\(\int \frac {-1+x^4}{\sqrt {-x-x^2+x^3} (1+x^4)} \, dx\) [1873]

Optimal result

Integrand size = 29, antiderivative size = 129 \[ \int \frac {-1+x^4}{\sqrt {-x-x^2+x^3} \left (1+x^4\right )} \, dx=-\sqrt {\frac {1}{3} \left (1+i \sqrt {2}\right )} \arctan \left (\frac {\sqrt {1-i \sqrt {2}} \sqrt {-x-x^2+x^3}}{-1-x+x^2}\right )-\sqrt {\frac {1}{3} \left (1-i \sqrt {2}\right )} \arctan \left (\frac {\sqrt {1+i \sqrt {2}} \sqrt {-x-x^2+x^3}}{-1-x+x^2}\right ) \]

[Out]

-1/3*(3+3*I*2^(1/2))^(1/2)*arctan((1-I*2^(1/2))^(1/2)*(x^3-x^2-x)^(1/2)/(x^2-x-1))-1/3*(3-3*I*2^(1/2))^(1/2)*a
rctan((1+I*2^(1/2))^(1/2)*(x^3-x^2-x)^(1/2)/(x^2-x-1))

Rubi [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 1.28 (sec) , antiderivative size = 625, normalized size of antiderivative = 4.84, number of steps used = 27, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2081, 6857, 730, 1112, 948, 174, 552, 551} \[ \int \frac {-1+x^4}{\sqrt {-x-x^2+x^3} \left (1+x^4\right )} \, dx=\frac {\sqrt {x} \sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x+2}{\left (1-\sqrt {5}\right ) x+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^3-x^2-x}}-\frac {\sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {2 x+\sqrt {5}-1} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \operatorname {EllipticPi}\left (-\frac {1}{2} \sqrt [4]{-1} \left (1+\sqrt {5}\right ),\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{2 \sqrt {x^3-x^2-x}}-\frac {\sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {2 x+\sqrt {5}-1} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \operatorname {EllipticPi}\left (\frac {1}{2} \sqrt [4]{-1} \left (1+\sqrt {5}\right ),\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{2 \sqrt {x^3-x^2-x}}-\frac {\sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {2 x+\sqrt {5}-1} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \operatorname {EllipticPi}\left (-\frac {1}{2} (-1)^{3/4} \left (1+\sqrt {5}\right ),\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{2 \sqrt {x^3-x^2-x}}-\frac {\sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {2 x+\sqrt {5}-1} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \operatorname {EllipticPi}\left (\frac {1}{2} (-1)^{3/4} \left (1+\sqrt {5}\right ),\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{2 \sqrt {x^3-x^2-x}} \]

[In]

Int[(-1 + x^4)/(Sqrt[-x - x^2 + x^3]*(1 + x^4)),x]

[Out]

(Sqrt[x]*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])/(5^(1/4)*Sqrt[(2 + (1 - Sqrt[5])*x)^(-1)]*S
qrt[-x - x^2 + x^3]) - (Sqrt[3 + Sqrt[5]]*Sqrt[x]*Sqrt[-1 + Sqrt[5] + 2*x]*Sqrt[1 - (2*x)/(1 + Sqrt[5])]*Ellip
ticPi[-1/2*((-1)^(1/4)*(1 + Sqrt[5])), ArcSin[Sqrt[2/(1 + Sqrt[5])]*Sqrt[x]], (-3 - Sqrt[5])/2])/(2*Sqrt[-x -
x^2 + x^3]) - (Sqrt[3 + Sqrt[5]]*Sqrt[x]*Sqrt[-1 + Sqrt[5] + 2*x]*Sqrt[1 - (2*x)/(1 + Sqrt[5])]*EllipticPi[((-
1)^(1/4)*(1 + Sqrt[5]))/2, ArcSin[Sqrt[2/(1 + Sqrt[5])]*Sqrt[x]], (-3 - Sqrt[5])/2])/(2*Sqrt[-x - x^2 + x^3])
- (Sqrt[3 + Sqrt[5]]*Sqrt[x]*Sqrt[-1 + Sqrt[5] + 2*x]*Sqrt[1 - (2*x)/(1 + Sqrt[5])]*EllipticPi[-1/2*((-1)^(3/4
)*(1 + Sqrt[5])), ArcSin[Sqrt[2/(1 + Sqrt[5])]*Sqrt[x]], (-3 - Sqrt[5])/2])/(2*Sqrt[-x - x^2 + x^3]) - (Sqrt[3
+ Sqrt[5]]*Sqrt[x]*Sqrt[-1 + Sqrt[5] + 2*x]*Sqrt[1 - (2*x)/(1 + Sqrt[5])]*EllipticPi[((-1)^(3/4)*(1 + Sqrt[5]
))/2, ArcSin[Sqrt[2/(1 + Sqrt[5])]*Sqrt[x]], (-3 - Sqrt[5])/2])/(2*Sqrt[-x - x^2 + x^3])

Rule 174

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Sym
bol] :> Dist[-2, Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d
*g - c*h)/d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && GtQ[(d*e - c
*f)/d, 0]

Rule 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b*(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-f/e, -d/c])

Rule 552

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 +
(d/c)*x^2]/Sqrt[c + d*x^2], Int[1/((a + b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[c, 0]

Rule 730

Int[(x_)^(m_)/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[x^(2*m + 1)/Sqrt[a + b*x^
2 + c*x^4], x], x, Sqrt[x]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[m^2, 1/4]

Rule 948

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Wi
th[{q = Rt[b^2 - 4*a*c, 2]}, Dist[Sqrt[b - q + 2*c*x]*(Sqrt[b + q + 2*c*x]/Sqrt[a + b*x + c*x^2]), Int[1/((d +
e*x)*Sqrt[f + g*x]*Sqrt[b - q + 2*c*x]*Sqrt[b + q + 2*c*x]), x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && Ne
Q[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 1112

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[Sqrt[(2*a + (
b - q)*x^2)/(2*a + (b + q)*x^2)]*(Sqrt[(2*a + (b + q)*x^2)/q]/(2*Sqrt[a + b*x^2 + c*x^4]*Sqrt[a/(2*a + (b + q)
*x^2)]))*EllipticF[ArcSin[x/Sqrt[(2*a + (b + q)*x^2)/(2*q)]], (b + q)/(2*q)], x]] /; FreeQ[{a, b, c}, x] && Gt
Q[b^2 - 4*a*c, 0] && LtQ[a, 0] && GtQ[c, 0]

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {-1+x^4}{\sqrt {x} \sqrt {-1-x+x^2} \left (1+x^4\right )} \, dx}{\sqrt {-x-x^2+x^3}} \\ & = \frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \left (\frac {1}{\sqrt {x} \sqrt {-1-x+x^2}}-\frac {2}{\sqrt {x} \sqrt {-1-x+x^2} \left (1+x^4\right )}\right ) \, dx}{\sqrt {-x-x^2+x^3}} \\ & = \frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {-1-x+x^2}} \, dx}{\sqrt {-x-x^2+x^3}}-\frac {\left (2 \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {-1-x+x^2} \left (1+x^4\right )} \, dx}{\sqrt {-x-x^2+x^3}} \\ & = -\frac {\left (2 \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \left (\frac {i}{2 \sqrt {x} \left (i-x^2\right ) \sqrt {-1-x+x^2}}+\frac {i}{2 \sqrt {x} \left (i+x^2\right ) \sqrt {-1-x+x^2}}\right ) \, dx}{\sqrt {-x-x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {-1-x+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}} \\ & = \frac {\sqrt {x} \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3}}-\frac {\left (i \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {1}{\sqrt {x} \left (i-x^2\right ) \sqrt {-1-x+x^2}} \, dx}{\sqrt {-x-x^2+x^3}}-\frac {\left (i \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {1}{\sqrt {x} \left (i+x^2\right ) \sqrt {-1-x+x^2}} \, dx}{\sqrt {-x-x^2+x^3}} \\ & = \frac {\sqrt {x} \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3}}-\frac {\left (i \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \left (-\frac {(-1)^{3/4}}{2 \left (\sqrt [4]{-1}-x\right ) \sqrt {x} \sqrt {-1-x+x^2}}-\frac {(-1)^{3/4}}{2 \sqrt {x} \left (\sqrt [4]{-1}+x\right ) \sqrt {-1-x+x^2}}\right ) \, dx}{\sqrt {-x-x^2+x^3}}-\frac {\left (i \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \left (-\frac {\sqrt [4]{-1}}{2 \left (-(-1)^{3/4}-x\right ) \sqrt {x} \sqrt {-1-x+x^2}}-\frac {\sqrt [4]{-1}}{2 \sqrt {x} \left (-(-1)^{3/4}+x\right ) \sqrt {-1-x+x^2}}\right ) \, dx}{\sqrt {-x-x^2+x^3}} \\ & = \frac {\sqrt {x} \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3}}-\frac {\left (\sqrt [4]{-1} \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {1}{\left (\sqrt [4]{-1}-x\right ) \sqrt {x} \sqrt {-1-x+x^2}} \, dx}{2 \sqrt {-x-x^2+x^3}}-\frac {\left (\sqrt [4]{-1} \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {1}{\sqrt {x} \left (\sqrt [4]{-1}+x\right ) \sqrt {-1-x+x^2}} \, dx}{2 \sqrt {-x-x^2+x^3}}+\frac {\left ((-1)^{3/4} \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {1}{\left (-(-1)^{3/4}-x\right ) \sqrt {x} \sqrt {-1-x+x^2}} \, dx}{2 \sqrt {-x-x^2+x^3}}+\frac {\left ((-1)^{3/4} \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {1}{\sqrt {x} \left (-(-1)^{3/4}+x\right ) \sqrt {-1-x+x^2}} \, dx}{2 \sqrt {-x-x^2+x^3}} \\ & = \frac {\sqrt {x} \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3}}-\frac {\left (\sqrt [4]{-1} \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \int \frac {1}{\left (\sqrt [4]{-1}-x\right ) \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}} \, dx}{2 \sqrt {-x-x^2+x^3}}-\frac {\left (\sqrt [4]{-1} \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \int \frac {1}{\sqrt {x} \left (\sqrt [4]{-1}+x\right ) \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}} \, dx}{2 \sqrt {-x-x^2+x^3}}+\frac {\left ((-1)^{3/4} \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \int \frac {1}{\left (-(-1)^{3/4}-x\right ) \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}} \, dx}{2 \sqrt {-x-x^2+x^3}}+\frac {\left ((-1)^{3/4} \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \int \frac {1}{\sqrt {x} \left (-(-1)^{3/4}+x\right ) \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}} \, dx}{2 \sqrt {-x-x^2+x^3}} \\ & = \frac {\sqrt {x} \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3}}+\frac {\left (\sqrt [4]{-1} \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [4]{-1}-x^2\right ) \sqrt {-1-\sqrt {5}+2 x^2} \sqrt {-1+\sqrt {5}+2 x^2}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}}+\frac {\left (\sqrt [4]{-1} \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [4]{-1}+x^2\right ) \sqrt {-1-\sqrt {5}+2 x^2} \sqrt {-1+\sqrt {5}+2 x^2}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}}-\frac {\left ((-1)^{3/4} \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \text {Subst}\left (\int \frac {1}{\left ((-1)^{3/4}-x^2\right ) \sqrt {-1-\sqrt {5}+2 x^2} \sqrt {-1+\sqrt {5}+2 x^2}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}}-\frac {\left ((-1)^{3/4} \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \text {Subst}\left (\int \frac {1}{\left ((-1)^{3/4}+x^2\right ) \sqrt {-1-\sqrt {5}+2 x^2} \sqrt {-1+\sqrt {5}+2 x^2}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}} \\ & = \frac {\sqrt {x} \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3}}+\frac {\left (\sqrt [4]{-1} \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [4]{-1}-x^2\right ) \sqrt {-1+\sqrt {5}+2 x^2} \sqrt {1+\frac {2 x^2}{-1-\sqrt {5}}}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}}+\frac {\left (\sqrt [4]{-1} \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [4]{-1}+x^2\right ) \sqrt {-1+\sqrt {5}+2 x^2} \sqrt {1+\frac {2 x^2}{-1-\sqrt {5}}}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}}-\frac {\left ((-1)^{3/4} \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}}\right ) \text {Subst}\left (\int \frac {1}{\left ((-1)^{3/4}-x^2\right ) \sqrt {-1+\sqrt {5}+2 x^2} \sqrt {1+\frac {2 x^2}{-1-\sqrt {5}}}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}}-\frac {\left ((-1)^{3/4} \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}}\right ) \text {Subst}\left (\int \frac {1}{\left ((-1)^{3/4}+x^2\right ) \sqrt {-1+\sqrt {5}+2 x^2} \sqrt {1+\frac {2 x^2}{-1-\sqrt {5}}}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}} \\ & = \frac {\sqrt {x} \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3}}-\frac {\sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \operatorname {EllipticPi}\left (-\frac {1}{2} \sqrt [4]{-1} \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 {-x-x^2+x^3}}-\frac {\sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \operatorname {EllipticPi}\left (\frac {1}{2} \sqrt [4]{-1} \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 {-x-x^2+x^3}}-\frac {\sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \operatorname {EllipticPi}\left (-\frac {1}{2} (-1)^{3/4} \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 {-x-x^2+x^3}}-\frac {\sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \operatorname {EllipticPi}\left (\frac {1}{2} (-1)^{3/4} \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 {-x-x^2+x^3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.73 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.08 \[ \int \frac {-1+x^4}{\sqrt {-x-x^2+x^3} \left (1+x^4\right )} \, dx=-\frac {\sqrt {x} \sqrt {-1-x+x^2} \left (\sqrt {1+i \sqrt {2}} \arctan \left (\frac {\sqrt {1-i \sqrt {2}} \sqrt {x}}{\sqrt {-1-x+x^2}}\right )+\sqrt {1-i \sqrt {2}} \arctan \left (\frac {\sqrt {1+i \sqrt {2}} \sqrt {x}}{\sqrt {-1-x+x^2}}\right )\right )}{\sqrt {3} \sqrt {x \left (-1-x+x^2\right )}} \]

[In]

Integrate[(-1 + x^4)/(Sqrt[-x - x^2 + x^3]*(1 + x^4)),x]

[Out]

-((Sqrt[x]*Sqrt[-1 - x + x^2]*(Sqrt[1 + I*Sqrt[2]]*ArcTan[(Sqrt[1 - I*Sqrt[2]]*Sqrt[x])/Sqrt[-1 - x + x^2]] +
Sqrt[1 - I*Sqrt[2]]*ArcTan[(Sqrt[1 + I*Sqrt[2]]*Sqrt[x])/Sqrt[-1 - x + x^2]]))/(Sqrt[3]*Sqrt[x*(-1 - x + x^2)]
))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(208\) vs. \(2(97)=194\).

Time = 4.13 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.62

method	result	size
default	\(\frac {\sqrt {3}\, \sqrt {2 \sqrt {3}+2}\, \left (\ln \left (\frac {x \sqrt {3}-\sqrt {x \left (x^{2}-x -1\right )}\, \sqrt {2 \sqrt {3}-2}+x^{2}-x -1}{x}\right )-\ln \left (\frac {x \sqrt {3}+\sqrt {x \left (x^{2}-x -1\right )}\, \sqrt {2 \sqrt {3}-2}+x^{2}-x -1}{x}\right )\right ) \sqrt {2 \sqrt {3}-2}-4 \left (3+\sqrt {3}\right ) \left (-\arctan \left (\frac {\sqrt {2 \sqrt {3}-2}\, x +2 \sqrt {x \left (x^{2}-x -1\right )}}{x \sqrt {2 \sqrt {3}+2}}\right )+\arctan \left (\frac {\sqrt {2 \sqrt {3}-2}\, x -2 \sqrt {x \left (x^{2}-x -1\right )}}{x \sqrt {2 \sqrt {3}+2}}\right )\right )}{12 \sqrt {2 \sqrt {3}+2}}\)	\(209\)
pseudoelliptic	\(\frac {\sqrt {3}\, \sqrt {2 \sqrt {3}+2}\, \left (\ln \left (\frac {x \sqrt {3}-\sqrt {x \left (x^{2}-x -1\right )}\, \sqrt {2 \sqrt {3}-2}+x^{2}-x -1}{x}\right )-\ln \left (\frac {x \sqrt {3}+\sqrt {x \left (x^{2}-x -1\right )}\, \sqrt {2 \sqrt {3}-2}+x^{2}-x -1}{x}\right )\right ) \sqrt {2 \sqrt {3}-2}-4 \left (3+\sqrt {3}\right ) \left (-\arctan \left (\frac {\sqrt {2 \sqrt {3}-2}\, x +2 \sqrt {x \left (x^{2}-x -1\right )}}{x \sqrt {2 \sqrt {3}+2}}\right )+\arctan \left (\frac {\sqrt {2 \sqrt {3}-2}\, x -2 \sqrt {x \left (x^{2}-x -1\right )}}{x \sqrt {2 \sqrt {3}+2}}\right )\right )}{12 \sqrt {2 \sqrt {3}+2}}\)	\(209\)
elliptic	\(\frac {2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \sqrt {-5 \left (x -\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{5 \sqrt {x^{3}-x^{2}-x}}+\frac {5^{\frac {3}{4}} \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \left (\sqrt {5}-1\right ) \sqrt {\frac {-1+2 x +\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {1-2 x +\sqrt {5}}\, \sqrt {-\frac {x}{\sqrt {5}-1}}\, \left (3 \underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2}+2 \underline {\hspace {1.25 ex}}\alpha +1+\sqrt {5}\, \left (\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2}-1\right )\right ) \operatorname {EllipticPi}\left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, -\frac {\underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {5}}{6}-\frac {\underline {\hspace {1.25 ex}}\alpha ^{3}}{6}+\frac {\underline {\hspace {1.25 ex}}\alpha ^{2}}{3}+\frac {\underline {\hspace {1.25 ex}}\alpha }{6}-\frac {\sqrt {5}}{6}+\frac {1}{2}-\frac {\underline {\hspace {1.25 ex}}\alpha \sqrt {5}}{6}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{\sqrt {x \left (x^{2}-x -1\right )}}\right )}{60}\)	\(288\)
trager	\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right ) \ln \left (\frac {-496 \operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right ) \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{4} x^{2}+496 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{4} \operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right )-120 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right ) x^{2}+248 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right ) x +224 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2} \sqrt {x^{3}-x^{2}-x}+120 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right ) x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right ) x -68 \sqrt {x^{3}-x^{2}-x}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right )}{12 x^{2} \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}-12 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+x^{2}+2 x -1}\right )}{6}+\operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right ) \ln \left (-\frac {4464 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{5} x^{2}-4464 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{5}+408 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{3} x^{2}+2232 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{3} x +336 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2} \sqrt {x^{3}-x^{2}-x}-408 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{3}-65 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right ) x^{2}+390 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right ) x +158 \sqrt {x^{3}-x^{2}-x}+65 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )}{12 x^{2} \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}-12 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+x^{2}-2 x -1}\right )\)	\(662\)

[In]

int((x^4-1)/(x^3-x^2-x)^(1/2)/(x^4+1),x,method=_RETURNVERBOSE)

[Out]

1/12/(2*3^(1/2)+2)^(1/2)*(3^(1/2)*(2*3^(1/2)+2)^(1/2)*(ln((x*3^(1/2)-(x*(x^2-x-1))^(1/2)*(2*3^(1/2)-2)^(1/2)+x
^2-x-1)/x)-ln((x*3^(1/2)+(x*(x^2-x-1))^(1/2)*(2*3^(1/2)-2)^(1/2)+x^2-x-1)/x))*(2*3^(1/2)-2)^(1/2)-4*(3+3^(1/2)
)*(-arctan(((2*3^(1/2)-2)^(1/2)*x+2*(x*(x^2-x-1))^(1/2))/x/(2*3^(1/2)+2)^(1/2))+arctan(((2*3^(1/2)-2)^(1/2)*x-
2*(x*(x^2-x-1))^(1/2))/x/(2*3^(1/2)+2)^(1/2))))

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 451 vs. \(2 (93) = 186\).

Time = 0.27 (sec) , antiderivative size = 451, normalized size of antiderivative = 3.50 \[ \int \frac {-1+x^4}{\sqrt {-x-x^2+x^3} \left (1+x^4\right )} \, dx=-\frac {1}{12} \, \sqrt {3} \sqrt {i \, \sqrt {2} - 1} \log \left (\frac {3 \, x^{4} - 6 \, x^{3} - 2 \, \sqrt {3} \sqrt {x^{3} - x^{2} - x} {\left (x^{2} + \sqrt {2} {\left (i \, x^{2} - i \, x - i\right )} + 2 \, x - 1\right )} \sqrt {i \, \sqrt {2} - 1} - 12 \, x^{2} - 6 \, \sqrt {2} {\left (i \, x^{3} - i \, x^{2} - i \, x\right )} + 6 \, x + 3}{x^{4} + 1}\right ) + \frac {1}{12} \, \sqrt {3} \sqrt {i \, \sqrt {2} - 1} \log \left (\frac {3 \, x^{4} - 6 \, x^{3} + 2 \, \sqrt {3} \sqrt {x^{3} - x^{2} - x} {\left (x^{2} - \sqrt {2} {\left (-i \, x^{2} + i \, x + i\right )} + 2 \, x - 1\right )} \sqrt {i \, \sqrt {2} - 1} - 12 \, x^{2} - 6 \, \sqrt {2} {\left (i \, x^{3} - i \, x^{2} - i \, x\right )} + 6 \, x + 3}{x^{4} + 1}\right ) + \frac {1}{12} \, \sqrt {3} \sqrt {-i \, \sqrt {2} - 1} \log \left (\frac {3 \, x^{4} - 6 \, x^{3} + 2 \, \sqrt {3} \sqrt {x^{3} - x^{2} - x} {\left (x^{2} - \sqrt {2} {\left (i \, x^{2} - i \, x - i\right )} + 2 \, x - 1\right )} \sqrt {-i \, \sqrt {2} - 1} - 12 \, x^{2} - 6 \, \sqrt {2} {\left (-i \, x^{3} + i \, x^{2} + i \, x\right )} + 6 \, x + 3}{x^{4} + 1}\right ) - \frac {1}{12} \, \sqrt {3} \sqrt {-i \, \sqrt {2} - 1} \log \left (\frac {3 \, x^{4} - 6 \, x^{3} - 2 \, \sqrt {3} \sqrt {x^{3} - x^{2} - x} {\left (x^{2} + \sqrt {2} {\left (-i \, x^{2} + i \, x + i\right )} + 2 \, x - 1\right )} \sqrt {-i \, \sqrt {2} - 1} - 12 \, x^{2} - 6 \, \sqrt {2} {\left (-i \, x^{3} + i \, x^{2} + i \, x\right )} + 6 \, x + 3}{x^{4} + 1}\right ) \]

[In]

integrate((x^4-1)/(x^3-x^2-x)^(1/2)/(x^4+1),x, algorithm="fricas")

[Out]

-1/12*sqrt(3)*sqrt(I*sqrt(2) - 1)*log((3*x^4 - 6*x^3 - 2*sqrt(3)*sqrt(x^3 - x^2 - x)*(x^2 + sqrt(2)*(I*x^2 - I
*x - I) + 2*x - 1)*sqrt(I*sqrt(2) - 1) - 12*x^2 - 6*sqrt(2)*(I*x^3 - I*x^2 - I*x) + 6*x + 3)/(x^4 + 1)) + 1/12
*sqrt(3)*sqrt(I*sqrt(2) - 1)*log((3*x^4 - 6*x^3 + 2*sqrt(3)*sqrt(x^3 - x^2 - x)*(x^2 - sqrt(2)*(-I*x^2 + I*x +
I) + 2*x - 1)*sqrt(I*sqrt(2) - 1) - 12*x^2 - 6*sqrt(2)*(I*x^3 - I*x^2 - I*x) + 6*x + 3)/(x^4 + 1)) + 1/12*sqr
t(3)*sqrt(-I*sqrt(2) - 1)*log((3*x^4 - 6*x^3 + 2*sqrt(3)*sqrt(x^3 - x^2 - x)*(x^2 - sqrt(2)*(I*x^2 - I*x - I)
+ 2*x - 1)*sqrt(-I*sqrt(2) - 1) - 12*x^2 - 6*sqrt(2)*(-I*x^3 + I*x^2 + I*x) + 6*x + 3)/(x^4 + 1)) - 1/12*sqrt(
3)*sqrt(-I*sqrt(2) - 1)*log((3*x^4 - 6*x^3 - 2*sqrt(3)*sqrt(x^3 - x^2 - x)*(x^2 + sqrt(2)*(-I*x^2 + I*x + I) +
2*x - 1)*sqrt(-I*sqrt(2) - 1) - 12*x^2 - 6*sqrt(2)*(-I*x^3 + I*x^2 + I*x) + 6*x + 3)/(x^4 + 1))

Sympy [F]

\[ \int \frac {-1+x^4}{\sqrt {-x-x^2+x^3} \left (1+x^4\right )} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{\sqrt {x \left (x^{2} - x - 1\right )} \left (x^{4} + 1\right )}\, dx \]

[In]

integrate((x**4-1)/(x**3-x**2-x)**(1/2)/(x**4+1),x)

[Out]

Integral((x - 1)*(x + 1)*(x**2 + 1)/(sqrt(x*(x**2 - x - 1))*(x**4 + 1)), x)

Maxima [F]

\[ \int \frac {-1+x^4}{\sqrt {-x-x^2+x^3} \left (1+x^4\right )} \, dx=\int { \frac {x^{4} - 1}{{\left (x^{4} + 1\right )} \sqrt {x^{3} - x^{2} - x}} \,d x } \]

[In]

integrate((x^4-1)/(x^3-x^2-x)^(1/2)/(x^4+1),x, algorithm="maxima")

[Out]

integrate((x^4 - 1)/((x^4 + 1)*sqrt(x^3 - x^2 - x)), x)

Giac [F]

\[ \int \frac {-1+x^4}{\sqrt {-x-x^2+x^3} \left (1+x^4\right )} \, dx=\int { \frac {x^{4} - 1}{{\left (x^{4} + 1\right )} \sqrt {x^{3} - x^{2} - x}} \,d x } \]

[In]

integrate((x^4-1)/(x^3-x^2-x)^(1/2)/(x^4+1),x, algorithm="giac")

[Out]

integrate((x^4 - 1)/((x^4 + 1)*sqrt(x^3 - x^2 - x)), x)

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 682, normalized size of antiderivative = 5.29 \[ \int \frac {-1+x^4}{\sqrt {-x-x^2+x^3} \left (1+x^4\right )} \, dx=\text {Too large to display} \]

[In]

int((x^4 - 1)/((x^4 + 1)*(x^3 - x^2 - x)^(1/2)),x)

[Out]

(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)*ellipticF(asi
n((x/(5^(1/2)/2 + 1/2))^(1/2)), -(5^(1/2)/2 + 1/2)/(5^(1/2)/2 - 1/2))*((5^(1/2)/2 - x + 1/2)/(5^(1/2)/2 + 1/2)
)^(1/2))/(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)*elliptic
Pi(2^(1/2)*(5^(1/2)/2 + 1/2)*(- 1/2 + 1i/2), asin((x/(5^(1/2)/2 + 1/2))^(1/2)), -(5^(1/2)/2 + 1/2)/(5^(1/2)/2
- 1/2)))/(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)*elliptic
Pi(2^(1/2)*(5^(1/2)/2 + 1/2)*(1/2 - 1i/2), asin((x/(5^(1/2)/2 + 1/2))^(1/2)), -(5^(1/2)/2 + 1/2)/(5^(1/2)/2 -
1/2)))/(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
(2^(1/2)*(5^(1/2)/2 + 1/2)*(1/2 + 1i/2), asin((x/(5^(1/2)/2 + 1/2))^(1/2)), -(5^(1/2)/2 + 1/2)/(5^(1/2)/2 - 1/
2)))/(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(2
^(1/2)*(5^(1/2)/2 + 1/2)*(- 1/2 - 1i/2), asin((x/(5^(1/2)/2 + 1/2))^(1/2)), -(5^(1/2)/2 + 1/2)/(5^(1/2)/2 - 1/
2)))/(x^3 - x^2 - x*(5^(1/2)/2 - 1/2)*(5^(1/2)/2 + 1/2))^(1/2)