\(\int \frac {1+x^6}{\sqrt {x+x^2+x^3} (1-x^6)} \, dx\) [1816]

Optimal result

Integrand size = 27, antiderivative size = 123 \[ \int \frac {1+x^6}{\sqrt {x+x^2+x^3} \left (1-x^6\right )} \, dx=\frac {2 \sqrt {x+x^2+x^3}}{3 \left (1+x+x^2\right )}+\frac {1}{3} \arctan \left (\frac {\sqrt {x+x^2+x^3}}{1+x+x^2}\right )+\frac {1}{3} \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {x+x^2+x^3}}{1+x+x^2}\right )+\frac {\text {arctanh}\left (\frac {\sqrt {3} \sqrt {x+x^2+x^3}}{1+x+x^2}\right )}{3 \sqrt {3}} \]

[Out]

2*(x^3+x^2+x)^(1/2)/(3*x^2+3*x+3)+1/3*arctan((x^3+x^2+x)^(1/2)/(x^2+x+1))+1/3*2^(1/2)*arctanh(2^(1/2)*(x^3+x^2
+x)^(1/2)/(x^2+x+1))+1/9*arctanh(3^(1/2)*(x^3+x^2+x)^(1/2)/(x^2+x+1))*3^(1/2)

Rubi [C] (verified)

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

Time = 1.56 (sec) , antiderivative size = 785, normalized size of antiderivative = 6.38, number of steps used = 43, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {2081, 6857, 730, 1117, 948, 175, 552, 551, 863, 836, 853, 1211, 1209} \[ \int \frac {1+x^6}{\sqrt {x+x^2+x^3} \left (1-x^6\right )} \, dx=\frac {4 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \operatorname {EllipticPi}\left (-1,\arcsin \left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right ),\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{3 \left (1-i \sqrt {3}\right ) \sqrt {x^3+x^2+x}}+\frac {4 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \operatorname {EllipticPi}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ),\arcsin \left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right ),\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{3 \left (1-i \sqrt {3}\right ) \sqrt {x^3+x^2+x}}+\frac {4 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \operatorname {EllipticPi}\left (\frac {1}{2} \left (-1+i \sqrt {3}\right ),\arcsin \left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right ),\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{3 \left (1-i \sqrt {3}\right ) \sqrt {x^3+x^2+x}}+\frac {4 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \operatorname {EllipticPi}\left (\frac {1}{2} \left (1+i \sqrt {3}\right ),\arcsin \left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right ),\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{3 \left (1-i \sqrt {3}\right ) \sqrt {x^3+x^2+x}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt {x} (x+1) \sqrt {\frac {x^2+x+1}{(x+1)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {x}\right ),\frac {1}{4}\right )}{6 \sqrt {x^3+x^2+x}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt {x} (x+1) \sqrt {\frac {x^2+x+1}{(x+1)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {x}\right ),\frac {1}{4}\right )}{6 \sqrt {x^3+x^2+x}}-\frac {\sqrt {x} (x+1) \sqrt {\frac {x^2+x+1}{(x+1)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {x}\right ),\frac {1}{4}\right )}{\sqrt {x^3+x^2+x}}+\frac {2 x \left (-i \sqrt {3} x-(-1)^{2/3}+1\right )}{9 \sqrt {x^3+x^2+x}}+\frac {2 x \left (i \sqrt {3} x+\sqrt [3]{-1}+1\right )}{9 \sqrt {x^3+x^2+x}} \]

[In]

Int[(1 + x^6)/(Sqrt[x + x^2 + x^3]*(1 - x^6)),x]

[Out]

(2*x*(1 - (-1)^(2/3) - I*Sqrt[3]*x))/(9*Sqrt[x + x^2 + x^3]) + (2*x*(1 + (-1)^(1/3) + I*Sqrt[3]*x))/(9*Sqrt[x
+ x^2 + x^3]) - (Sqrt[x]*(1 + x)*Sqrt[(1 + x + x^2)/(1 + x)^2]*EllipticF[2*ArcTan[Sqrt[x]], 1/4])/Sqrt[x + x^2
+ x^3] + ((1 - I*Sqrt[3])*Sqrt[x]*(1 + x)*Sqrt[(1 + x + x^2)/(1 + x)^2]*EllipticF[2*ArcTan[Sqrt[x]], 1/4])/(6
*Sqrt[x + x^2 + x^3]) + ((1 + I*Sqrt[3])*Sqrt[x]*(1 + x)*Sqrt[(1 + x + x^2)/(1 + x)^2]*EllipticF[2*ArcTan[Sqrt
[x]], 1/4])/(6*Sqrt[x + x^2 + x^3]) + (4*Sqrt[x]*Sqrt[1 + (2*x)/(1 - I*Sqrt[3])]*Sqrt[1 + (2*x)/(1 + I*Sqrt[3]
)]*EllipticPi[-1, ArcSin[((1 - I*Sqrt[3])*Sqrt[x])/2], (I + Sqrt[3])/(I - Sqrt[3])])/(3*(1 - I*Sqrt[3])*Sqrt[x
+ x^2 + x^3]) + (4*Sqrt[x]*Sqrt[1 + (2*x)/(1 - I*Sqrt[3])]*Sqrt[1 + (2*x)/(1 + I*Sqrt[3])]*EllipticPi[(1 - I*
Sqrt[3])/2, ArcSin[((1 - I*Sqrt[3])*Sqrt[x])/2], (I + Sqrt[3])/(I - Sqrt[3])])/(3*(1 - I*Sqrt[3])*Sqrt[x + x^2
+ x^3]) + (4*Sqrt[x]*Sqrt[1 + (2*x)/(1 - I*Sqrt[3])]*Sqrt[1 + (2*x)/(1 + I*Sqrt[3])]*EllipticPi[(-1 + I*Sqrt[
3])/2, ArcSin[((1 - I*Sqrt[3])*Sqrt[x])/2], (I + Sqrt[3])/(I - Sqrt[3])])/(3*(1 - I*Sqrt[3])*Sqrt[x + x^2 + x^
3]) + (4*Sqrt[x]*Sqrt[1 + (2*x)/(1 - I*Sqrt[3])]*Sqrt[1 + (2*x)/(1 + I*Sqrt[3])]*EllipticPi[(1 + I*Sqrt[3])/2,
ArcSin[((1 - I*Sqrt[3])*Sqrt[x])/2], (I + Sqrt[3])/(I - Sqrt[3])])/(3*(1 - I*Sqrt[3])*Sqrt[x + x^2 + x^3])

Rule 175

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] &&  !SimplerQ[e
+ f*x, c + d*x] &&  !SimplerQ[g + h*x, c + d*x]

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 836

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)
*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 853

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

Rule 863

Int[((x_)^(n_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_))/((d_) + (e_.)*(x_)), x_Symbol] :> Int[x^n*(a/d + c*(
x/e))*(a + b*x + c*x^2)^(p - 1), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b
*d*e + a*e^2, 0] &&  !IntegerQ[p] && ( !IntegerQ[n] ||  !IntegerQ[2*p] || IGtQ[n, 2] || (GtQ[p, 0] && NeQ[n, 2
]))

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 1117

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

Rule 1209

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(
-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 +
q^2*x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c))], x] /; EqQ[e + d*q^2,
0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rule 1211

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

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^6}{\sqrt {x} \sqrt {1+x+x^2} \left (1-x^6\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^6\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^6\right )} \, dx}{\sqrt {x+x^2+x^3}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {1+x+x^2}\right ) \int \left (\frac {1}{2 \sqrt {x} \sqrt {1+x+x^2} \left (1-x^3\right )}+\frac {1}{2 \sqrt {x} \sqrt {1+x+x^2} \left (1+x^3\right )}\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} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {x}\right ),\frac {1}{4}\right )}{\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} \left (1-x^3\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} \left (1+x^3\right )} \, dx}{\sqrt {x+x^2+x^3}} \\ & = -\frac {\sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {x}\right ),\frac {1}{4}\right )}{\sqrt {x+x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \left (-\frac {1}{3 (-1-x) \sqrt {x} \sqrt {1+x+x^2}}-\frac {1}{3 \sqrt {x} \left (-1+\sqrt [3]{-1} x\right ) \sqrt {1+x+x^2}}-\frac {1}{3 \sqrt {x} \left (-1-(-1)^{2/3} x\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 \left (\frac {1}{3 (1-x) \sqrt {x} \sqrt {1+x+x^2}}+\frac {1}{3 \sqrt {x} \left (1+\sqrt [3]{-1} x\right ) \sqrt {1+x+x^2}}+\frac {1}{3 \sqrt {x} \left (1-(-1)^{2/3} x\right ) \sqrt {1+x+x^2}}\right ) \, dx}{\sqrt {x+x^2+x^3}} \\ & = -\frac {\sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {x}\right ),\frac {1}{4}\right )}{\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}{3 \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}{3 \sqrt {x+x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1}{\sqrt {x} \left (-1+\sqrt [3]{-1} x\right ) \sqrt {1+x+x^2}} \, dx}{3 \sqrt {x+x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1}{\sqrt {x} \left (1+\sqrt [3]{-1} x\right ) \sqrt {1+x+x^2}} \, dx}{3 \sqrt {x+x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1}{\sqrt {x} \left (-1-(-1)^{2/3} x\right ) \sqrt {1+x+x^2}} \, dx}{3 \sqrt {x+x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1}{\sqrt {x} \left (1-(-1)^{2/3} x\right ) \sqrt {1+x+x^2}} \, dx}{3 \sqrt {x+x^2+x^3}} \\ & = -\frac {\sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {x}\right ),\frac {1}{4}\right )}{\sqrt {x+x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}\right ) \int \frac {1}{(-1-x) \sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}} \, dx}{3 \sqrt {x+x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}\right ) \int \frac {1}{(1-x) \sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}} \, dx}{3 \sqrt {x+x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}\right ) \int \frac {1}{\sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x} \left (-1+\sqrt [3]{-1} x\right )} \, dx}{3 \sqrt {x+x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}\right ) \int \frac {1}{\sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x} \left (-1-(-1)^{2/3} x\right )} \, dx}{3 \sqrt {x+x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1+\sqrt [3]{-1} x}{\sqrt {x} \left (1+x+x^2\right )^{3/2}} \, dx}{3 \sqrt {x+x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1-(-1)^{2/3} x}{\sqrt {x} \left (1+x+x^2\right )^{3/2}} \, dx}{3 \sqrt {x+x^2+x^3}} \\ & = \frac {2 x \left (1-(-1)^{2/3}-i \sqrt {3} x\right )}{9 \sqrt {x+x^2+x^3}}+\frac {2 x \left (1+\sqrt [3]{-1}+i \sqrt {3} x\right )}{9 \sqrt {x+x^2+x^3}}-\frac {\sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {x}\right ),\frac {1}{4}\right )}{\sqrt {x+x^2+x^3}}-\frac {\left (2 \sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+x^2\right ) \sqrt {1-i \sqrt {3}+2 x^2} \sqrt {1+i \sqrt {3}+2 x^2}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}\right ) \text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {1-i \sqrt {3}+2 x^2} \sqrt {1+i \sqrt {3}+2 x^2}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-i \sqrt {3}+2 x^2} \sqrt {1+i \sqrt {3}+2 x^2} \left (1-\sqrt [3]{-1} x^2\right )} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-i \sqrt {3}+2 x^2} \sqrt {1+i \sqrt {3}+2 x^2} \left (1+(-1)^{2/3} x^2\right )} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {\frac {1}{2} \left (2-\sqrt [3]{-1}\right )-\frac {1}{2} i \sqrt {3} x}{\sqrt {x} \sqrt {1+x+x^2}} \, dx}{9 \sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {\frac {1}{2} \left (2+(-1)^{2/3}\right )+\frac {1}{2} i \sqrt {3} x}{\sqrt {x} \sqrt {1+x+x^2}} \, dx}{9 \sqrt {x+x^2+x^3}} \\ & = \frac {2 x \left (1-(-1)^{2/3}-i \sqrt {3} x\right )}{9 \sqrt {x+x^2+x^3}}+\frac {2 x \left (1+\sqrt [3]{-1}+i \sqrt {3} x\right )}{9 \sqrt {x+x^2+x^3}}-\frac {\sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {x}\right ),\frac {1}{4}\right )}{\sqrt {x+x^2+x^3}}-\frac {\left (2 \sqrt {x} \sqrt {1+i \sqrt {3}+2 x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+x^2\right ) \sqrt {1+i \sqrt {3}+2 x^2} \sqrt {1+\frac {2 x^2}{1-i \sqrt {3}}}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1+i \sqrt {3}+2 x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}}\right ) \text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {1+i \sqrt {3}+2 x^2} \sqrt {1+\frac {2 x^2}{1-i \sqrt {3}}}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1+i \sqrt {3}+2 x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+i \sqrt {3}+2 x^2} \left (1-\sqrt [3]{-1} x^2\right ) \sqrt {1+\frac {2 x^2}{1-i \sqrt {3}}}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1+i \sqrt {3}+2 x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+i \sqrt {3}+2 x^2} \left (1+(-1)^{2/3} x^2\right ) \sqrt {1+\frac {2 x^2}{1-i \sqrt {3}}}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x+x^2+x^3}}+\frac {\left (4 \sqrt {x} \sqrt {1+x+x^2}\right ) \text {Subst}\left (\int \frac {\frac {1}{2} \left (2-\sqrt [3]{-1}\right )-\frac {1}{2} i \sqrt {3} x^2}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{9 \sqrt {x+x^2+x^3}}+\frac {\left (4 \sqrt {x} \sqrt {1+x+x^2}\right ) \text {Subst}\left (\int \frac {\frac {1}{2} \left (2+(-1)^{2/3}\right )+\frac {1}{2} i \sqrt {3} x^2}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{9 \sqrt {x+x^2+x^3}} \\ & = \frac {2 x \left (1-(-1)^{2/3}-i \sqrt {3} x\right )}{9 \sqrt {x+x^2+x^3}}+\frac {2 x \left (1+\sqrt [3]{-1}+i \sqrt {3} x\right )}{9 \sqrt {x+x^2+x^3}}-\frac {\sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {x}\right ),\frac {1}{4}\right )}{\sqrt {x+x^2+x^3}}-\frac {\left (2 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+x^2\right ) \sqrt {1+\frac {2 x^2}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x^2}{1+i \sqrt {3}}}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}}\right ) \text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {1+\frac {2 x^2}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x^2}{1+i \sqrt {3}}}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\sqrt [3]{-1} x^2\right ) \sqrt {1+\frac {2 x^2}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x^2}{1+i \sqrt {3}}}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}}\right ) \text {Subst}\left (\int \frac {1}{\left (1+(-1)^{2/3} x^2\right ) \sqrt {1+\frac {2 x^2}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x^2}{1+i \sqrt {3}}}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x+x^2+x^3}}+\frac {\left (\left (1-i \sqrt {3}\right ) \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 )}{3 \sqrt {x+x^2+x^3}}+\frac {\left (\left (1+i \sqrt {3}\right ) \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 )}{3 \sqrt {x+x^2+x^3}} \\ & = \frac {2 x \left (1-(-1)^{2/3}-i \sqrt {3} x\right )}{9 \sqrt {x+x^2+x^3}}+\frac {2 x \left (1+\sqrt [3]{-1}+i \sqrt {3} x\right )}{9 \sqrt {x+x^2+x^3}}-\frac {\sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {x}\right ),\frac {1}{4}\right )}{\sqrt {x+x^2+x^3}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {x}\right ),\frac {1}{4}\right )}{6 \sqrt {x+x^2+x^3}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {x}\right ),\frac {1}{4}\right )}{6 \sqrt {x+x^2+x^3}}+\frac {4 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \operatorname {EllipticPi}\left (-1,\arcsin \left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right ),\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{3 \left (1-i \sqrt {3}\right ) \sqrt {x+x^2+x^3}}+\frac {4 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \operatorname {EllipticPi}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ),\arcsin \left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right ),\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{3 \left (1-i \sqrt {3}\right ) \sqrt {x+x^2+x^3}}+\frac {4 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \operatorname {EllipticPi}\left (\frac {1}{2} \left (-1+i \sqrt {3}\right ),\arcsin \left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right ),\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{3 \left (1-i \sqrt {3}\right ) \sqrt {x+x^2+x^3}}+\frac {4 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \operatorname {EllipticPi}\left (\frac {1}{2} \left (1+i \sqrt {3}\right ),\arcsin \left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right ),\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{3 \left (1-i \sqrt {3}\right ) \sqrt {x+x^2+x^3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.10 \[ \int \frac {1+x^6}{\sqrt {x+x^2+x^3} \left (1-x^6\right )} \, dx=\frac {\sqrt {x} \left (6 \sqrt {x}+3 \sqrt {1+x+x^2} \arctan \left (\frac {\sqrt {x}}{\sqrt {1+x+x^2}}\right )+3 \sqrt {2} \sqrt {1+x+x^2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {1+x+x^2}}\right )+\sqrt {3} \sqrt {1+x+x^2} \text {arctanh}\left (\frac {\sqrt {3} \sqrt {x}}{\sqrt {1+x+x^2}}\right )\right )}{9 \sqrt {x \left (1+x+x^2\right )}} \]

[In]

Integrate[(1 + x^6)/(Sqrt[x + x^2 + x^3]*(1 - x^6)),x]

[Out]

(Sqrt[x]*(6*Sqrt[x] + 3*Sqrt[1 + x + x^2]*ArcTan[Sqrt[x]/Sqrt[1 + x + x^2]] + 3*Sqrt[2]*Sqrt[1 + x + x^2]*ArcT
anh[(Sqrt[2]*Sqrt[x])/Sqrt[1 + x + x^2]] + Sqrt[3]*Sqrt[1 + x + x^2]*ArcTanh[(Sqrt[3]*Sqrt[x])/Sqrt[1 + x + x^
2]]))/(9*Sqrt[x*(1 + x + x^2)])

Maple [A] (verified)

Time = 2.18 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.65

method	result	size
risch	\(\frac {2 x}{3 \sqrt {x \left (x^{2}+x +1\right )}}+\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (x^{2}+x +1\right )}\, \sqrt {3}}{3 x}\right )}{9}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (x^{2}+x +1\right )}\, \sqrt {2}}{2 x}\right )}{3}-\frac {\arctan \left (\frac {\sqrt {x \left (x^{2}+x +1\right )}}{x}\right )}{3}\)	\(80\)
default	\(-\frac {-\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (x^{2}+x +1\right )}\, \sqrt {2}}{2 x}\right ) \sqrt {x \left (x^{2}+x +1\right )}-\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (x^{2}+x +1\right )}\, \sqrt {3}}{3 x}\right ) \sqrt {x \left (x^{2}+x +1\right )}}{3}+\arctan \left (\frac {\sqrt {x \left (x^{2}+x +1\right )}}{x}\right ) \sqrt {x \left (x^{2}+x +1\right )}-2 x}{3 \sqrt {x \left (x^{2}+x +1\right )}}\)	\(111\)
pseudoelliptic	\(\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (x^{2}+x +1\right )}\, \sqrt {3}}{3 x}\right ) \sqrt {x \left (x^{2}+x +1\right )}+3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (x^{2}+x +1\right )}\, \sqrt {2}}{2 x}\right ) \sqrt {x \left (x^{2}+x +1\right )}-3 \arctan \left (\frac {\sqrt {x \left (x^{2}+x +1\right )}}{x}\right ) \sqrt {x \left (x^{2}+x +1\right )}+6 x}{9 \sqrt {x \left (x^{2}+x +1\right )}}\)	\(111\)
trager	\(\frac {2 \sqrt {x^{3}+x^{2}+x}}{3 \left (x^{2}+x +1\right )}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \sqrt {x^{3}+x^{2}+x}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{\left (1+x \right )^{2}}\right )}{6}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+6 \sqrt {x^{3}+x^{2}+x}}{\left (-1+x \right )^{2}}\right )}{18}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +4 \sqrt {x^{3}+x^{2}+x}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )}{x^{2}-x +1}\right )}{6}\)	\(180\)
elliptic	\(\text {Expression too large to display}\)	\(1330\)

[In]

int((x^6+1)/(x^3+x^2+x)^(1/2)/(-x^6+1),x,method=_RETURNVERBOSE)

[Out]

2/3*x/(x*(x^2+x+1))^(1/2)+1/9*3^(1/2)*arctanh(1/3*(x*(x^2+x+1))^(1/2)/x*3^(1/2))+1/3*2^(1/2)*arctanh(1/2*(x*(x
^2+x+1))^(1/2)/x*2^(1/2))-1/3*arctan((x*(x^2+x+1))^(1/2)/x)

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.59 \[ \int \frac {1+x^6}{\sqrt {x+x^2+x^3} \left (1-x^6\right )} \, dx=\frac {3 \, \sqrt {2} {\left (x^{2} + x + 1\right )} \log \left (\frac {x^{4} + 14 \, x^{3} + 4 \, \sqrt {2} \sqrt {x^{3} + x^{2} + x} {\left (x^{2} + 3 \, x + 1\right )} + 19 \, x^{2} + 14 \, x + 1}{x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1}\right ) + \sqrt {3} {\left (x^{2} + x + 1\right )} \log \left (\frac {x^{4} + 20 \, x^{3} + 4 \, \sqrt {3} \sqrt {x^{3} + x^{2} + x} {\left (x^{2} + 4 \, x + 1\right )} + 30 \, x^{2} + 20 \, x + 1}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1}\right ) - 6 \, {\left (x^{2} + x + 1\right )} \arctan \left (\frac {x^{2} + 1}{2 \, \sqrt {x^{3} + x^{2} + x}}\right ) + 24 \, \sqrt {x^{3} + x^{2} + x}}{36 \, {\left (x^{2} + x + 1\right )}} \]

[In]

integrate((x^6+1)/(x^3+x^2+x)^(1/2)/(-x^6+1),x, algorithm="fricas")

[Out]

1/36*(3*sqrt(2)*(x^2 + x + 1)*log((x^4 + 14*x^3 + 4*sqrt(2)*sqrt(x^3 + x^2 + x)*(x^2 + 3*x + 1) + 19*x^2 + 14*
x + 1)/(x^4 - 2*x^3 + 3*x^2 - 2*x + 1)) + sqrt(3)*(x^2 + x + 1)*log((x^4 + 20*x^3 + 4*sqrt(3)*sqrt(x^3 + x^2 +
x)*(x^2 + 4*x + 1) + 30*x^2 + 20*x + 1)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1)) - 6*(x^2 + x + 1)*arctan(1/2*(x^2 +
1)/sqrt(x^3 + x^2 + x)) + 24*sqrt(x^3 + x^2 + x))/(x^2 + x + 1)

Sympy [F]

\[ \int \frac {1+x^6}{\sqrt {x+x^2+x^3} \left (1-x^6\right )} \, dx=- \int \frac {x^{6}}{x^{6} \sqrt {x^{3} + x^{2} + x} - \sqrt {x^{3} + x^{2} + x}}\, dx - \int \frac {1}{x^{6} \sqrt {x^{3} + x^{2} + x} - \sqrt {x^{3} + x^{2} + x}}\, dx \]

[In]

integrate((x**6+1)/(x**3+x**2+x)**(1/2)/(-x**6+1),x)

[Out]

-Integral(x**6/(x**6*sqrt(x**3 + x**2 + x) - sqrt(x**3 + x**2 + x)), x) - Integral(1/(x**6*sqrt(x**3 + x**2 +
x) - sqrt(x**3 + x**2 + x)), x)

Maxima [F]

\[ \int \frac {1+x^6}{\sqrt {x+x^2+x^3} \left (1-x^6\right )} \, dx=\int { -\frac {x^{6} + 1}{{\left (x^{6} - 1\right )} \sqrt {x^{3} + x^{2} + x}} \,d x } \]

[In]

integrate((x^6+1)/(x^3+x^2+x)^(1/2)/(-x^6+1),x, algorithm="maxima")

[Out]

-integrate((x^6 + 1)/((x^6 - 1)*sqrt(x^3 + x^2 + x)), x)

Giac [F]

\[ \int \frac {1+x^6}{\sqrt {x+x^2+x^3} \left (1-x^6\right )} \, dx=\int { -\frac {x^{6} + 1}{{\left (x^{6} - 1\right )} \sqrt {x^{3} + x^{2} + x}} \,d x } \]

[In]

integrate((x^6+1)/(x^3+x^2+x)^(1/2)/(-x^6+1),x, algorithm="giac")

[Out]

integrate(-(x^6 + 1)/((x^6 - 1)*sqrt(x^3 + x^2 + x)), x)

Mupad [B] (verification not implemented)

Time = 5.50 (sec) , antiderivative size = 1195, normalized size of antiderivative = 9.72 \[ \int \frac {1+x^6}{\sqrt {x+x^2+x^3} \left (1-x^6\right )} \, dx=\text {Too large to display} \]

[In]

int(-(x^6 + 1)/((x^6 - 1)*(x + x^2 + x^3)^(1/2)),x)

[Out]

(2*((3^(1/2)*1i)/6 - 1/6)*(x/((3^(1/2)*1i)/2 - 1/2))^(1/2)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 - 1/2)
)^(1/2)*((x + (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 1/2))^(1/2)*ellipticPi(-1, asin((x/((3^(1/2)*1i)/2 - 1/2
))^(1/2)), -((3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 + 1/2)))/(x^2 + x^3 - x*((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i
)/2 + 1/2))^(1/2) - (2*((3^(1/2)*1i)/2 - 1/2)*(x/((3^(1/2)*1i)/2 - 1/2))^(1/2)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((
3^(1/2)*1i)/2 - 1/2))^(1/2)*((x + (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 1/2))^(1/2)*ellipticF(asin((x/((3^(1
/2)*1i)/2 - 1/2))^(1/2)), -((3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 + 1/2)))/(x^2 + x^3 - x*((3^(1/2)*1i)/2 - 1/
2)*((3^(1/2)*1i)/2 + 1/2))^(1/2) - (2*((3^(1/2)*1i)/6 - 1/6)*(x/((3^(1/2)*1i)/2 - 1/2))^(1/2)*((ellipticE(asin
((x/((3^(1/2)*1i)/2 - 1/2))^(1/2)), -((3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 + 1/2)) - ((x/((3^(1/2)*1i)/2 - 1/
2))^(1/2)*(x/((3^(1/2)*1i)/2 + 1/2) + 1)^(1/2))/(1 - x/((3^(1/2)*1i)/2 - 1/2))^(1/2))/(((3^(1/2)*1i)/2 - 1/2)/
((3^(1/2)*1i)/2 + 1/2) + 1) - ellipticF(asin((x/((3^(1/2)*1i)/2 - 1/2))^(1/2)), -((3^(1/2)*1i)/2 - 1/2)/((3^(1
/2)*1i)/2 + 1/2)))*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 - 1/2))^(1/2)*((x + (3^(1/2)*1i)/2 + 1/2)/((3^
(1/2)*1i)/2 + 1/2))^(1/2))/(x^2 + x^3 - x*((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2) + (2*((3^(1/2)*
1i)/2 - 1/2)*(x/((3^(1/2)*1i)/2 - 1/2))^(1/2)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 - 1/2))^(1/2)*((x +
(3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 1/2))^(1/2)*ellipticPi(1/2 - (3^(1/2)*1i)/2, asin((x/((3^(1/2)*1i)/2
- 1/2))^(1/2)), -((3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 + 1/2)))/(3*(x^2 + x^3 - x*((3^(1/2)*1i)/2 - 1/2)*((3^
(1/2)*1i)/2 + 1/2))^(1/2)) + (2*((3^(1/2)*1i)/2 - 1/2)*(x/((3^(1/2)*1i)/2 - 1/2))^(1/2)*(-(x - (3^(1/2)*1i)/2
+ 1/2)/((3^(1/2)*1i)/2 - 1/2))^(1/2)*((x + (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 1/2))^(1/2)*ellipticPi((3^(
1/2)*1i)/2 - 1/2, asin((x/((3^(1/2)*1i)/2 - 1/2))^(1/2)), -((3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 + 1/2)))/(3*
(x^2 + x^3 - x*((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2)) + (2*((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)
/6 + 1/6)*(x/((3^(1/2)*1i)/2 - 1/2))^(1/2)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 - 1/2))^(1/2)*((x + (3
^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 1/2))^(1/2)*ellipticPi(((3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 + 1/2), as
in((x/((3^(1/2)*1i)/2 - 1/2))^(1/2)), -((3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 + 1/2)))/(((3^(1/2)*1i)/2 + 1/2)
*(x^2 + x^3 - x*((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2)) + (2*((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i
)/6 + 1/6)*(x/((3^(1/2)*1i)/2 - 1/2))^(1/2)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 - 1/2))^(1/2)*((x + (
3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 1/2))^(1/2)*(ellipticE(asin((x/((3^(1/2)*1i)/2 - 1/2))^(1/2)), -((3^(1/
2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 + 1/2)) + (sin(2*asin((x/((3^(1/2)*1i)/2 - 1/2))^(1/2)))*((3^(1/2)*1i)/2 - 1/2
))/(2*((3^(1/2)*1i)/2 + 1/2)*(x/((3^(1/2)*1i)/2 + 1/2) + 1)^(1/2))))/(((3^(1/2)*1i)/2 + 1/2)*(((3^(1/2)*1i)/2
- 1/2)/((3^(1/2)*1i)/2 + 1/2) + 1)*(x^2 + x^3 - x*((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2))