Integrand size = 23, antiderivative size = 103 \[ \int \frac {1+x}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}} \, dx=\frac {1}{6} \sqrt {-3+2 \sqrt {3}} \arctan \left (\frac {\sqrt {3+2 \sqrt {3}} \sqrt {1+x^3}}{1-x+x^2}\right )-\frac {1}{6} \sqrt {3+2 \sqrt {3}} \text {arctanh}\left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt {1+x^3}}{1-x+x^2}\right ) \]
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 = 103, 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 = 0.71 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.90, 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 |
| \(\displaystyle \frac {\arctan \left (\frac {\sqrt {3+2 \sqrt {3}} (x+1)}{\sqrt {x^3+1}}\right )}{2 \sqrt {3 \left (3+2 \sqrt {3}\right )}}-\frac {\text {arctanh}\left (\frac {\sqrt {2 \sqrt {3}-3} (x+1)}{\sqrt {x^3+1}}\right )}{2 \sqrt {3 \left (2 \sqrt {3}-3\right )}}\) |
ArcTan[(Sqrt[3 + 2*Sqrt[3]]*(1 + x))/Sqrt[1 + x^3]]/(2*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])])
3.15.58.3.1 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 = 0.61 (sec) , antiderivative size = 694, normalized size of antiderivative = 6.74
| method | result | size |
| default | \(-\frac {\sqrt {\frac {1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}-\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}{2 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}+\frac {i \sqrt {3}}{-3+i \sqrt {3}}}\, \sqrt {3}\, \operatorname {EllipticPi}\left (\sqrt {\frac {1+x}{\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 {i \sqrt {\frac {1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}-\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}{2 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}+\frac {i \sqrt {3}}{-3+i \sqrt {3}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {1+x}{\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 {1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}-\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}{2 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}+\frac {i \sqrt {3}}{-3+i \sqrt {3}}}\, \sqrt {3}\, \operatorname {EllipticPi}\left (\sqrt {\frac {1+x}{\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 {i \sqrt {\frac {1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}-\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}{2 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}+\frac {i \sqrt {3}}{-3+i \sqrt {3}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {1+x}{\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}}\) | \(694\) |
| elliptic | \(-\frac {\sqrt {\frac {1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}-\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}{2 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}+\frac {i \sqrt {3}}{-3+i \sqrt {3}}}\, \sqrt {3}\, \operatorname {EllipticPi}\left (\sqrt {\frac {1+x}{\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 {i \sqrt {\frac {1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}-\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}{2 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}+\frac {i \sqrt {3}}{-3+i \sqrt {3}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {1+x}{\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 {1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}-\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}{2 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}+\frac {i \sqrt {3}}{-3+i \sqrt {3}}}\, \sqrt {3}\, \operatorname {EllipticPi}\left (\sqrt {\frac {1+x}{\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 {i \sqrt {\frac {1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}-\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}{2 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}+\frac {i \sqrt {3}}{-3+i \sqrt {3}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {1+x}{\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}}\) | \(694\) |
-1/2*(1/(3/2-1/2*I*3^(1/2))+1/(3/2-1/2*I*3^(1/2))*x)^(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)*3^(1/2)*EllipticPi(((1+x)/(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*I*(1/(3/2-1/2*I*3^(1/2))+1/(3/2-1/2*I*3^(1/ 2))*x)^(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)*EllipticP i(((1+x)/(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))+ 1/(3/2-1/2*I*3^(1/2))*x)^(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)*3^(1/2)*EllipticPi(((1+x)/(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*I*(1/(3/2-1/2*I*3^(1/2))+1/(3/2-1/2*I*3^(1/2))*x)^(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(((1+x)/(3/2-1/2*I*3^(1/2...
Leaf count of result is larger than twice the leaf count of optimal. 383 vs. \(2 (79) = 158\).
Time = 0.29 (sec) , antiderivative size = 383, normalized size of antiderivative = 3.72 \[ \int \frac {1+x}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}} \, dx=-\frac {1}{24} \, \sqrt {2 \, \sqrt {3} + 3} \log \left (\frac {x^{4} - 2 \, x^{3} + 6 \, x^{2} + 2 \, \sqrt {x^{3} + 1} {\left (2 \, x^{2} - \sqrt {3} {\left (x^{2} - 2 \, x\right )} - 2 \, x + 2\right )} \sqrt {2 \, \sqrt {3} + 3} + 4 \, \sqrt {3} {\left (x^{3} + 1\right )} + 4 \, x + 4}{x^{4} + 4 \, x^{3} - 8 \, x + 4}\right ) + \frac {1}{24} \, \sqrt {2 \, \sqrt {3} + 3} \log \left (\frac {x^{4} - 2 \, x^{3} + 6 \, x^{2} - 2 \, \sqrt {x^{3} + 1} {\left (2 \, x^{2} - \sqrt {3} {\left (x^{2} - 2 \, x\right )} - 2 \, x + 2\right )} \sqrt {2 \, \sqrt {3} + 3} + 4 \, \sqrt {3} {\left (x^{3} + 1\right )} + 4 \, x + 4}{x^{4} + 4 \, x^{3} - 8 \, x + 4}\right ) - \frac {1}{24} \, \sqrt {-2 \, \sqrt {3} + 3} \log \left (\frac {x^{4} - 2 \, x^{3} + 6 \, x^{2} + 2 \, \sqrt {x^{3} + 1} {\left (2 \, x^{2} + \sqrt {3} {\left (x^{2} - 2 \, x\right )} - 2 \, x + 2\right )} \sqrt {-2 \, \sqrt {3} + 3} - 4 \, \sqrt {3} {\left (x^{3} + 1\right )} + 4 \, x + 4}{x^{4} + 4 \, x^{3} - 8 \, x + 4}\right ) + \frac {1}{24} \, \sqrt {-2 \, \sqrt {3} + 3} \log \left (\frac {x^{4} - 2 \, x^{3} + 6 \, x^{2} - 2 \, \sqrt {x^{3} + 1} {\left (2 \, x^{2} + \sqrt {3} {\left (x^{2} - 2 \, x\right )} - 2 \, x + 2\right )} \sqrt {-2 \, \sqrt {3} + 3} - 4 \, \sqrt {3} {\left (x^{3} + 1\right )} + 4 \, x + 4}{x^{4} + 4 \, x^{3} - 8 \, x + 4}\right ) \]
-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 = 0.24 (sec) , antiderivative size = 220, normalized size of antiderivative = 2.14 \[ \int \frac {1+x}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}} \, dx=-\frac {\left (\Pi \left (\sqrt {3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right );\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )-\Pi \left (-\sqrt {3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right );\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )\right )\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (\sqrt {3}+1{}\mathrm {i}\right )\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}}{2\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]
-((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)) - ellipt icPi(-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 + 1)/((3^(1/2)*1i)/2 + 3/2) )^(1/2)*(3^(1/2) + 1i)*(((3^(1/2)*1i)/2 - x + 1/2)/((3^(1/2)*1i)/2 + 3/2)) ^(1/2))/(2*(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))