\(\int \frac {(2+x^3) \sqrt {-1-x^2+x^3}}{1-x^2-2 x^3-x^4+x^5+x^6} \, dx\) [1291]
Optimal result
Integrand size = 45, antiderivative size = 93 \[
\int \frac {\left (2+x^3\right ) \sqrt {-1-x^2+x^3}}{1-x^2-2 x^3-x^4+x^5+x^6} \, dx=\frac {1}{5} \left (-5-\sqrt {5}\right ) \arctan \left (\frac {\sqrt {\frac {3}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt {-1-x^2+x^3}}\right )+\frac {1}{5} \left (5-\sqrt {5}\right ) \arctan \left (\frac {\sqrt {\frac {2}{3+\sqrt {5}}} x}{\sqrt {-1-x^2+x^3}}\right )
\]
[Out]
1/5*(-5-5^(1/2))*arctan((1/2+1/2*5^(1/2))*x/(x^3-x^2-1)^(1/2))+1/5*(5-5^(1/2))*arctan(2^(1/2)/(3+5^(1/2))^(1/2
)*x/(x^3-x^2-1)^(1/2))
Rubi [F]
\[
\int \frac {\left (2+x^3\right ) \sqrt {-1-x^2+x^3}}{1-x^2-2 x^3-x^4+x^5+x^6} \, dx=\int \frac {\left (2+x^3\right ) \sqrt {-1-x^2+x^3}}{1-x^2-2 x^3-x^4+x^5+x^6} \, dx
\]
[In]
Int[((2 + x^3)*Sqrt[-1 - x^2 + x^3])/(1 - x^2 - 2*x^3 - x^4 + x^5 + x^6),x]
[Out]
2*Defer[Int][Sqrt[-1 - x^2 + x^3]/(1 - x^2 - 2*x^3 - x^4 + x^5 + x^6), x] + Defer[Int][(x^3*Sqrt[-1 - x^2 + x^
3])/(1 - x^2 - 2*x^3 - x^4 + x^5 + x^6), x]
Rubi steps \begin{align*}
\text {integral}& = \int \left (\frac {2 \sqrt {-1-x^2+x^3}}{1-x^2-2 x^3-x^4+x^5+x^6}+\frac {x^3 \sqrt {-1-x^2+x^3}}{1-x^2-2 x^3-x^4+x^5+x^6}\right ) \, dx \\ & = 2 \int \frac {\sqrt {-1-x^2+x^3}}{1-x^2-2 x^3-x^4+x^5+x^6} \, dx+\int \frac {x^3 \sqrt {-1-x^2+x^3}}{1-x^2-2 x^3-x^4+x^5+x^6} \, dx \\
\end{align*}
Mathematica [A] (verified)
Time = 0.42 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.83
\[
\int \frac {\left (2+x^3\right ) \sqrt {-1-x^2+x^3}}{1-x^2-2 x^3-x^4+x^5+x^6} \, dx=\frac {1}{5} \left (-\left (\left (-5+\sqrt {5}\right ) \arctan \left (\frac {\left (-1+\sqrt {5}\right ) x}{2 \sqrt {-1-x^2+x^3}}\right )\right )-\left (5+\sqrt {5}\right ) \arctan \left (\frac {\left (1+\sqrt {5}\right ) x}{2 \sqrt {-1-x^2+x^3}}\right )\right )
\]
[In]
Integrate[((2 + x^3)*Sqrt[-1 - x^2 + x^3])/(1 - x^2 - 2*x^3 - x^4 + x^5 + x^6),x]
[Out]
(-((-5 + Sqrt[5])*ArcTan[((-1 + Sqrt[5])*x)/(2*Sqrt[-1 - x^2 + x^3])]) - (5 + Sqrt[5])*ArcTan[((1 + Sqrt[5])*x
)/(2*Sqrt[-1 - x^2 + x^3])])/5
Maple [A] (verified)
Time = 11.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.76
| | |
| method | result | size |
| | |
| default |
\(-\frac {\left (\left (-\sqrt {5}-1\right ) \arctan \left (\frac {2 \sqrt {x^{3}-x^{2}-1}}{x \left (\sqrt {5}+1\right )}\right )+\left (\sqrt {5}-1\right ) \arctan \left (\frac {2 \sqrt {x^{3}-x^{2}-1}}{x \left (\sqrt {5}-1\right )}\right )\right ) \sqrt {5}}{5}\) |
\(71\) |
| pseudoelliptic |
\(-\frac {\left (\left (-\sqrt {5}-1\right ) \arctan \left (\frac {2 \sqrt {x^{3}-x^{2}-1}}{x \left (\sqrt {5}+1\right )}\right )+\left (\sqrt {5}-1\right ) \arctan \left (\frac {2 \sqrt {x^{3}-x^{2}-1}}{x \left (\sqrt {5}-1\right )}\right )\right ) \sqrt {5}}{5}\) |
\(71\) |
| trager |
\(5 \ln \left (\frac {175 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{5} x^{2}-35 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{3} x^{3}+230 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{3} x^{2}+30 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{2} \sqrt {x^{3}-x^{2}-1}\, x -18 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right ) x^{3}+35 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{3}+72 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right ) x^{2}+16 \sqrt {x^{3}-x^{2}-1}\, x +18 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )}{5 x^{2} \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{2}+x^{3}+2 x^{2}-1}\right ) \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{3}+3 \ln \left (\frac {175 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{5} x^{2}-35 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{3} x^{3}+230 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{3} x^{2}+30 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{2} \sqrt {x^{3}-x^{2}-1}\, x -18 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right ) x^{3}+35 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{3}+72 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right ) x^{2}+16 \sqrt {x^{3}-x^{2}-1}\, x +18 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )}{5 x^{2} \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{2}+x^{3}+2 x^{2}-1}\right ) \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )-\operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right ) \ln \left (-\frac {75 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{5} x^{2}+15 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{3} x^{3}-5 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{3} x^{2}+30 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{2} \sqrt {x^{3}-x^{2}-1}\, x +2 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right ) x^{3}-15 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{3}-2 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right ) x^{2}+2 \sqrt {x^{3}-x^{2}-1}\, x -2 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )}{5 x^{2} \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{2}-x^{3}+x^{2}+1}\right )\) |
\(674\) |
| elliptic |
\(\text {Expression too large to display}\) |
\(2877\) |
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[In]
int((x^3+2)*(x^3-x^2-1)^(1/2)/(x^6+x^5-x^4-2*x^3-x^2+1),x,method=_RETURNVERBOSE)
[Out]
-1/5*((-5^(1/2)-1)*arctan(2*(x^3-x^2-1)^(1/2)/x/(5^(1/2)+1))+(5^(1/2)-1)*arctan(2*(x^3-x^2-1)^(1/2)/x/(5^(1/2)
-1)))*5^(1/2)
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 497 vs. \(2 (61) = 122\).
Time = 0.31 (sec) , antiderivative size = 497, normalized size of antiderivative = 5.34
\[
\int \frac {\left (2+x^3\right ) \sqrt {-1-x^2+x^3}}{1-x^2-2 x^3-x^4+x^5+x^6} \, dx=-\frac {1}{20} \, \sqrt {5} \sqrt {2 \, \sqrt {5} - 6} \log \left (\frac {2 \, x^{6} - 4 \, x^{5} - 4 \, x^{3} + 4 \, x^{2} + {\left (2 \, x^{4} + \sqrt {5} x^{3} + x^{3} - 2 \, x\right )} \sqrt {x^{3} - x^{2} - 1} \sqrt {2 \, \sqrt {5} - 6} + 2 \, \sqrt {5} {\left (x^{5} - x^{4} - x^{2}\right )} + 2}{x^{6} + x^{5} - x^{4} - 2 \, x^{3} - x^{2} + 1}\right ) + \frac {1}{20} \, \sqrt {5} \sqrt {2 \, \sqrt {5} - 6} \log \left (\frac {2 \, x^{6} - 4 \, x^{5} - 4 \, x^{3} + 4 \, x^{2} - {\left (2 \, x^{4} + \sqrt {5} x^{3} + x^{3} - 2 \, x\right )} \sqrt {x^{3} - x^{2} - 1} \sqrt {2 \, \sqrt {5} - 6} + 2 \, \sqrt {5} {\left (x^{5} - x^{4} - x^{2}\right )} + 2}{x^{6} + x^{5} - x^{4} - 2 \, x^{3} - x^{2} + 1}\right ) + \frac {1}{20} \, \sqrt {5} \sqrt {-2 \, \sqrt {5} - 6} \log \left (\frac {2 \, x^{6} - 4 \, x^{5} - 4 \, x^{3} + 4 \, x^{2} + {\left (2 \, x^{4} - \sqrt {5} x^{3} + x^{3} - 2 \, x\right )} \sqrt {x^{3} - x^{2} - 1} \sqrt {-2 \, \sqrt {5} - 6} - 2 \, \sqrt {5} {\left (x^{5} - x^{4} - x^{2}\right )} + 2}{x^{6} + x^{5} - x^{4} - 2 \, x^{3} - x^{2} + 1}\right ) - \frac {1}{20} \, \sqrt {5} \sqrt {-2 \, \sqrt {5} - 6} \log \left (\frac {2 \, x^{6} - 4 \, x^{5} - 4 \, x^{3} + 4 \, x^{2} - {\left (2 \, x^{4} - \sqrt {5} x^{3} + x^{3} - 2 \, x\right )} \sqrt {x^{3} - x^{2} - 1} \sqrt {-2 \, \sqrt {5} - 6} - 2 \, \sqrt {5} {\left (x^{5} - x^{4} - x^{2}\right )} + 2}{x^{6} + x^{5} - x^{4} - 2 \, x^{3} - x^{2} + 1}\right )
\]
[In]
integrate((x^3+2)*(x^3-x^2-1)^(1/2)/(x^6+x^5-x^4-2*x^3-x^2+1),x, algorithm="fricas")
[Out]
-1/20*sqrt(5)*sqrt(2*sqrt(5) - 6)*log((2*x^6 - 4*x^5 - 4*x^3 + 4*x^2 + (2*x^4 + sqrt(5)*x^3 + x^3 - 2*x)*sqrt(
x^3 - x^2 - 1)*sqrt(2*sqrt(5) - 6) + 2*sqrt(5)*(x^5 - x^4 - x^2) + 2)/(x^6 + x^5 - x^4 - 2*x^3 - x^2 + 1)) + 1
/20*sqrt(5)*sqrt(2*sqrt(5) - 6)*log((2*x^6 - 4*x^5 - 4*x^3 + 4*x^2 - (2*x^4 + sqrt(5)*x^3 + x^3 - 2*x)*sqrt(x^
3 - x^2 - 1)*sqrt(2*sqrt(5) - 6) + 2*sqrt(5)*(x^5 - x^4 - x^2) + 2)/(x^6 + x^5 - x^4 - 2*x^3 - x^2 + 1)) + 1/2
0*sqrt(5)*sqrt(-2*sqrt(5) - 6)*log((2*x^6 - 4*x^5 - 4*x^3 + 4*x^2 + (2*x^4 - sqrt(5)*x^3 + x^3 - 2*x)*sqrt(x^3
- x^2 - 1)*sqrt(-2*sqrt(5) - 6) - 2*sqrt(5)*(x^5 - x^4 - x^2) + 2)/(x^6 + x^5 - x^4 - 2*x^3 - x^2 + 1)) - 1/2
0*sqrt(5)*sqrt(-2*sqrt(5) - 6)*log((2*x^6 - 4*x^5 - 4*x^3 + 4*x^2 - (2*x^4 - sqrt(5)*x^3 + x^3 - 2*x)*sqrt(x^3
- x^2 - 1)*sqrt(-2*sqrt(5) - 6) - 2*sqrt(5)*(x^5 - x^4 - x^2) + 2)/(x^6 + x^5 - x^4 - 2*x^3 - x^2 + 1))
Sympy [F]
\[
\int \frac {\left (2+x^3\right ) \sqrt {-1-x^2+x^3}}{1-x^2-2 x^3-x^4+x^5+x^6} \, dx=\int \frac {\left (x^{3} + 2\right ) \sqrt {x^{3} - x^{2} - 1}}{x^{6} + x^{5} - x^{4} - 2 x^{3} - x^{2} + 1}\, dx
\]
[In]
integrate((x**3+2)*(x**3-x**2-1)**(1/2)/(x**6+x**5-x**4-2*x**3-x**2+1),x)
[Out]
Integral((x**3 + 2)*sqrt(x**3 - x**2 - 1)/(x**6 + x**5 - x**4 - 2*x**3 - x**2 + 1), x)
Maxima [F]
\[
\int \frac {\left (2+x^3\right ) \sqrt {-1-x^2+x^3}}{1-x^2-2 x^3-x^4+x^5+x^6} \, dx=\int { \frac {\sqrt {x^{3} - x^{2} - 1} {\left (x^{3} + 2\right )}}{x^{6} + x^{5} - x^{4} - 2 \, x^{3} - x^{2} + 1} \,d x }
\]
[In]
integrate((x^3+2)*(x^3-x^2-1)^(1/2)/(x^6+x^5-x^4-2*x^3-x^2+1),x, algorithm="maxima")
[Out]
integrate(sqrt(x^3 - x^2 - 1)*(x^3 + 2)/(x^6 + x^5 - x^4 - 2*x^3 - x^2 + 1), x)
Giac [F]
\[
\int \frac {\left (2+x^3\right ) \sqrt {-1-x^2+x^3}}{1-x^2-2 x^3-x^4+x^5+x^6} \, dx=\int { \frac {\sqrt {x^{3} - x^{2} - 1} {\left (x^{3} + 2\right )}}{x^{6} + x^{5} - x^{4} - 2 \, x^{3} - x^{2} + 1} \,d x }
\]
[In]
integrate((x^3+2)*(x^3-x^2-1)^(1/2)/(x^6+x^5-x^4-2*x^3-x^2+1),x, algorithm="giac")
[Out]
integrate(sqrt(x^3 - x^2 - 1)*(x^3 + 2)/(x^6 + x^5 - x^4 - 2*x^3 - x^2 + 1), x)
Mupad [B] (verification not implemented)
Time = 8.11 (sec) , antiderivative size = 2803, normalized size of antiderivative = 30.14
\[
\int \frac {\left (2+x^3\right ) \sqrt {-1-x^2+x^3}}{1-x^2-2 x^3-x^4+x^5+x^6} \, dx=\text {Too large to display}
\]
[In]
int(-((x^3 + 2)*(x^3 - x^2 - 1)^(1/2))/(x^2 + 2*x^3 + x^4 - x^5 - x^6 - 1),x)
[Out]
symsum(-(2*((x - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)
^(1/3))*1i)/2 + 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2 -
1/3)/(1/(6*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3
)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + (3*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))/2))^(1/2)*
((1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - x + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3) + 1/3)/(1/(6*(
(31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/
2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + (3*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))/2))^(1/2)*ellipticPi((1
/(6*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((3
1^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + (3*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))/2)/(root(z^6 + z^5
- z^4 - 2*z^3 - z^2 + 1, z, k) - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1
/2))/108 + 29/54)^(1/3))*1i)/2 + 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 +
29/54)^(1/3)/2 - 1/3), asin(((x - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(
1/2))/108 + 29/54)^(1/3))*1i)/2 + 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108
+ 29/54)^(1/3)/2 - 1/3)/(1/(6*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))
/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + (3*((31^(1/2)*108^(1/2))/108 + 29/54)
^(1/3))/2))^(1/2)), (3^(1/2)*(1/(6*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - (3^(1/2)*(1/(9*((31^(1/2)*108^(
1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + (3*((31^(1/2)*108^(1/2))/108 + 2
9/54)^(1/3))/2)*1i)/(3*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1
/3))))*(1/(6*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/
3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + (3*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))/2)*(-(3^(
1/2)*(x + (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))
*1i)/2 + 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2 - 1/3)*1
i)/(3*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))))^(1/2)*(roo
t(z^6 + z^5 - z^4 - 2*z^3 - z^2 + 1, z, k)^2 - 3*root(z^6 + z^5 - z^4 - 2*z^3 - z^2 + 1, z, k)^3 - root(z^6 +
z^5 - z^4 - 2*z^3 - z^2 + 1, z, k)^4 + 2*root(z^6 + z^5 - z^4 - 2*z^3 - z^2 + 1, z, k)^5 + 3))/((x^3 - x^2 - x
*(((3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2
- 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2 + 1/3)*((3^(1/2
)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + 1/(18*((
31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2 - 1/3) - (1/(9*((31^(1/2)
*108^(1/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3) + 1/3)*((3^(1/2)*(1/(9*((31^(1/2)*1
08^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 - 1/(18*((31^(1/2)*108^(1/2))/
108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2 + 1/3) + (1/(9*((31^(1/2)*108^(1/2))/108 + 29
/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3) + 1/3)*((3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/5
4)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))
+ ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2 - 1/3)) + (1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) + ((31^
(1/2)*108^(1/2))/108 + 29/54)^(1/3) + 1/3)*((3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1
/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 - 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1
/2))/108 + 29/54)^(1/3)/2 + 1/3)*((3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1
/2))/108 + 29/54)^(1/3))*1i)/2 + 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 +
29/54)^(1/3)/2 - 1/3))^(1/2)*(root(z^6 + z^5 - z^4 - 2*z^3 - z^2 + 1, z, k) - (3^(1/2)*(1/(9*((31^(1/2)*108^(
1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + 1/(18*((31^(1/2)*108^(1/2))/108
+ 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2 - 1/3)*(6*root(z^6 + z^5 - z^4 - 2*z^3 - z^2 + 1,
z, k)^2 + 4*root(z^6 + z^5 - z^4 - 2*z^3 - z^2 + 1, z, k)^3 - 5*root(z^6 + z^5 - z^4 - 2*z^3 - z^2 + 1, z, k)
^4 - 6*root(z^6 + z^5 - z^4 - 2*z^3 - z^2 + 1, z, k)^5 + 2*root(z^6 + z^5 - z^4 - 2*z^3 - z^2 + 1, z, k))), k,
1, 6) + (2*((x - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54
)^(1/3))*1i)/2 + 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2
- 1/3)/(1/(6*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/
3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + (3*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))/2))^(1/2)
*((1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - x + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3) + 1/3)/(1/(6*
((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1
/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + (3*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))/2))^(1/2)*ellipticF(as
in(((x - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*
1i)/2 + 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2 - 1/3)/(1
/(6*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((3
1^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + (3*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))/2))^(1/2)), (3^(1/
2)*(1/(6*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))
- ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + (3*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))/2)*1i)/(3*(1/
(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))))*(1/(6*((31^(1/2)*10
8^(1/2))/108 + 29/54)^(1/3)) - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2)
)/108 + 29/54)^(1/3))*1i)/2 + (3*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))/2)*(-(3^(1/2)*(x + (3^(1/2)*(1/(9*(
(31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + 1/(18*((31^(1/2)*
108^(1/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2 - 1/3)*1i)/(3*(1/(9*((31^(1/2)*108
^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))))^(1/2))/(x^3 - x^2 - x*(((3^(1/2)*(1/
(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 - 1/(18*((31^(1
/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2 + 1/3)*((3^(1/2)*(1/(9*((31^(1
/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + 1/(18*((31^(1/2)*108^(1
/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2 - 1/3) - (1/(9*((31^(1/2)*108^(1/2))/108
+ 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3) + 1/3)*((3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 +
29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 - 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1
/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2 + 1/3) + (1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) + (
(31^(1/2)*108^(1/2))/108 + 29/54)^(1/3) + 1/3)*((3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((3
1^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*10
8^(1/2))/108 + 29/54)^(1/3)/2 - 1/3)) + (1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*108^(1/2)
)/108 + 29/54)^(1/3) + 1/3)*((3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/
108 + 29/54)^(1/3))*1i)/2 - 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/5
4)^(1/3)/2 + 1/3)*((3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/5
4)^(1/3))*1i)/2 + 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2
- 1/3))^(1/2)