Integrand size = 28, antiderivative size = 111 \[ \int \frac {1-x+x^2}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}} \, dx=-\frac {1}{2} \sqrt {\frac {1}{3} \left (3+2 \sqrt {3}\right )} \arctan \left (\frac {\sqrt {3+2 \sqrt {3}} \sqrt {1+x^3}}{1-x+x^2}\right )-\frac {1}{2} \sqrt {\frac {1}{3} \left (-3+2 \sqrt {3}\right )} \text {arctanh}\left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt {1+x^3}}{1-x+x^2}\right ) \]
-1/6*(9+6*3^(1/2))^(1/2)*arctan((3+2*3^(1/2))^(1/2)*(x^3+1)^(1/2)/(x^2-x+1 ))-1/6*(-9+6*3^(1/2))^(1/2)*arctanh((-3+2*3^(1/2))^(1/2)*(x^3+1)^(1/2)/(x^ 2-x+1))
Time = 1.15 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.95 \[ \int \frac {1-x+x^2}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}} \, dx=-\frac {\sqrt {3+2 \sqrt {3}} \arctan \left (\frac {\sqrt {3+2 \sqrt {3}} \sqrt {1+x^3}}{1-x+x^2}\right )+\sqrt {-3+2 \sqrt {3}} \text {arctanh}\left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt {1+x^3}}{1-x+x^2}\right )}{2 \sqrt {3}} \]
-1/2*(Sqrt[3 + 2*Sqrt[3]]*ArcTan[(Sqrt[3 + 2*Sqrt[3]]*Sqrt[1 + x^3])/(1 - x + x^2)] + Sqrt[-3 + 2*Sqrt[3]]*ArcTanh[(Sqrt[-3 + 2*Sqrt[3]]*Sqrt[1 + x^ 3])/(1 - x + x^2)])/Sqrt[3]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.08 (sec) , antiderivative size = 412, normalized size of antiderivative = 3.71, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, 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.
| \(\displaystyle \int \frac {x^2-x+1}{\left (x^2+2 x-2\right ) \sqrt {x^3+1}} \, dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {1}{\sqrt {x^3+1}}+\frac {3 (1-x)}{\left (x^2+2 x-2\right ) \sqrt {x^3+1}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\left (2+\sqrt {3}\right )^{3/2} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{2 \sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}+\frac {2 \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {\sqrt {2-\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{2 \sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {1}{2} \sqrt {\frac {1}{3} \left (3+2 \sqrt {3}\right )} \arctan \left (\frac {\sqrt {3+2 \sqrt {3}} (x+1)}{\sqrt {x^3+1}}\right )-\frac {1}{2} \sqrt {\frac {1}{3} \left (2 \sqrt {3}-3\right )} \text {arctanh}\left (\frac {\sqrt {2 \sqrt {3}-3} (x+1)}{\sqrt {x^3+1}}\right )\) |
-1/2*(Sqrt[(3 + 2*Sqrt[3])/3]*ArcTan[(Sqrt[3 + 2*Sqrt[3]]*(1 + x))/Sqrt[1 + x^3]]) - (Sqrt[(-3 + 2*Sqrt[3])/3]*ArcTanh[(Sqrt[-3 + 2*Sqrt[3]]*(1 + x) )/Sqrt[1 + x^3]])/2 - (Sqrt[2 - Sqrt[3]]*(1 + x)*Sqrt[(1 - x + x^2)/(1 + S qrt[3] + x)^2]*EllipticF[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]])/(2*3^(1/4)*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3]) + (2*Sqrt[2 + Sqrt[3]]*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*Elli pticF[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]])/(3^(1/ 4)*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3]) - ((2 + Sqrt[3])^(3/2) *(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*EllipticF[ArcSin[(1 - Sqr t[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]])/(2*3^(1/4)*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3])
3.17.34.3.1 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 9 vs. order 3.
Time = 3.81 (sec) , antiderivative size = 584, normalized size of antiderivative = 5.26
| method | result | size |
| trager | \(-\frac {\operatorname {RootOf}\left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right ) \ln \left (-\frac {240 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{5} x^{2}-480 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{5} x +232 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{3} x^{2}-304 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{3} x +160 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{3}+32 \sqrt {x^{3}+1}\, \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{2}+55 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right ) x^{2}-22 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right ) x +88 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )+16 \sqrt {x^{3}+1}}{{\left (12 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{2} x +5 x -4\right )}^{2}}\right )}{2}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{2}+2\right ) \ln \left (-\frac {240 \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{2}+2\right ) \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{4} x^{2}-480 \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{2}+2\right ) \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{4} x +8 \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{2}+2\right ) \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{2} x^{2}-176 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{2}+2\right ) x -64 \sqrt {x^{3}+1}\, \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{2}-160 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{2}+2\right )-\operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{2}+2\right ) x^{2}+10 \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{2}+2\right ) x +8 \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{2}+2\right )}{{\left (12 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{2} x +x +4\right )}^{2}}\right )}{4}\) | \(584\) |
| default | \(\text {Expression too large to display}\) | \(1501\) |
| elliptic | \(\text {Expression too large to display}\) | \(1706\) |
-1/2*RootOf(48*_Z^4+24*_Z^2-1)*ln(-(240*RootOf(48*_Z^4+24*_Z^2-1)^5*x^2-48 0*RootOf(48*_Z^4+24*_Z^2-1)^5*x+232*RootOf(48*_Z^4+24*_Z^2-1)^3*x^2-304*Ro otOf(48*_Z^4+24*_Z^2-1)^3*x+160*RootOf(48*_Z^4+24*_Z^2-1)^3+32*(x^3+1)^(1/ 2)*RootOf(48*_Z^4+24*_Z^2-1)^2+55*RootOf(48*_Z^4+24*_Z^2-1)*x^2-22*RootOf( 48*_Z^4+24*_Z^2-1)*x+88*RootOf(48*_Z^4+24*_Z^2-1)+16*(x^3+1)^(1/2))/(12*Ro otOf(48*_Z^4+24*_Z^2-1)^2*x+5*x-4)^2)-1/4*RootOf(_Z^2+4*RootOf(48*_Z^4+24* _Z^2-1)^2+2)*ln(-(240*RootOf(_Z^2+4*RootOf(48*_Z^4+24*_Z^2-1)^2+2)*RootOf( 48*_Z^4+24*_Z^2-1)^4*x^2-480*RootOf(_Z^2+4*RootOf(48*_Z^4+24*_Z^2-1)^2+2)* RootOf(48*_Z^4+24*_Z^2-1)^4*x+8*RootOf(_Z^2+4*RootOf(48*_Z^4+24*_Z^2-1)^2+ 2)*RootOf(48*_Z^4+24*_Z^2-1)^2*x^2-176*RootOf(48*_Z^4+24*_Z^2-1)^2*RootOf( _Z^2+4*RootOf(48*_Z^4+24*_Z^2-1)^2+2)*x-64*(x^3+1)^(1/2)*RootOf(48*_Z^4+24 *_Z^2-1)^2-160*RootOf(48*_Z^4+24*_Z^2-1)^2*RootOf(_Z^2+4*RootOf(48*_Z^4+24 *_Z^2-1)^2+2)-RootOf(_Z^2+4*RootOf(48*_Z^4+24*_Z^2-1)^2+2)*x^2+10*RootOf(_ Z^2+4*RootOf(48*_Z^4+24*_Z^2-1)^2+2)*x+8*RootOf(_Z^2+4*RootOf(48*_Z^4+24*_ Z^2-1)^2+2))/(12*RootOf(48*_Z^4+24*_Z^2-1)^2*x+x+4)^2)
Leaf count of result is larger than twice the leaf count of optimal. 373 vs. \(2 (79) = 158\).
Time = 0.31 (sec) , antiderivative size = 373, normalized size of antiderivative = 3.36 \[ \int \frac {1-x+x^2}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}} \, dx=-\frac {1}{24} \, \sqrt {3} \sqrt {2 \, \sqrt {3} - 3} \log \left (\frac {x^{4} - 2 \, x^{3} + 6 \, x^{2} + 2 \, \sqrt {x^{3} + 1} {\left (x^{2} + 2 \, \sqrt {3} {\left (x + 1\right )} + 2 \, x + 4\right )} \sqrt {2 \, \sqrt {3} - 3} + 4 \, \sqrt {3} {\left (x^{3} + 1\right )} + 4 \, x + 4}{x^{4} + 4 \, x^{3} - 8 \, x + 4}\right ) + \frac {1}{24} \, \sqrt {3} \sqrt {2 \, \sqrt {3} - 3} \log \left (\frac {x^{4} - 2 \, x^{3} + 6 \, x^{2} - 2 \, \sqrt {x^{3} + 1} {\left (x^{2} + 2 \, \sqrt {3} {\left (x + 1\right )} + 2 \, x + 4\right )} \sqrt {2 \, \sqrt {3} - 3} + 4 \, \sqrt {3} {\left (x^{3} + 1\right )} + 4 \, x + 4}{x^{4} + 4 \, x^{3} - 8 \, x + 4}\right ) + \frac {1}{24} \, \sqrt {3} \sqrt {-2 \, \sqrt {3} - 3} \log \left (\frac {x^{4} - 2 \, x^{3} + 6 \, x^{2} + 2 \, \sqrt {x^{3} + 1} {\left (x^{2} - 2 \, \sqrt {3} {\left (x + 1\right )} + 2 \, x + 4\right )} \sqrt {-2 \, \sqrt {3} - 3} - 4 \, \sqrt {3} {\left (x^{3} + 1\right )} + 4 \, x + 4}{x^{4} + 4 \, x^{3} - 8 \, x + 4}\right ) - \frac {1}{24} \, \sqrt {3} \sqrt {-2 \, \sqrt {3} - 3} \log \left (\frac {x^{4} - 2 \, x^{3} + 6 \, x^{2} - 2 \, \sqrt {x^{3} + 1} {\left (x^{2} - 2 \, \sqrt {3} {\left (x + 1\right )} + 2 \, x + 4\right )} \sqrt {-2 \, \sqrt {3} - 3} - 4 \, \sqrt {3} {\left (x^{3} + 1\right )} + 4 \, x + 4}{x^{4} + 4 \, x^{3} - 8 \, x + 4}\right ) \]
-1/24*sqrt(3)*sqrt(2*sqrt(3) - 3)*log((x^4 - 2*x^3 + 6*x^2 + 2*sqrt(x^3 + 1)*(x^2 + 2*sqrt(3)*(x + 1) + 2*x + 4)*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(3)*sqrt(2*sqrt(3) - 3)*log((x^4 - 2*x^3 + 6*x^2 - 2*sqrt(x^3 + 1)*(x^2 + 2*sqrt(3)*(x + 1) + 2*x + 4)*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(3)*sqrt(-2*sqrt(3) - 3)*log((x^4 - 2*x^3 + 6*x^2 + 2*sqrt(x^3 + 1)*(x^2 - 2*sqrt(3)*(x + 1) + 2*x + 4)*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(3)* sqrt(-2*sqrt(3) - 3)*log((x^4 - 2*x^3 + 6*x^2 - 2*sqrt(x^3 + 1)*(x^2 - 2*s qrt(3)*(x + 1) + 2*x + 4)*sqrt(-2*sqrt(3) - 3) - 4*sqrt(3)*(x^3 + 1) + 4*x + 4)/(x^4 + 4*x^3 - 8*x + 4))
\[ \int \frac {1-x+x^2}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}} \, dx=\int \frac {x^{2} - x + 1}{\sqrt {\left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x^{2} + 2 x - 2\right )}\, dx \]
\[ \int \frac {1-x+x^2}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}} \, dx=\int { \frac {x^{2} - x + 1}{\sqrt {x^{3} + 1} {\left (x^{2} + 2 \, x - 2\right )}} \,d x } \]
\[ \int \frac {1-x+x^2}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}} \, dx=\int { \frac {x^{2} - x + 1}{\sqrt {x^{3} + 1} {\left (x^{2} + 2 \, x - 2\right )}} \,d x } \]
Time = 0.22 (sec) , antiderivative size = 509, normalized size of antiderivative = 4.59 \[ \int \frac {1-x+x^2}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}} \, dx=\frac {2\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\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}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\mathrm {F}\left (\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 )}{\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 )}}+\frac {\left (3\,\sqrt {3}-6\right )\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\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}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (\frac {\sqrt {3}\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3};\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 )}{3\,\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 )}}-\frac {\left (3\,\sqrt {3}+6\right )\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\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}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (-\frac {\sqrt {3}\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3};\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 )}{3\,\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 )}} \]
(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)*ellipticF(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 - 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) + ((3*3^(1/2) - 6)*((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)*((3^(1/2)*1i)/2 + 3/2))/3, 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*(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)) - ((3*3^(1/2) + 6)*((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)*((3^(1/2)*1i)/2 + 3/2))/3, 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*(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))