Integrand size = 23, antiderivative size = 99 \[ \int \frac {-1+x}{\left (-2-2 x+x^2\right ) \sqrt {-1+x^3}} \, dx=\frac {1}{6} \sqrt {3+2 \sqrt {3}} \arctan \left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt {-1+x^3}}{1+x+x^2}\right )-\frac {1}{6} \sqrt {-3+2 \sqrt {3}} \text {arctanh}\left (\frac {\sqrt {3+2 \sqrt {3}} \sqrt {-1+x^3}}{1+x+x^2}\right ) \]
1/6*(3+2*3^(1/2))^(1/2)*arctan((-3+2*3^(1/2))^(1/2)*(x^3-1)^(1/2)/(x^2+x+1 ))-1/6*(-3+2*3^(1/2))^(1/2)*arctanh((3+2*3^(1/2))^(1/2)*(x^3-1)^(1/2)/(x^2 +x+1))
Time = 1.28 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x}{\left (-2-2 x+x^2\right ) \sqrt {-1+x^3}} \, dx=\frac {1}{6} \sqrt {3+2 \sqrt {3}} \arctan \left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt {-1+x^3}}{1+x+x^2}\right )-\frac {1}{6} \sqrt {-3+2 \sqrt {3}} \text {arctanh}\left (\frac {\sqrt {3+2 \sqrt {3}} \sqrt {-1+x^3}}{1+x+x^2}\right ) \]
(Sqrt[3 + 2*Sqrt[3]]*ArcTan[(Sqrt[-3 + 2*Sqrt[3]]*Sqrt[-1 + x^3])/(1 + x + x^2)])/6 - (Sqrt[-3 + 2*Sqrt[3]]*ArcTanh[(Sqrt[3 + 2*Sqrt[3]]*Sqrt[-1 + x ^3])/(1 + x + x^2)])/6
Time = 0.74 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.98, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {7279, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x-1}{\left (x^2-2 x-2\right ) \sqrt {x^3-1}} \, dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {1}{\left (2 x-2 \sqrt {3}-2\right ) \sqrt {x^3-1}}+\frac {1}{\left (2 x+2 \sqrt {3}-2\right ) \sqrt {x^3-1}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\text {arctanh}\left (\frac {\sqrt {3+2 \sqrt {3}} (1-x)}{\sqrt {x^3-1}}\right )}{2 \sqrt {3 \left (3+2 \sqrt {3}\right )}}-\frac {\arctan \left (\frac {\sqrt {2 \sqrt {3}-3} (1-x)}{\sqrt {x^3-1}}\right )}{2 \sqrt {3 \left (2 \sqrt {3}-3\right )}}\) |
-1/2*ArcTan[(Sqrt[-3 + 2*Sqrt[3]]*(1 - x))/Sqrt[-1 + x^3]]/Sqrt[3*(-3 + 2* Sqrt[3])] + ArcTanh[(Sqrt[3 + 2*Sqrt[3]]*(1 - x))/Sqrt[-1 + x^3]]/(2*Sqrt[ 3*(3 + 2*Sqrt[3])])
3.14.70.3.1 Defintions of rubi rules used
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 6.24 (sec) , antiderivative size = 702, normalized size of antiderivative = 7.09
| method | result | size |
| default | \(\frac {\sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {1}{3-i \sqrt {3}}-\frac {i \sqrt {3}}{2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{i \sqrt {3}+3}+\frac {i \sqrt {3}}{i \sqrt {3}+3}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right ) \sqrt {3}}{2 \sqrt {x^{3}-1}}+\frac {i \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {1}{3-i \sqrt {3}}-\frac {i \sqrt {3}}{2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{i \sqrt {3}+3}+\frac {i \sqrt {3}}{i \sqrt {3}+3}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {x^{3}-1}}-\frac {\sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {1}{3-i \sqrt {3}}-\frac {i \sqrt {3}}{2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{i \sqrt {3}+3}+\frac {i \sqrt {3}}{i \sqrt {3}+3}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right ) \sqrt {3}}{2 \sqrt {x^{3}-1}}-\frac {i \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {1}{3-i \sqrt {3}}-\frac {i \sqrt {3}}{2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{i \sqrt {3}+3}+\frac {i \sqrt {3}}{i \sqrt {3}+3}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {x^{3}-1}}\) | \(702\) |
| elliptic | \(\frac {\sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {1}{3-i \sqrt {3}}-\frac {i \sqrt {3}}{2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{i \sqrt {3}+3}+\frac {i \sqrt {3}}{i \sqrt {3}+3}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right ) \sqrt {3}}{2 \sqrt {x^{3}-1}}+\frac {i \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {1}{3-i \sqrt {3}}-\frac {i \sqrt {3}}{2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{i \sqrt {3}+3}+\frac {i \sqrt {3}}{i \sqrt {3}+3}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {x^{3}-1}}-\frac {\sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {1}{3-i \sqrt {3}}-\frac {i \sqrt {3}}{2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{i \sqrt {3}+3}+\frac {i \sqrt {3}}{i \sqrt {3}+3}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right ) \sqrt {3}}{2 \sqrt {x^{3}-1}}-\frac {i \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {1}{3-i \sqrt {3}}-\frac {i \sqrt {3}}{2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{i \sqrt {3}+3}+\frac {i \sqrt {3}}{i \sqrt {3}+3}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {x^{3}-1}}\) | \(702\) |
1/2*(1/(-3/2-1/2*I*3^(1/2))*x-1/(-3/2-1/2*I*3^(1/2)))^(1/2)*(1/(3/2-1/2*I* 3^(1/2))*x+1/2/(3/2-1/2*I*3^(1/2))-1/2*I/(3/2-1/2*I*3^(1/2))*3^(1/2))^(1/2 )*(1/(3/2+1/2*I*3^(1/2))*x+1/2/(3/2+1/2*I*3^(1/2))+1/2*I/(3/2+1/2*I*3^(1/2 ))*3^(1/2))^(1/2)/(x^3-1)^(1/2)*EllipticPi(((x-1)/(-3/2-1/2*I*3^(1/2)))^(1 /2),-1/3*(3/2+1/2*I*3^(1/2))*3^(1/2),((3/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/ 2)))^(1/2))*3^(1/2)+1/2*I*(1/(-3/2-1/2*I*3^(1/2))*x-1/(-3/2-1/2*I*3^(1/2)) )^(1/2)*(1/(3/2-1/2*I*3^(1/2))*x+1/2/(3/2-1/2*I*3^(1/2))-1/2*I/(3/2-1/2*I* 3^(1/2))*3^(1/2))^(1/2)*(1/(3/2+1/2*I*3^(1/2))*x+1/2/(3/2+1/2*I*3^(1/2))+1 /2*I/(3/2+1/2*I*3^(1/2))*3^(1/2))^(1/2)/(x^3-1)^(1/2)*EllipticPi(((x-1)/(- 3/2-1/2*I*3^(1/2)))^(1/2),-1/3*(3/2+1/2*I*3^(1/2))*3^(1/2),((3/2+1/2*I*3^( 1/2))/(3/2-1/2*I*3^(1/2)))^(1/2))-1/2*(1/(-3/2-1/2*I*3^(1/2))*x-1/(-3/2-1/ 2*I*3^(1/2)))^(1/2)*(1/(3/2-1/2*I*3^(1/2))*x+1/2/(3/2-1/2*I*3^(1/2))-1/2*I /(3/2-1/2*I*3^(1/2))*3^(1/2))^(1/2)*(1/(3/2+1/2*I*3^(1/2))*x+1/2/(3/2+1/2* I*3^(1/2))+1/2*I/(3/2+1/2*I*3^(1/2))*3^(1/2))^(1/2)/(x^3-1)^(1/2)*Elliptic Pi(((x-1)/(-3/2-1/2*I*3^(1/2)))^(1/2),1/3*(3/2+1/2*I*3^(1/2))*3^(1/2),((3/ 2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2))*3^(1/2)-1/2*I*(1/(-3/2-1/2*I* 3^(1/2))*x-1/(-3/2-1/2*I*3^(1/2)))^(1/2)*(1/(3/2-1/2*I*3^(1/2))*x+1/2/(3/2 -1/2*I*3^(1/2))-1/2*I/(3/2-1/2*I*3^(1/2))*3^(1/2))^(1/2)*(1/(3/2+1/2*I*3^( 1/2))*x+1/2/(3/2+1/2*I*3^(1/2))+1/2*I/(3/2+1/2*I*3^(1/2))*3^(1/2))^(1/2)/( x^3-1)^(1/2)*EllipticPi(((x-1)/(-3/2-1/2*I*3^(1/2)))^(1/2),1/3*(3/2+1/2...
Leaf count of result is larger than twice the leaf count of optimal. 383 vs. \(2 (75) = 150\).
Time = 0.28 (sec) , antiderivative size = 383, normalized size of antiderivative = 3.87 \[ \int \frac {-1+x}{\left (-2-2 x+x^2\right ) \sqrt {-1+x^3}} \, dx=-\frac {1}{24} \, \sqrt {2 \, \sqrt {3} - 3} \log \left (\frac {x^{4} + 2 \, x^{3} + 6 \, x^{2} + 2 \, \sqrt {x^{3} - 1} {\left (2 \, x^{2} + \sqrt {3} {\left (x^{2} + 2 \, x\right )} + 2 \, x + 2\right )} \sqrt {2 \, \sqrt {3} - 3} + 4 \, \sqrt {3} {\left (x^{3} - 1\right )} - 4 \, x + 4}{x^{4} - 4 \, x^{3} + 8 \, x + 4}\right ) + \frac {1}{24} \, \sqrt {2 \, \sqrt {3} - 3} \log \left (\frac {x^{4} + 2 \, x^{3} + 6 \, x^{2} - 2 \, \sqrt {x^{3} - 1} {\left (2 \, x^{2} + \sqrt {3} {\left (x^{2} + 2 \, x\right )} + 2 \, x + 2\right )} \sqrt {2 \, \sqrt {3} - 3} + 4 \, \sqrt {3} {\left (x^{3} - 1\right )} - 4 \, x + 4}{x^{4} - 4 \, x^{3} + 8 \, x + 4}\right ) - \frac {1}{24} \, \sqrt {-2 \, \sqrt {3} - 3} \log \left (\frac {x^{4} + 2 \, x^{3} + 6 \, x^{2} + 2 \, \sqrt {x^{3} - 1} {\left (2 \, x^{2} - \sqrt {3} {\left (x^{2} + 2 \, x\right )} + 2 \, x + 2\right )} \sqrt {-2 \, \sqrt {3} - 3} - 4 \, \sqrt {3} {\left (x^{3} - 1\right )} - 4 \, x + 4}{x^{4} - 4 \, x^{3} + 8 \, x + 4}\right ) + \frac {1}{24} \, \sqrt {-2 \, \sqrt {3} - 3} \log \left (\frac {x^{4} + 2 \, x^{3} + 6 \, x^{2} - 2 \, \sqrt {x^{3} - 1} {\left (2 \, x^{2} - \sqrt {3} {\left (x^{2} + 2 \, x\right )} + 2 \, x + 2\right )} \sqrt {-2 \, \sqrt {3} - 3} - 4 \, \sqrt {3} {\left (x^{3} - 1\right )} - 4 \, x + 4}{x^{4} - 4 \, x^{3} + 8 \, x + 4}\right ) \]
-1/24*sqrt(2*sqrt(3) - 3)*log((x^4 + 2*x^3 + 6*x^2 + 2*sqrt(x^3 - 1)*(2*x^ 2 + sqrt(3)*(x^2 + 2*x) + 2*x + 2)*sqrt(2*sqrt(3) - 3) + 4*sqrt(3)*(x^3 - 1) - 4*x + 4)/(x^4 - 4*x^3 + 8*x + 4)) + 1/24*sqrt(2*sqrt(3) - 3)*log((x^4 + 2*x^3 + 6*x^2 - 2*sqrt(x^3 - 1)*(2*x^2 + sqrt(3)*(x^2 + 2*x) + 2*x + 2) *sqrt(2*sqrt(3) - 3) + 4*sqrt(3)*(x^3 - 1) - 4*x + 4)/(x^4 - 4*x^3 + 8*x + 4)) - 1/24*sqrt(-2*sqrt(3) - 3)*log((x^4 + 2*x^3 + 6*x^2 + 2*sqrt(x^3 - 1 )*(2*x^2 - sqrt(3)*(x^2 + 2*x) + 2*x + 2)*sqrt(-2*sqrt(3) - 3) - 4*sqrt(3) *(x^3 - 1) - 4*x + 4)/(x^4 - 4*x^3 + 8*x + 4)) + 1/24*sqrt(-2*sqrt(3) - 3) *log((x^4 + 2*x^3 + 6*x^2 - 2*sqrt(x^3 - 1)*(2*x^2 - sqrt(3)*(x^2 + 2*x) + 2*x + 2)*sqrt(-2*sqrt(3) - 3) - 4*sqrt(3)*(x^3 - 1) - 4*x + 4)/(x^4 - 4*x ^3 + 8*x + 4))
\[ \int \frac {-1+x}{\left (-2-2 x+x^2\right ) \sqrt {-1+x^3}} \, dx=\int \frac {x - 1}{\sqrt {\left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x^{2} - 2 x - 2\right )}\, dx \]
\[ \int \frac {-1+x}{\left (-2-2 x+x^2\right ) \sqrt {-1+x^3}} \, dx=\int { \frac {x - 1}{\sqrt {x^{3} - 1} {\left (x^{2} - 2 \, x - 2\right )}} \,d x } \]
\[ \int \frac {-1+x}{\left (-2-2 x+x^2\right ) \sqrt {-1+x^3}} \, dx=\int { \frac {x - 1}{\sqrt {x^{3} - 1} {\left (x^{2} - 2 \, x - 2\right )}} \,d x } \]
Time = 0.25 (sec) , antiderivative size = 221, normalized size of antiderivative = 2.23 \[ \int \frac {-1+x}{\left (-2-2 x+x^2\right ) \sqrt {-1+x^3}} \, dx=-\frac {\left (\Pi \left (\sqrt {3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right );\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )-\Pi \left (-\sqrt {3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right );\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (\sqrt {3}+1{}\mathrm {i}\right )}{2\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]
-((ellipticPi(3^(1/2)*((3^(1/2)*1i)/6 + 1/2), 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)) - ellip ticPi(-3^(1/2)*((3^(1/2)*1i)/6 + 1/2), 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)))*(-(x - (3^(1/ 2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2)*((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*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) - x*(((3^(1/ 2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) + x^3)^(1/2))