3.20.10 \(\int \frac {x^2}{\sqrt {-x-x^2+x^3} (-1+x^4)} \, dx\) [1910]

Optimal. Leaf size=133 \[ -\frac {1}{2} \text {ArcTan}\left (\frac {\sqrt {-x-x^2+x^3}}{-1-x+x^2}\right )+\frac {1}{4} \sqrt {\frac {1}{5}+\frac {2 i}{5}} \text {ArcTan}\left (\frac {\sqrt {1-2 i} \sqrt {-x-x^2+x^3}}{-1-x+x^2}\right )+\frac {1}{4} \sqrt {\frac {1}{5}-\frac {2 i}{5}} \text {ArcTan}\left (\frac {\sqrt {1+2 i} \sqrt {-x-x^2+x^3}}{-1-x+x^2}\right ) \]

[Out]

-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))

________________________________________________________________________________________

Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 2.02, antiderivative size = 467, normalized size of antiderivative = 3.51, number of steps used = 55, number of rules used = 11, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {2081, 6857, 971, 730, 1112, 948, 174, 552, 551, 1148, 1198} \begin {gather*} -\frac {\sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {2 x+\sqrt {5}-1} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \Pi \left (\frac {1}{2} \left (-1-\sqrt {5}\right );\text {ArcSin}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{4 \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}}} \Pi \left (-\frac {1}{2} i \left (1+\sqrt {5}\right );\text {ArcSin}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{4 \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}}} \Pi \left (\frac {1}{2} i \left (1+\sqrt {5}\right );\text {ArcSin}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{4 \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}}} \Pi \left (\frac {1}{2} \left (1+\sqrt {5}\right );\text {ArcSin}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{4 \sqrt {x^3-x^2-x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(Sqrt[3 + Sqrt[5]]*Sqrt[x]*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])/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*I)*(1 + Sqrt[5]), ArcSin[Sqrt[2/(1
+ Sqrt[5])]*Sqrt[x]], (-3 - Sqrt[5])/2])/(4*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[(I/2)*(1 + Sqrt[5]), ArcSin[Sqrt[2/(1 + Sqrt[5])]*Sqrt[x]]
, (-3 - Sqrt[5])/2])/(4*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 + Sqrt[5])/2, ArcSin[Sqrt[2/(1 + Sqrt[5])]*Sqrt[x]], (-3 - Sqrt[5])/2])/(4*S
qrt[-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 971

Int[((f_.) + (g_.)*(x_))^(n_)/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Int
[ExpandIntegrand[1/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), (f + g*x)^(n + 1/2)/(d + e*x), x], x] /; FreeQ[{a, b
, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[
n + 1/2]

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 1148

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

Rule 1198

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}
, Simp[e*x*((b + q + 2*c*x^2)/(2*c*Sqrt[a + b*x^2 + c*x^4])), x] - Simp[e*q*Sqrt[(2*a + (b - q)*x^2)/(2*a + (b
+ q)*x^2)]*(Sqrt[(2*a + (b + q)*x^2)/q]/(2*c*Sqrt[a + b*x^2 + c*x^4]*Sqrt[a/(2*a + (b + q)*x^2)]))*EllipticE[
ArcSin[x/Sqrt[(2*a + (b + q)*x^2)/(2*q)]], (b + q)/(2*q)], x] /; EqQ[2*c*d - e*(b - q), 0]] /; FreeQ[{a, b, c,
d, e}, x] && GtQ[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*} \int \frac {x^2}{\sqrt {-x-x^2+x^3} \left (-1+x^4\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {x^{3/2}}{\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 {x^{3/2}}{2 \left (1-x^2\right ) \sqrt {-1-x+x^2}}-\frac {x^{3/2}}{2 \left (1+x^2\right ) \sqrt {-1-x+x^2}}\right ) \, dx}{\sqrt {-x-x^2+x^3}}\\ &=-\frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {x^{3/2}}{\left (1-x^2\right ) \sqrt {-1-x+x^2}} \, dx}{2 \sqrt {-x-x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {x^{3/2}}{\left (1+x^2\right ) \sqrt {-1-x+x^2}} \, dx}{2 \sqrt {-x-x^2+x^3}}\\ &=-\frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \left (\frac {i x^{3/2}}{2 (i-x) \sqrt {-1-x+x^2}}+\frac {i x^{3/2}}{2 (i+x) \sqrt {-1-x+x^2}}\right ) \, dx}{2 \sqrt {-x-x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \left (\frac {x^{3/2}}{2 (1-x) \sqrt {-1-x+x^2}}+\frac {x^{3/2}}{2 (1+x) \sqrt {-1-x+x^2}}\right ) \, dx}{2 \sqrt {-x-x^2+x^3}}\\ &=-\frac {\left (i \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {x^{3/2}}{(i-x) \sqrt {-1-x+x^2}} \, dx}{4 \sqrt {-x-x^2+x^3}}-\frac {\left (i \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {x^{3/2}}{(i+x) \sqrt {-1-x+x^2}} \, dx}{4 \sqrt {-x-x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {x^{3/2}}{(1-x) \sqrt {-1-x+x^2}} \, dx}{4 \sqrt {-x-x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {x^{3/2}}{(1+x) \sqrt {-1-x+x^2}} \, dx}{4 \sqrt {-x-x^2+x^3}}\\ &=-\frac {\left (i \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \left (-\frac {i}{\sqrt {x} \sqrt {-1-x+x^2}}+\frac {1}{(-i-x) \sqrt {x} \sqrt {-1-x+x^2}}+\frac {\sqrt {x}}{\sqrt {-1-x+x^2}}\right ) \, dx}{4 \sqrt {-x-x^2+x^3}}-\frac {\left (i \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \left (-\frac {i}{\sqrt {x} \sqrt {-1-x+x^2}}-\frac {\sqrt {x}}{\sqrt {-1-x+x^2}}+\frac {1}{\sqrt {x} (-i+x) \sqrt {-1-x+x^2}}\right ) \, dx}{4 \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 {1}{(1-x) \sqrt {x} \sqrt {-1-x+x^2}}-\frac {\sqrt {x}}{\sqrt {-1-x+x^2}}\right ) \, dx}{4 \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 {\sqrt {x}}{\sqrt {-1-x+x^2}}+\frac {1}{\sqrt {x} (1+x) \sqrt {-1-x+x^2}}\right ) \, dx}{4 \sqrt {-x-x^2+x^3}}\\ &=-\frac {\left (i \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {1}{(-i-x) \sqrt {x} \sqrt {-1-x+x^2}} \, dx}{4 \sqrt {-x-x^2+x^3}}-\frac {\left (i \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {1}{\sqrt {x} (-i+x) \sqrt {-1-x+x^2}} \, dx}{4 \sqrt {-x-x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {1}{(1-x) \sqrt {x} \sqrt {-1-x+x^2}} \, dx}{4 \sqrt {-x-x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {1}{\sqrt {x} (1+x) \sqrt {-1-x+x^2}} \, dx}{4 \sqrt {-x-x^2+x^3}}\\ &=-\frac {\left (i \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \int \frac {1}{(-i-x) \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}} \, dx}{4 \sqrt {-x-x^2+x^3}}-\frac {\left (i \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \int \frac {1}{\sqrt {x} (-i+x) \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}} \, dx}{4 \sqrt {-x-x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \int \frac {1}{(1-x) \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}} \, dx}{4 \sqrt {-x-x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \int \frac {1}{\sqrt {x} (1+x) \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}} \, dx}{4 \sqrt {-x-x^2+x^3}}\\ &=\frac {\left (i \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \text {Subst}\left (\int \frac {1}{\left (i-x^2\right ) \sqrt {-1-\sqrt {5}+2 x^2} \sqrt {-1+\sqrt {5}+2 x^2}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-x-x^2+x^3}}+\frac {\left (i \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \text {Subst}\left (\int \frac {1}{\left (i+x^2\right ) \sqrt {-1-\sqrt {5}+2 x^2} \sqrt {-1+\sqrt {5}+2 x^2}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-x-x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \text {Subst}\left (\int \frac {1}{\left (-1-x^2\right ) \sqrt {-1-\sqrt {5}+2 x^2} \sqrt {-1+\sqrt {5}+2 x^2}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-x-x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+x^2\right ) \sqrt {-1-\sqrt {5}+2 x^2} \sqrt {-1+\sqrt {5}+2 x^2}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-x-x^2+x^3}}\\ &=\frac {\left (i \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}}\right ) \text {Subst}\left (\int \frac {1}{\left (i-x^2\right ) \sqrt {-1+\sqrt {5}+2 x^2} \sqrt {1+\frac {2 x^2}{-1-\sqrt {5}}}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-x-x^2+x^3}}+\frac {\left (i \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}}\right ) \text {Subst}\left (\int \frac {1}{\left (i+x^2\right ) \sqrt {-1+\sqrt {5}+2 x^2} \sqrt {1+\frac {2 x^2}{-1-\sqrt {5}}}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-x-x^2+x^3}}+\frac {\left (\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-x^2\right ) \sqrt {-1+\sqrt {5}+2 x^2} \sqrt {1+\frac {2 x^2}{-1-\sqrt {5}}}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-x-x^2+x^3}}+\frac {\left (\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+x^2\right ) \sqrt {-1+\sqrt {5}+2 x^2} \sqrt {1+\frac {2 x^2}{-1-\sqrt {5}}}} \, dx,x,\sqrt {x}\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}}} \Pi \left (\frac {1}{2} \left (-1-\sqrt {5}\right );\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{4 \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}}} \Pi \left (-\frac {1}{2} i \left (1+\sqrt {5}\right );\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{4 \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}}} \Pi \left (\frac {1}{2} i \left (1+\sqrt {5}\right );\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{4 \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}}} \Pi \left (\frac {1}{2} \left (1+\sqrt {5}\right );\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{4 \sqrt {-x-x^2+x^3}}\\ \end {align*}

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Mathematica [A]
time = 0.56, size = 135, normalized size = 1.02 \begin {gather*} \frac {\sqrt {x} \sqrt {-1-x+x^2} \left (-2 \sqrt {5} \text {ArcTan}\left (\frac {\sqrt {x}}{\sqrt {-1-x+x^2}}\right )+\sqrt {1+2 i} \text {ArcTan}\left (\frac {\sqrt {1-2 i} \sqrt {x}}{\sqrt {-1-x+x^2}}\right )+\sqrt {1-2 i} \text {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 )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(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)])

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 1.75, size = 898, normalized size = 6.75

method	result	size
default	\(\text {Expression too large to display}\)	\(898\)
trager	\(\text {Expression too large to display}\)	\(914\)
elliptic	\(\text {Expression too large to display}\)	\(1436\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*(1/2*5^(1/2)-1/2)*((x-1/2+1/2*5^(1/2))/(1/2*5^(1/2)-1/2))^(1/2)*(-5*(x-1/2-1/2*5^(1/2))*5^(1/2))^(1/2)*(
-x/(1/2*5^(1/2)-1/2))^(1/2)/(x^3-x^2-x)^(1/2)/(3/2-1/2*5^(1/2))*EllipticPi(((x-1/2+1/2*5^(1/2))/(1/2*5^(1/2)-1
/2))^(1/2),(1/2-1/2*5^(1/2))/(3/2-1/2*5^(1/2)),1/5*5^(1/2)*((1/2*5^(1/2)-1/2)*5^(1/2))^(1/2))+1/10*(1/2*5^(1/2
)-1/2)*((x-1/2+1/2*5^(1/2))/(1/2*5^(1/2)-1/2))^(1/2)*(-5*(x-1/2-1/2*5^(1/2))*5^(1/2))^(1/2)*(-x/(1/2*5^(1/2)-1
/2))^(1/2)/(x^3-x^2-x)^(1/2)/(-1/2-1/2*5^(1/2))*EllipticPi(((x-1/2+1/2*5^(1/2))/(1/2*5^(1/2)-1/2))^(1/2),(1/2-
1/2*5^(1/2))/(-1/2-1/2*5^(1/2)),1/5*5^(1/2)*((1/2*5^(1/2)-1/2)*5^(1/2))^(1/2))-1/20*I*(x/(1/2*5^(1/2)-1/2)-1/2
/(1/2*5^(1/2)-1/2)+1/2/(1/2*5^(1/2)-1/2)*5^(1/2))^(1/2)*(-5*x*5^(1/2)+5/2*5^(1/2)+25/2)^(1/2)*(-x/(1/2*5^(1/2)
-1/2))^(1/2)/(x^3-x^2-x)^(1/2)/(1/2-I-1/2*5^(1/2))*EllipticPi(((x-1/2+1/2*5^(1/2))/(1/2*5^(1/2)-1/2))^(1/2),(1
/2-1/2*5^(1/2))/(1/2-I-1/2*5^(1/2)),1/5*5^(1/2)*((1/2*5^(1/2)-1/2)*5^(1/2))^(1/2))*5^(1/2)+1/20*I*(x/(1/2*5^(1
/2)-1/2)-1/2/(1/2*5^(1/2)-1/2)+1/2/(1/2*5^(1/2)-1/2)*5^(1/2))^(1/2)*(-5*x*5^(1/2)+5/2*5^(1/2)+25/2)^(1/2)*(-x/
(1/2*5^(1/2)-1/2))^(1/2)/(x^3-x^2-x)^(1/2)/(1/2-I-1/2*5^(1/2))*EllipticPi(((x-1/2+1/2*5^(1/2))/(1/2*5^(1/2)-1/
2))^(1/2),(1/2-1/2*5^(1/2))/(1/2-I-1/2*5^(1/2)),1/5*5^(1/2)*((1/2*5^(1/2)-1/2)*5^(1/2))^(1/2))+1/20*I*(x/(1/2*
5^(1/2)-1/2)-1/2/(1/2*5^(1/2)-1/2)+1/2/(1/2*5^(1/2)-1/2)*5^(1/2))^(1/2)*(-5*x*5^(1/2)+5/2*5^(1/2)+25/2)^(1/2)*
(-x/(1/2*5^(1/2)-1/2))^(1/2)/(x^3-x^2-x)^(1/2)/(1/2+I-1/2*5^(1/2))*EllipticPi(((x-1/2+1/2*5^(1/2))/(1/2*5^(1/2
)-1/2))^(1/2),(1/2-1/2*5^(1/2))/(1/2+I-1/2*5^(1/2)),1/5*5^(1/2)*((1/2*5^(1/2)-1/2)*5^(1/2))^(1/2))*5^(1/2)-1/2
0*I*(x/(1/2*5^(1/2)-1/2)-1/2/(1/2*5^(1/2)-1/2)+1/2/(1/2*5^(1/2)-1/2)*5^(1/2))^(1/2)*(-5*x*5^(1/2)+5/2*5^(1/2)+
25/2)^(1/2)*(-x/(1/2*5^(1/2)-1/2))^(1/2)/(x^3-x^2-x)^(1/2)/(1/2+I-1/2*5^(1/2))*EllipticPi(((x-1/2+1/2*5^(1/2))
/(1/2*5^(1/2)-1/2))^(1/2),(1/2-1/2*5^(1/2))/(1/2+I-1/2*5^(1/2)),1/5*5^(1/2)*((1/2*5^(1/2)-1/2)*5^(1/2))^(1/2))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2485 vs. \(2 (97) = 194\).
time = 0.67, size = 2485, normalized size = 18.68 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/320*5^(1/4)*(sqrt(5)*sqrt(2) - sqrt(2))*sqrt(sqrt(5) + 5)*log(5*(5*x^4 - 20*x^3 + 5^(1/4)*sqrt(x^3 - x^2 -
x)*(sqrt(5)*sqrt(2)*(x^2 - 6*x - 1) - 5*sqrt(2)*(x^2 - 2*x - 1))*sqrt(sqrt(5) + 5) + 30*x^2 + 20*sqrt(5)*(x^3
- x^2 - x) + 20*x + 5)/(x^4 + 2*x^2 + 1)) + 1/320*5^(1/4)*(sqrt(5)*sqrt(2) - sqrt(2))*sqrt(sqrt(5) + 5)*log(5*
(5*x^4 - 20*x^3 - 5^(1/4)*sqrt(x^3 - x^2 - x)*(sqrt(5)*sqrt(2)*(x^2 - 6*x - 1) - 5*sqrt(2)*(x^2 - 2*x - 1))*sq
rt(sqrt(5) + 5) + 30*x^2 + 20*sqrt(5)*(x^3 - x^2 - x) + 20*x + 5)/(x^4 + 2*x^2 + 1)) + 1/40*5^(1/4)*sqrt(2)*sq
rt(sqrt(5) + 5)*arctan(-1/200*(100*x^11 + 1300*x^10 - 6700*x^9 - 4400*x^8 + 28400*x^7 + 1400*x^6 - 28400*x^5 -
4400*x^4 + 6700*x^3 + 1300*x^2 + 5*sqrt(x^3 - x^2 - x)*(5^(3/4)*(sqrt(5)*sqrt(2)*(x^10 - 4*x^9 - 17*x^8 + 56*
x^7 + 78*x^6 - 136*x^5 - 78*x^4 + 56*x^3 + 17*x^2 - 4*x - 1) - sqrt(2)*(x^10 + 8*x^9 - 57*x^8 - 24*x^7 + 294*x
^6 + 64*x^5 - 294*x^4 - 24*x^3 + 57*x^2 + 8*x - 1)) + 4*5^(1/4)*(sqrt(5)*sqrt(2)*(3*x^9 + 7*x^8 + 14*x^7 - 81*
x^6 - 10*x^5 + 81*x^4 + 14*x^3 - 7*x^2 + 3*x) - 5*sqrt(2)*(x^9 + 9*x^8 - 18*x^7 - 15*x^6 + 26*x^5 + 15*x^4 - 1
8*x^3 - 9*x^2 + x)))*sqrt(sqrt(5) + 5) - sqrt(5)*(240*x^10 + 160*x^9 - 1680*x^8 - 480*x^7 + 3200*x^6 + 480*x^5
- 1680*x^4 - 160*x^3 + 240*x^2 + sqrt(x^3 - x^2 - x)*(5^(3/4)*(sqrt(5)*sqrt(2)*(x^10 - 6*x^9 - 25*x^8 + 120*x
^7 - 58*x^6 - 196*x^5 + 58*x^4 + 120*x^3 + 25*x^2 - 6*x - 1) - sqrt(2)*(x^10 - 2*x^9 - 17*x^8 + 216*x^7 - 306*
x^6 - 396*x^5 + 306*x^4 + 216*x^3 + 17*x^2 - 2*x - 1)) + 4*5^(1/4)*(sqrt(5)*sqrt(2)*(3*x^9 - 3*x^8 - 56*x^7 +
69*x^6 + 90*x^5 - 69*x^4 - 56*x^3 + 3*x^2 + 3*x) - 5*sqrt(2)*(x^9 + 3*x^8 - 28*x^7 + 11*x^6 + 54*x^5 - 11*x^4
- 28*x^3 - 3*x^2 + x)))*sqrt(sqrt(5) + 5) + 4*sqrt(5)*(5*x^11 - 25*x^10 - 105*x^9 + 440*x^8 + 50*x^7 - 830*x^6
- 50*x^5 + 440*x^4 + 105*x^3 - 25*x^2 - sqrt(5)*(x^11 + 3*x^10 - 37*x^9 + 96*x^8 - 6*x^7 - 166*x^6 + 6*x^5 +
96*x^4 + 37*x^3 + 3*x^2 - x) - 5*x) - 80*sqrt(5)*(x^10 + 6*x^9 - 23*x^8 - 2*x^7 + 40*x^6 + 2*x^5 - 23*x^4 - 6*
x^3 + x^2))*sqrt((5*x^4 - 20*x^3 + 5^(1/4)*sqrt(x^3 - x^2 - x)*(sqrt(5)*sqrt(2)*(x^2 - 6*x - 1) - 5*sqrt(2)*(x
^2 - 2*x - 1))*sqrt(sqrt(5) + 5) + 30*x^2 + 20*sqrt(5)*(x^3 - x^2 - x) + 20*x + 5)/(x^4 + 2*x^2 + 1)) + 20*sqr
t(5)*(5*x^11 - 15*x^10 - 15*x^9 + 20*x^8 - 20*x^7 + 70*x^6 + 20*x^5 + 20*x^4 + 15*x^3 - 15*x^2 - sqrt(5)*(x^11
+ 13*x^10 - 67*x^9 - 44*x^8 + 284*x^7 + 14*x^6 - 284*x^5 - 44*x^4 + 67*x^3 + 13*x^2 - x) - 5*x) - 100*sqrt(5)
*(x^11 - 3*x^10 - 3*x^9 + 4*x^8 - 4*x^7 + 14*x^6 + 4*x^5 + 4*x^4 + 3*x^3 - 3*x^2 - x) - 100*x)/(x^11 - 9*x^10
- 45*x^9 + 180*x^8 + 18*x^7 - 326*x^6 - 18*x^5 + 180*x^4 + 45*x^3 - 9*x^2 - x)) + 1/40*5^(1/4)*sqrt(2)*sqrt(sq
rt(5) + 5)*arctan(1/200*(100*x^11 + 1300*x^10 - 6700*x^9 - 4400*x^8 + 28400*x^7 + 1400*x^6 - 28400*x^5 - 4400*
x^4 + 6700*x^3 + 1300*x^2 - 5*sqrt(x^3 - x^2 - x)*(5^(3/4)*(sqrt(5)*sqrt(2)*(x^10 - 4*x^9 - 17*x^8 + 56*x^7 +
78*x^6 - 136*x^5 - 78*x^4 + 56*x^3 + 17*x^2 - 4*x - 1) - sqrt(2)*(x^10 + 8*x^9 - 57*x^8 - 24*x^7 + 294*x^6 + 6
4*x^5 - 294*x^4 - 24*x^3 + 57*x^2 + 8*x - 1)) + 4*5^(1/4)*(sqrt(5)*sqrt(2)*(3*x^9 + 7*x^8 + 14*x^7 - 81*x^6 -
10*x^5 + 81*x^4 + 14*x^3 - 7*x^2 + 3*x) - 5*sqrt(2)*(x^9 + 9*x^8 - 18*x^7 - 15*x^6 + 26*x^5 + 15*x^4 - 18*x^3
- 9*x^2 + x)))*sqrt(sqrt(5) + 5) - sqrt(5)*(240*x^10 + 160*x^9 - 1680*x^8 - 480*x^7 + 3200*x^6 + 480*x^5 - 168
0*x^4 - 160*x^3 + 240*x^2 - sqrt(x^3 - x^2 - x)*(5^(3/4)*(sqrt(5)*sqrt(2)*(x^10 - 6*x^9 - 25*x^8 + 120*x^7 - 5
8*x^6 - 196*x^5 + 58*x^4 + 120*x^3 + 25*x^2 - 6*x - 1) - sqrt(2)*(x^10 - 2*x^9 - 17*x^8 + 216*x^7 - 306*x^6 -
396*x^5 + 306*x^4 + 216*x^3 + 17*x^2 - 2*x - 1)) + 4*5^(1/4)*(sqrt(5)*sqrt(2)*(3*x^9 - 3*x^8 - 56*x^7 + 69*x^6
+ 90*x^5 - 69*x^4 - 56*x^3 + 3*x^2 + 3*x) - 5*sqrt(2)*(x^9 + 3*x^8 - 28*x^7 + 11*x^6 + 54*x^5 - 11*x^4 - 28*x
^3 - 3*x^2 + x)))*sqrt(sqrt(5) + 5) + 4*sqrt(5)*(5*x^11 - 25*x^10 - 105*x^9 + 440*x^8 + 50*x^7 - 830*x^6 - 50*
x^5 + 440*x^4 + 105*x^3 - 25*x^2 - sqrt(5)*(x^11 + 3*x^10 - 37*x^9 + 96*x^8 - 6*x^7 - 166*x^6 + 6*x^5 + 96*x^4
+ 37*x^3 + 3*x^2 - x) - 5*x) - 80*sqrt(5)*(x^10 + 6*x^9 - 23*x^8 - 2*x^7 + 40*x^6 + 2*x^5 - 23*x^4 - 6*x^3 +
x^2))*sqrt((5*x^4 - 20*x^3 - 5^(1/4)*sqrt(x^3 - x^2 - x)*(sqrt(5)*sqrt(2)*(x^2 - 6*x - 1) - 5*sqrt(2)*(x^2 - 2
*x - 1))*sqrt(sqrt(5) + 5) + 30*x^2 + 20*sqrt(5)*(x^3 - x^2 - x) + 20*x + 5)/(x^4 + 2*x^2 + 1)) + 20*sqrt(5)*(
5*x^11 - 15*x^10 - 15*x^9 + 20*x^8 - 20*x^7 + 70*x^6 + 20*x^5 + 20*x^4 + 15*x^3 - 15*x^2 - sqrt(5)*(x^11 + 13*
x^10 - 67*x^9 - 44*x^8 + 284*x^7 + 14*x^6 - 284*x^5 - 44*x^4 + 67*x^3 + 13*x^2 - x) - 5*x) - 100*sqrt(5)*(x^11
- 3*x^10 - 3*x^9 + 4*x^8 - 4*x^7 + 14*x^6 + 4*x^5 + 4*x^4 + 3*x^3 - 3*x^2 - x) - 100*x)/(x^11 - 9*x^10 - 45*x
^9 + 180*x^8 + 18*x^7 - 326*x^6 - 18*x^5 + 180*x^4 + 45*x^3 - 9*x^2 - x)) + 1/4*arctan(1/2*(x^2 - 2*x - 1)/sqr
t(x^3 - x^2 - x))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.06, size = 533, normalized size = 4.01 \begin {gather*} -\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((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)) - ((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)*e
llipticPi(- 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))

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