\(\int \frac {x^2 (-2+x^3) \sqrt {1+x^3}}{4+12 x^3+13 x^6+4 x^9} \, dx\) [203]
Optimal result
Integrand size = 37, antiderivative size = 21 \[
\int \frac {x^2 \left (-2+x^3\right ) \sqrt {1+x^3}}{4+12 x^3+13 x^6+4 x^9} \, dx=-\frac {1}{3} \arctan \left (\frac {x^3}{2 \left (1+x^3\right )^{3/2}}\right )
\]
[Out]
-1/3*arctan(1/2*x^3/(x^3+1)^(3/2))
Rubi [A] (verified)
Time = 0.34 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90, number of
steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {6847, 2119, 209}
\[
\int \frac {x^2 \left (-2+x^3\right ) \sqrt {1+x^3}}{4+12 x^3+13 x^6+4 x^9} \, dx=\frac {1}{3} \arctan \left (\frac {2 \left (x^3+1\right )^{3/2}}{x^3}\right )
\]
[In]
Int[(x^2*(-2 + x^3)*Sqrt[1 + x^3])/(4 + 12*x^3 + 13*x^6 + 4*x^9),x]
[Out]
ArcTan[(2*(1 + x^3)^(3/2))/x^3]/3
Rule 209
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])
Rule 2119
Int[((x_)^(m_.)*((A_) + (B_.)*(x_)^(n_.)))/((a_) + (b_.)*(x_)^(k_.) + (c_.)*(x_)^(n_.) + (d_.)*(x_)^(n2_)), x_
Symbol] :> Dist[A^2*((m - n + 1)/(m + 1)), Subst[Int[1/(a + A^2*b*(m - n + 1)^2*x^2), x], x, x^(m + 1)/(A*(m -
n + 1) + B*(m + 1)*x^n)], x] /; FreeQ[{a, b, c, d, A, B, m, n}, x] && EqQ[n2, 2*n] && EqQ[k, 2*(m + 1)] && Eq
Q[a*B^2*(m + 1)^2 - A^2*d*(m - n + 1)^2, 0] && EqQ[B*c*(m + 1) - 2*A*d*(m - n + 1), 0]
Rule 6847
Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]
Rubi steps \begin{align*}
\text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {(-2+x) \sqrt {1+x}}{4+12 x+13 x^2+4 x^3} \, dx,x,x^3\right ) \\ & = \frac {2}{3} \text {Subst}\left (\int \frac {x^2 \left (-3+x^2\right )}{1-2 x^2+x^4+4 x^6} \, dx,x,\sqrt {1+x^3}\right ) \\ & = 2 \text {Subst}\left (\int \frac {1}{1+36 x^2} \, dx,x,\frac {\left (1+x^3\right )^{3/2}}{3 x^3}\right ) \\ & = \frac {1}{3} \arctan \left (\frac {2 \left (1+x^3\right )^{3/2}}{x^3}\right ) \\
\end{align*}
Mathematica [A] (verified)
Time = 0.04 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.76
\[
\int \frac {x^2 \left (-2+x^3\right ) \sqrt {1+x^3}}{4+12 x^3+13 x^6+4 x^9} \, dx=\frac {1}{3} \arctan \left (\sqrt {1+x^3}\right )-\frac {1}{3} \arctan \left (\frac {1+2 x^3}{\sqrt {1+x^3}}\right )
\]
[In]
Integrate[(x^2*(-2 + x^3)*Sqrt[1 + x^3])/(4 + 12*x^3 + 13*x^6 + 4*x^9),x]
[Out]
ArcTan[Sqrt[1 + x^3]]/3 - ArcTan[(1 + 2*x^3)/Sqrt[1 + x^3]]/3
Maple [B] (verified)
Leaf count of result is larger than twice the leaf count of optimal. \(45\) vs. \(2(15)=30\).
Time = 7.12 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.19
| | |
| method | result | size |
| | |
| default |
\(\frac {\arctan \left (\sqrt {x^{3}+1}\right )}{3}-\frac {\arctan \left (4 \sqrt {x^{3}+1}-\sqrt {7}\right )}{3}-\frac {\arctan \left (4 \sqrt {x^{3}+1}+\sqrt {7}\right )}{3}\) |
\(46\) |
| pseudoelliptic |
\(-\frac {\arctan \left (\frac {3 \sqrt {x^{3}+1}+\sqrt {7}}{4 x^{3}+5+\sqrt {x^{3}+1}\, \sqrt {7}}\right )}{3}-\frac {\arctan \left (4 \sqrt {x^{3}+1}-\sqrt {7}\right )}{3}\) |
\(57\) |
| trager |
\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{9}+4 \sqrt {x^{3}+1}\, x^{6}-11 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{6}+4 \sqrt {x^{3}+1}\, x^{3}-12 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{\left (4 x^{6}+5 x^{3}+2\right ) \left (x^{3}+2\right )}\right )}{6}\) |
\(99\) |
| elliptic |
\(\text {Expression too large to display}\) |
\(2469\) |
| | |
|
|
|
|
[In]
int(x^2*(x^3-2)*(x^3+1)^(1/2)/(4*x^9+13*x^6+12*x^3+4),x,method=_RETURNVERBOSE)
[Out]
1/3*arctan((x^3+1)^(1/2))-1/3*arctan(4*(x^3+1)^(1/2)-7^(1/2))-1/3*arctan(4*(x^3+1)^(1/2)+7^(1/2))
Fricas [A] (verification not implemented)
none
Time = 0.24 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71
\[
\int \frac {x^2 \left (-2+x^3\right ) \sqrt {1+x^3}}{4+12 x^3+13 x^6+4 x^9} \, dx=\frac {1}{3} \, \arctan \left (\frac {2 \, {\left (x^{3} + 1\right )}^{\frac {3}{2}}}{x^{3}}\right )
\]
[In]
integrate(x^2*(x^3-2)*(x^3+1)^(1/2)/(4*x^9+13*x^6+12*x^3+4),x, algorithm="fricas")
[Out]
1/3*arctan(2*(x^3 + 1)^(3/2)/x^3)
Sympy [F(-1)]
Timed out. \[
\int \frac {x^2 \left (-2+x^3\right ) \sqrt {1+x^3}}{4+12 x^3+13 x^6+4 x^9} \, dx=\text {Timed out}
\]
[In]
integrate(x**2*(x**3-2)*(x**3+1)**(1/2)/(4*x**9+13*x**6+12*x**3+4),x)
[Out]
Timed out
Maxima [F]
\[
\int \frac {x^2 \left (-2+x^3\right ) \sqrt {1+x^3}}{4+12 x^3+13 x^6+4 x^9} \, dx=\int { \frac {\sqrt {x^{3} + 1} {\left (x^{3} - 2\right )} x^{2}}{4 \, x^{9} + 13 \, x^{6} + 12 \, x^{3} + 4} \,d x }
\]
[In]
integrate(x^2*(x^3-2)*(x^3+1)^(1/2)/(4*x^9+13*x^6+12*x^3+4),x, algorithm="maxima")
[Out]
integrate(sqrt(x^3 + 1)*(x^3 - 2)*x^2/(4*x^9 + 13*x^6 + 12*x^3 + 4), x)
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (15) = 30\).
Time = 0.28 (sec) , antiderivative size = 67, normalized size of antiderivative = 3.19
\[
\int \frac {x^2 \left (-2+x^3\right ) \sqrt {1+x^3}}{4+12 x^3+13 x^6+4 x^9} \, dx=-\frac {1}{3} \, \arctan \left (2 \, \sqrt {2} \left (\frac {1}{4}\right )^{\frac {3}{4}} {\left (\sqrt {14} \left (\frac {1}{4}\right )^{\frac {1}{4}} + 4 \, \sqrt {x^{3} + 1}\right )}\right ) - \frac {1}{3} \, \arctan \left (-2 \, \sqrt {2} \left (\frac {1}{4}\right )^{\frac {3}{4}} {\left (\sqrt {14} \left (\frac {1}{4}\right )^{\frac {1}{4}} - 4 \, \sqrt {x^{3} + 1}\right )}\right ) + \frac {1}{3} \, \arctan \left (\sqrt {x^{3} + 1}\right )
\]
[In]
integrate(x^2*(x^3-2)*(x^3+1)^(1/2)/(4*x^9+13*x^6+12*x^3+4),x, algorithm="giac")
[Out]
-1/3*arctan(2*sqrt(2)*(1/4)^(3/4)*(sqrt(14)*(1/4)^(1/4) + 4*sqrt(x^3 + 1))) - 1/3*arctan(-2*sqrt(2)*(1/4)^(3/4
)*(sqrt(14)*(1/4)^(1/4) - 4*sqrt(x^3 + 1))) + 1/3*arctan(sqrt(x^3 + 1))
Mupad [B] (verification not implemented)
Time = 0.12 (sec) , antiderivative size = 2737, normalized size of antiderivative = 130.33
\[
\int \frac {x^2 \left (-2+x^3\right ) \sqrt {1+x^3}}{4+12 x^3+13 x^6+4 x^9} \, dx=\text {Too large to display}
\]
[In]
int((x^2*(x^3 + 1)^(1/2)*(x^3 - 2))/(12*x^3 + 13*x^6 + 4*x^9 + 4),x)
[Out]
(2*((3^(1/2)*1i)/2 + 3/2)*((x + (3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2)*((x + 1)/((3^(1/2)*1i)/2 +
3/2))^(1/2)*(((3^(1/2)*1i)/2 - x + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*ellipticPi(((3^(1/2)*1i)/2 + 3/2)/((- (
7^(1/2)*1i)/8 - 5/8)^(1/3) + 1), asin(((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/
2)*1i)/2 - 3/2))*(2*(- (7^(1/2)*1i)/8 - 5/8)^(2/3) + (- (7^(1/2)*1i)/8 - 5/8)^(5/3) - (- (7^(1/2)*1i)/8 - 5/8)
^(8/3)))/(((- (7^(1/2)*1i)/8 - 5/8)^(1/3) + 1)*(x^3 - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) -
((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2)*(36*(- (7^(1/2)*1i)/8 - 5/8)^(2/3) + 78*(- (7^(1/2)*1i)/8
- 5/8)^(5/3) + 36*(- (7^(1/2)*1i)/8 - 5/8)^(8/3))) - (((3^(1/2)*1i)/2 + 3/2)*((x + (3^(1/2)*1i)/2 - 1/2)/((3^
(1/2)*1i)/2 - 3/2))^(1/2)*((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(((3^(1/2)*1i)/2 - x + 1/2)/((3^(1/2)*1i)/2 +
3/2))^(1/2)*ellipticPi(((3^(1/2)*1i)/2 + 3/2)/((-1)^(1/3)*2^(1/3) + 1), asin(((x + 1)/((3^(1/2)*1i)/2 + 3/2))
^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2)))/(3*((-1)^(1/3)*2^(1/3) + 1)*(x^3 - x*(((3^(1/2)*1i)/
2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) - ((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2)) + (2*((3^(1/2)*1i
)/2 + 3/2)*((x + (3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2)*((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*((
(3^(1/2)*1i)/2 - x + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*ellipticPi(((3^(1/2)*1i)/2 + 3/2)/(((7^(1/2)*1i)/8 - 5
/8)^(1/3) + 1), asin(((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2))*
(2*((7^(1/2)*1i)/8 - 5/8)^(2/3) + ((7^(1/2)*1i)/8 - 5/8)^(5/3) - ((7^(1/2)*1i)/8 - 5/8)^(8/3)))/((((7^(1/2)*1i
)/8 - 5/8)^(1/3) + 1)*(x^3 - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) - ((3^(1/2)*1i)/2 - 1/2)*((
3^(1/2)*1i)/2 + 1/2))^(1/2)*(36*((7^(1/2)*1i)/8 - 5/8)^(2/3) + 78*((7^(1/2)*1i)/8 - 5/8)^(5/3) + 36*((7^(1/2)*
1i)/8 - 5/8)^(8/3))) + (2*((3^(1/2)*1i)/2 + 3/2)*((x + (3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2)*((x
+ 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(((3^(1/2)*1i)/2 - x + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*ellipticPi(((3^(
1/2)*1i)/2 + 3/2)/(((3^(1/2)*1i)/2 - 1/2)*(- (7^(1/2)*1i)/8 - 5/8)^(1/3) + 1), asin(((x + 1)/((3^(1/2)*1i)/2 +
3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2))*(2*((3^(1/2)*1i)/2 - 1/2)^2*(- (7^(1/2)*1i)/8 -
5/8)^(2/3) + ((3^(1/2)*1i)/2 - 1/2)^5*(- (7^(1/2)*1i)/8 - 5/8)^(5/3) - ((3^(1/2)*1i)/2 - 1/2)^8*(- (7^(1/2)*1
i)/8 - 5/8)^(8/3)))/((((3^(1/2)*1i)/2 - 1/2)*(- (7^(1/2)*1i)/8 - 5/8)^(1/3) + 1)*(x^3 - x*(((3^(1/2)*1i)/2 - 1
/2)*((3^(1/2)*1i)/2 + 1/2) + 1) - ((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2)*(36*((3^(1/2)*1i)/2 - 1
/2)^2*(- (7^(1/2)*1i)/8 - 5/8)^(2/3) + 78*((3^(1/2)*1i)/2 - 1/2)^5*(- (7^(1/2)*1i)/8 - 5/8)^(5/3) + 36*((3^(1/
2)*1i)/2 - 1/2)^8*(- (7^(1/2)*1i)/8 - 5/8)^(8/3))) + (2*((3^(1/2)*1i)/2 + 3/2)*((x + (3^(1/2)*1i)/2 - 1/2)/((3
^(1/2)*1i)/2 - 3/2))^(1/2)*((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(((3^(1/2)*1i)/2 - x + 1/2)/((3^(1/2)*1i)/2
+ 3/2))^(1/2)*ellipticPi(((3^(1/2)*1i)/2 + 3/2)/(((3^(1/2)*1i)/2 - 1/2)*((7^(1/2)*1i)/8 - 5/8)^(1/3) + 1), asi
n(((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2))*(2*((3^(1/2)*1i)/2
- 1/2)^2*((7^(1/2)*1i)/8 - 5/8)^(2/3) + ((3^(1/2)*1i)/2 - 1/2)^5*((7^(1/2)*1i)/8 - 5/8)^(5/3) - ((3^(1/2)*1i)/
2 - 1/2)^8*((7^(1/2)*1i)/8 - 5/8)^(8/3)))/((((3^(1/2)*1i)/2 - 1/2)*((7^(1/2)*1i)/8 - 5/8)^(1/3) + 1)*(x^3 - x*
(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) - ((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2)*(36
*((3^(1/2)*1i)/2 - 1/2)^2*((7^(1/2)*1i)/8 - 5/8)^(2/3) + 78*((3^(1/2)*1i)/2 - 1/2)^5*((7^(1/2)*1i)/8 - 5/8)^(5
/3) + 36*((3^(1/2)*1i)/2 - 1/2)^8*((7^(1/2)*1i)/8 - 5/8)^(8/3))) + (2*((3^(1/2)*1i)/2 + 3/2)*((x + (3^(1/2)*1i
)/2 - 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2)*((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(((3^(1/2)*1i)/2 - x + 1/2)/((
3^(1/2)*1i)/2 + 3/2))^(1/2)*ellipticPi(-((3^(1/2)*1i)/2 + 3/2)/(((3^(1/2)*1i)/2 + 1/2)*(- (7^(1/2)*1i)/8 - 5/8
)^(1/3) - 1), asin(((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2))*((
(3^(1/2)*1i)/2 + 1/2)^5*(- (7^(1/2)*1i)/8 - 5/8)^(5/3) - 2*((3^(1/2)*1i)/2 + 1/2)^2*(- (7^(1/2)*1i)/8 - 5/8)^(
2/3) + ((3^(1/2)*1i)/2 + 1/2)^8*(- (7^(1/2)*1i)/8 - 5/8)^(8/3)))/((((3^(1/2)*1i)/2 + 1/2)*(- (7^(1/2)*1i)/8 -
5/8)^(1/3) - 1)*(x^3 - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) - ((3^(1/2)*1i)/2 - 1/2)*((3^(1/2
)*1i)/2 + 1/2))^(1/2)*(36*((3^(1/2)*1i)/2 + 1/2)^2*(- (7^(1/2)*1i)/8 - 5/8)^(2/3) - 78*((3^(1/2)*1i)/2 + 1/2)^
5*(- (7^(1/2)*1i)/8 - 5/8)^(5/3) + 36*((3^(1/2)*1i)/2 + 1/2)^8*(- (7^(1/2)*1i)/8 - 5/8)^(8/3))) + (2*((3^(1/2)
*1i)/2 + 3/2)*((x + (3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2)*((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)
*(((3^(1/2)*1i)/2 - x + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*ellipticPi(-((3^(1/2)*1i)/2 + 3/2)/(((3^(1/2)*1i)/2
+ 1/2)*((7^(1/2)*1i)/8 - 5/8)^(1/3) - 1), asin(((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/
2)/((3^(1/2)*1i)/2 - 3/2))*(((3^(1/2)*1i)/2 + 1/2)^5*((7^(1/2)*1i)/8 - 5/8)^(5/3) - 2*((3^(1/2)*1i)/2 + 1/2)^2
*((7^(1/2)*1i)/8 - 5/8)^(2/3) + ((3^(1/2)*1i)/2 + 1/2)^8*((7^(1/2)*1i)/8 - 5/8)^(8/3)))/((((3^(1/2)*1i)/2 + 1/
2)*((7^(1/2)*1i)/8 - 5/8)^(1/3) - 1)*(x^3 - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) - ((3^(1/2)*
1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2)*(36*((3^(1/2)*1i)/2 + 1/2)^2*((7^(1/2)*1i)/8 - 5/8)^(2/3) - 78*((3^
(1/2)*1i)/2 + 1/2)^5*((7^(1/2)*1i)/8 - 5/8)^(5/3) + 36*((3^(1/2)*1i)/2 + 1/2)^8*((7^(1/2)*1i)/8 - 5/8)^(8/3)))
- (2*((3^(1/2)*1i)/2 + 3/2)*((x + (3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2)*((x + 1)/((3^(1/2)*1i)/
2 + 3/2))^(1/2)*(((3^(1/2)*1i)/2 - x + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(2*(-1)^(2/3)*2^(2/3)*((3^(1/2)*1i)/
2 - 1/2)^5 - 2*(-1)^(2/3)*2^(2/3)*((3^(1/2)*1i)/2 - 1/2)^2 + 4*(-1)^(2/3)*2^(2/3)*((3^(1/2)*1i)/2 - 1/2)^8)*el
lipticPi(((3^(1/2)*1i)/2 + 3/2)/((-1)^(1/3)*2^(1/3)*((3^(1/2)*1i)/2 - 1/2) + 1), asin(((x + 1)/((3^(1/2)*1i)/2
+ 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2)))/(((-1)^(1/3)*2^(1/3)*((3^(1/2)*1i)/2 - 1/2)
+ 1)*(x^3 - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) - ((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1
/2))^(1/2)*(36*(-1)^(2/3)*2^(2/3)*((3^(1/2)*1i)/2 - 1/2)^2 - 156*(-1)^(2/3)*2^(2/3)*((3^(1/2)*1i)/2 - 1/2)^5 +
144*(-1)^(2/3)*2^(2/3)*((3^(1/2)*1i)/2 - 1/2)^8)) - (2*((3^(1/2)*1i)/2 + 3/2)*((x + (3^(1/2)*1i)/2 - 1/2)/((3
^(1/2)*1i)/2 - 3/2))^(1/2)*((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(((3^(1/2)*1i)/2 - x + 1/2)/((3^(1/2)*1i)/2
+ 3/2))^(1/2)*(2*(-1)^(2/3)*2^(2/3)*((3^(1/2)*1i)/2 + 1/2)^2 + 2*(-1)^(2/3)*2^(2/3)*((3^(1/2)*1i)/2 + 1/2)^5 -
4*(-1)^(2/3)*2^(2/3)*((3^(1/2)*1i)/2 + 1/2)^8)*ellipticPi(-((3^(1/2)*1i)/2 + 3/2)/((-1)^(1/3)*2^(1/3)*((3^(1/
2)*1i)/2 + 1/2) - 1), asin(((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 -
3/2)))/(((-1)^(1/3)*2^(1/3)*((3^(1/2)*1i)/2 + 1/2) - 1)*(x^3 - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2
) + 1) - ((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2)*(36*(-1)^(2/3)*2^(2/3)*((3^(1/2)*1i)/2 + 1/2)^2
+ 156*(-1)^(2/3)*2^(2/3)*((3^(1/2)*1i)/2 + 1/2)^5 + 144*(-1)^(2/3)*2^(2/3)*((3^(1/2)*1i)/2 + 1/2)^8))