Integrand size = 24, antiderivative size = 125 \[ \int \frac {\sqrt {-x-x^2+x^3}}{-1+x^4} \, dx=\frac {1}{2} \arctan \left (\frac {\sqrt {-x-x^2+x^3}}{-1-x+x^2}\right )-\frac {1}{4} \sqrt {1-2 i} \arctan \left (\frac {\sqrt {1-2 i} \sqrt {-x-x^2+x^3}}{-1-x+x^2}\right )-\frac {1}{4} \sqrt {1+2 i} \arctan \left (\frac {\sqrt {1+2 i} \sqrt {-x-x^2+x^3}}{-1-x+x^2}\right ) \]
1/2*arctan((x^3-x^2-x)^(1/2)/(x^2-x-1))-1/4*(1-2*I)^(1/2)*arctan((1-2*I)^( 1/2)*(x^3-x^2-x)^(1/2)/(x^2-x-1))-1/4*(1+2*I)^(1/2)*arctan((1+2*I)^(1/2)*( x^3-x^2-x)^(1/2)/(x^2-x-1))
Time = 0.39 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt {-x-x^2+x^3}}{-1+x^4} \, dx=\frac {\sqrt {x} \sqrt {-1-x+x^2} \left (2 \arctan \left (\frac {\sqrt {x}}{\sqrt {-1-x+x^2}}\right )-\sqrt {1-2 i} \arctan \left (\frac {\sqrt {1-2 i} \sqrt {x}}{\sqrt {-1-x+x^2}}\right )-\sqrt {1+2 i} \arctan \left (\frac {\sqrt {1+2 i} \sqrt {x}}{\sqrt {-1-x+x^2}}\right )\right )}{4 \sqrt {x \left (-1-x+x^2\right )}} \]
(Sqrt[x]*Sqrt[-1 - x + x^2]*(2*ArcTan[Sqrt[x]/Sqrt[-1 - x + x^2]] - Sqrt[1 - 2*I]*ArcTan[(Sqrt[1 - 2*I]*Sqrt[x])/Sqrt[-1 - x + x^2]] - Sqrt[1 + 2*I] *ArcTan[(Sqrt[1 + 2*I]*Sqrt[x])/Sqrt[-1 - x + x^2]]))/(4*Sqrt[x*(-1 - x + x^2)])
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 2.71 (sec) , antiderivative size = 1571, normalized size of antiderivative = 12.57, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {2467, 25, 2035, 7276, 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.
| \(\displaystyle \int \frac {\sqrt {x^3-x^2-x}}{x^4-1} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x^3-x^2-x} \int -\frac {\sqrt {x} \sqrt {x^2-x-1}}{1-x^4}dx}{\sqrt {x} \sqrt {x^2-x-1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x^3-x^2-x} \int \frac {\sqrt {x} \sqrt {x^2-x-1}}{1-x^4}dx}{\sqrt {x} \sqrt {x^2-x-1}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x^3-x^2-x} \int \frac {x \sqrt {x^2-x-1}}{1-x^4}d\sqrt {x}}{\sqrt {x} \sqrt {x^2-x-1}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {2 \sqrt {x^3-x^2-x} \int \left (\frac {x \sqrt {x^2-x-1}}{2 \left (x^2+1\right )}-\frac {x \sqrt {x^2-x-1}}{2 \left (x^2-1\right )}\right )d\sqrt {x}}{\sqrt {x} \sqrt {x^2-x-1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {x^3-x^2-x} \left (\frac {1}{8} \sqrt {1-2 i} \arctan \left (\frac {\sqrt {1-2 i} \sqrt {x}}{\sqrt {x^2-x-1}}\right )+\frac {1}{8} \sqrt {1+2 i} \arctan \left (\frac {\sqrt {1+2 i} \sqrt {x}}{\sqrt {x^2-x-1}}\right )+\frac {\left (1+\sqrt {5}\right ) \sqrt {2 x+\sqrt {5}-1} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{4 \sqrt {2} \left (3+\sqrt {5}\right ) \sqrt {x^2-x-1}}+\frac {\left (1+\sqrt {5}\right ) \sqrt {2 x+\sqrt {5}-1} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{4 \sqrt {2} \left (1-\sqrt {5}\right ) \sqrt {x^2-x-1}}-\frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x+2}{\left (1-\sqrt {5}\right ) x+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{4 \sqrt [4]{5} \left (3+\sqrt {5}\right ) \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^2-x-1}}+\frac {\left ((1+2 i)+\sqrt {5}\right ) \sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x+2}{\left (1-\sqrt {5}\right ) x+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{16 \sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^2-x-1}}+\frac {\left ((1-2 i)+\sqrt {5}\right ) \sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x+2}{\left (1-\sqrt {5}\right ) x+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{16 \sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^2-x-1}}-\frac {\left (1+\sqrt {5}\right ) \sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x+2}{\left (1-\sqrt {5}\right ) x+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{16 \sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^2-x-1}}+\frac {\left (3-\sqrt {5}\right ) \sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x+2}{\left (1-\sqrt {5}\right ) x+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{16 \sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^2-x-1}}-\frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x+2}{\left (1-\sqrt {5}\right ) x+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{4 \sqrt [4]{5} \left (1-\sqrt {5}\right ) \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^2-x-1}}-\frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x+2}{\left (1-\sqrt {5}\right ) x+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{8 \sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^2-x-1}}-\frac {\left (2+\sqrt {5}\right ) \sqrt {2 x+\sqrt {5}-1} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \operatorname {EllipticPi}\left (\frac {1}{2} \left (-1-\sqrt {5}\right ),\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{2 \sqrt {2} \left (3+\sqrt {5}\right ) \sqrt {x^2-x-1}}+\frac {\sqrt {2 x+\sqrt {5}-1} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \operatorname {EllipticPi}\left (\frac {1}{2} \left (1+\sqrt {5}\right ),\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{2 \sqrt {2} \left (1-\sqrt {5}\right ) \sqrt {x^2-x-1}}\right )}{\sqrt {x} \sqrt {x^2-x-1}}\) |
(-2*Sqrt[-x - x^2 + x^3]*((Sqrt[1 - 2*I]*ArcTan[(Sqrt[1 - 2*I]*Sqrt[x])/Sq rt[-1 - x + x^2]])/8 + (Sqrt[1 + 2*I]*ArcTan[(Sqrt[1 + 2*I]*Sqrt[x])/Sqrt[ -1 - x + x^2]])/8 + ((1 + Sqrt[5])*Sqrt[-1 + Sqrt[5] + 2*x]*Sqrt[1 - (2*x) /(1 + Sqrt[5])]*EllipticF[ArcSin[Sqrt[2/(1 + Sqrt[5])]*Sqrt[x]], (-3 - Sqr t[5])/2])/(4*Sqrt[2]*(1 - Sqrt[5])*Sqrt[-1 - x + x^2]) + ((1 + Sqrt[5])*Sq rt[-1 + Sqrt[5] + 2*x]*Sqrt[1 - (2*x)/(1 + Sqrt[5])]*EllipticF[ArcSin[Sqrt [2/(1 + Sqrt[5])]*Sqrt[x]], (-3 - Sqrt[5])/2])/(4*Sqrt[2]*(3 + Sqrt[5])*Sq rt[-1 - x + x^2]) - (Sqrt[-2 - (1 - Sqrt[5])*x]*Sqrt[(2 + (1 + Sqrt[5])*x) /(2 + (1 - Sqrt[5])*x)]*EllipticF[ArcSin[(Sqrt[2]*5^(1/4)*Sqrt[x])/Sqrt[-2 - (1 - Sqrt[5])*x]], (5 - Sqrt[5])/10])/(8*5^(1/4)*Sqrt[(2 + (1 - Sqrt[5] )*x)^(-1)]*Sqrt[-1 - x + x^2]) - (Sqrt[-2 - (1 - Sqrt[5])*x]*Sqrt[(2 + (1 + Sqrt[5])*x)/(2 + (1 - Sqrt[5])*x)]*EllipticF[ArcSin[(Sqrt[2]*5^(1/4)*Sqr t[x])/Sqrt[-2 - (1 - Sqrt[5])*x]], (5 - Sqrt[5])/10])/(4*5^(1/4)*(1 - Sqrt [5])*Sqrt[(2 + (1 - Sqrt[5])*x)^(-1)]*Sqrt[-1 - x + x^2]) + ((3 - Sqrt[5]) *Sqrt[-2 - (1 - Sqrt[5])*x]*Sqrt[(2 + (1 + Sqrt[5])*x)/(2 + (1 - Sqrt[5])* x)]*EllipticF[ArcSin[(Sqrt[2]*5^(1/4)*Sqrt[x])/Sqrt[-2 - (1 - Sqrt[5])*x]] , (5 - Sqrt[5])/10])/(16*5^(1/4)*Sqrt[(2 + (1 - Sqrt[5])*x)^(-1)]*Sqrt[-1 - x + x^2]) - ((1 + Sqrt[5])*Sqrt[-2 - (1 - Sqrt[5])*x]*Sqrt[(2 + (1 + Sqr t[5])*x)/(2 + (1 - Sqrt[5])*x)]*EllipticF[ArcSin[(Sqrt[2]*5^(1/4)*Sqrt[x]) /Sqrt[-2 - (1 - Sqrt[5])*x]], (5 - Sqrt[5])/10])/(16*5^(1/4)*Sqrt[(2 + ...
3.19.38.3.1 Defintions of rubi rules used
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(217\) vs. \(2(105)=210\).
Time = 3.51 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.74
| method | result | size |
| default | \(\frac {-\ln \left (\frac {-\sqrt {x \left (x^{2}-x -1\right )}\, \sqrt {-2+2 \sqrt {5}}+x \sqrt {5}+x^{2}-x -1}{x}\right )+\ln \left (\frac {\sqrt {x \left (x^{2}-x -1\right )}\, \sqrt {-2+2 \sqrt {5}}+x \sqrt {5}+x^{2}-x -1}{x}\right )+\left (-\sqrt {5}-1\right ) \arctan \left (\frac {\sqrt {-2+2 \sqrt {5}}\, x -2 \sqrt {x \left (x^{2}-x -1\right )}}{x \sqrt {2+2 \sqrt {5}}}\right )+\left (\sqrt {5}+1\right ) \arctan \left (\frac {\sqrt {-2+2 \sqrt {5}}\, x +2 \sqrt {x \left (x^{2}-x -1\right )}}{x \sqrt {2+2 \sqrt {5}}}\right )-2 \arctan \left (\frac {\sqrt {x \left (x^{2}-x -1\right )}}{x}\right ) \sqrt {2+2 \sqrt {5}}}{4 \sqrt {2+2 \sqrt {5}}}\) | \(218\) |
| pseudoelliptic | \(\frac {-\ln \left (\frac {-\sqrt {x \left (x^{2}-x -1\right )}\, \sqrt {-2+2 \sqrt {5}}+x \sqrt {5}+x^{2}-x -1}{x}\right )+\ln \left (\frac {\sqrt {x \left (x^{2}-x -1\right )}\, \sqrt {-2+2 \sqrt {5}}+x \sqrt {5}+x^{2}-x -1}{x}\right )+\left (-\sqrt {5}-1\right ) \arctan \left (\frac {\sqrt {-2+2 \sqrt {5}}\, x -2 \sqrt {x \left (x^{2}-x -1\right )}}{x \sqrt {2+2 \sqrt {5}}}\right )+\left (\sqrt {5}+1\right ) \arctan \left (\frac {\sqrt {-2+2 \sqrt {5}}\, x +2 \sqrt {x \left (x^{2}-x -1\right )}}{x \sqrt {2+2 \sqrt {5}}}\right )-2 \arctan \left (\frac {\sqrt {x \left (x^{2}-x -1\right )}}{x}\right ) \sqrt {2+2 \sqrt {5}}}{4 \sqrt {2+2 \sqrt {5}}}\) | \(218\) |
| trager | \(\text {Expression too large to display}\) | \(913\) |
| elliptic | \(\text {Expression too large to display}\) | \(1430\) |
1/4*(-ln((-(x*(x^2-x-1))^(1/2)*(-2+2*5^(1/2))^(1/2)+x*5^(1/2)+x^2-x-1)/x)+ ln(((x*(x^2-x-1))^(1/2)*(-2+2*5^(1/2))^(1/2)+x*5^(1/2)+x^2-x-1)/x)+(-5^(1/ 2)-1)*arctan(((-2+2*5^(1/2))^(1/2)*x-2*(x*(x^2-x-1))^(1/2))/x/(2+2*5^(1/2) )^(1/2))+(5^(1/2)+1)*arctan(((-2+2*5^(1/2))^(1/2)*x+2*(x*(x^2-x-1))^(1/2)) /x/(2+2*5^(1/2))^(1/2))-2*arctan((x*(x^2-x-1))^(1/2)/x)*(2+2*5^(1/2))^(1/2 ))/(2+2*5^(1/2))^(1/2)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 284 vs. \(2 (97) = 194\).
Time = 0.30 (sec) , antiderivative size = 284, normalized size of antiderivative = 2.27 \[ \int \frac {\sqrt {-x-x^2+x^3}}{-1+x^4} \, dx=\frac {1}{16} \, \sqrt {2 i - 1} \log \left (\frac {x^{4} + \left (4 i - 2\right ) \, x^{3} + 2 \, \sqrt {2 i - 1} \sqrt {x^{3} - x^{2} - x} {\left (x^{2} + 2 i \, x - 1\right )} - \left (4 i + 6\right ) \, x^{2} - \left (4 i - 2\right ) \, x + 1}{x^{4} + 2 \, x^{2} + 1}\right ) - \frac {1}{16} \, \sqrt {2 i - 1} \log \left (\frac {x^{4} + \left (4 i - 2\right ) \, x^{3} - 2 \, \sqrt {2 i - 1} \sqrt {x^{3} - x^{2} - x} {\left (x^{2} + 2 i \, x - 1\right )} - \left (4 i + 6\right ) \, x^{2} - \left (4 i - 2\right ) \, x + 1}{x^{4} + 2 \, x^{2} + 1}\right ) + \frac {1}{16} \, \sqrt {-2 i - 1} \log \left (\frac {x^{4} - \left (4 i + 2\right ) \, x^{3} + 2 \, \sqrt {-2 i - 1} \sqrt {x^{3} - x^{2} - x} {\left (x^{2} - 2 i \, x - 1\right )} + \left (4 i - 6\right ) \, x^{2} + \left (4 i + 2\right ) \, x + 1}{x^{4} + 2 \, x^{2} + 1}\right ) - \frac {1}{16} \, \sqrt {-2 i - 1} \log \left (\frac {x^{4} - \left (4 i + 2\right ) \, x^{3} - 2 \, \sqrt {-2 i - 1} \sqrt {x^{3} - x^{2} - x} {\left (x^{2} - 2 i \, x - 1\right )} + \left (4 i - 6\right ) \, x^{2} + \left (4 i + 2\right ) \, x + 1}{x^{4} + 2 \, x^{2} + 1}\right ) - \frac {1}{4} \, \arctan \left (\frac {x^{2} - 2 \, x - 1}{2 \, \sqrt {x^{3} - x^{2} - x}}\right ) \]
1/16*sqrt(2*I - 1)*log((x^4 + (4*I - 2)*x^3 + 2*sqrt(2*I - 1)*sqrt(x^3 - x ^2 - x)*(x^2 + 2*I*x - 1) - (4*I + 6)*x^2 - (4*I - 2)*x + 1)/(x^4 + 2*x^2 + 1)) - 1/16*sqrt(2*I - 1)*log((x^4 + (4*I - 2)*x^3 - 2*sqrt(2*I - 1)*sqrt (x^3 - x^2 - x)*(x^2 + 2*I*x - 1) - (4*I + 6)*x^2 - (4*I - 2)*x + 1)/(x^4 + 2*x^2 + 1)) + 1/16*sqrt(-2*I - 1)*log((x^4 - (4*I + 2)*x^3 + 2*sqrt(-2*I - 1)*sqrt(x^3 - x^2 - x)*(x^2 - 2*I*x - 1) + (4*I - 6)*x^2 + (4*I + 2)*x + 1)/(x^4 + 2*x^2 + 1)) - 1/16*sqrt(-2*I - 1)*log((x^4 - (4*I + 2)*x^3 - 2 *sqrt(-2*I - 1)*sqrt(x^3 - x^2 - x)*(x^2 - 2*I*x - 1) + (4*I - 6)*x^2 + (4 *I + 2)*x + 1)/(x^4 + 2*x^2 + 1)) - 1/4*arctan(1/2*(x^2 - 2*x - 1)/sqrt(x^ 3 - x^2 - x))
\[ \int \frac {\sqrt {-x-x^2+x^3}}{-1+x^4} \, dx=\int \frac {\sqrt {x \left (x^{2} - x - 1\right )}}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}\, dx \]
\[ \int \frac {\sqrt {-x-x^2+x^3}}{-1+x^4} \, dx=\int { \frac {\sqrt {x^{3} - x^{2} - x}}{x^{4} - 1} \,d x } \]
\[ \int \frac {\sqrt {-x-x^2+x^3}}{-1+x^4} \, dx=\int { \frac {\sqrt {x^{3} - x^{2} - x}}{x^{4} - 1} \,d x } \]
Time = 0.09 (sec) , antiderivative size = 537, normalized size of antiderivative = 4.30 \[ \int \frac {\sqrt {-x-x^2+x^3}}{-1+x^4} \, dx=\frac {\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\sqrt {\frac {x+\frac {\sqrt {5}}{2}-\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\,\sqrt {\frac {\frac {\sqrt {5}}{2}-x+\frac {1}{2}}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\Pi \left (-\frac {\sqrt {5}}{2}-\frac {1}{2};\mathrm {asin}\left (\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\right )\middle |-\frac {\frac {\sqrt {5}}{2}+\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}\right )}{2\,\sqrt {x^3-x^2-\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )\,\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,x}}+\frac {\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\sqrt {\frac {x+\frac {\sqrt {5}}{2}-\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\,\sqrt {\frac {\frac {\sqrt {5}}{2}-x+\frac {1}{2}}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\Pi \left (\frac {\sqrt {5}}{2}+\frac {1}{2};\mathrm {asin}\left (\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\right )\middle |-\frac {\frac {\sqrt {5}}{2}+\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}\right )}{2\,\sqrt {x^3-x^2-\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )\,\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,x}}+\frac {\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\sqrt {\frac {x+\frac {\sqrt {5}}{2}-\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\,\sqrt {\frac {\frac {\sqrt {5}}{2}-x+\frac {1}{2}}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\Pi \left (-\frac {\sqrt {5}\,1{}\mathrm {i}}{2}-\frac {1}{2}{}\mathrm {i};\mathrm {asin}\left (\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\right )\middle |-\frac {\frac {\sqrt {5}}{2}+\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}\right )\,\left (-\frac {1}{2}+1{}\mathrm {i}\right )}{\sqrt {x^3-x^2-\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )\,\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,x}}+\frac {\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\sqrt {\frac {x+\frac {\sqrt {5}}{2}-\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\,\sqrt {\frac {\frac {\sqrt {5}}{2}-x+\frac {1}{2}}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\Pi \left (\frac {\sqrt {5}\,1{}\mathrm {i}}{2}+\frac {1}{2}{}\mathrm {i};\mathrm {asin}\left (\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\right )\middle |-\frac {\frac {\sqrt {5}}{2}+\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}\right )\,\left (-\frac {1}{2}-\mathrm {i}\right )}{\sqrt {x^3-x^2-\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )\,\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,x}} \]
((5^(1/2)/2 + 1/2)*(x/(5^(1/2)/2 + 1/2))^(1/2)*((x + 5^(1/2)/2 - 1/2)/(5^( 1/2)/2 - 1/2))^(1/2)*((5^(1/2)/2 - x + 1/2)/(5^(1/2)/2 + 1/2))^(1/2)*ellip ticPi(- 5^(1/2)/2 - 1/2, asin((x/(5^(1/2)/2 + 1/2))^(1/2)), -(5^(1/2)/2 + 1/2)/(5^(1/2)/2 - 1/2)))/(2*(x^3 - x^2 - x*(5^(1/2)/2 - 1/2)*(5^(1/2)/2 + 1/2))^(1/2)) + ((5^(1/2)/2 + 1/2)*(x/(5^(1/2)/2 + 1/2))^(1/2)*((x + 5^(1/2 )/2 - 1/2)/(5^(1/2)/2 - 1/2))^(1/2)*((5^(1/2)/2 - x + 1/2)/(5^(1/2)/2 + 1/ 2))^(1/2)*ellipticPi(5^(1/2)/2 + 1/2, asin((x/(5^(1/2)/2 + 1/2))^(1/2)), - (5^(1/2)/2 + 1/2)/(5^(1/2)/2 - 1/2)))/(2*(x^3 - x^2 - x*(5^(1/2)/2 - 1/2)* (5^(1/2)/2 + 1/2))^(1/2)) - ((5^(1/2)/2 + 1/2)*(x/(5^(1/2)/2 + 1/2))^(1/2) *((x + 5^(1/2)/2 - 1/2)/(5^(1/2)/2 - 1/2))^(1/2)*((5^(1/2)/2 - x + 1/2)/(5 ^(1/2)/2 + 1/2))^(1/2)*ellipticPi(- (5^(1/2)*1i)/2 - 1i/2, asin((x/(5^(1/2 )/2 + 1/2))^(1/2)), -(5^(1/2)/2 + 1/2)/(5^(1/2)/2 - 1/2))*(1/2 - 1i))/(x^3 - x^2 - x*(5^(1/2)/2 - 1/2)*(5^(1/2)/2 + 1/2))^(1/2) - ((5^(1/2)/2 + 1/2) *(x/(5^(1/2)/2 + 1/2))^(1/2)*((x + 5^(1/2)/2 - 1/2)/(5^(1/2)/2 - 1/2))^(1/ 2)*((5^(1/2)/2 - x + 1/2)/(5^(1/2)/2 + 1/2))^(1/2)*ellipticPi((5^(1/2)*1i) /2 + 1i/2, asin((x/(5^(1/2)/2 + 1/2))^(1/2)), -(5^(1/2)/2 + 1/2)/(5^(1/2)/ 2 - 1/2))*(1/2 + 1i))/(x^3 - x^2 - x*(5^(1/2)/2 - 1/2)*(5^(1/2)/2 + 1/2))^ (1/2)