I/6)*(1 - (-1)^(5/6))*(-x^2 + x^3)^(1/3)*Log[-1 + x^(1/3)/(-1 + x)^(1/
3)])/((-1 + x)^(1/3)*x^(2/3)) + ((I/6)*(1 + (-1)^(5/6))*(-x^2 + x^3)^(1/3)*Log[-1 + x^(1/3)/(-1 + x)^(1/3)])/(
(-1 + x)^(1/3)*x^(2/3)) - (5*(-1)^(1/6)*((1 - 6*I) - 6*Sqrt[3])*(-x^2 + x^3)^(1/3)*Log[-1 + x^(1/3)/(-1 + x)^(
1/3)])/(486*(-1 + x)^(1/3)*x^(2/3)) - (((5*I)/72)*((-2 - I) + Sqrt[3])*(-x^2 + x^3)^(1/3)*Log[-1 + x^(1/3)/(-1
+ x)^(1/3)])/((-1 + x)^(1/3)*x^(2/3)) + ((I/12)*((-2 + I) + Sqrt[3])*(-x^2 + x^3)^(1/3)*Log[-1 + x^(1/3)/(-1
+ x)^(1/3)])/((-1 + x)^(1/3)*x^(2/3)) + ((-1)^(2/3)*((-2 + I) + Sqrt[3])*(-x^2 + x^3)^(1/3)*Log[-1 + x^(1/3)/(
-1 + x)^(1/3)])/(8*(-1 + x)^(1/3)*x^(2/3)) + ((5/72 + (5*I)/72)*(-1)^(1/3)*(1 + Sqrt[3])*(-x^2 + x^3)^(1/3)*Lo
g[-1 + x^(1/3)/(-1 + x)^(1/3)])/((-1 + x)^(1/3)*x^(2/3)) + ((I/12)*((2 + I) + Sqrt[3])*(-x^2 + x^3)^(1/3)*Log[
-1 + x^(1/3)/(-1 + x)^(1/3)])/((-1 + x)^(1/3)*x^(2/3)) - (5*(-1)^(1/3)*((2 + I) + Sqrt[3])*(-x^2 + x^3)^(1/3)*
Log[-1 + x^(1/3)/(-1 + x)^(1/3)])/(72*(-1 + x)^(1/3)*x^(2/3)) - ((-1)^(2/3)*((2 + I) + Sqrt[3])*(-x^2 + x^3)^(
1/3)*Log[-1 + x^(1/3)/(-1 + x)^(1/3)])/(8*(-1 + x)^(1/3)*x^(2/3)) + (5*(-1)^(1/6)*((1 + 6*I) + 6*Sqrt[3])*(-x^
2 + x^3)^(1/3)*Log[-1 + x^(1/3)/(-1 + x)^(1/3)])/(486*(-1 + x)^(1/3)*x^(2/3)) + ((-(1 + (-1)^(5/6))^(-1))^(2/3
)*((-2 + I) + Sqrt[3])*(-x^2 + x^3)^(1/3)*Log[(-1)^(1/6) - x])/(24*(-1 + x)^(1/3)*x^(2/3)) + ((-1)^(5/6)*(-1 +
(-1)^(1/6))^(1/3)*(-x^2 + x^3)^(1/3)*Log[-(-1)^(5/6) - x])/(12*(-1 + x)^(1/3)*x^(2/3)) - (11*(-x^2 + x^3)^(1/
3)*Log[-1 + x])/(243*(-1 + x)^(1/3)*x^(2/3)) - (((5*I)/108)*(1 - (-1)^(1/6))*(-x^2 + x^3)^(1/3)*Log[-1 + x])/(
(-1 + x)^(1/3)*x^(2/3)) - (5*(-1)^(2/3)*(12 - (-1)^(1/6))*(-x^2 + x^3)^(1/3)*Log[-1 + x])/(1458*(-1 + x)^(1/3)
*x^(2/3)) + ((-1)^(1/6)*(1 + (-1)^(1/6))*(-x^2 + x^3)^(1/3)*Log[-1 + x])/(12*(-1 + x)^(1/3)*x^(2/3)) - (5*(-1)
^(2/3)*(12 + (-1)^(1/6))*(-x^2 + x^3)^(1/3)*Log[-1 + x])/(1458*(-1 + x)^(1/3)*x^(2/3)) - ((-1)^(5/6)*(I - (-1)
^(1/3))*(-x^2 + x^3)^(1/3)*Log[-1 + x])/(12*(-1 + x)^(1/3)*x^(2/3)) - ((I/18)*(1 - (-1)^(5/6))*(-x^2 + x^3)^(1
/3)*Log[-1 + x])/((-1 + x)^(1/3)*x^(2/3)) + ((I/18)*(1 + (-1)^(5/6))*(-x^2 + x^3)^(1/3)*Log[-1 + x])/((-1 + x)
^(1/3)*x^(2/3)) - (5*(-1)^(1/6)*((1 - 6*I) - 6*Sqrt[3])*(-x^2 + x^3)^(1/3)*Log[-1 + x])/(1458*(-1 + x)^(1/3)*x
^(2/3)) - (((5*I)/216)*((-2 - I) + Sqrt[3])*(-x^2 + x^3)^(1/3)*Log[-1 + x])/((-1 + x)^(1/3)*x^(2/3)) + ((I/36)
*((-2 + I) + Sqrt[3])*(-x^2 + x^3)^(1/3)*Log[-1 + x])/((-1 + x)^(1/3)*x^(2/3)) + ((-1)^(2/3)*((-2 + I) + Sqrt[
3])*(-x^2 + x^3)^(1/3)*Log[-1 + x])/(24*(-1 + x)^(1/3)*x^(2/3)) + (((11*I)/216)*((2 + I) + Sqrt[3])*(-x^2 + x^
3)^(1/3)*Log[-1 + x])/((-1 + x)^(1/3)*x^(2/3)) - (5*(-1)^(1/3)*((2 + I) + Sqrt[3])*(-x^2 + x^3)^(1/3)*Log[-1 +
x])/(216*(-1 + x)^(1/3)*x^(2/3)) - ((-1)^(2/3)*((2 + I) + Sqrt[3])*(-x^2 + x^3)^(1/3)*Log[-1 + x])/(24*(-1 +
x)^(1/3)*x^(2/3)) + (5*(-1)^(1/6)*((1 + 6*I) + 6*Sqrt[3])*(-x^2 + x^3)^(1/3)*Log[-1 + x])/(1458*(-1 + x)^(1/3)
*x^(2/3)) - ((-1)^(1/6)*(-(1 + (-1)^(1/6))^(-1))^(2/3)*(I + (-1)^(1/3))*(-x^2 + x^3)^(1/3)*Log[(-1)^(1/6) + (-
1)^(1/3)*x])/(12*(-1 + x)^(1/3)*x^(2/3)) + ((I/12)*(1 + I)^(1/3)*(-x^2 + x^3)^(1/3)*Log[-(-1)^(5/6) + (-1)^(1/
3)*x])/((-1 + x)^(1/3)*x^(2/3)) - ((-1)^(1/6)*(-1 + I)^(1/3)*(-x^2 + x^3)^(1/3)*Log[(-1)^(1/6) - (-1)^(2/3)*x]
)/(12*(-1 + x)^(1/3)*x^(2/3)) + ((-1)^(1/3)*((2 + I) + Sqrt[3])*(-x^2 + x^3)^(1/3)*Log[-(-1)^(5/6) - (-1)^(2/3
)*x])/(24*(-1 + (-1)^(5/6))^(2/3)*(-1 + x)^(1/3)*x^(2/3))
Rule 50
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]
Rule 59
Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[d/b, 3]}, -Simp[(Sqrt
[3]*q*ArcTan[(2*q*(a + b*x)^(1/3))/(Sqrt[3]*(c + d*x)^(1/3)) + 1/Sqrt[3]])/d, x] + (-Simp[(3*q*Log[(q*(a + b*x
)^(1/3))/(c + d*x)^(1/3) - 1])/(2*d), x] - Simp[(q*Log[c + d*x])/(2*d), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[
b*c - a*d, 0] && PosQ[d/b]
Rule 91
Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)*((e_.) + (f_.)*(x_))), x_Symbol] :> With[{q = Rt[
(d*e - c*f)/(b*e - a*f), 3]}, -Simp[(Sqrt[3]*q*ArcTan[1/Sqrt[3] + (2*q*(a + b*x)^(1/3))/(Sqrt[3]*(c + d*x)^(1/
3))])/(d*e - c*f), x] + (Simp[(q*Log[e + f*x])/(2*(d*e - c*f)), x] - Simp[(3*q*Log[q*(a + b*x)^(1/3) - (c + d*
x)^(1/3)])/(2*(d*e - c*f)), x])] /; FreeQ[{a, b, c, d, e, f}, x]
Rule 101
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a +
b*x)^m*(c + d*x)^n*(e + f*x)^(p + 1))/(f*(m + n + p + 1)), x] - Dist[1/(f*(m + n + p + 1)), Int[(a + b*x)^(m -
1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a*f) + b*n*(d*e - c*f))
*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (Integ
ersQ[2*m, 2*n, 2*p] || (IntegersQ[m, n + p] || IntegersQ[p, m + n]))
Rule 105
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Dist[b/f, Int[(a
+ b*x)^(m - 1)*(c + d*x)^n, x], x] - Dist[(b*e - a*f)/f, Int[((a + b*x)^(m - 1)*(c + d*x)^n)/(e + f*x), x], x]
/; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[Simplify[m + n + 1], 0] && (GtQ[m, 0] || ( !RationalQ[m] && (Su
mSimplerQ[m, -1] || !SumSimplerQ[n, -1])))
Rule 157
Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
Rule 2056
Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] && !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] && !PolyQ[P, x, 2]
Rule 6725
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]
Rubi steps
\begin {align*} \int \frac {x^3 \sqrt [3]{-x^2+x^3}}{1+x^6} \, dx &=\frac {\sqrt [3]{-x^2+x^3} \int \frac {\sqrt [3]{-1+x} x^{11/3}}{1+x^6} \, dx}{\sqrt [3]{-1+x} x^{2/3}}\\ &=\frac {\sqrt [3]{-x^2+x^3} \int \left (\frac {i \sqrt [3]{-1+x} x^{11/3}}{2 \left (i-x^3\right )}+\frac {i \sqrt [3]{-1+x} x^{11/3}}{2 \left (i+x^3\right )}\right ) \, dx}{\sqrt [3]{-1+x} x^{2/3}}\\ &=\frac {\left (i \sqrt [3]{-x^2+x^3}\right ) \int \frac {\sqrt [3]{-1+x} x^{11/3}}{i-x^3} \, dx}{2 \sqrt [3]{-1+x} x^{2/3}}+\frac {\left (i \sqrt [3]{-x^2+x^3}\right ) \int \frac {\sqrt [3]{-1+x} x^{11/3}}{i+x^3} \, dx}{2 \sqrt [3]{-1+x} x^{2/3}}\\ &=\text {rest of steps removed due to Latex formating problem} \end {align*}
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Mathematica [C] time = 0.61, size = 393, normalized size = 1.87 \begin {gather*} \frac {\sqrt [3]{(x-1) x^2} \left (-i \left (\sqrt {3}+(-2-i)\right ) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) (x-1)}{x}\right )+i \left (\sqrt {3}+(-2-i)\right ) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) (x-1)}{x}\right )-\sqrt {3} \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};-\frac {2 (x-1)}{\left ((-2+i)+\sqrt {3}\right ) x}\right )+(2-i) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};-\frac {2 (x-1)}{\left ((-2+i)+\sqrt {3}\right ) x}\right )-(1+i) \sqrt {3} \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\left (i+\sqrt {3}\right ) (x-1)}{\left ((-2+i)+\sqrt {3}\right ) x}\right )+(1+i) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\left (i+\sqrt {3}\right ) (x-1)}{\left ((-2+i)+\sqrt {3}\right ) x}\right )+\sqrt {3} \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {2 (x-1)}{\left ((2+i)+\sqrt {3}\right ) x}\right )-(2-i) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {2 (x-1)}{\left ((2+i)+\sqrt {3}\right ) x}\right )+(1+i) \sqrt {3} \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\left (i+\sqrt {3}\right ) (x-1)}{\left ((2+i)+\sqrt {3}\right ) x}\right )-(1+i) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\left (i+\sqrt {3}\right ) (x-1)}{\left ((2+i)+\sqrt {3}\right ) x}\right )\right )}{4 \left (1+(-1)^{5/6}\right ) x} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(x^3*(-x^2 + x^3)^(1/3))/(1 + x^6),x]
[Out]
(((-1 + x)*x^2)^(1/3)*((-I)*((-2 - I) + Sqrt[3])*Hypergeometric2F1[1/3, 1, 4/3, ((1/2 - I/2)*(-1 + x))/x] + I*
((-2 - I) + Sqrt[3])*Hypergeometric2F1[1/3, 1, 4/3, ((1/2 + I/2)*(-1 + x))/x] + (2 - I)*Hypergeometric2F1[1/3,
1, 4/3, (-2*(-1 + x))/(((-2 + I) + Sqrt[3])*x)] - Sqrt[3]*Hypergeometric2F1[1/3, 1, 4/3, (-2*(-1 + x))/(((-2
+ I) + Sqrt[3])*x)] + (1 + I)*Hypergeometric2F1[1/3, 1, 4/3, ((I + Sqrt[3])*(-1 + x))/(((-2 + I) + Sqrt[3])*x)
] - (1 + I)*Sqrt[3]*Hypergeometric2F1[1/3, 1, 4/3, ((I + Sqrt[3])*(-1 + x))/(((-2 + I) + Sqrt[3])*x)] - (2 - I
)*Hypergeometric2F1[1/3, 1, 4/3, (2*(-1 + x))/(((2 + I) + Sqrt[3])*x)] + Sqrt[3]*Hypergeometric2F1[1/3, 1, 4/3
, (2*(-1 + x))/(((2 + I) + Sqrt[3])*x)] - (1 + I)*Hypergeometric2F1[1/3, 1, 4/3, ((I + Sqrt[3])*(-1 + x))/(((2
+ I) + Sqrt[3])*x)] + (1 + I)*Sqrt[3]*Hypergeometric2F1[1/3, 1, 4/3, ((I + Sqrt[3])*(-1 + x))/(((2 + I) + Sqr
t[3])*x)]))/(4*(1 + (-1)^(5/6))*x)
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IntegrateAlgebraic [A] time = 0.52, size = 210, normalized size = 1.00 \begin {gather*} \frac {1}{6} \text {RootSum}\left [2-2 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^3}\&\right ]-\frac {1}{6} \text {RootSum}\left [1-2 \text {$\#$1}^3+5 \text {$\#$1}^6-4 \text {$\#$1}^9+\text {$\#$1}^{12}\&,\frac {\log (x) \text {$\#$1}-\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}+2 \log (x) \text {$\#$1}^4-2 \log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^4-\log (x) \text {$\#$1}^7+\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^7}{-1+5 \text {$\#$1}^3-6 \text {$\#$1}^6+2 \text {$\#$1}^9}\&\right ] \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(x^3*(-x^2 + x^3)^(1/3))/(1 + x^6),x]
[Out]
RootSum[2 - 2*#1^3 + #1^6 & , (-(Log[x]*#1) + Log[(-x^2 + x^3)^(1/3) - x*#1]*#1)/(-1 + #1^3) & ]/6 - RootSum[1
- 2*#1^3 + 5*#1^6 - 4*#1^9 + #1^12 & , (Log[x]*#1 - Log[(-x^2 + x^3)^(1/3) - x*#1]*#1 + 2*Log[x]*#1^4 - 2*Log
[(-x^2 + x^3)^(1/3) - x*#1]*#1^4 - Log[x]*#1^7 + Log[(-x^2 + x^3)^(1/3) - x*#1]*#1^7)/(-1 + 5*#1^3 - 6*#1^6 +
2*#1^9) & ]/6
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fricas [B] time = 1.15, size = 9711, normalized size = 46.24
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3*(x^3-x^2)^(1/3)/(x^6+1),x, algorithm="fricas")
[Out]
1/12*2^(2/3)*(4*sqrt(3) + 8)^(1/6)*cos(2/3*arctan(2*sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)*sqrt(sqrt(3) + 2) - sq
rt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)))*log(8*(sqrt(3)*2^(2/3)*(x^3 - x^2)^(1/3)*x*(4*sqrt(3) + 8)^(1/6)*cos(2/3*a
rctan(2*sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)*sqrt(sqrt(3) + 2) - sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7))) - (x^3 -
x^2)^(1/3)*(sqrt(3)*2^(2/3)*x - 2*2^(2/3)*x)*(4*sqrt(3) + 8)^(1/6)*sqrt(4*sqrt(3) + 7)*sin(2/3*arctan(2*sqrt(
4*sqrt(3) + 7)*(4*sqrt(3) - 7)*sqrt(sqrt(3) + 2) - sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7))) + 2^(1/3)*x^2*(4*sqrt
(3) + 8)^(1/3) + 2*(x^3 - x^2)^(2/3))/x^2) + 1/12*2^(2/3)*(-4*sqrt(3) + 8)^(1/6)*cos(2/3*arctan((4*sqrt(3) + 7
)*sqrt(-4*sqrt(3) + 8)*sqrt(-4*sqrt(3) + 7) - (4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 7)))*log(-8*(sqrt(3)*2^(2/3)*(
x^3 - x^2)^(1/3)*x*(-4*sqrt(3) + 8)^(1/6)*cos(2/3*arctan((4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 8)*sqrt(-4*sqrt(3)
+ 7) - (4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 7))) + (x^3 - x^2)^(1/3)*(sqrt(3)*2^(2/3)*x + 2*2^(2/3)*x)*(-4*sqrt(3
) + 8)^(1/6)*sqrt(-4*sqrt(3) + 7)*sin(2/3*arctan((4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 8)*sqrt(-4*sqrt(3) + 7) - (
4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 7))) - 2^(1/3)*x^2*(-4*sqrt(3) + 8)^(1/3) - 2*(x^3 - x^2)^(2/3))/x^2) + 1/3*2
^(2/3)*(4*sqrt(3) + 8)^(1/6)*arctan(1/4*(2*(x^3 - x^2)^(1/3)*(4*sqrt(3)*2^(1/3) - 7*2^(1/3))*(4*sqrt(3) + 8)^(
5/6)*sqrt(4*sqrt(3) + 7)*cos(2/3*arctan(2*sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)*sqrt(sqrt(3) + 2) - sqrt(4*sqrt(
3) + 7)*(4*sqrt(3) - 7))) - sqrt(2)*((4*sqrt(3)*2^(1/3)*x - 7*2^(1/3)*x)*(4*sqrt(3) + 8)^(5/6)*sqrt(4*sqrt(3)
+ 7)*cos(2/3*arctan(2*sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)*sqrt(sqrt(3) + 2) - sqrt(4*sqrt(3) + 7)*(4*sqrt(3) -
7))) - (2*sqrt(3)*2^(1/3)*x - 3*2^(1/3)*x)*(4*sqrt(3) + 8)^(5/6)*sin(2/3*arctan(2*sqrt(4*sqrt(3) + 7)*(4*sqrt
(3) - 7)*sqrt(sqrt(3) + 2) - sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7))))*sqrt((sqrt(3)*2^(2/3)*(x^3 - x^2)^(1/3)*x*
(4*sqrt(3) + 8)^(1/6)*cos(2/3*arctan(2*sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)*sqrt(sqrt(3) + 2) - sqrt(4*sqrt(3)
+ 7)*(4*sqrt(3) - 7))) - (x^3 - x^2)^(1/3)*(sqrt(3)*2^(2/3)*x - 2*2^(2/3)*x)*(4*sqrt(3) + 8)^(1/6)*sqrt(4*sqrt
(3) + 7)*sin(2/3*arctan(2*sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)*sqrt(sqrt(3) + 2) - sqrt(4*sqrt(3) + 7)*(4*sqrt(
3) - 7))) + 2^(1/3)*x^2*(4*sqrt(3) + 8)^(1/3) + 2*(x^3 - x^2)^(2/3))/x^2) - 2*((x^3 - x^2)^(1/3)*(2*sqrt(3)*2^
(1/3) - 3*2^(1/3))*(4*sqrt(3) + 8)^(5/6) + 8*x*cos(2/3*arctan(2*sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)*sqrt(sqrt(
3) + 2) - sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7))))*sin(2/3*arctan(2*sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)*sqrt(sqr
t(3) + 2) - sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7))) - 4*(2*sqrt(3)*x - 3*x)*sqrt(4*sqrt(3) + 7))/(4*x*cos(2/3*ar
ctan(2*sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)*sqrt(sqrt(3) + 2) - sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)))^2 - 3*x))
*sin(2/3*arctan(2*sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)*sqrt(sqrt(3) + 2) - sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7))
) - 1/3*2^(2/3)*(-4*sqrt(3) + 8)^(1/6)*arctan(-1/4*(2*(x^3 - x^2)^(1/3)*(4*sqrt(3)*2^(1/3) + 7*2^(1/3))*(-4*sq
rt(3) + 8)^(5/6)*sqrt(-4*sqrt(3) + 7)*cos(2/3*arctan((4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 8)*sqrt(-4*sqrt(3) + 7)
- (4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 7))) - sqrt(2)*((4*sqrt(3)*2^(1/3)*x + 7*2^(1/3)*x)*(-4*sqrt(3) + 8)^(5/6
)*sqrt(-4*sqrt(3) + 7)*cos(2/3*arctan((4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 8)*sqrt(-4*sqrt(3) + 7) - (4*sqrt(3) +
7)*sqrt(-4*sqrt(3) + 7))) + (2*sqrt(3)*2^(1/3)*x + 3*2^(1/3)*x)*(-4*sqrt(3) + 8)^(5/6)*sin(2/3*arctan((4*sqrt
(3) + 7)*sqrt(-4*sqrt(3) + 8)*sqrt(-4*sqrt(3) + 7) - (4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 7))))*sqrt(-(sqrt(3)*2^
(2/3)*(x^3 - x^2)^(1/3)*x*(-4*sqrt(3) + 8)^(1/6)*cos(2/3*arctan((4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 8)*sqrt(-4*s
qrt(3) + 7) - (4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 7))) + (x^3 - x^2)^(1/3)*(sqrt(3)*2^(2/3)*x + 2*2^(2/3)*x)*(-4
*sqrt(3) + 8)^(1/6)*sqrt(-4*sqrt(3) + 7)*sin(2/3*arctan((4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 8)*sqrt(-4*sqrt(3) +
7) - (4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 7))) - 2^(1/3)*x^2*(-4*sqrt(3) + 8)^(1/3) - 2*(x^3 - x^2)^(2/3))/x^2)
+ 2*((x^3 - x^2)^(1/3)*(2*sqrt(3)*2^(1/3) + 3*2^(1/3))*(-4*sqrt(3) + 8)^(5/6) - 8*x*cos(2/3*arctan((4*sqrt(3)
+ 7)*sqrt(-4*sqrt(3) + 8)*sqrt(-4*sqrt(3) + 7) - (4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 7))))*sin(2/3*arctan((4*sqr
t(3) + 7)*sqrt(-4*sqrt(3) + 8)*sqrt(-4*sqrt(3) + 7) - (4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 7))) - 4*(2*sqrt(3)*x
+ 3*x)*sqrt(-4*sqrt(3) + 7))/(4*x*cos(2/3*arctan((4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 8)*sqrt(-4*sqrt(3) + 7) - (
4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 7)))^2 - 3*x))*sin(2/3*arctan((4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 8)*sqrt(-4*sq
rt(3) + 7) - (4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 7))) + 1/6*2^(1/6)*cos(2/3*arctan(sqrt(2) + 1))*log((2*2^(1/6)*
(x^3 - x^2)^(1/3)*x*sin(2/3*arctan(sqrt(2) + 1)) + 2^(1/3)*x^2 + (x^3 - x^2)^(2/3))/x^2) + 2/3*2^(1/6)*arctan(
1/2*(2^(5/6)*x*sqrt((2*2^(1/6)*(x^3 - x^2)^(1/3)*x*sin(2/3*arctan(sqrt(2) + 1)) + 2^(1/3)*x^2 + (x^3 - x^2)^(2
/3))/x^2) - 2*x*sin(2/3*arctan(sqrt(2) + 1)) - 2^(5/6)*(x^3 - x^2)^(1/3))/(x*cos(2/3*arctan(sqrt(2) + 1))))*si
n(2/3*arctan(sqrt(2) + 1)) - 1/6*(sqrt(3)*2^(2/3)*(-4*sqrt(3) + 8)^(1/6)*cos(2/3*arctan((4*sqrt(3) + 7)*sqrt(-
4*sqrt(3) + 8)*sqrt(-4*sqrt(3) + 7) - (4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 7))) - 2^(2/3)*(-4*sqrt(3) + 8)^(1/6)*
sin(2/3*arctan((4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 8)*sqrt(-4*sqrt(3) + 7) - (4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 7
))))*arctan(1/16*(4*(x^3 - x^2)^(1/3)*((4*sqrt(3)*2^(1/3) + 7*2^(1/3))*sqrt(-4*sqrt(3) + 7) + 3*sqrt(3)*2^(1/3
) + 6*2^(1/3))*(-4*sqrt(3) + 8)^(5/6)*cos(2/3*arctan((4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 8)*sqrt(-4*sqrt(3) + 7)
- (4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 7)))^2 - 6*(x^3 - x^2)^(1/3)*((4*sqrt(3)*2^(1/3) + 7*2^(1/3))*sqrt(-4*sqr
t(3) + 7) + sqrt(3)*2^(1/3) + 2*2^(1/3))*(-4*sqrt(3) + 8)^(5/6) + 8*(sqrt(3)*x - (2*sqrt(3)*x + 3*x)*sqrt(-4*s
qrt(3) + 7))*cos(2/3*arctan((4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 8)*sqrt(-4*sqrt(3) + 7) - (4*sqrt(3) + 7)*sqrt(-
4*sqrt(3) + 7))) - 4*((x^3 - x^2)^(1/3)*((7*sqrt(3)*2^(1/3) + 12*2^(1/3))*sqrt(-4*sqrt(3) + 7) - 2*sqrt(3)*2^(
1/3) - 3*2^(1/3))*(-4*sqrt(3) + 8)^(5/6)*cos(2/3*arctan((4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 8)*sqrt(-4*sqrt(3) +
7) - (4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 7))) - 16*x*cos(2/3*arctan((4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 8)*sqrt(-
4*sqrt(3) + 7) - (4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 7)))^2 + 6*(sqrt(3)*x + 2*x)*sqrt(-4*sqrt(3) + 7) + 6*x)*si
n(2/3*arctan((4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 8)*sqrt(-4*sqrt(3) + 7) - (4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 7))
) - (2*(3*sqrt(3)*2^(1/3)*x + (4*sqrt(3)*2^(1/3)*x + 7*2^(1/3)*x)*sqrt(-4*sqrt(3) + 7) + 6*2^(1/3)*x)*(-4*sqrt
(3) + 8)^(5/6)*cos(2/3*arctan((4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 8)*sqrt(-4*sqrt(3) + 7) - (4*sqrt(3) + 7)*sqrt
(-4*sqrt(3) + 7)))^2 + 2*(2*sqrt(3)*2^(1/3)*x - (7*sqrt(3)*2^(1/3)*x + 12*2^(1/3)*x)*sqrt(-4*sqrt(3) + 7) + 3*
2^(1/3)*x)*(-4*sqrt(3) + 8)^(5/6)*cos(2/3*arctan((4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 8)*sqrt(-4*sqrt(3) + 7) - (
4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 7)))*sin(2/3*arctan((4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 8)*sqrt(-4*sqrt(3) + 7)
- (4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 7))) - 3*(sqrt(3)*2^(1/3)*x + (4*sqrt(3)*2^(1/3)*x + 7*2^(1/3)*x)*sqrt(-4
*sqrt(3) + 7) + 2*2^(1/3)*x)*(-4*sqrt(3) + 8)^(5/6))*sqrt((2*2^(1/3)*x^2*(-4*sqrt(3) + 8)^(1/3) + (x^3 - x^2)^
(1/3)*(sqrt(3)*2^(2/3)*x - (2*sqrt(3)*2^(2/3)*x + 3*2^(2/3)*x)*sqrt(-4*sqrt(3) + 7))*(-4*sqrt(3) + 8)^(1/6)*co
s(2/3*arctan((4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 8)*sqrt(-4*sqrt(3) + 7) - (4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 7))
) + (x^3 - x^2)^(1/3)*(3*2^(2/3)*x + (sqrt(3)*2^(2/3)*x + 2*2^(2/3)*x)*sqrt(-4*sqrt(3) + 7))*(-4*sqrt(3) + 8)^
(1/6)*sin(2/3*arctan((4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 8)*sqrt(-4*sqrt(3) + 7) - (4*sqrt(3) + 7)*sqrt(-4*sqrt(
3) + 7))) + 4*(x^3 - x^2)^(2/3))/x^2))/(4*x*cos(2/3*arctan((4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 8)*sqrt(-4*sqrt(3
) + 7) - (4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 7)))^3 - 3*x*cos(2/3*arctan((4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 8)*sq
rt(-4*sqrt(3) + 7) - (4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 7))))) - 1/6*(sqrt(3)*2^(2/3)*(-4*sqrt(3) + 8)^(1/6)*co
s(2/3*arctan((4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 8)*sqrt(-4*sqrt(3) + 7) - (4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 7))
) + 2^(2/3)*(-4*sqrt(3) + 8)^(1/6)*sin(2/3*arctan((4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 8)*sqrt(-4*sqrt(3) + 7) -
(4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 7))))*arctan(-1/16*(4*(x^3 - x^2)^(1/3)*((4*sqrt(3)*2^(1/3) + 7*2^(1/3))*sqr
t(-4*sqrt(3) + 7) - 3*sqrt(3)*2^(1/3) - 6*2^(1/3))*(-4*sqrt(3) + 8)^(5/6)*cos(2/3*arctan((4*sqrt(3) + 7)*sqrt(
-4*sqrt(3) + 8)*sqrt(-4*sqrt(3) + 7) - (4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 7)))^2 - 6*(x^3 - x^2)^(1/3)*((4*sqrt
(3)*2^(1/3) + 7*2^(1/3))*sqrt(-4*sqrt(3) + 7) - sqrt(3)*2^(1/3) - 2*2^(1/3))*(-4*sqrt(3) + 8)^(5/6) - 8*(sqrt(
3)*x + (2*sqrt(3)*x + 3*x)*sqrt(-4*sqrt(3) + 7))*cos(2/3*arctan((4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 8)*sqrt(-4*s
qrt(3) + 7) - (4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 7))) + 4*((x^3 - x^2)^(1/3)*((7*sqrt(3)*2^(1/3) + 12*2^(1/3))*
sqrt(-4*sqrt(3) + 7) + 2*sqrt(3)*2^(1/3) + 3*2^(1/3))*(-4*sqrt(3) + 8)^(5/6)*cos(2/3*arctan((4*sqrt(3) + 7)*sq
rt(-4*sqrt(3) + 8)*sqrt(-4*sqrt(3) + 7) - (4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 7))) + 16*x*cos(2/3*arctan((4*sqrt
(3) + 7)*sqrt(-4*sqrt(3) + 8)*sqrt(-4*sqrt(3) + 7) - (4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 7)))^2 + 6*(sqrt(3)*x +
2*x)*sqrt(-4*sqrt(3) + 7) - 6*x)*sin(2/3*arctan((4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 8)*sqrt(-4*sqrt(3) + 7) - (
4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 7))) + (2*(3*sqrt(3)*2^(1/3)*x - (4*sqrt(3)*2^(1/3)*x + 7*2^(1/3)*x)*sqrt(-4*
sqrt(3) + 7) + 6*2^(1/3)*x)*(-4*sqrt(3) + 8)^(5/6)*cos(2/3*arctan((4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 8)*sqrt(-4
*sqrt(3) + 7) - (4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 7)))^2 - 2*(2*sqrt(3)*2^(1/3)*x + (7*sqrt(3)*2^(1/3)*x + 12*
2^(1/3)*x)*sqrt(-4*sqrt(3) + 7) + 3*2^(1/3)*x)*(-4*sqrt(3) + 8)^(5/6)*cos(2/3*arctan((4*sqrt(3) + 7)*sqrt(-4*s
qrt(3) + 8)*sqrt(-4*sqrt(3) + 7) - (4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 7)))*sin(2/3*arctan((4*sqrt(3) + 7)*sqrt(
-4*sqrt(3) + 8)*sqrt(-4*sqrt(3) + 7) - (4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 7))) - 3*(sqrt(3)*2^(1/3)*x - (4*sqrt
(3)*2^(1/3)*x + 7*2^(1/3)*x)*sqrt(-4*sqrt(3) + 7) + 2*2^(1/3)*x)*(-4*sqrt(3) + 8)^(5/6))*sqrt((2*2^(1/3)*x^2*(
-4*sqrt(3) + 8)^(1/3) + (x^3 - x^2)^(1/3)*(sqrt(3)*2^(2/3)*x + (2*sqrt(3)*2^(2/3)*x + 3*2^(2/3)*x)*sqrt(-4*sqr
t(3) + 7))*(-4*sqrt(3) + 8)^(1/6)*cos(2/3*arctan((4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 8)*sqrt(-4*sqrt(3) + 7) - (
4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 7))) - (x^3 - x^2)^(1/3)*(3*2^(2/3)*x - (sqrt(3)*2^(2/3)*x + 2*2^(2/3)*x)*sqr
t(-4*sqrt(3) + 7))*(-4*sqrt(3) + 8)^(1/6)*sin(2/3*arctan((4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 8)*sqrt(-4*sqrt(3)
+ 7) - (4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 7))) + 4*(x^3 - x^2)^(2/3))/x^2))/(4*x*cos(2/3*arctan((4*sqrt(3) + 7)
*sqrt(-4*sqrt(3) + 8)*sqrt(-4*sqrt(3) + 7) - (4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 7)))^3 - 3*x*cos(2/3*arctan((4*
sqrt(3) + 7)*sqrt(-4*sqrt(3) + 8)*sqrt(-4*sqrt(3) + 7) - (4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 7))))) + 1/6*(sqrt(
3)*2^(2/3)*(4*sqrt(3) + 8)^(1/6)*cos(2/3*arctan(2*sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)*sqrt(sqrt(3) + 2) - sqrt
(4*sqrt(3) + 7)*(4*sqrt(3) - 7))) - 2^(2/3)*(4*sqrt(3) + 8)^(1/6)*sin(2/3*arctan(2*sqrt(4*sqrt(3) + 7)*(4*sqrt
(3) - 7)*sqrt(sqrt(3) + 2) - sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7))))*arctan(-1/16*(4*((x^3 - x^2)^(1/3)*(4*sqrt
(3)*2^(1/3) - 7*2^(1/3))*sqrt(4*sqrt(3) + 7) + 3*(x^3 - x^2)^(1/3)*(sqrt(3)*2^(1/3) - 2*2^(1/3)))*(4*sqrt(3) +
8)^(5/6)*cos(2/3*arctan(2*sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)*sqrt(sqrt(3) + 2) - sqrt(4*sqrt(3) + 7)*(4*sqrt
(3) - 7)))^2 + 8*(sqrt(3)*x - (2*sqrt(3)*x - 3*x)*sqrt(4*sqrt(3) + 7))*cos(2/3*arctan(2*sqrt(4*sqrt(3) + 7)*(4
*sqrt(3) - 7)*sqrt(sqrt(3) + 2) - sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7))) + 4*(16*x*cos(2/3*arctan(2*sqrt(4*sqrt
(3) + 7)*(4*sqrt(3) - 7)*sqrt(sqrt(3) + 2) - sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)))^2 + ((x^3 - x^2)^(1/3)*(7*s
qrt(3)*2^(1/3) - 12*2^(1/3))*sqrt(4*sqrt(3) + 7) - (x^3 - x^2)^(1/3)*(2*sqrt(3)*2^(1/3) - 3*2^(1/3)))*(4*sqrt(
3) + 8)^(5/6)*cos(2/3*arctan(2*sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)*sqrt(sqrt(3) + 2) - sqrt(4*sqrt(3) + 7)*(4*
sqrt(3) - 7))) + 6*(sqrt(3)*x - 2*x)*sqrt(4*sqrt(3) + 7) - 6*x)*sin(2/3*arctan(2*sqrt(4*sqrt(3) + 7)*(4*sqrt(3
) - 7)*sqrt(sqrt(3) + 2) - sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7))) - 6*((x^3 - x^2)^(1/3)*(4*sqrt(3)*2^(1/3) - 7
*2^(1/3))*sqrt(4*sqrt(3) + 7) + (x^3 - x^2)^(1/3)*(sqrt(3)*2^(1/3) - 2*2^(1/3)))*(4*sqrt(3) + 8)^(5/6) - (2*(3
*sqrt(3)*2^(1/3)*x + (4*sqrt(3)*2^(1/3)*x - 7*2^(1/3)*x)*sqrt(4*sqrt(3) + 7) - 6*2^(1/3)*x)*(4*sqrt(3) + 8)^(5
/6)*cos(2/3*arctan(2*sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)*sqrt(sqrt(3) + 2) - sqrt(4*sqrt(3) + 7)*(4*sqrt(3) -
7)))^2 - 2*(2*sqrt(3)*2^(1/3)*x - (7*sqrt(3)*2^(1/3)*x - 12*2^(1/3)*x)*sqrt(4*sqrt(3) + 7) - 3*2^(1/3)*x)*(4*s
qrt(3) + 8)^(5/6)*cos(2/3*arctan(2*sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)*sqrt(sqrt(3) + 2) - sqrt(4*sqrt(3) + 7)
*(4*sqrt(3) - 7)))*sin(2/3*arctan(2*sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)*sqrt(sqrt(3) + 2) - sqrt(4*sqrt(3) + 7
)*(4*sqrt(3) - 7))) - 3*(sqrt(3)*2^(1/3)*x + (4*sqrt(3)*2^(1/3)*x - 7*2^(1/3)*x)*sqrt(4*sqrt(3) + 7) - 2*2^(1/
3)*x)*(4*sqrt(3) + 8)^(5/6))*sqrt((2*2^(1/3)*x^2*(4*sqrt(3) + 8)^(1/3) - (sqrt(3)*2^(2/3)*(x^3 - x^2)^(1/3)*x
- (x^3 - x^2)^(1/3)*(2*sqrt(3)*2^(2/3)*x - 3*2^(2/3)*x)*sqrt(4*sqrt(3) + 7))*(4*sqrt(3) + 8)^(1/6)*cos(2/3*arc
tan(2*sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)*sqrt(sqrt(3) + 2) - sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7))) - (3*2^(2/
3)*(x^3 - x^2)^(1/3)*x - (x^3 - x^2)^(1/3)*(sqrt(3)*2^(2/3)*x - 2*2^(2/3)*x)*sqrt(4*sqrt(3) + 7))*(4*sqrt(3) +
8)^(1/6)*sin(2/3*arctan(2*sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)*sqrt(sqrt(3) + 2) - sqrt(4*sqrt(3) + 7)*(4*sqrt
(3) - 7))) + 4*(x^3 - x^2)^(2/3))/x^2))/(4*x*cos(2/3*arctan(2*sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)*sqrt(sqrt(3)
+ 2) - sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)))^3 - 3*x*cos(2/3*arctan(2*sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)*sqr
t(sqrt(3) + 2) - sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7))))) + 1/6*(sqrt(3)*2^(2/3)*(4*sqrt(3) + 8)^(1/6)*cos(2/3*
arctan(2*sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)*sqrt(sqrt(3) + 2) - sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7))) + 2^(2/
3)*(4*sqrt(3) + 8)^(1/6)*sin(2/3*arctan(2*sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)*sqrt(sqrt(3) + 2) - sqrt(4*sqrt(
3) + 7)*(4*sqrt(3) - 7))))*arctan(1/16*(4*((x^3 - x^2)^(1/3)*(4*sqrt(3)*2^(1/3) - 7*2^(1/3))*sqrt(4*sqrt(3) +
7) - 3*(x^3 - x^2)^(1/3)*(sqrt(3)*2^(1/3) - 2*2^(1/3)))*(4*sqrt(3) + 8)^(5/6)*cos(2/3*arctan(2*sqrt(4*sqrt(3)
+ 7)*(4*sqrt(3) - 7)*sqrt(sqrt(3) + 2) - sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)))^2 - 8*(sqrt(3)*x + (2*sqrt(3)*x
- 3*x)*sqrt(4*sqrt(3) + 7))*cos(2/3*arctan(2*sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)*sqrt(sqrt(3) + 2) - sqrt(4*s
qrt(3) + 7)*(4*sqrt(3) - 7))) + 4*(16*x*cos(2/3*arctan(2*sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)*sqrt(sqrt(3) + 2)
- sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)))^2 - ((x^3 - x^2)^(1/3)*(7*sqrt(3)*2^(1/3) - 12*2^(1/3))*sqrt(4*sqrt(3
) + 7) + (x^3 - x^2)^(1/3)*(2*sqrt(3)*2^(1/3) - 3*2^(1/3)))*(4*sqrt(3) + 8)^(5/6)*cos(2/3*arctan(2*sqrt(4*sqrt
(3) + 7)*(4*sqrt(3) - 7)*sqrt(sqrt(3) + 2) - sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7))) - 6*(sqrt(3)*x - 2*x)*sqrt(
4*sqrt(3) + 7) - 6*x)*sin(2/3*arctan(2*sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)*sqrt(sqrt(3) + 2) - sqrt(4*sqrt(3)
+ 7)*(4*sqrt(3) - 7))) - 6*((x^3 - x^2)^(1/3)*(4*sqrt(3)*2^(1/3) - 7*2^(1/3))*sqrt(4*sqrt(3) + 7) - (x^3 - x^2
)^(1/3)*(sqrt(3)*2^(1/3) - 2*2^(1/3)))*(4*sqrt(3) + 8)^(5/6) + (2*(3*sqrt(3)*2^(1/3)*x - (4*sqrt(3)*2^(1/3)*x
- 7*2^(1/3)*x)*sqrt(4*sqrt(3) + 7) - 6*2^(1/3)*x)*(4*sqrt(3) + 8)^(5/6)*cos(2/3*arctan(2*sqrt(4*sqrt(3) + 7)*(
4*sqrt(3) - 7)*sqrt(sqrt(3) + 2) - sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)))^2 + 2*(2*sqrt(3)*2^(1/3)*x + (7*sqrt(
3)*2^(1/3)*x - 12*2^(1/3)*x)*sqrt(4*sqrt(3) + 7) - 3*2^(1/3)*x)*(4*sqrt(3) + 8)^(5/6)*cos(2/3*arctan(2*sqrt(4*
sqrt(3) + 7)*(4*sqrt(3) - 7)*sqrt(sqrt(3) + 2) - sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)))*sin(2/3*arctan(2*sqrt(4
*sqrt(3) + 7)*(4*sqrt(3) - 7)*sqrt(sqrt(3) + 2) - sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7))) - 3*(sqrt(3)*2^(1/3)*x
- (4*sqrt(3)*2^(1/3)*x - 7*2^(1/3)*x)*sqrt(4*sqrt(3) + 7) - 2*2^(1/3)*x)*(4*sqrt(3) + 8)^(5/6))*sqrt((2*2^(1/
3)*x^2*(4*sqrt(3) + 8)^(1/3) - (sqrt(3)*2^(2/3)*(x^3 - x^2)^(1/3)*x + (x^3 - x^2)^(1/3)*(2*sqrt(3)*2^(2/3)*x -
3*2^(2/3)*x)*sqrt(4*sqrt(3) + 7))*(4*sqrt(3) + 8)^(1/6)*cos(2/3*arctan(2*sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)*
sqrt(sqrt(3) + 2) - sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7))) + (3*2^(2/3)*(x^3 - x^2)^(1/3)*x + (x^3 - x^2)^(1/3)
*(sqrt(3)*2^(2/3)*x - 2*2^(2/3)*x)*sqrt(4*sqrt(3) + 7))*(4*sqrt(3) + 8)^(1/6)*sin(2/3*arctan(2*sqrt(4*sqrt(3)
+ 7)*(4*sqrt(3) - 7)*sqrt(sqrt(3) + 2) - sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7))) + 4*(x^3 - x^2)^(2/3))/x^2))/(4
*x*cos(2/3*arctan(2*sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)*sqrt(sqrt(3) + 2) - sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7
)))^3 - 3*x*cos(2/3*arctan(2*sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)*sqrt(sqrt(3) + 2) - sqrt(4*sqrt(3) + 7)*(4*sq
rt(3) - 7))))) + 1/3*(sqrt(3)*2^(1/6)*cos(2/3*arctan(sqrt(2) + 1)) - 2^(1/6)*sin(2/3*arctan(sqrt(2) + 1)))*arc
tan((2^(5/6)*(x^3 - x^2)^(1/3)*cos(2/3*arctan(sqrt(2) + 1)) - (4*x*cos(2/3*arctan(sqrt(2) + 1)) + sqrt(3)*2^(5
/6)*(x^3 - x^2)^(1/3))*sin(2/3*arctan(sqrt(2) + 1)) + sqrt(3)*x + (sqrt(3)*2^(5/6)*x*sin(2/3*arctan(sqrt(2) +
1)) - 2^(5/6)*x*cos(2/3*arctan(sqrt(2) + 1)))*sqrt((sqrt(3)*2^(1/6)*(x^3 - x^2)^(1/3)*x*cos(2/3*arctan(sqrt(2)
+ 1)) - 2^(1/6)*(x^3 - x^2)^(1/3)*x*sin(2/3*arctan(sqrt(2) + 1)) + 2^(1/3)*x^2 + (x^3 - x^2)^(2/3))/x^2))/(4*
x*cos(2/3*arctan(sqrt(2) + 1))^2 - 3*x)) + 1/3*(sqrt(3)*2^(1/6)*cos(2/3*arctan(sqrt(2) + 1)) + 2^(1/6)*sin(2/3
*arctan(sqrt(2) + 1)))*arctan(-(2^(5/6)*(x^3 - x^2)^(1/3)*cos(2/3*arctan(sqrt(2) + 1)) - (4*x*cos(2/3*arctan(s
qrt(2) + 1)) - sqrt(3)*2^(5/6)*(x^3 - x^2)^(1/3))*sin(2/3*arctan(sqrt(2) + 1)) - sqrt(3)*x - (sqrt(3)*2^(5/6)*
x*sin(2/3*arctan(sqrt(2) + 1)) + 2^(5/6)*x*cos(2/3*arctan(sqrt(2) + 1)))*sqrt(-(sqrt(3)*2^(1/6)*(x^3 - x^2)^(1
/3)*x*cos(2/3*arctan(sqrt(2) + 1)) + 2^(1/6)*(x^3 - x^2)^(1/3)*x*sin(2/3*arctan(sqrt(2) + 1)) - 2^(1/3)*x^2 -
(x^3 - x^2)^(2/3))/x^2))/(4*x*cos(2/3*arctan(sqrt(2) + 1))^2 - 3*x)) + 1/12*(sqrt(3)*2^(1/6)*sin(2/3*arctan(sq
rt(2) + 1)) - 2^(1/6)*cos(2/3*arctan(sqrt(2) + 1)))*log(-4*(sqrt(3)*2^(1/6)*(x^3 - x^2)^(1/3)*x*cos(2/3*arctan
(sqrt(2) + 1)) + 2^(1/6)*(x^3 - x^2)^(1/3)*x*sin(2/3*arctan(sqrt(2) + 1)) - 2^(1/3)*x^2 - (x^3 - x^2)^(2/3))/x
^2) - 1/12*(sqrt(3)*2^(1/6)*sin(2/3*arctan(sqrt(2) + 1)) + 2^(1/6)*cos(2/3*arctan(sqrt(2) + 1)))*log(4*(sqrt(3
)*2^(1/6)*(x^3 - x^2)^(1/3)*x*cos(2/3*arctan(sqrt(2) + 1)) - 2^(1/6)*(x^3 - x^2)^(1/3)*x*sin(2/3*arctan(sqrt(2
) + 1)) + 2^(1/3)*x^2 + (x^3 - x^2)^(2/3))/x^2) + 1/24*(sqrt(3)*2^(2/3)*(4*sqrt(3) + 8)^(1/6)*sin(2/3*arctan(2
*sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)*sqrt(sqrt(3) + 2) - sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7))) - 2^(2/3)*(4*sq
rt(3) + 8)^(1/6)*cos(2/3*arctan(2*sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)*sqrt(sqrt(3) + 2) - sqrt(4*sqrt(3) + 7)*
(4*sqrt(3) - 7))))*log(16*(2*2^(1/3)*x^2*(4*sqrt(3) + 8)^(1/3) - (sqrt(3)*2^(2/3)*(x^3 - x^2)^(1/3)*x + (x^3 -
x^2)^(1/3)*(2*sqrt(3)*2^(2/3)*x - 3*2^(2/3)*x)*sqrt(4*sqrt(3) + 7))*(4*sqrt(3) + 8)^(1/6)*cos(2/3*arctan(2*sq
rt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)*sqrt(sqrt(3) + 2) - sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7))) + (3*2^(2/3)*(x^3
- x^2)^(1/3)*x + (x^3 - x^2)^(1/3)*(sqrt(3)*2^(2/3)*x - 2*2^(2/3)*x)*sqrt(4*sqrt(3) + 7))*(4*sqrt(3) + 8)^(1/6
)*sin(2/3*arctan(2*sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)*sqrt(sqrt(3) + 2) - sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)
)) + 4*(x^3 - x^2)^(2/3))/x^2) - 1/24*(sqrt(3)*2^(2/3)*(4*sqrt(3) + 8)^(1/6)*sin(2/3*arctan(2*sqrt(4*sqrt(3) +
7)*(4*sqrt(3) - 7)*sqrt(sqrt(3) + 2) - sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7))) + 2^(2/3)*(4*sqrt(3) + 8)^(1/6)*
cos(2/3*arctan(2*sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)*sqrt(sqrt(3) + 2) - sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)))
)*log(16*(2*2^(1/3)*x^2*(4*sqrt(3) + 8)^(1/3) - (sqrt(3)*2^(2/3)*(x^3 - x^2)^(1/3)*x - (x^3 - x^2)^(1/3)*(2*sq
rt(3)*2^(2/3)*x - 3*2^(2/3)*x)*sqrt(4*sqrt(3) + 7))*(4*sqrt(3) + 8)^(1/6)*cos(2/3*arctan(2*sqrt(4*sqrt(3) + 7)
*(4*sqrt(3) - 7)*sqrt(sqrt(3) + 2) - sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7))) - (3*2^(2/3)*(x^3 - x^2)^(1/3)*x -
(x^3 - x^2)^(1/3)*(sqrt(3)*2^(2/3)*x - 2*2^(2/3)*x)*sqrt(4*sqrt(3) + 7))*(4*sqrt(3) + 8)^(1/6)*sin(2/3*arctan(
2*sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7)*sqrt(sqrt(3) + 2) - sqrt(4*sqrt(3) + 7)*(4*sqrt(3) - 7))) + 4*(x^3 - x^2
)^(2/3))/x^2) + 1/24*(sqrt(3)*2^(2/3)*(-4*sqrt(3) + 8)^(1/6)*sin(2/3*arctan((4*sqrt(3) + 7)*sqrt(-4*sqrt(3) +
8)*sqrt(-4*sqrt(3) + 7) - (4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 7))) - 2^(2/3)*(-4*sqrt(3) + 8)^(1/6)*cos(2/3*arct
an((4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 8)*sqrt(-4*sqrt(3) + 7) - (4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 7))))*log(16*
(2*2^(1/3)*x^2*(-4*sqrt(3) + 8)^(1/3) + (x^3 - x^2)^(1/3)*(sqrt(3)*2^(2/3)*x + (2*sqrt(3)*2^(2/3)*x + 3*2^(2/3
)*x)*sqrt(-4*sqrt(3) + 7))*(-4*sqrt(3) + 8)^(1/6)*cos(2/3*arctan((4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 8)*sqrt(-4*
sqrt(3) + 7) - (4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 7))) - (x^3 - x^2)^(1/3)*(3*2^(2/3)*x - (sqrt(3)*2^(2/3)*x +
2*2^(2/3)*x)*sqrt(-4*sqrt(3) + 7))*(-4*sqrt(3) + 8)^(1/6)*sin(2/3*arctan((4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 8)*
sqrt(-4*sqrt(3) + 7) - (4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 7))) + 4*(x^3 - x^2)^(2/3))/x^2) - 1/24*(sqrt(3)*2^(2
/3)*(-4*sqrt(3) + 8)^(1/6)*sin(2/3*arctan((4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 8)*sqrt(-4*sqrt(3) + 7) - (4*sqrt(
3) + 7)*sqrt(-4*sqrt(3) + 7))) + 2^(2/3)*(-4*sqrt(3) + 8)^(1/6)*cos(2/3*arctan((4*sqrt(3) + 7)*sqrt(-4*sqrt(3)
+ 8)*sqrt(-4*sqrt(3) + 7) - (4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 7))))*log(16*(2*2^(1/3)*x^2*(-4*sqrt(3) + 8)^(1
/3) + (x^3 - x^2)^(1/3)*(sqrt(3)*2^(2/3)*x - (2*sqrt(3)*2^(2/3)*x + 3*2^(2/3)*x)*sqrt(-4*sqrt(3) + 7))*(-4*sqr
t(3) + 8)^(1/6)*cos(2/3*arctan((4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 8)*sqrt(-4*sqrt(3) + 7) - (4*sqrt(3) + 7)*sqr
t(-4*sqrt(3) + 7))) + (x^3 - x^2)^(1/3)*(3*2^(2/3)*x + (sqrt(3)*2^(2/3)*x + 2*2^(2/3)*x)*sqrt(-4*sqrt(3) + 7))
*(-4*sqrt(3) + 8)^(1/6)*sin(2/3*arctan((4*sqrt(3) + 7)*sqrt(-4*sqrt(3) + 8)*sqrt(-4*sqrt(3) + 7) - (4*sqrt(3)
+ 7)*sqrt(-4*sqrt(3) + 7))) + 4*(x^3 - x^2)^(2/3))/x^2)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} x^{3}}{x^{6} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3*(x^3-x^2)^(1/3)/(x^6+1),x, algorithm="giac")
[Out]
integrate((x^3 - x^2)^(1/3)*x^3/(x^6 + 1), x)
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maple [B] time = 173.96, size = 84511, normalized size = 402.43
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| method |
result |
size |
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| trager |
\(\text {Expression too large to display}\) |
\(84511\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^3*(x^3-x^2)^(1/3)/(x^6+1),x,method=_RETURNVERBOSE)
[Out]
result too large to display
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} x^{3}}{x^{6} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3*(x^3-x^2)^(1/3)/(x^6+1),x, algorithm="maxima")
[Out]
integrate((x^3 - x^2)^(1/3)*x^3/(x^6 + 1), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^3\,{\left (x^3-x^2\right )}^{1/3}}{x^6+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^3*(x^3 - x^2)^(1/3))/(x^6 + 1),x)
[Out]
int((x^3*(x^3 - x^2)^(1/3))/(x^6 + 1), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} \sqrt [3]{x^{2} \left (x - 1\right )}}{\left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**3*(x**3-x**2)**(1/3)/(x**6+1),x)
[Out]
Integral(x**3*(x**2*(x - 1))**(1/3)/((x**2 + 1)*(x**4 - x**2 + 1)), x)
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