\(\int \frac {2+x-x^3}{\sqrt {1+x+x^3} (1+x-x^2+x^3)} \, dx\) [67]
Optimal result
Integrand size = 32, antiderivative size = 15 \[
\int \frac {2+x-x^3}{\sqrt {1+x+x^3} \left (1+x-x^2+x^3\right )} \, dx=2 \text {arctanh}\left (\frac {x}{\sqrt {1+x+x^3}}\right )
\]
[Out]
2*arctanh(x/(x^3+x+1)^(1/2))
Rubi [F]
\[
\int \frac {2+x-x^3}{\sqrt {1+x+x^3} \left (1+x-x^2+x^3\right )} \, dx=\int \frac {2+x-x^3}{\sqrt {1+x+x^3} \left (1+x-x^2+x^3\right )} \, dx
\]
[In]
Int[(2 + x - x^3)/(Sqrt[1 + x + x^3]*(1 + x - x^2 + x^3)),x]
[Out]
(((-2*I)/3)*Sqrt[(2*(3/(-9 + Sqrt[93]))^(1/3) - (2*(-9 + Sqrt[93]))^(1/3) + 6^(2/3)*x)/(6*(3/(-9 + Sqrt[93]))^
(1/3) - 3*(2*(-9 + Sqrt[93]))^(1/3) - I*6^(1/6)*Sqrt[12 + 6*3^(1/3)*(2/(-9 + Sqrt[93]))^(2/3) + 2^(1/3)*(3*(-9
+ Sqrt[93]))^(2/3)])]*Sqrt[6 + 6*3^(1/3)*(2/(-9 + Sqrt[93]))^(2/3) + 2^(1/3)*(3*(-9 + Sqrt[93]))^(2/3) - 6*3^
(1/3)*((6/(-9 + Sqrt[93]))^(1/3) - ((-9 + Sqrt[93])/2)^(1/3))*x + 18*x^2]*EllipticF[ArcSin[Sqrt[I*(6^(1/3)*(2*
(3/(-9 + Sqrt[93]))^(1/3) - (2*(-9 + Sqrt[93]))^(1/3) - I*6^(1/6)*Sqrt[12 + 6*3^(1/3)*(2/(-9 + Sqrt[93]))^(2/3
) + 2^(1/3)*(3*(-9 + Sqrt[93]))^(2/3)]) - 12*x)]/(2^(3/4)*(3*(12 + 6*3^(1/3)*(2/(-9 + Sqrt[93]))^(2/3) + 2^(1/
3)*(3*(-9 + Sqrt[93]))^(2/3)))^(1/4))], (2*6^(1/6)*Sqrt[12 + 6*3^(1/3)*(2/(-9 + Sqrt[93]))^(2/3) + 2^(1/3)*(3*
(-9 + Sqrt[93]))^(2/3)])/(I*(6*(3/(-9 + Sqrt[93]))^(1/3) - 3*(2*(-9 + Sqrt[93]))^(1/3)) + 6^(1/6)*Sqrt[12 + 6*
3^(1/3)*(2/(-9 + Sqrt[93]))^(2/3) + 2^(1/3)*(3*(-9 + Sqrt[93]))^(2/3)])])/Sqrt[1 + x + x^3] + 3*Defer[Int][1/(
Sqrt[1 + x + x^3]*(1 + x - x^2 + x^3)), x] + 2*Defer[Int][x/(Sqrt[1 + x + x^3]*(1 + x - x^2 + x^3)), x] - Defe
r[Int][x^2/(Sqrt[1 + x + x^3]*(1 + x - x^2 + x^3)), x]
Rubi steps \begin{align*}
\text {integral}& = \int \left (-\frac {1}{\sqrt {1+x+x^3}}+\frac {3+2 x-x^2}{\sqrt {1+x+x^3} \left (1+x-x^2+x^3\right )}\right ) \, dx \\ & = -\int \frac {1}{\sqrt {1+x+x^3}} \, dx+\int \frac {3+2 x-x^2}{\sqrt {1+x+x^3} \left (1+x-x^2+x^3\right )} \, dx \\ & = -\frac {\left (\sqrt {\frac {2 \sqrt [3]{\frac {3}{-9+\sqrt {93}}}-\sqrt [3]{2 \left (-9+\sqrt {93}\right )}}{6^{2/3}}+x} \sqrt {\frac {1}{18} \left (6+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}\right )-\frac {\left (\sqrt [3]{\frac {6}{-9+\sqrt {93}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {93}\right )}\right ) x}{3^{2/3}}+x^2}\right ) \int \frac {1}{\sqrt {\frac {2 \sqrt [3]{\frac {3}{-9+\sqrt {93}}}-\sqrt [3]{2 \left (-9+\sqrt {93}\right )}}{6^{2/3}}+x} \sqrt {\frac {1}{18} \left (6+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}\right )-\frac {\left (\sqrt [3]{\frac {6}{-9+\sqrt {93}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {93}\right )}\right ) x}{3^{2/3}}+x^2}} \, dx}{\sqrt {1+x+x^3}}+\int \left (\frac {3}{\sqrt {1+x+x^3} \left (1+x-x^2+x^3\right )}+\frac {2 x}{\sqrt {1+x+x^3} \left (1+x-x^2+x^3\right )}-\frac {x^2}{\sqrt {1+x+x^3} \left (1+x-x^2+x^3\right )}\right ) \, dx \\ & = 2 \int \frac {x}{\sqrt {1+x+x^3} \left (1+x-x^2+x^3\right )} \, dx+3 \int \frac {1}{\sqrt {1+x+x^3} \left (1+x-x^2+x^3\right )} \, dx-\frac {\left (2 \sqrt {\frac {1}{3} \left (-12-6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}-\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}\right )} \sqrt {\frac {\frac {2 \sqrt [3]{\frac {3}{-9+\sqrt {93}}}-\sqrt [3]{2 \left (-9+\sqrt {93}\right )}}{6^{2/3}}+x}{\frac {\sqrt [3]{\frac {6}{-9+\sqrt {93}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {93}\right )}}{3^{2/3}}+\frac {\sqrt [3]{2} \left (2 \sqrt [3]{\frac {3}{-9+\sqrt {93}}}-\sqrt [3]{2 \left (-9+\sqrt {93}\right )}\right )}{3^{2/3}}-\sqrt {\frac {1}{6} \left (-12-6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}-\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}\right )}}} \sqrt {-\frac {\frac {1}{18} \left (6+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}\right )-\frac {\left (\sqrt [3]{\frac {6}{-9+\sqrt {93}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {93}\right )}\right ) x}{3^{2/3}}+x^2}{\frac {\left (\sqrt [3]{\frac {6}{-9+\sqrt {93}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {93}\right )}\right )^2}{3 \sqrt [3]{3}}-\frac {2}{9} \left (6+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}\right )}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {\sqrt {\frac {2}{3} \left (-12-6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}-\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}\right )} x^2}{\frac {\sqrt [3]{\frac {6}{-9+\sqrt {93}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {93}\right )}}{3^{2/3}}+\frac {\sqrt [3]{2} \left (2 \sqrt [3]{\frac {3}{-9+\sqrt {93}}}-\sqrt [3]{2 \left (-9+\sqrt {93}\right )}\right )}{3^{2/3}}-\sqrt {\frac {1}{6} \left (-12-6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}-\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}\right )}}}} \, dx,x,\sqrt {\frac {-\frac {\sqrt [3]{\frac {6}{-9+\sqrt {93}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {93}\right )}}{3^{2/3}}+\sqrt {\frac {1}{6} \left (-12-6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}-\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}\right )}+2 x}{\sqrt {\frac {2}{3} \left (-12-6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}-\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}\right )}}}\right )}{\sqrt {1+x+x^3}}-\int \frac {x^2}{\sqrt {1+x+x^3} \left (1+x-x^2+x^3\right )} \, dx \\ & = -\frac {2 i \sqrt {\frac {2 \sqrt [3]{\frac {3}{-9+\sqrt {93}}}-\sqrt [3]{2 \left (-9+\sqrt {93}\right )}+6^{2/3} x}{6 \sqrt [3]{\frac {3}{-9+\sqrt {93}}}-3 \sqrt [3]{2 \left (-9+\sqrt {93}\right )}-i \sqrt [6]{6} \sqrt {12+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}}}} \sqrt {6+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}-6 \sqrt [3]{3} \left (\sqrt [3]{\frac {6}{-9+\sqrt {93}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {93}\right )}\right ) x+18 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {i \left (\sqrt [3]{6} \left (2 \sqrt [3]{\frac {3}{-9+\sqrt {93}}}-\sqrt [3]{2 \left (-9+\sqrt {93}\right )}-i \sqrt [6]{6} \sqrt {12+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}}\right )-12 x\right )}}{2^{3/4} \sqrt [4]{3 \left (12+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}\right )}}\right ),\frac {2 \sqrt [6]{6} \sqrt {12+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}}}{i \left (6 \sqrt [3]{\frac {3}{-9+\sqrt {93}}}-3 \sqrt [3]{2 \left (-9+\sqrt {93}\right )}\right )+\sqrt [6]{6} \sqrt {12+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}}}\right )}{3 \sqrt {1+x+x^3}}+2 \int \frac {x}{\sqrt {1+x+x^3} \left (1+x-x^2+x^3\right )} \, dx+3 \int \frac {1}{\sqrt {1+x+x^3} \left (1+x-x^2+x^3\right )} \, dx-\int \frac {x^2}{\sqrt {1+x+x^3} \left (1+x-x^2+x^3\right )} \, dx \\
\end{align*}
Mathematica [A] (verified)
Time = 0.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00
\[
\int \frac {2+x-x^3}{\sqrt {1+x+x^3} \left (1+x-x^2+x^3\right )} \, dx=2 \text {arctanh}\left (\frac {x}{\sqrt {1+x+x^3}}\right )
\]
[In]
Integrate[(2 + x - x^3)/(Sqrt[1 + x + x^3]*(1 + x - x^2 + x^3)),x]
[Out]
2*ArcTanh[x/Sqrt[1 + x + x^3]]
Maple [A] (verified)
Time = 2.58 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07
| | |
| method | result | size |
| | |
| default |
\(2 \,\operatorname {arctanh}\left (\frac {\sqrt {x^{3}+x +1}}{x}\right )\) |
\(16\) |
| pseudoelliptic |
\(2 \,\operatorname {arctanh}\left (\frac {\sqrt {x^{3}+x +1}}{x}\right )\) |
\(16\) |
| trager |
\(-\ln \left (-\frac {-x^{3}+2 \sqrt {x^{3}+x +1}\, x -x^{2}-x -1}{x^{3}-x^{2}+x +1}\right )\) |
\(45\) |
| elliptic |
\(\text {Expression too large to display}\) |
\(1905\) |
| | |
|
|
|
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[In]
int((-x^3+x+2)/(x^3+x+1)^(1/2)/(x^3-x^2+x+1),x,method=_RETURNVERBOSE)
[Out]
2*arctanh((x^3+x+1)^(1/2)/x)
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (13) = 26\).
Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 2.33
\[
\int \frac {2+x-x^3}{\sqrt {1+x+x^3} \left (1+x-x^2+x^3\right )} \, dx=\log \left (\frac {x^{3} + x^{2} + 2 \, \sqrt {x^{3} + x + 1} x + x + 1}{x^{3} - x^{2} + x + 1}\right )
\]
[In]
integrate((-x^3+x+2)/(x^3+x+1)^(1/2)/(x^3-x^2+x+1),x, algorithm="fricas")
[Out]
log((x^3 + x^2 + 2*sqrt(x^3 + x + 1)*x + x + 1)/(x^3 - x^2 + x + 1))
Sympy [F]
\[
\int \frac {2+x-x^3}{\sqrt {1+x+x^3} \left (1+x-x^2+x^3\right )} \, dx=- \int \left (- \frac {x}{x^{3} \sqrt {x^{3} + x + 1} - x^{2} \sqrt {x^{3} + x + 1} + x \sqrt {x^{3} + x + 1} + \sqrt {x^{3} + x + 1}}\right )\, dx - \int \frac {x^{3}}{x^{3} \sqrt {x^{3} + x + 1} - x^{2} \sqrt {x^{3} + x + 1} + x \sqrt {x^{3} + x + 1} + \sqrt {x^{3} + x + 1}}\, dx - \int \left (- \frac {2}{x^{3} \sqrt {x^{3} + x + 1} - x^{2} \sqrt {x^{3} + x + 1} + x \sqrt {x^{3} + x + 1} + \sqrt {x^{3} + x + 1}}\right )\, dx
\]
[In]
integrate((-x**3+x+2)/(x**3+x+1)**(1/2)/(x**3-x**2+x+1),x)
[Out]
-Integral(-x/(x**3*sqrt(x**3 + x + 1) - x**2*sqrt(x**3 + x + 1) + x*sqrt(x**3 + x + 1) + sqrt(x**3 + x + 1)),
x) - Integral(x**3/(x**3*sqrt(x**3 + x + 1) - x**2*sqrt(x**3 + x + 1) + x*sqrt(x**3 + x + 1) + sqrt(x**3 + x +
1)), x) - Integral(-2/(x**3*sqrt(x**3 + x + 1) - x**2*sqrt(x**3 + x + 1) + x*sqrt(x**3 + x + 1) + sqrt(x**3 +
x + 1)), x)
Maxima [F]
\[
\int \frac {2+x-x^3}{\sqrt {1+x+x^3} \left (1+x-x^2+x^3\right )} \, dx=\int { -\frac {x^{3} - x - 2}{{\left (x^{3} - x^{2} + x + 1\right )} \sqrt {x^{3} + x + 1}} \,d x }
\]
[In]
integrate((-x^3+x+2)/(x^3+x+1)^(1/2)/(x^3-x^2+x+1),x, algorithm="maxima")
[Out]
-integrate((x^3 - x - 2)/((x^3 - x^2 + x + 1)*sqrt(x^3 + x + 1)), x)
Giac [F]
\[
\int \frac {2+x-x^3}{\sqrt {1+x+x^3} \left (1+x-x^2+x^3\right )} \, dx=\int { -\frac {x^{3} - x - 2}{{\left (x^{3} - x^{2} + x + 1\right )} \sqrt {x^{3} + x + 1}} \,d x }
\]
[In]
integrate((-x^3+x+2)/(x^3+x+1)^(1/2)/(x^3-x^2+x+1),x, algorithm="giac")
[Out]
integrate(-(x^3 - x - 2)/((x^3 - x^2 + x + 1)*sqrt(x^3 + x + 1)), x)
Mupad [B] (verification not implemented)
Time = 6.89 (sec) , antiderivative size = 2490, normalized size of antiderivative = 166.00
\[
\int \frac {2+x-x^3}{\sqrt {1+x+x^3} \left (1+x-x^2+x^3\right )} \, dx=\text {Too large to display}
\]
[In]
int((x - x^3 + 2)/((x + x^3 + 1)^(1/2)*(x - x^2 + x^3 + 1)),x)
[Out]
symsum(-(2*(-(x + 1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) - ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))/((3^(
1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(2*((31
^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + (3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))/2))^(1/2)*((x + (3^(1/2)*(1/(
3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(6*((31^(1/2)*10
8^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)/2)/((3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/1
08 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(2*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)
) + (3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))/2))^(1/2)*((3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3
)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(2*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + (3*((31^(1
/2)*108^(1/2))/108 - 1/2)^(1/3))/2)*ellipticPi(((3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^
(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(2*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + (3*((31^(1/2)*108^(1
/2))/108 - 1/2)^(1/3))/2)/(root(z^3 - z^2 + z + 1, z, k) + (3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/
3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(6*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2
)*108^(1/2))/108 - 1/2)^(1/3)/2), asin(((x + (3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/
2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(6*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/1
08 - 1/2)^(1/3)/2)/((3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^
(1/3))*1i)/2 - 1/(2*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + (3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))/2))^(
1/2)), -(3^(1/2)*((3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1
/3))*1i)/2 - 1/(2*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + (3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))/2)*1i)/
(3*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))))*(2*root(z^3 - z^2
+ z + 1, z, k) - root(z^3 - z^2 + z + 1, z, k)^2 + 3)*((3^(1/2)*(x - (3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108
- 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(6*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))
+ ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)/2)*1i)/(3*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*
108^(1/2))/108 - 1/2)^(1/3))))^(1/2))/((x^3 - x*((1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) - ((31^(1/2)*10
8^(1/2))/108 - 1/2)^(1/3))*((3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108
- 1/2)^(1/3))*1i)/2 + 1/(6*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) - ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)/2
) - (1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) - ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*((3^(1/2)*(1/(3*((
31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(6*((31^(1/2)*108^(1
/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)/2) + ((3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108
- 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(6*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))
+ ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)/2)*((3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/
2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 + 1/(6*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) - ((31^(1/2)*108^(1/2))/1
08 - 1/2)^(1/3)/2)) - (1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) - ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*
((3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(6
*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)/2)*((3^(1/2)*(1/(3*((31^(1/2
)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 + 1/(6*((31^(1/2)*108^(1/2))/10
8 - 1/2)^(1/3)) - ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)/2))^(1/2)*(3*root(z^3 - z^2 + z + 1, z, k)^2 - 2*root
(z^3 - z^2 + z + 1, z, k) + 1)*(root(z^3 - z^2 + z + 1, z, k) + (3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2
)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(6*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31
^(1/2)*108^(1/2))/108 - 1/2)^(1/3)/2)), k, 1, 3) - (2*(-(x + 1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) - ((
31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))/((3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108
^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(2*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + (3*((31^(1/2)*108^(1/2))/108
- 1/2)^(1/3))/2))^(1/2)*ellipticF(asin(((x + (3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/
2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(6*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/1
08 - 1/2)^(1/3)/2)/((3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^
(1/3))*1i)/2 - 1/(2*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + (3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))/2))^(
1/2)), -(3^(1/2)*((3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1
/3))*1i)/2 - 1/(2*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + (3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))/2)*1i)/
(3*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))))*((x + (3^(1/2)*(1
/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(6*((31^(1/2)*
108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)/2)/((3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))
/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(2*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/
3)) + (3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))/2))^(1/2)*((3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1
/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(2*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + (3*((31^
(1/2)*108^(1/2))/108 - 1/2)^(1/3))/2)*((3^(1/2)*(x - (3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) +
((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(6*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^
(1/2))/108 - 1/2)^(1/3)/2)*1i)/(3*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 -
1/2)^(1/3))))^(1/2))/(x^3 - x*((1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) - ((31^(1/2)*108^(1/2))/108 - 1/2
)^(1/3))*((3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)
/2 + 1/(6*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) - ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)/2) - (1/(3*((31^(1/
2)*108^(1/2))/108 - 1/2)^(1/3)) - ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*((3^(1/2)*(1/(3*((31^(1/2)*108^(1/2)
)/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(6*((31^(1/2)*108^(1/2))/108 - 1/2)^(1
/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)/2) + ((3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + (
(31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(6*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(
1/2))/108 - 1/2)^(1/3)/2)*((3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108
- 1/2)^(1/3))*1i)/2 + 1/(6*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) - ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)/2)
) - (1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) - ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*((3^(1/2)*(1/(3*((
31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(6*((31^(1/2)*108^(1
/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)/2)*((3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 -
1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 + 1/(6*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) -
((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)/2))^(1/2)