Integrand size = 27, antiderivative size = 133 \[ \int \frac {x^2}{\sqrt {-x-x^2+x^3} \left (-1+x^4\right )} \, dx=-\frac {1}{2} \arctan \left (\frac {\sqrt {-x-x^2+x^3}}{-1-x+x^2}\right )+\frac {1}{4} \sqrt {\frac {1}{5}+\frac {2 i}{5}} \arctan \left (\frac {\sqrt {1-2 i} \sqrt {-x-x^2+x^3}}{-1-x+x^2}\right )+\frac {1}{4} \sqrt {\frac {1}{5}-\frac {2 i}{5}} \arctan \left (\frac {\sqrt {1+2 i} \sqrt {-x-x^2+x^3}}{-1-x+x^2}\right ) \]
-1/2*arctan((x^3-x^2-x)^(1/2)/(x^2-x-1))+1/20*(5+10*I)^(1/2)*arctan((1-2*I )^(1/2)*(x^3-x^2-x)^(1/2)/(x^2-x-1))+1/20*(5-10*I)^(1/2)*arctan((1+2*I)^(1 /2)*(x^3-x^2-x)^(1/2)/(x^2-x-1))
Time = 0.55 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.02 \[ \int \frac {x^2}{\sqrt {-x-x^2+x^3} \left (-1+x^4\right )} \, dx=\frac {\sqrt {x} \sqrt {-1-x+x^2} \left (-2 \sqrt {5} \arctan \left (\frac {\sqrt {x}}{\sqrt {-1-x+x^2}}\right )+\sqrt {1+2 i} \arctan \left (\frac {\sqrt {1-2 i} \sqrt {x}}{\sqrt {-1-x+x^2}}\right )+\sqrt {1-2 i} \arctan \left (\frac {\sqrt {1+2 i} \sqrt {x}}{\sqrt {-1-x+x^2}}\right )\right )}{4 \sqrt {5} \sqrt {x \left (-1-x+x^2\right )}} \]
(Sqrt[x]*Sqrt[-1 - x + x^2]*(-2*Sqrt[5]*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[5]* Sqrt[x*(-1 - x + x^2)])
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.73 (sec) , antiderivative size = 985, normalized size of antiderivative = 7.41, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, 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.
| \(\displaystyle \int \frac {x^2}{\sqrt {x^3-x^2-x} \left (x^4-1\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {x^2-x-1} \int -\frac {x^{3/2}}{\sqrt {x^2-x-1} \left (1-x^4\right )}dx}{\sqrt {x^3-x^2-x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {x^2-x-1} \int \frac {x^{3/2}}{\sqrt {x^2-x-1} \left (1-x^4\right )}dx}{\sqrt {x^3-x^2-x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^2-x-1} \int \frac {x^2}{\sqrt {x^2-x-1} \left (1-x^4\right )}d\sqrt {x}}{\sqrt {x^3-x^2-x}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^2-x-1} \int \left (-\frac {1}{4 (x-1) \sqrt {x^2-x-1}}+\frac {1}{4 (x+1) \sqrt {x^2-x-1}}-\frac {1}{2 \left (x^2+1\right ) \sqrt {x^2-x-1}}\right )d\sqrt {x}}{\sqrt {x^3-x^2-x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^2-x-1} \left (-\frac {\arctan \left (\frac {\sqrt {1-2 i} \sqrt {x}}{\sqrt {x^2-x-1}}\right )}{8 \sqrt {1-2 i}}-\frac {\arctan \left (\frac {\sqrt {1+2 i} \sqrt {x}}{\sqrt {x^2-x-1}}\right )}{8 \sqrt {1+2 i}}-\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 {\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^3-x^2-x}}\) |
(-2*Sqrt[x]*Sqrt[-1 - x + x^2]*(-1/8*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]]/(8*Sqrt[1 + 2*I]) - ((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 - Sqrt[5])/2])/(4*Sqrt[2]*(1 - Sqrt[5])*Sqrt[-1 - x + x^2]) - ((1 + Sqrt [5])*Sqrt[-1 + Sqrt[5] + 2*x]*Sqrt[1 - (2*x)/(1 + Sqrt[5])]*EllipticF[ArcS in[Sqrt[2/(1 + Sqrt[5])]*Sqrt[x]], (-3 - Sqrt[5])/2])/(4*Sqrt[2]*(3 + Sqrt [5])*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)*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)*Sqrt[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]) + (Sqrt[- 2 - (1 - Sqrt[5])*x]*Sqrt[(2 + (1 + Sqrt[5])*x)/(2 + (1 - Sqrt[5])*x)]*Ell ipticF[ArcSin[(Sqrt[2]*5^(1/4)*Sqrt[x])/Sqrt[-2 - (1 - Sqrt[5])*x]], (5 - Sqrt[5])/10])/(4*5^(1/4)*(3 + Sqrt[5])*Sqrt[(2 + (1 - Sqrt[5])*x)^(-1)]*Sq rt[-1 - x + x^2]) + ((2 + Sqrt[5])*Sqrt[-1 + Sqrt[5] + 2*x]*Sqrt[1 - (2*x) /(1 + Sqrt[5])]*EllipticPi[(-1 - Sqrt[5])/2, ArcSin[Sqrt[2/(1 + Sqrt[5])]* Sqrt[x]], (-3 - Sqrt[5])/2])/(2*Sqrt[2]*(3 + Sqrt[5])*Sqrt[-1 - x + x^2...
3.20.10.3.1 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. \(224\) vs. \(2(105)=210\).
Time = 5.37 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.69
| method | result | size |
| default | \(\frac {-\sqrt {5}\, \ln \left (\frac {-\sqrt {x \left (x^{2}-x -1\right )}\, \sqrt {-2+2 \sqrt {5}}+x \sqrt {5}+x^{2}-x -1}{x}\right )+\sqrt {5}\, \ln \left (\frac {\sqrt {x \left (x^{2}-x -1\right )}\, \sqrt {-2+2 \sqrt {5}}+x \sqrt {5}+x^{2}-x -1}{x}\right )+\left (-5-\sqrt {5}\right ) \arctan \left (\frac {\sqrt {-2+2 \sqrt {5}}\, x +2 \sqrt {x \left (x^{2}-x -1\right )}}{x \sqrt {2+2 \sqrt {5}}}\right )+10 \arctan \left (\frac {\sqrt {x \left (x^{2}-x -1\right )}}{x}\right ) \sqrt {2+2 \sqrt {5}}+\left (5+\sqrt {5}\right ) \arctan \left (\frac {\sqrt {-2+2 \sqrt {5}}\, x -2 \sqrt {x \left (x^{2}-x -1\right )}}{x \sqrt {2+2 \sqrt {5}}}\right )}{20 \sqrt {2+2 \sqrt {5}}}\) | \(225\) |
| pseudoelliptic | \(\frac {-\sqrt {5}\, \ln \left (\frac {-\sqrt {x \left (x^{2}-x -1\right )}\, \sqrt {-2+2 \sqrt {5}}+x \sqrt {5}+x^{2}-x -1}{x}\right )+\sqrt {5}\, \ln \left (\frac {\sqrt {x \left (x^{2}-x -1\right )}\, \sqrt {-2+2 \sqrt {5}}+x \sqrt {5}+x^{2}-x -1}{x}\right )+\left (-5-\sqrt {5}\right ) \arctan \left (\frac {\sqrt {-2+2 \sqrt {5}}\, x +2 \sqrt {x \left (x^{2}-x -1\right )}}{x \sqrt {2+2 \sqrt {5}}}\right )+10 \arctan \left (\frac {\sqrt {x \left (x^{2}-x -1\right )}}{x}\right ) \sqrt {2+2 \sqrt {5}}+\left (5+\sqrt {5}\right ) \arctan \left (\frac {\sqrt {-2+2 \sqrt {5}}\, x -2 \sqrt {x \left (x^{2}-x -1\right )}}{x \sqrt {2+2 \sqrt {5}}}\right )}{20 \sqrt {2+2 \sqrt {5}}}\) | \(225\) |
| trager | \(\text {Expression too large to display}\) | \(911\) |
| elliptic | \(\text {Expression too large to display}\) | \(1436\) |
1/20/(2+2*5^(1/2))^(1/2)*(-5^(1/2)*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)*ln(((x*(x^2-x-1))^(1/2)*(-2+2*5^(1/2) )^(1/2)+x*5^(1/2)+x^2-x-1)/x)+(-5-5^(1/2))*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))+10*arctan((x*(x^2-x-1))^(1/2 )/x)*(2+2*5^(1/2))^(1/2)+(5+5^(1/2))*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)))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 324 vs. \(2 (97) = 194\).
Time = 0.29 (sec) , antiderivative size = 324, normalized size of antiderivative = 2.44 \[ \int \frac {x^2}{\sqrt {-x-x^2+x^3} \left (-1+x^4\right )} \, dx=\frac {1}{80} \, \sqrt {5} \sqrt {-2 i - 1} \log \left (\frac {5 \, x^{4} + \left (20 i - 10\right ) \, x^{3} - 2 \, \sqrt {5} \sqrt {-2 i - 1} \sqrt {x^{3} - x^{2} - x} {\left (-\left (2 i - 1\right ) \, x^{2} + \left (2 i + 4\right ) \, x + 2 i - 1\right )} - \left (20 i + 30\right ) \, x^{2} - \left (20 i - 10\right ) \, x + 5}{x^{4} + 2 \, x^{2} + 1}\right ) - \frac {1}{80} \, \sqrt {5} \sqrt {-2 i - 1} \log \left (\frac {5 \, x^{4} + \left (20 i - 10\right ) \, x^{3} - 2 \, \sqrt {5} \sqrt {-2 i - 1} \sqrt {x^{3} - x^{2} - x} {\left (\left (2 i - 1\right ) \, x^{2} - \left (2 i + 4\right ) \, x - 2 i + 1\right )} - \left (20 i + 30\right ) \, x^{2} - \left (20 i - 10\right ) \, x + 5}{x^{4} + 2 \, x^{2} + 1}\right ) + \frac {1}{80} \, \sqrt {5} \sqrt {2 i - 1} \log \left (\frac {5 \, x^{4} - \left (20 i + 10\right ) \, x^{3} - 2 \, \sqrt {5} \sqrt {2 i - 1} \sqrt {x^{3} - x^{2} - x} {\left (\left (2 i + 1\right ) \, x^{2} - \left (2 i - 4\right ) \, x - 2 i - 1\right )} + \left (20 i - 30\right ) \, x^{2} + \left (20 i + 10\right ) \, x + 5}{x^{4} + 2 \, x^{2} + 1}\right ) - \frac {1}{80} \, \sqrt {5} \sqrt {2 i - 1} \log \left (\frac {5 \, x^{4} - \left (20 i + 10\right ) \, x^{3} - 2 \, \sqrt {5} \sqrt {2 i - 1} \sqrt {x^{3} - x^{2} - x} {\left (-\left (2 i + 1\right ) \, x^{2} + \left (2 i - 4\right ) \, x + 2 i + 1\right )} + \left (20 i - 30\right ) \, x^{2} + \left (20 i + 10\right ) \, x + 5}{x^{4} + 2 \, x^{2} + 1}\right ) + \frac {1}{4} \, \arctan \left (\frac {x^{2} - 2 \, x - 1}{2 \, \sqrt {x^{3} - x^{2} - x}}\right ) \]
1/80*sqrt(5)*sqrt(-2*I - 1)*log((5*x^4 + (20*I - 10)*x^3 - 2*sqrt(5)*sqrt( -2*I - 1)*sqrt(x^3 - x^2 - x)*(-(2*I - 1)*x^2 + (2*I + 4)*x + 2*I - 1) - ( 20*I + 30)*x^2 - (20*I - 10)*x + 5)/(x^4 + 2*x^2 + 1)) - 1/80*sqrt(5)*sqrt (-2*I - 1)*log((5*x^4 + (20*I - 10)*x^3 - 2*sqrt(5)*sqrt(-2*I - 1)*sqrt(x^ 3 - x^2 - x)*((2*I - 1)*x^2 - (2*I + 4)*x - 2*I + 1) - (20*I + 30)*x^2 - ( 20*I - 10)*x + 5)/(x^4 + 2*x^2 + 1)) + 1/80*sqrt(5)*sqrt(2*I - 1)*log((5*x ^4 - (20*I + 10)*x^3 - 2*sqrt(5)*sqrt(2*I - 1)*sqrt(x^3 - x^2 - x)*((2*I + 1)*x^2 - (2*I - 4)*x - 2*I - 1) + (20*I - 30)*x^2 + (20*I + 10)*x + 5)/(x ^4 + 2*x^2 + 1)) - 1/80*sqrt(5)*sqrt(2*I - 1)*log((5*x^4 - (20*I + 10)*x^3 - 2*sqrt(5)*sqrt(2*I - 1)*sqrt(x^3 - x^2 - x)*(-(2*I + 1)*x^2 + (2*I - 4) *x + 2*I + 1) + (20*I - 30)*x^2 + (20*I + 10)*x + 5)/(x^4 + 2*x^2 + 1)) + 1/4*arctan(1/2*(x^2 - 2*x - 1)/sqrt(x^3 - x^2 - x))
\[ \int \frac {x^2}{\sqrt {-x-x^2+x^3} \left (-1+x^4\right )} \, dx=\int \frac {x^{2}}{\sqrt {x \left (x^{2} - x - 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}\, dx \]
\[ \int \frac {x^2}{\sqrt {-x-x^2+x^3} \left (-1+x^4\right )} \, dx=\int { \frac {x^{2}}{{\left (x^{4} - 1\right )} \sqrt {x^{3} - x^{2} - x}} \,d x } \]
\[ \int \frac {x^2}{\sqrt {-x-x^2+x^3} \left (-1+x^4\right )} \, dx=\int { \frac {x^{2}}{{\left (x^{4} - 1\right )} \sqrt {x^{3} - x^{2} - x}} \,d x } \]
Time = 0.06 (sec) , antiderivative size = 533, normalized size of antiderivative = 4.01 \[ \int \frac {x^2}{\sqrt {-x-x^2+x^3} \left (-1+x^4\right )} \, 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 )}{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 )}{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}} \]
((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)*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)))/(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)/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 )))/(2*(x^3 - x^2 - x*(5^(1/2)/2 - 1/2)*(5^(1/2)/2 + 1/2))^(1/2))