3.6.2 \(\int \frac {(-1+x^2) \sqrt {x+x^3}}{(1+x^2) (1+x+x^2)^2} \, dx\)

Optimal. Leaf size=39 \[ \frac {\sqrt {x^3+x}}{x^2+x+1}-\tan ^{-1}\left (\frac {\sqrt {x^3+x}}{x^2+1}\right ) \]

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Rubi [C]  time = 9.42, antiderivative size = 2062, normalized size of antiderivative = 52.87, number of steps used = 196, number of rules used = 18, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {2056, 6733, 6742, 1727, 1742, 12, 1248, 725, 206, 1715, 1196, 1709, 220, 1707, 1725, 1217, 2, 6728}

result too large to display

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[x + x^3])/(3*(1 + I*Sqrt[3])*(1 - I*Sqrt[3] - 2*Sqrt[x])*Sqrt[x]) + ((I + Sqrt[3])*Sqrt[x + x^3])/(3*(
I - Sqrt[3])*(1 - I*Sqrt[3] - 2*Sqrt[x])*Sqrt[x]) + (2*Sqrt[x + x^3])/(3*(1 - I*Sqrt[3])*(1 + I*Sqrt[3] - 2*Sq
rt[x])*Sqrt[x]) + ((I - Sqrt[3])*Sqrt[x + x^3])/(3*(I + Sqrt[3])*(1 + I*Sqrt[3] - 2*Sqrt[x])*Sqrt[x]) - (2*Sqr
t[x + x^3])/(3*(1 + I*Sqrt[3])*(1 - I*Sqrt[3] + 2*Sqrt[x])*Sqrt[x]) - ((I + Sqrt[3])*Sqrt[x + x^3])/(3*(I - Sq
rt[3])*(1 - I*Sqrt[3] + 2*Sqrt[x])*Sqrt[x]) - (2*Sqrt[x + x^3])/(3*(1 - I*Sqrt[3])*(1 + I*Sqrt[3] + 2*Sqrt[x])
*Sqrt[x]) - ((I - Sqrt[3])*Sqrt[x + x^3])/(3*(I + Sqrt[3])*(1 + I*Sqrt[3] + 2*Sqrt[x])*Sqrt[x]) + (2*Sqrt[x +
x^3])/(3*(1 - I*Sqrt[3])*(1 + x)) + (2*Sqrt[x + x^3])/(3*(1 + I*Sqrt[3])*(1 + x)) + ((I - Sqrt[3])*Sqrt[x + x^
3])/(3*(I + Sqrt[3])*(1 + x)) + ((I + Sqrt[3])*Sqrt[x + x^3])/(3*(I - Sqrt[3])*(1 + x)) - (2*Sqrt[x + x^3]*Arc
Tan[Sqrt[x]/Sqrt[1 + x^2]])/(3*Sqrt[x]*Sqrt[1 + x^2]) + (Sqrt[x + x^3]*ArcTan[Sqrt[x]/Sqrt[1 + x^2]])/(Sqrt[3]
*(I - Sqrt[3])*Sqrt[x]*Sqrt[1 + x^2]) - ((1 - I*Sqrt[3])*Sqrt[x + x^3]*ArcTan[Sqrt[x]/Sqrt[1 + x^2]])/(6*Sqrt[
x]*Sqrt[1 + x^2]) - ((1 + I*Sqrt[3])*Sqrt[x + x^3]*ArcTan[Sqrt[x]/Sqrt[1 + x^2]])/(6*Sqrt[x]*Sqrt[1 + x^2]) -
(Sqrt[x + x^3]*ArcTan[Sqrt[x]/Sqrt[1 + x^2]])/(Sqrt[3]*(I + Sqrt[3])*Sqrt[x]*Sqrt[1 + x^2]) - ((I - Sqrt[3])*S
qrt[x + x^3]*ArcTan[Sqrt[x]/Sqrt[1 + x^2]])/(2*(I + Sqrt[3])*Sqrt[x]*Sqrt[1 + x^2]) - ((I + Sqrt[3])*Sqrt[x +
x^3]*ArcTan[Sqrt[x]/Sqrt[1 + x^2]])/(2*(I - Sqrt[3])*Sqrt[x]*Sqrt[1 + x^2]) - (Sqrt[x + x^3]*ArcTanh[(2 - (1 -
I*Sqrt[3])*x)/(Sqrt[2*(1 - I*Sqrt[3])]*Sqrt[1 + x^2])])/(2*(I + Sqrt[3])*Sqrt[x]*Sqrt[1 + x^2]) + (Sqrt[x + x
^3]*ArcTanh[(4 + (1 + I*Sqrt[3])^2*x)/(2*Sqrt[2*(1 - I*Sqrt[3])]*Sqrt[1 + x^2])])/(2*(I + Sqrt[3])*Sqrt[x]*Sqr
t[1 + x^2]) - (2*(1 + x)*Sqrt[(1 + x^2)/(1 + x)^2]*Sqrt[x + x^3]*EllipticE[2*ArcTan[Sqrt[x]], 1/2])/(3*(1 - I*
Sqrt[3])*Sqrt[x]*(1 + x^2)) - (2*(1 + x)*Sqrt[(1 + x^2)/(1 + x)^2]*Sqrt[x + x^3]*EllipticE[2*ArcTan[Sqrt[x]],
1/2])/(3*(1 + I*Sqrt[3])*Sqrt[x]*(1 + x^2)) - ((I - Sqrt[3])*(1 + x)*Sqrt[(1 + x^2)/(1 + x)^2]*Sqrt[x + x^3]*E
llipticE[2*ArcTan[Sqrt[x]], 1/2])/(3*(I + Sqrt[3])*Sqrt[x]*(1 + x^2)) - ((I + Sqrt[3])*(1 + x)*Sqrt[(1 + x^2)/
(1 + x)^2]*Sqrt[x + x^3]*EllipticE[2*ArcTan[Sqrt[x]], 1/2])/(3*(I - Sqrt[3])*Sqrt[x]*(1 + x^2)) + ((1 + x)*Sqr
t[(1 + x^2)/(1 + x)^2]*Sqrt[x + x^3]*EllipticF[2*ArcTan[Sqrt[x]], 1/2])/(3*Sqrt[x]*(1 + x^2)) + ((1 + x)*Sqrt[
(1 + x^2)/(1 + x)^2]*Sqrt[x + x^3]*EllipticF[2*ArcTan[Sqrt[x]], 1/2])/(3*(1 - I*Sqrt[3])*Sqrt[x]*(1 + x^2)) -
((1 - I*Sqrt[3])*(1 + x)*Sqrt[(1 + x^2)/(1 + x)^2]*Sqrt[x + x^3]*EllipticF[2*ArcTan[Sqrt[x]], 1/2])/(4*Sqrt[x]
*(1 + x^2)) + ((1 + x)*Sqrt[(1 + x^2)/(1 + x)^2]*Sqrt[x + x^3]*EllipticF[2*ArcTan[Sqrt[x]], 1/2])/(3*(1 + I*Sq
rt[3])*Sqrt[x]*(1 + x^2)) - ((1 + I*Sqrt[3])*(1 + x)*Sqrt[(1 + x^2)/(1 + x)^2]*Sqrt[x + x^3]*EllipticF[2*ArcTa
n[Sqrt[x]], 1/2])/(4*Sqrt[x]*(1 + x^2)) + ((3 - I*Sqrt[3])*(1 + x)*Sqrt[(1 + x^2)/(1 + x)^2]*Sqrt[x + x^3]*Ell
ipticPi[1/4, 2*ArcTan[Sqrt[x]], 1/2])/(24*Sqrt[x]*(1 + x^2)) + ((3 + I*Sqrt[3])*(1 + x)*Sqrt[(1 + x^2)/(1 + x)
^2]*Sqrt[x + x^3]*EllipticPi[1/4, 2*ArcTan[Sqrt[x]], 1/2])/(24*Sqrt[x]*(1 + x^2)) + ((I - Sqrt[3])*(1 + x)*Sqr
t[(1 + x^2)/(1 + x)^2]*Sqrt[x + x^3]*EllipticPi[1/4, 2*ArcTan[Sqrt[x]], 1/2])/(4*(I + Sqrt[3])*Sqrt[x]*(1 + x^
2)) + ((I + Sqrt[3])*(1 + x)*Sqrt[(1 + x^2)/(1 + x)^2]*Sqrt[x + x^3]*EllipticPi[1/4, 2*ArcTan[Sqrt[x]], 1/2])/
(4*(I - Sqrt[3])*Sqrt[x]*(1 + x^2))

Rule 2

Int[(u_.)*((a_.) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[u*a^p, x] /; FreeQ[{a, b, n, p}, x] && EqQ[b, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 220

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

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 1196

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

Rule 1217

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

Rule 1248

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(d + e*x)^q
*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x]

Rule 1707

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

Rule 1709

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

Rule 1715

Int[(P4x_)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[c/a, 2], A = Coeff[P4x
, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, -Dist[C/(e*q), Int[(1 - q*x^2)/Sqrt[a + c*x^4], x], x] +
Dist[1/(c*e), Int[(A*c*e + a*C*d*q + (B*c*e - C*(c*d - a*e*q))*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]), x], x]] /;
FreeQ[{a, c, d, e}, x] && PolyQ[P4x, x^2, 2] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a]

Rule 1725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Dist[d, Int[1/((d^2 - e^2*x^2)*Sqrt[a + c*
x^4]), x], x] - Dist[e, Int[x/((d^2 - e^2*x^2)*Sqrt[a + c*x^4]), x], x] /; FreeQ[{a, c, d, e}, x]

Rule 1727

Int[((d_) + (e_.)*(x_))^(q_)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[(e^3*(d + e*x)^(q + 1)*Sqrt[a + c*x^
4])/((q + 1)*(c*d^4 + a*e^4)), x] + Dist[c/((q + 1)*(c*d^4 + a*e^4)), Int[((d + e*x)^(q + 1)*Simp[d^3*(q + 1)
- d^2*e*(q + 1)*x + d*e^2*(q + 1)*x^2 - e^3*(q + 3)*x^3, x])/Sqrt[a + c*x^4], x], x] /; FreeQ[{a, c, d, e}, x]
&& NeQ[c*d^4 + a*e^4, 0] && ILtQ[q, -1]

Rule 1742

Int[(Px_)/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[P
x, x, 1], C = Coeff[Px, x, 2], D = Coeff[Px, x, 3]}, Int[(x*(B*d - A*e + (d*D - C*e)*x^2))/((d^2 - e^2*x^2)*Sq
rt[a + c*x^4]), x] + Int[(A*d + (C*d - B*e)*x^2 - D*e*x^4)/((d^2 - e^2*x^2)*Sqrt[a + c*x^4]), x]] /; FreeQ[{a,
c, d, e}, x] && PolyQ[Px, x] && LeQ[Expon[Px, x], 3] && NeQ[c*d^4 + a*e^4, 0]

Rule 2056

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 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rule 6733

Int[(u_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k, Subst[Int[x^(k*(m + 1) - 1)*(u /. x -> x^k
), x], x, x^(1/k)], x]] /; FractionQ[m]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {\left (-1+x^2\right ) \sqrt {x+x^3}}{\left (1+x^2\right ) \left (1+x+x^2\right )^2} \, dx &=\frac {\sqrt {x+x^3} \int \frac {\sqrt {x} \left (-1+x^2\right )}{\sqrt {1+x^2} \left (1+x+x^2\right )^2} \, dx}{\sqrt {x} \sqrt {1+x^2}}\\ &=\frac {\left (2 \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (-1+x^4\right )}{\sqrt {1+x^4} \left (1+x^2+x^4\right )^2} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^2}}\\ &=\frac {\left (2 \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \left (\frac {-1-x}{4 \left (1-x+x^2\right )^2 \sqrt {1+x^4}}+\frac {1+x}{4 \left (1-x+x^2\right ) \sqrt {1+x^4}}+\frac {-1+x}{4 \left (1+x+x^2\right )^2 \sqrt {1+x^4}}+\frac {1-x}{4 \left (1+x+x^2\right ) \sqrt {1+x^4}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^2}}\\ &=\frac {\sqrt {x+x^3} \operatorname {Subst}\left (\int \frac {-1-x}{\left (1-x+x^2\right )^2 \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1+x^2}}+\frac {\sqrt {x+x^3} \operatorname {Subst}\left (\int \frac {1+x}{\left (1-x+x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1+x^2}}+\frac {\sqrt {x+x^3} \operatorname {Subst}\left (\int \frac {-1+x}{\left (1+x+x^2\right )^2 \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1+x^2}}+\frac {\sqrt {x+x^3} \operatorname {Subst}\left (\int \frac {1-x}{\left (1+x+x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1+x^2}}\\ &=\frac {\sqrt {x+x^3} \operatorname {Subst}\left (\int \left (\frac {1-i \sqrt {3}}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt {1+x^4}}+\frac {1+i \sqrt {3}}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt {1+x^4}}\right ) \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1+x^2}}+\frac {\sqrt {x+x^3} \operatorname {Subst}\left (\int \left (\frac {-1-i \sqrt {3}}{\left (1-i \sqrt {3}+2 x\right ) \sqrt {1+x^4}}+\frac {-1+i \sqrt {3}}{\left (1+i \sqrt {3}+2 x\right ) \sqrt {1+x^4}}\right ) \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1+x^2}}+\frac {\sqrt {x+x^3} \operatorname {Subst}\left (\int \left (-\frac {1}{\left (1-x+x^2\right )^2 \sqrt {1+x^4}}-\frac {x}{\left (1-x+x^2\right )^2 \sqrt {1+x^4}}\right ) \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1+x^2}}+\frac {\sqrt {x+x^3} \operatorname {Subst}\left (\int \left (-\frac {1}{\left (1+x+x^2\right )^2 \sqrt {1+x^4}}+\frac {x}{\left (1+x+x^2\right )^2 \sqrt {1+x^4}}\right ) \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1+x^2}}\\ &=-\frac {\sqrt {x+x^3} \operatorname {Subst}\left (\int \frac {1}{\left (1-x+x^2\right )^2 \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1+x^2}}-\frac {\sqrt {x+x^3} \operatorname {Subst}\left (\int \frac {x}{\left (1-x+x^2\right )^2 \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1+x^2}}-\frac {\sqrt {x+x^3} \operatorname {Subst}\left (\int \frac {1}{\left (1+x+x^2\right )^2 \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1+x^2}}+\frac {\sqrt {x+x^3} \operatorname {Subst}\left (\int \frac {x}{\left (1+x+x^2\right )^2 \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1+x^2}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-i \sqrt {3}+2 x\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1+x^2}}+\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1+x^2}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1+x^2}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1+x^2}}\\ &=-\frac {\sqrt {x+x^3} \operatorname {Subst}\left (\int \left (-\frac {2 \left (1+i \sqrt {3}\right )}{3 \left (1+i \sqrt {3}-2 x\right )^2 \sqrt {1+x^4}}+\frac {2 i}{3 \sqrt {3} \left (1+i \sqrt {3}-2 x\right ) \sqrt {1+x^4}}-\frac {2 \left (1-i \sqrt {3}\right )}{3 \left (-1+i \sqrt {3}+2 x\right )^2 \sqrt {1+x^4}}+\frac {2 i}{3 \sqrt {3} \left (-1+i \sqrt {3}+2 x\right ) \sqrt {1+x^4}}\right ) \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1+x^2}}-\frac {\sqrt {x+x^3} \operatorname {Subst}\left (\int \left (-\frac {4}{3 \left (1+i \sqrt {3}-2 x\right )^2 \sqrt {1+x^4}}+\frac {4 i}{3 \sqrt {3} \left (1+i \sqrt {3}-2 x\right ) \sqrt {1+x^4}}-\frac {4}{3 \left (-1+i \sqrt {3}+2 x\right )^2 \sqrt {1+x^4}}+\frac {4 i}{3 \sqrt {3} \left (-1+i \sqrt {3}+2 x\right ) \sqrt {1+x^4}}\right ) \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1+x^2}}+\frac {\sqrt {x+x^3} \operatorname {Subst}\left (\int \left (-\frac {2 \left (-1+i \sqrt {3}\right )}{3 \left (-1+i \sqrt {3}-2 x\right )^2 \sqrt {1+x^4}}-\frac {2 i}{3 \sqrt {3} \left (-1+i \sqrt {3}-2 x\right ) \sqrt {1+x^4}}-\frac {2 \left (-1-i \sqrt {3}\right )}{3 \left (1+i \sqrt {3}+2 x\right )^2 \sqrt {1+x^4}}-\frac {2 i}{3 \sqrt {3} \left (1+i \sqrt {3}+2 x\right ) \sqrt {1+x^4}}\right ) \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1+x^2}}-\frac {\sqrt {x+x^3} \operatorname {Subst}\left (\int \left (-\frac {4}{3 \left (-1+i \sqrt {3}-2 x\right )^2 \sqrt {1+x^4}}+\frac {4 i}{3 \sqrt {3} \left (-1+i \sqrt {3}-2 x\right ) \sqrt {1+x^4}}-\frac {4}{3 \left (1+i \sqrt {3}+2 x\right )^2 \sqrt {1+x^4}}+\frac {4 i}{3 \sqrt {3} \left (1+i \sqrt {3}+2 x\right ) \sqrt {1+x^4}}\right ) \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1+x^2}}-\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (\left (1-i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^2}}-\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (\left (-1-i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^2}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \left (1-i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (-1-i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1+x^2}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \left (1-i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (1-i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1+x^2}}-\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (\left (1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^2}}-\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (\left (-1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^2}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \left (1+i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (-1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1+x^2}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \left (1+i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1+x^2}}\\ &=\frac {\left (2 \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+i \sqrt {3}-2 x\right )^2 \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x} \sqrt {1+x^2}}+\frac {\left (2 \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+i \sqrt {3}-2 x\right )^2 \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x} \sqrt {1+x^2}}+\frac {\left (2 \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+i \sqrt {3}+2 x\right )^2 \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x} \sqrt {1+x^2}}+\frac {\left (2 \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+i \sqrt {3}+2 x\right )^2 \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x} \sqrt {1+x^2}}-\frac {\left (i \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+i \sqrt {3}-2 x\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {3} \sqrt {x} \sqrt {1+x^2}}-\frac {\left (i \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+i \sqrt {3}-2 x\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {3} \sqrt {x} \sqrt {1+x^2}}-\frac {\left (i \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {3} \sqrt {x} \sqrt {1+x^2}}-\frac {\left (i \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {3} \sqrt {x} \sqrt {1+x^2}}-\frac {\left (2 i \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+i \sqrt {3}-2 x\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {3} \sqrt {x} \sqrt {1+x^2}}-\frac {\left (2 i \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+i \sqrt {3}-2 x\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {3} \sqrt {x} \sqrt {1+x^2}}-\frac {\left (2 i \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {3} \sqrt {x} \sqrt {1+x^2}}-\frac {\left (2 i \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {3} \sqrt {x} \sqrt {1+x^2}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{4 \sqrt {x} \sqrt {1+x^2}}-\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (1-i \sqrt {3}\right )^2-4 x\right ) \sqrt {1+x^2}} \, dx,x,x\right )}{2 \sqrt {x} \sqrt {1+x^2}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{\left (\left (1-i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^2}}+\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+i \sqrt {3}-2 x\right )^2 \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x} \sqrt {1+x^2}}+\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+i \sqrt {3}+2 x\right )^2 \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x} \sqrt {1+x^2}}-\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (-1-i \sqrt {3}\right )^2-4 x\right ) \sqrt {1+x^2}} \, dx,x,x\right )}{2 \sqrt {x} \sqrt {1+x^2}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{4 \sqrt {x} \sqrt {1+x^2}}-\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (1+i \sqrt {3}\right )^2-4 x\right ) \sqrt {1+x^2}} \, dx,x,x\right )}{2 \sqrt {x} \sqrt {1+x^2}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{\left (\left (1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^2}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \left (1-i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{4 \left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \left (1-i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{\left (\left (-1-i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{\left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+i \sqrt {3}-2 x\right )^2 \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x} \sqrt {1+x^2}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+i \sqrt {3}+2 x\right )^2 \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x} \sqrt {1+x^2}}-\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (-1+i \sqrt {3}\right )^2-4 x\right ) \sqrt {1+x^2}} \, dx,x,x\right )}{2 \sqrt {x} \sqrt {1+x^2}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \left (1+i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{4 \left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \left (1+i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{\left (\left (-1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{\left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}\\ &=\frac {2 \sqrt {x+x^3}}{3 \left (1+i \sqrt {3}\right ) \left (1-i \sqrt {3}-2 \sqrt {x}\right ) \sqrt {x}}+\frac {\left (i+\sqrt {3}\right ) \sqrt {x+x^3}}{3 \left (i-\sqrt {3}\right ) \left (1-i \sqrt {3}-2 \sqrt {x}\right ) \sqrt {x}}+\frac {2 \sqrt {x+x^3}}{3 \left (1-i \sqrt {3}\right ) \left (1+i \sqrt {3}-2 \sqrt {x}\right ) \sqrt {x}}+\frac {\left (i-\sqrt {3}\right ) \sqrt {x+x^3}}{3 \left (i+\sqrt {3}\right ) \left (1+i \sqrt {3}-2 \sqrt {x}\right ) \sqrt {x}}-\frac {2 \sqrt {x+x^3}}{3 \left (1+i \sqrt {3}\right ) \left (1-i \sqrt {3}+2 \sqrt {x}\right ) \sqrt {x}}-\frac {\left (i+\sqrt {3}\right ) \sqrt {x+x^3}}{3 \left (i-\sqrt {3}\right ) \left (1-i \sqrt {3}+2 \sqrt {x}\right ) \sqrt {x}}-\frac {2 \sqrt {x+x^3}}{3 \left (1-i \sqrt {3}\right ) \left (1+i \sqrt {3}+2 \sqrt {x}\right ) \sqrt {x}}-\frac {\left (i-\sqrt {3}\right ) \sqrt {x+x^3}}{3 \left (i+\sqrt {3}\right ) \left (1+i \sqrt {3}+2 \sqrt {x}\right ) \sqrt {x}}-\frac {\left (i-\sqrt {3}\right ) \sqrt {x+x^3} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {1+x^2}}\right )}{2 \left (i+\sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}-\frac {\left (i+\sqrt {3}\right ) \sqrt {x+x^3} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {1+x^2}}\right )}{2 \left (i-\sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}-\frac {\left (1-i \sqrt {3}\right ) (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{4 \sqrt {x} \left (1+x^2\right )}-\frac {\left (1+i \sqrt {3}\right ) (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{4 \sqrt {x} \left (1+x^2\right )}+\frac {\left (3-i \sqrt {3}\right ) (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} \Pi \left (\frac {1}{4};2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{8 \sqrt {x} \left (1+x^2\right )}+\frac {\left (3+i \sqrt {3}\right ) (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} \Pi \left (\frac {1}{4};2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{8 \sqrt {x} \left (1+x^2\right )}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{16+\left (1-i \sqrt {3}\right )^4-x^2} \, dx,x,\frac {-4-\left (1-i \sqrt {3}\right )^2 x}{\sqrt {1+x^2}}\right )}{2 \sqrt {x} \sqrt {1+x^2}}-\frac {\sqrt {x+x^3} \operatorname {Subst}\left (\int \frac {8+2 \left (1+i \sqrt {3}\right )^2 x-4 \left (1+i \sqrt {3}\right ) x^2-8 x^3}{\left (1+i \sqrt {3}+2 x\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{12 \left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}-\frac {\sqrt {x+x^3} \operatorname {Subst}\left (\int \frac {8-2 \left (1+i \sqrt {3}\right )^2 x-4 \left (1+i \sqrt {3}\right ) x^2+8 x^3}{\left (1+i \sqrt {3}-2 x\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{12 \left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}+\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{16+\left (-1-i \sqrt {3}\right )^4-x^2} \, dx,x,\frac {-4-\left (-1-i \sqrt {3}\right )^2 x}{\sqrt {1+x^2}}\right )}{2 \sqrt {x} \sqrt {1+x^2}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{16+\left (1+i \sqrt {3}\right )^4-x^2} \, dx,x,\frac {-4-\left (1+i \sqrt {3}\right )^2 x}{\sqrt {1+x^2}}\right )}{2 \sqrt {x} \sqrt {1+x^2}}-2 \frac {\left (i \left (-1+i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (-1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {3} \sqrt {x} \sqrt {1+x^2}}-2 \frac {\left (2 i \left (-1+i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (-1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {3} \sqrt {x} \sqrt {1+x^2}}-\frac {\sqrt {x+x^3} \operatorname {Subst}\left (\int \frac {-8-4 \left (1+i \sqrt {3}\right ) x+4 \left (1-i \sqrt {3}\right ) x^2-8 x^3}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{12 \left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}-\frac {\sqrt {x+x^3} \operatorname {Subst}\left (\int \frac {-8+4 \left (1+i \sqrt {3}\right ) x+4 \left (1-i \sqrt {3}\right ) x^2+8 x^3}{\left (-1+i \sqrt {3}-2 x\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{12 \left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}-\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {-8-4 \left (1+i \sqrt {3}\right ) x+4 \left (1-i \sqrt {3}\right ) x^2-8 x^3}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{24 \left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}-\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {-8+4 \left (1+i \sqrt {3}\right ) x+4 \left (1-i \sqrt {3}\right ) x^2+8 x^3}{\left (-1+i \sqrt {3}-2 x\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{24 \left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{16+\left (-1+i \sqrt {3}\right )^4-x^2} \, dx,x,\frac {-4-\left (-1+i \sqrt {3}\right )^2 x}{\sqrt {1+x^2}}\right )}{2 \sqrt {x} \sqrt {1+x^2}}-2 \frac {\left (i \left (1+i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {3} \sqrt {x} \sqrt {1+x^2}}-2 \frac {\left (2 i \left (1+i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {3} \sqrt {x} \sqrt {1+x^2}}-\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {8+2 \left (1+i \sqrt {3}\right )^2 x-4 \left (1+i \sqrt {3}\right ) x^2-8 x^3}{\left (1+i \sqrt {3}+2 x\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{24 \left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}-\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {8-2 \left (1+i \sqrt {3}\right )^2 x-4 \left (1+i \sqrt {3}\right ) x^2+8 x^3}{\left (1+i \sqrt {3}-2 x\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{24 \left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}\\ &=\frac {2 \sqrt {x+x^3}}{3 \left (1+i \sqrt {3}\right ) \left (1-i \sqrt {3}-2 \sqrt {x}\right ) \sqrt {x}}+\frac {\left (i+\sqrt {3}\right ) \sqrt {x+x^3}}{3 \left (i-\sqrt {3}\right ) \left (1-i \sqrt {3}-2 \sqrt {x}\right ) \sqrt {x}}+\frac {2 \sqrt {x+x^3}}{3 \left (1-i \sqrt {3}\right ) \left (1+i \sqrt {3}-2 \sqrt {x}\right ) \sqrt {x}}+\frac {\left (i-\sqrt {3}\right ) \sqrt {x+x^3}}{3 \left (i+\sqrt {3}\right ) \left (1+i \sqrt {3}-2 \sqrt {x}\right ) \sqrt {x}}-\frac {2 \sqrt {x+x^3}}{3 \left (1+i \sqrt {3}\right ) \left (1-i \sqrt {3}+2 \sqrt {x}\right ) \sqrt {x}}-\frac {\left (i+\sqrt {3}\right ) \sqrt {x+x^3}}{3 \left (i-\sqrt {3}\right ) \left (1-i \sqrt {3}+2 \sqrt {x}\right ) \sqrt {x}}-\frac {2 \sqrt {x+x^3}}{3 \left (1-i \sqrt {3}\right ) \left (1+i \sqrt {3}+2 \sqrt {x}\right ) \sqrt {x}}-\frac {\left (i-\sqrt {3}\right ) \sqrt {x+x^3}}{3 \left (i+\sqrt {3}\right ) \left (1+i \sqrt {3}+2 \sqrt {x}\right ) \sqrt {x}}-\frac {\left (i-\sqrt {3}\right ) \sqrt {x+x^3} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {1+x^2}}\right )}{2 \left (i+\sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}-\frac {\left (i+\sqrt {3}\right ) \sqrt {x+x^3} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {1+x^2}}\right )}{2 \left (i-\sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}-\frac {\sqrt {x+x^3} \tanh ^{-1}\left (\frac {2-\left (1-i \sqrt {3}\right ) x}{\sqrt {2 \left (1-i \sqrt {3}\right )} \sqrt {1+x^2}}\right )}{2 \left (i+\sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}+\frac {\sqrt {x+x^3} \tanh ^{-1}\left (\frac {2-\left (1+i \sqrt {3}\right ) x}{\sqrt {2 \left (1+i \sqrt {3}\right )} \sqrt {1+x^2}}\right )}{2 \left (i-\sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}+\frac {\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )} \sqrt {x+x^3} \tanh ^{-1}\left (\frac {2-\left (1+i \sqrt {3}\right ) x}{\sqrt {2 \left (1+i \sqrt {3}\right )} \sqrt {1+x^2}}\right )}{4 \sqrt {x} \sqrt {1+x^2}}+\frac {\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )} \sqrt {x+x^3} \tanh ^{-1}\left (\frac {4+\left (1+i \sqrt {3}\right )^2 x}{2 \sqrt {2 \left (1-i \sqrt {3}\right )} \sqrt {1+x^2}}\right )}{4 \sqrt {x} \sqrt {1+x^2}}-\frac {\left (1-i \sqrt {3}\right ) (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{4 \sqrt {x} \left (1+x^2\right )}-\frac {\left (1+i \sqrt {3}\right ) (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{4 \sqrt {x} \left (1+x^2\right )}+\frac {\left (3-i \sqrt {3}\right ) (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} \Pi \left (\frac {1}{4};2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{8 \sqrt {x} \left (1+x^2\right )}+\frac {\left (3+i \sqrt {3}\right ) (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} \Pi \left (\frac {1}{4};2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{8 \sqrt {x} \left (1+x^2\right )}-2 \left (\frac {\left (i \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {3} \sqrt {x} \sqrt {1+x^2}}+\frac {\left (2 i \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{\left (\left (1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {3} \sqrt {x} \sqrt {1+x^2}}\right )-2 \left (\frac {\left (i \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {3} \sqrt {x} \sqrt {1+x^2}}+\frac {\left (4 i \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{\left (\left (1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {3} \sqrt {x} \sqrt {1+x^2}}\right )-\frac {\sqrt {x+x^3} \operatorname {Subst}\left (\int \frac {\left (16-2 \left (1+i \sqrt {3}\right )^3\right ) x}{\left (\left (1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{12 \left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}-\frac {\sqrt {x+x^3} \operatorname {Subst}\left (\int \frac {\left (-16+2 \left (1+i \sqrt {3}\right )^3\right ) x}{\left (\left (1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{12 \left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}-2 \frac {\sqrt {x+x^3} \operatorname {Subst}\left (\int \frac {8 \left (1+i \sqrt {3}\right )-8 \left (1+i \sqrt {3}\right )^2 x^2+16 x^4}{\left (\left (1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{12 \left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}-2 \left (\frac {\left (i \left (-1+i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {3} \left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}+\frac {\left (2 i \left (-1+i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{\left (\left (-1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {3} \left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}\right )-2 \left (\frac {\left (i \left (-1+i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {3} \left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}+\frac {\left (4 i \left (-1+i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{\left (\left (-1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {3} \left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}\right )-\frac {\sqrt {x+x^3} \operatorname {Subst}\left (\int \frac {x \left (16-4 \left (-1+i \sqrt {3}\right ) \left (1+i \sqrt {3}\right )+\left (-8 \left (1-i \sqrt {3}\right )-8 \left (-1+i \sqrt {3}\right )\right ) x^2\right )}{\left (\left (-1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{12 \left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}-\frac {\sqrt {x+x^3} \operatorname {Subst}\left (\int \frac {x \left (-16+4 \left (-1+i \sqrt {3}\right ) \left (1+i \sqrt {3}\right )+\left (8 \left (1-i \sqrt {3}\right )+8 \left (-1+i \sqrt {3}\right )\right ) x^2\right )}{\left (\left (-1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{12 \left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}-2 \frac {\sqrt {x+x^3} \operatorname {Subst}\left (\int \frac {-8 \left (-1+i \sqrt {3}\right )+\left (4 \left (1-i \sqrt {3}\right ) \left (-1+i \sqrt {3}\right )+8 \left (1+i \sqrt {3}\right )\right ) x^2+16 x^4}{\left (\left (-1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{12 \left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}-\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {x \left (16-4 \left (-1+i \sqrt {3}\right ) \left (1+i \sqrt {3}\right )+\left (-8 \left (1-i \sqrt {3}\right )-8 \left (-1+i \sqrt {3}\right )\right ) x^2\right )}{\left (\left (-1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{24 \left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}-\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {x \left (-16+4 \left (-1+i \sqrt {3}\right ) \left (1+i \sqrt {3}\right )+\left (8 \left (1-i \sqrt {3}\right )+8 \left (-1+i \sqrt {3}\right )\right ) x^2\right )}{\left (\left (-1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{24 \left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}-2 \frac {\left (\left (1-i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {-8 \left (-1+i \sqrt {3}\right )+\left (4 \left (1-i \sqrt {3}\right ) \left (-1+i \sqrt {3}\right )+8 \left (1+i \sqrt {3}\right )\right ) x^2+16 x^4}{\left (\left (-1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{24 \left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}-\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {\left (16-2 \left (1+i \sqrt {3}\right )^3\right ) x}{\left (\left (1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{24 \left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}-\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {\left (-16+2 \left (1+i \sqrt {3}\right )^3\right ) x}{\left (\left (1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{24 \left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}-2 \frac {\left (\left (1+i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {8 \left (1+i \sqrt {3}\right )-8 \left (1+i \sqrt {3}\right )^2 x^2+16 x^4}{\left (\left (1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{24 \left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}\\ &=\frac {2 \sqrt {x+x^3}}{3 \left (1+i \sqrt {3}\right ) \left (1-i \sqrt {3}-2 \sqrt {x}\right ) \sqrt {x}}+\frac {\left (i+\sqrt {3}\right ) \sqrt {x+x^3}}{3 \left (i-\sqrt {3}\right ) \left (1-i \sqrt {3}-2 \sqrt {x}\right ) \sqrt {x}}+\frac {2 \sqrt {x+x^3}}{3 \left (1-i \sqrt {3}\right ) \left (1+i \sqrt {3}-2 \sqrt {x}\right ) \sqrt {x}}+\frac {\left (i-\sqrt {3}\right ) \sqrt {x+x^3}}{3 \left (i+\sqrt {3}\right ) \left (1+i \sqrt {3}-2 \sqrt {x}\right ) \sqrt {x}}-\frac {2 \sqrt {x+x^3}}{3 \left (1+i \sqrt {3}\right ) \left (1-i \sqrt {3}+2 \sqrt {x}\right ) \sqrt {x}}-\frac {\left (i+\sqrt {3}\right ) \sqrt {x+x^3}}{3 \left (i-\sqrt {3}\right ) \left (1-i \sqrt {3}+2 \sqrt {x}\right ) \sqrt {x}}-\frac {2 \sqrt {x+x^3}}{3 \left (1-i \sqrt {3}\right ) \left (1+i \sqrt {3}+2 \sqrt {x}\right ) \sqrt {x}}-\frac {\left (i-\sqrt {3}\right ) \sqrt {x+x^3}}{3 \left (i+\sqrt {3}\right ) \left (1+i \sqrt {3}+2 \sqrt {x}\right ) \sqrt {x}}-\frac {\left (i-\sqrt {3}\right ) \sqrt {x+x^3} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {1+x^2}}\right )}{2 \left (i+\sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}-\frac {\left (i+\sqrt {3}\right ) \sqrt {x+x^3} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {1+x^2}}\right )}{2 \left (i-\sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}-\frac {\sqrt {x+x^3} \tanh ^{-1}\left (\frac {2-\left (1-i \sqrt {3}\right ) x}{\sqrt {2 \left (1-i \sqrt {3}\right )} \sqrt {1+x^2}}\right )}{2 \left (i+\sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}+\frac {\sqrt {x+x^3} \tanh ^{-1}\left (\frac {2-\left (1+i \sqrt {3}\right ) x}{\sqrt {2 \left (1+i \sqrt {3}\right )} \sqrt {1+x^2}}\right )}{2 \left (i-\sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}+\frac {\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )} \sqrt {x+x^3} \tanh ^{-1}\left (\frac {2-\left (1+i \sqrt {3}\right ) x}{\sqrt {2 \left (1+i \sqrt {3}\right )} \sqrt {1+x^2}}\right )}{4 \sqrt {x} \sqrt {1+x^2}}+\frac {\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )} \sqrt {x+x^3} \tanh ^{-1}\left (\frac {4+\left (1+i \sqrt {3}\right )^2 x}{2 \sqrt {2 \left (1-i \sqrt {3}\right )} \sqrt {1+x^2}}\right )}{4 \sqrt {x} \sqrt {1+x^2}}-\frac {\left (1-i \sqrt {3}\right ) (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{4 \sqrt {x} \left (1+x^2\right )}-\frac {\left (1+i \sqrt {3}\right ) (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{4 \sqrt {x} \left (1+x^2\right )}+\frac {\left (3-i \sqrt {3}\right ) (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} \Pi \left (\frac {1}{4};2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{8 \sqrt {x} \left (1+x^2\right )}+\frac {\left (3+i \sqrt {3}\right ) (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} \Pi \left (\frac {1}{4};2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{8 \sqrt {x} \left (1+x^2\right )}-2 \left (-\frac {\sqrt {x+x^3} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {1+x^2}}\right )}{3 \sqrt {3} \left (i-\sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}+\frac {i (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{6 \sqrt {3} \sqrt {x} \left (1+x^2\right )}-\frac {\left (i-\sqrt {3}\right ) (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} \Pi \left (\frac {1}{4};2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{12 \left (i+\sqrt {3}\right ) \sqrt {x} \left (1+x^2\right )}\right )-2 \left (-\frac {\sqrt {x+x^3} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {1+x^2}}\right )}{6 \sqrt {3} \left (i-\sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}+\frac {i (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{12 \sqrt {3} \sqrt {x} \left (1+x^2\right )}-\frac {\left (i-\sqrt {3}\right ) (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} \Pi \left (\frac {1}{4};2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{24 \left (i+\sqrt {3}\right ) \sqrt {x} \left (1+x^2\right )}\right )-2 \left (\frac {\sqrt {x+x^3} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {1+x^2}}\right )}{3 \sqrt {3} \left (i+\sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}-\frac {i (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{6 \sqrt {3} \sqrt {x} \left (1+x^2\right )}-\frac {\left (i+\sqrt {3}\right ) (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} \Pi \left (\frac {1}{4};2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{12 \left (i-\sqrt {3}\right ) \sqrt {x} \left (1+x^2\right )}\right )-2 \left (\frac {\sqrt {x+x^3} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {1+x^2}}\right )}{6 \sqrt {3} \left (i+\sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}-\frac {i (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{12 \sqrt {3} \sqrt {x} \left (1+x^2\right )}-\frac {\left (i+\sqrt {3}\right ) (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} \Pi \left (\frac {1}{4};2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{24 \left (i-\sqrt {3}\right ) \sqrt {x} \left (1+x^2\right )}\right )-2 \left (-\frac {\sqrt {x+x^3} \operatorname {Subst}\left (\int \frac {-32 \left (1+i \sqrt {3}\right )+16 \left (1+i \sqrt {3}\right )^2+\left (32 \left (1+i \sqrt {3}\right )^2-16 \left (4+\left (1+i \sqrt {3}\right )^2\right )\right ) x^2}{\left (\left (1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{48 \left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}+\frac {\sqrt {x+x^3} \operatorname {Subst}\left (\int \frac {1-x^2}{\sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{3 \left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}\right )-\frac {\sqrt {x+x^3} \operatorname {Subst}\left (\int \frac {\left (16-4 \left (-1+i \sqrt {3}\right ) \left (1+i \sqrt {3}\right )\right ) x}{\left (\left (-1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{12 \left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}-\frac {\sqrt {x+x^3} \operatorname {Subst}\left (\int \frac {\left (-16+4 \left (-1+i \sqrt {3}\right ) \left (1+i \sqrt {3}\right )\right ) x}{\left (\left (-1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{12 \left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}-2 \left (-\frac {\sqrt {x+x^3} \operatorname {Subst}\left (\int \frac {32 \left (-1+i \sqrt {3}\right )+16 \left (-1+i \sqrt {3}\right )^2+\left (-16 \left (4+\left (-1+i \sqrt {3}\right )^2\right )-4 \left (4 \left (1-i \sqrt {3}\right ) \left (-1+i \sqrt {3}\right )+8 \left (1+i \sqrt {3}\right )\right )\right ) x^2}{\left (\left (-1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{48 \left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}+\frac {\sqrt {x+x^3} \operatorname {Subst}\left (\int \frac {1-x^2}{\sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{3 \left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}\right )-\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {\left (16-4 \left (-1+i \sqrt {3}\right ) \left (1+i \sqrt {3}\right )\right ) x}{\left (\left (-1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{24 \left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}-\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {\left (-16+4 \left (-1+i \sqrt {3}\right ) \left (1+i \sqrt {3}\right )\right ) x}{\left (\left (-1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{24 \left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}-2 \left (-\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {32 \left (-1+i \sqrt {3}\right )+16 \left (-1+i \sqrt {3}\right )^2+\left (-16 \left (4+\left (-1+i \sqrt {3}\right )^2\right )-4 \left (4 \left (1-i \sqrt {3}\right ) \left (-1+i \sqrt {3}\right )+8 \left (1+i \sqrt {3}\right )\right )\right ) x^2}{\left (\left (-1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{96 \left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}+\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{\sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{6 \left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}\right )-2 \left (-\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {-32 \left (1+i \sqrt {3}\right )+16 \left (1+i \sqrt {3}\right )^2+\left (32 \left (1+i \sqrt {3}\right )^2-16 \left (4+\left (1+i \sqrt {3}\right )^2\right )\right ) x^2}{\left (\left (1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{96 \left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{\sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{6 \left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}\right )\\ &=\frac {2 \sqrt {x+x^3}}{3 \left (1+i \sqrt {3}\right ) \left (1-i \sqrt {3}-2 \sqrt {x}\right ) \sqrt {x}}+\frac {\left (i+\sqrt {3}\right ) \sqrt {x+x^3}}{3 \left (i-\sqrt {3}\right ) \left (1-i \sqrt {3}-2 \sqrt {x}\right ) \sqrt {x}}+\frac {2 \sqrt {x+x^3}}{3 \left (1-i \sqrt {3}\right ) \left (1+i \sqrt {3}-2 \sqrt {x}\right ) \sqrt {x}}+\frac {\left (i-\sqrt {3}\right ) \sqrt {x+x^3}}{3 \left (i+\sqrt {3}\right ) \left (1+i \sqrt {3}-2 \sqrt {x}\right ) \sqrt {x}}-\frac {2 \sqrt {x+x^3}}{3 \left (1+i \sqrt {3}\right ) \left (1-i \sqrt {3}+2 \sqrt {x}\right ) \sqrt {x}}-\frac {\left (i+\sqrt {3}\right ) \sqrt {x+x^3}}{3 \left (i-\sqrt {3}\right ) \left (1-i \sqrt {3}+2 \sqrt {x}\right ) \sqrt {x}}-\frac {2 \sqrt {x+x^3}}{3 \left (1-i \sqrt {3}\right ) \left (1+i \sqrt {3}+2 \sqrt {x}\right ) \sqrt {x}}-\frac {\left (i-\sqrt {3}\right ) \sqrt {x+x^3}}{3 \left (i+\sqrt {3}\right ) \left (1+i \sqrt {3}+2 \sqrt {x}\right ) \sqrt {x}}-\frac {\left (i-\sqrt {3}\right ) \sqrt {x+x^3} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {1+x^2}}\right )}{2 \left (i+\sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}-\frac {\left (i+\sqrt {3}\right ) \sqrt {x+x^3} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {1+x^2}}\right )}{2 \left (i-\sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}-\frac {\sqrt {x+x^3} \tanh ^{-1}\left (\frac {2-\left (1-i \sqrt {3}\right ) x}{\sqrt {2 \left (1-i \sqrt {3}\right )} \sqrt {1+x^2}}\right )}{2 \left (i+\sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}+\frac {\sqrt {x+x^3} \tanh ^{-1}\left (\frac {2-\left (1+i \sqrt {3}\right ) x}{\sqrt {2 \left (1+i \sqrt {3}\right )} \sqrt {1+x^2}}\right )}{2 \left (i-\sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}+\frac {\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )} \sqrt {x+x^3} \tanh ^{-1}\left (\frac {2-\left (1+i \sqrt {3}\right ) x}{\sqrt {2 \left (1+i \sqrt {3}\right )} \sqrt {1+x^2}}\right )}{4 \sqrt {x} \sqrt {1+x^2}}+\frac {\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )} \sqrt {x+x^3} \tanh ^{-1}\left (\frac {4+\left (1+i \sqrt {3}\right )^2 x}{2 \sqrt {2 \left (1-i \sqrt {3}\right )} \sqrt {1+x^2}}\right )}{4 \sqrt {x} \sqrt {1+x^2}}-\frac {\left (1-i \sqrt {3}\right ) (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{4 \sqrt {x} \left (1+x^2\right )}-\frac {\left (1+i \sqrt {3}\right ) (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{4 \sqrt {x} \left (1+x^2\right )}+\frac {\left (3-i \sqrt {3}\right ) (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} \Pi \left (\frac {1}{4};2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{8 \sqrt {x} \left (1+x^2\right )}+\frac {\left (3+i \sqrt {3}\right ) (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} \Pi \left (\frac {1}{4};2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{8 \sqrt {x} \left (1+x^2\right )}-2 \left (-\frac {\sqrt {x+x^3} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {1+x^2}}\right )}{3 \sqrt {3} \left (i-\sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}+\frac {i (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{6 \sqrt {3} \sqrt {x} \left (1+x^2\right )}-\frac {\left (i-\sqrt {3}\right ) (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} \Pi \left (\frac {1}{4};2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{12 \left (i+\sqrt {3}\right ) \sqrt {x} \left (1+x^2\right )}\right )-2 \left (-\frac {\sqrt {x+x^3} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {1+x^2}}\right )}{6 \sqrt {3} \left (i-\sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}+\frac {i (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{12 \sqrt {3} \sqrt {x} \left (1+x^2\right )}-\frac {\left (i-\sqrt {3}\right ) (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} \Pi \left (\frac {1}{4};2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{24 \left (i+\sqrt {3}\right ) \sqrt {x} \left (1+x^2\right )}\right )-2 \left (\frac {\sqrt {x+x^3} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {1+x^2}}\right )}{3 \sqrt {3} \left (i+\sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}-\frac {i (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{6 \sqrt {3} \sqrt {x} \left (1+x^2\right )}-\frac {\left (i+\sqrt {3}\right ) (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} \Pi \left (\frac {1}{4};2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{12 \left (i-\sqrt {3}\right ) \sqrt {x} \left (1+x^2\right )}\right )-2 \left (\frac {\sqrt {x+x^3} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {1+x^2}}\right )}{6 \sqrt {3} \left (i+\sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}-\frac {i (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{12 \sqrt {3} \sqrt {x} \left (1+x^2\right )}-\frac {\left (i+\sqrt {3}\right ) (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} \Pi \left (\frac {1}{4};2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{24 \left (i-\sqrt {3}\right ) \sqrt {x} \left (1+x^2\right )}\right )-2 \left (-\frac {\sqrt {x+x^3}}{3 \left (1+i \sqrt {3}\right ) (1+x)}+\frac {(1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} E\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{3 \left (1+i \sqrt {3}\right ) \sqrt {x} \left (1+x^2\right )}+\frac {\left (8 \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{\left (\left (-1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{3 \left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}-\frac {\left (\left (i-\sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{3 \left (1+i \sqrt {3}\right ) \left (i+\sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}\right )-2 \left (-\frac {\left (i+\sqrt {3}\right ) \sqrt {x+x^3}}{6 \left (i-\sqrt {3}\right ) (1+x)}+\frac {\left (i+\sqrt {3}\right ) (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} E\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{6 \left (i-\sqrt {3}\right ) \sqrt {x} \left (1+x^2\right )}+\frac {\left (4 \left (1-i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{\left (\left (-1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{3 \left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}-\frac {\left (\left (i-\sqrt {3}\right ) \left (1-i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{6 \left (1+i \sqrt {3}\right ) \left (i+\sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}\right )-2 \left (-\frac {\sqrt {x+x^3}}{3 \left (1-i \sqrt {3}\right ) (1+x)}+\frac {(1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} E\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{3 \left (1-i \sqrt {3}\right ) \sqrt {x} \left (1+x^2\right )}+\frac {\left (8 \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{\left (\left (1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{3 \left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}-\frac {\left (\left (i+\sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{3 \left (i-\sqrt {3}\right ) \left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}\right )-2 \left (-\frac {\left (i-\sqrt {3}\right ) \sqrt {x+x^3}}{6 \left (i+\sqrt {3}\right ) (1+x)}+\frac {\left (i-\sqrt {3}\right ) (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} E\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{6 \left (i+\sqrt {3}\right ) \sqrt {x} \left (1+x^2\right )}+\frac {\left (4 \left (1+i \sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{\left (\left (1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{3 \left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}-\frac {\left (\left (1+i \sqrt {3}\right ) \left (i+\sqrt {3}\right ) \sqrt {x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{6 \left (i-\sqrt {3}\right ) \left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}\right )\\ &=\frac {2 \sqrt {x+x^3}}{3 \left (1+i \sqrt {3}\right ) \left (1-i \sqrt {3}-2 \sqrt {x}\right ) \sqrt {x}}+\frac {\left (i+\sqrt {3}\right ) \sqrt {x+x^3}}{3 \left (i-\sqrt {3}\right ) \left (1-i \sqrt {3}-2 \sqrt {x}\right ) \sqrt {x}}+\frac {2 \sqrt {x+x^3}}{3 \left (1-i \sqrt {3}\right ) \left (1+i \sqrt {3}-2 \sqrt {x}\right ) \sqrt {x}}+\frac {\left (i-\sqrt {3}\right ) \sqrt {x+x^3}}{3 \left (i+\sqrt {3}\right ) \left (1+i \sqrt {3}-2 \sqrt {x}\right ) \sqrt {x}}-\frac {2 \sqrt {x+x^3}}{3 \left (1+i \sqrt {3}\right ) \left (1-i \sqrt {3}+2 \sqrt {x}\right ) \sqrt {x}}-\frac {\left (i+\sqrt {3}\right ) \sqrt {x+x^3}}{3 \left (i-\sqrt {3}\right ) \left (1-i \sqrt {3}+2 \sqrt {x}\right ) \sqrt {x}}-\frac {2 \sqrt {x+x^3}}{3 \left (1-i \sqrt {3}\right ) \left (1+i \sqrt {3}+2 \sqrt {x}\right ) \sqrt {x}}-\frac {\left (i-\sqrt {3}\right ) \sqrt {x+x^3}}{3 \left (i+\sqrt {3}\right ) \left (1+i \sqrt {3}+2 \sqrt {x}\right ) \sqrt {x}}-\frac {\left (i-\sqrt {3}\right ) \sqrt {x+x^3} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {1+x^2}}\right )}{2 \left (i+\sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}-\frac {\left (i+\sqrt {3}\right ) \sqrt {x+x^3} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {1+x^2}}\right )}{2 \left (i-\sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}-\frac {\sqrt {x+x^3} \tanh ^{-1}\left (\frac {2-\left (1-i \sqrt {3}\right ) x}{\sqrt {2 \left (1-i \sqrt {3}\right )} \sqrt {1+x^2}}\right )}{2 \left (i+\sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}+\frac {\sqrt {x+x^3} \tanh ^{-1}\left (\frac {2-\left (1+i \sqrt {3}\right ) x}{\sqrt {2 \left (1+i \sqrt {3}\right )} \sqrt {1+x^2}}\right )}{2 \left (i-\sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}+\frac {\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )} \sqrt {x+x^3} \tanh ^{-1}\left (\frac {2-\left (1+i \sqrt {3}\right ) x}{\sqrt {2 \left (1+i \sqrt {3}\right )} \sqrt {1+x^2}}\right )}{4 \sqrt {x} \sqrt {1+x^2}}+\frac {\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )} \sqrt {x+x^3} \tanh ^{-1}\left (\frac {4+\left (1+i \sqrt {3}\right )^2 x}{2 \sqrt {2 \left (1-i \sqrt {3}\right )} \sqrt {1+x^2}}\right )}{4 \sqrt {x} \sqrt {1+x^2}}-\frac {\left (1-i \sqrt {3}\right ) (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{4 \sqrt {x} \left (1+x^2\right )}-\frac {\left (1+i \sqrt {3}\right ) (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{4 \sqrt {x} \left (1+x^2\right )}+\frac {\left (3-i \sqrt {3}\right ) (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} \Pi \left (\frac {1}{4};2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{8 \sqrt {x} \left (1+x^2\right )}+\frac {\left (3+i \sqrt {3}\right ) (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} \Pi \left (\frac {1}{4};2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{8 \sqrt {x} \left (1+x^2\right )}-2 \left (-\frac {\sqrt {x+x^3}}{3 \left (1+i \sqrt {3}\right ) (1+x)}+\frac {\sqrt {x+x^3} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {1+x^2}}\right )}{6 \sqrt {x} \sqrt {1+x^2}}+\frac {(1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} E\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{3 \left (1+i \sqrt {3}\right ) \sqrt {x} \left (1+x^2\right )}-\frac {(1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{6 \left (1-i \sqrt {3}\right ) \sqrt {x} \left (1+x^2\right )}+\frac {i (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} \Pi \left (\frac {1}{4};2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{4 \sqrt {3} \sqrt {x} \left (1+x^2\right )}\right )-2 \left (-\frac {\left (i-\sqrt {3}\right ) \sqrt {x+x^3}}{6 \left (i+\sqrt {3}\right ) (1+x)}+\frac {\sqrt {x+x^3} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {1+x^2}}\right )}{3 \left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}+\frac {\left (i-\sqrt {3}\right ) (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} E\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{6 \left (i+\sqrt {3}\right ) \sqrt {x} \left (1+x^2\right )}-\frac {(1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{12 \sqrt {x} \left (1+x^2\right )}+\frac {\left (3 i-\sqrt {3}\right ) (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} \Pi \left (\frac {1}{4};2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{12 \left (i-\sqrt {3}\right ) \sqrt {x} \left (1+x^2\right )}\right )-2 \left (-\frac {\sqrt {x+x^3} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {1+x^2}}\right )}{3 \sqrt {3} \left (i-\sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}+\frac {i (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{6 \sqrt {3} \sqrt {x} \left (1+x^2\right )}-\frac {\left (i-\sqrt {3}\right ) (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} \Pi \left (\frac {1}{4};2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{12 \left (i+\sqrt {3}\right ) \sqrt {x} \left (1+x^2\right )}\right )-2 \left (-\frac {\sqrt {x+x^3} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {1+x^2}}\right )}{6 \sqrt {3} \left (i-\sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}+\frac {i (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{12 \sqrt {3} \sqrt {x} \left (1+x^2\right )}-\frac {\left (i-\sqrt {3}\right ) (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} \Pi \left (\frac {1}{4};2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{24 \left (i+\sqrt {3}\right ) \sqrt {x} \left (1+x^2\right )}\right )-2 \left (\frac {\sqrt {x+x^3} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {1+x^2}}\right )}{3 \sqrt {3} \left (i+\sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}-\frac {i (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{6 \sqrt {3} \sqrt {x} \left (1+x^2\right )}-\frac {\left (i+\sqrt {3}\right ) (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} \Pi \left (\frac {1}{4};2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{12 \left (i-\sqrt {3}\right ) \sqrt {x} \left (1+x^2\right )}\right )-2 \left (\frac {\sqrt {x+x^3} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {1+x^2}}\right )}{6 \sqrt {3} \left (i+\sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}-\frac {i (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{12 \sqrt {3} \sqrt {x} \left (1+x^2\right )}-\frac {\left (i+\sqrt {3}\right ) (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} \Pi \left (\frac {1}{4};2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{24 \left (i-\sqrt {3}\right ) \sqrt {x} \left (1+x^2\right )}\right )-2 \left (-\frac {\sqrt {x+x^3}}{3 \left (1-i \sqrt {3}\right ) (1+x)}+\frac {\sqrt {x+x^3} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {1+x^2}}\right )}{6 \sqrt {x} \sqrt {1+x^2}}+\frac {(1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} E\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{3 \left (1-i \sqrt {3}\right ) \sqrt {x} \left (1+x^2\right )}-\frac {(1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{6 \left (1+i \sqrt {3}\right ) \sqrt {x} \left (1+x^2\right )}+\frac {\left (3 i+\sqrt {3}\right ) (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} \Pi \left (\frac {1}{4};2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{12 \left (i-\sqrt {3}\right ) \sqrt {x} \left (1+x^2\right )}\right )-2 \left (-\frac {\left (i+\sqrt {3}\right ) \sqrt {x+x^3}}{6 \left (i-\sqrt {3}\right ) (1+x)}+\frac {\sqrt {x+x^3} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {1+x^2}}\right )}{3 \left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x^2}}+\frac {\left (i+\sqrt {3}\right ) (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} E\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{6 \left (i-\sqrt {3}\right ) \sqrt {x} \left (1+x^2\right )}-\frac {(1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{12 \sqrt {x} \left (1+x^2\right )}+\frac {\left (3 i+\sqrt {3}\right ) (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \sqrt {x+x^3} \Pi \left (\frac {1}{4};2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{2}\right )}{12 \left (i+\sqrt {3}\right ) \sqrt {x} \left (1+x^2\right )}\right )\\ \end {align*}

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Mathematica [C]  time = 1.03, size = 117, normalized size = 3.00 \begin {gather*} \sqrt {x^3+x} \left (\frac {1}{x^2+x+1}+\frac {\sqrt [4]{-1} \sqrt {\frac {1}{x^2}+1} \sqrt {x} \left (-F\left (\left .i \sinh ^{-1}\left (\frac {\sqrt [4]{-1}}{\sqrt {x}}\right )\right |-1\right )+\Pi \left (-\sqrt [6]{-1};\left .i \sinh ^{-1}\left (\frac {\sqrt [4]{-1}}{\sqrt {x}}\right )\right |-1\right )+\Pi \left (-(-1)^{5/6};\left .i \sinh ^{-1}\left (\frac {\sqrt [4]{-1}}{\sqrt {x}}\right )\right |-1\right )\right )}{x^2+1}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[x + x^3]*((1 + x + x^2)^(-1) + ((-1)^(1/4)*Sqrt[1 + x^(-2)]*Sqrt[x]*(-EllipticF[I*ArcSinh[(-1)^(1/4)/Sqrt
[x]], -1] + EllipticPi[-(-1)^(1/6), I*ArcSinh[(-1)^(1/4)/Sqrt[x]], -1] + EllipticPi[-(-1)^(5/6), I*ArcSinh[(-1
)^(1/4)/Sqrt[x]], -1]))/(1 + x^2))

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IntegrateAlgebraic [A]  time = 0.25, size = 39, normalized size = 1.00 \begin {gather*} \frac {\sqrt {x+x^3}}{1+x+x^2}-\tan ^{-1}\left (\frac {\sqrt {x+x^3}}{1+x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.47, size = 45, normalized size = 1.15 \begin {gather*} \frac {{\left (x^{2} + x + 1\right )} \arctan \left (\frac {x^{2} - x + 1}{2 \, \sqrt {x^{3} + x}}\right ) + 2 \, \sqrt {x^{3} + x}}{2 \, {\left (x^{2} + x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{3} + x} {\left (x^{2} - 1\right )}}{{\left (x^{2} + x + 1\right )}^{2} {\left (x^{2} + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.41, size = 71, normalized size = 1.82

method	result	size
trager	\(\frac {\sqrt {x^{3}+x}}{x^{2}+x +1}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x +\RootOf \left (\textit {\_Z}^{2}+1\right )+2 \sqrt {x^{3}+x}}{x^{2}+x +1}\right )}{2}\)	\(71\)
elliptic	\(\frac {\sqrt {x^{3}+x}}{x^{2}+x +1}+\frac {i \sqrt {-i \left (i+x \right )}\, \sqrt {2}\, \sqrt {i \left (-i+x \right )}\, \sqrt {i x}\, \EllipticF \left (\sqrt {-i \left (i+x \right )}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {x^{3}+x}}+\frac {i \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {-i \left (i+x \right )}\, \sqrt {2}\, \sqrt {i \left (-i+x \right )}\, \sqrt {i x}\, \left (i \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )+1+i\right ) \EllipticPi \left (\sqrt {-i \left (i+x \right )}, \frac {1}{2}-i+\frac {i \sqrt {3}}{2}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {x^{3}+x}}+\frac {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {-i \left (i+x \right )}\, \sqrt {2}\, \sqrt {i \left (-i+x \right )}\, \sqrt {i x}\, \left (i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )+1+i\right ) \EllipticPi \left (\sqrt {-i \left (i+x \right )}, \frac {1}{2}-i-\frac {i \sqrt {3}}{2}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {x^{3}+x}}\)	\(240\)
risch	\(\frac {\left (x^{2}+1\right ) x}{\left (x^{2}+x +1\right ) \sqrt {\left (x^{2}+1\right ) x}}+\frac {i \sqrt {-i \left (i+x \right )}\, \sqrt {2}\, \sqrt {i \left (-i+x \right )}\, \sqrt {i x}\, \EllipticF \left (\sqrt {-i \left (i+x \right )}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {x^{3}+x}}+\frac {i \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {-i \left (i+x \right )}\, \sqrt {2}\, \sqrt {i \left (-i+x \right )}\, \sqrt {i x}\, \left (i \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )+1+i\right ) \EllipticPi \left (\sqrt {-i \left (i+x \right )}, \frac {1}{2}-i+\frac {i \sqrt {3}}{2}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {x^{3}+x}}+\frac {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {-i \left (i+x \right )}\, \sqrt {2}\, \sqrt {i \left (-i+x \right )}\, \sqrt {i x}\, \left (i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )+1+i\right ) \EllipticPi \left (\sqrt {-i \left (i+x \right )}, \frac {1}{2}-i-\frac {i \sqrt {3}}{2}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {x^{3}+x}}\)	\(248\)
default	\(\frac {i \sqrt {-i \left (i+x \right )}\, \sqrt {2}\, \sqrt {i \left (-i+x \right )}\, \sqrt {i x}\, \EllipticF \left (\sqrt {-i \left (i+x \right )}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {x^{3}+x}}-\frac {2 i \left (\frac {1}{2}-\frac {i \sqrt {3}}{6}\right ) \sqrt {-i \left (i+x \right )}\, \sqrt {2}\, \sqrt {i \left (-i+x \right )}\, \sqrt {i x}\, \left (i \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )+1+i\right ) \EllipticPi \left (\sqrt {-i \left (i+x \right )}, \frac {1}{2}-i+\frac {i \sqrt {3}}{2}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x^{3}+x}}-\frac {2 i \left (\frac {i \sqrt {3}}{6}+\frac {1}{2}\right ) \sqrt {-i \left (i+x \right )}\, \sqrt {2}\, \sqrt {i \left (-i+x \right )}\, \sqrt {i x}\, \left (i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )+1+i\right ) \EllipticPi \left (\sqrt {-i \left (i+x \right )}, \frac {1}{2}-i-\frac {i \sqrt {3}}{2}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x^{3}+x}}+\frac {\sqrt {x^{3}+x}}{x^{2}+x +1}-\frac {i \left (-\frac {3}{4}+\frac {i \sqrt {3}}{12}\right ) \sqrt {-i \left (i+x \right )}\, \sqrt {2}\, \sqrt {i \left (-i+x \right )}\, \sqrt {i x}\, \left (i \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )+1+i\right ) \EllipticPi \left (\sqrt {-i \left (i+x \right )}, \frac {1}{2}-i+\frac {i \sqrt {3}}{2}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x^{3}+x}}-\frac {i \left (-\frac {3}{4}-\frac {i \sqrt {3}}{12}\right ) \sqrt {-i \left (i+x \right )}\, \sqrt {2}\, \sqrt {i \left (-i+x \right )}\, \sqrt {i x}\, \left (i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )+1+i\right ) \EllipticPi \left (\sqrt {-i \left (i+x \right )}, \frac {1}{2}-i-\frac {i \sqrt {3}}{2}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x^{3}+x}}\)	\(410\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^3+x)^(1/2)/(x^2+x+1)-1/2*RootOf(_Z^2+1)*ln((RootOf(_Z^2+1)*x^2-RootOf(_Z^2+1)*x+RootOf(_Z^2+1)+2*(x^3+x)^(1
/2))/(x^2+x+1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{3} + x} {\left (x^{2} - 1\right )}}{{\left (x^{2} + x + 1\right )}^{2} {\left (x^{2} + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.73, size = 49, normalized size = 1.26 \begin {gather*} \frac {\sqrt {x^3+x}}{x^2+x+1}-\frac {\ln \left (x^2+x+1\right )\,1{}\mathrm {i}}{2}+\frac {\ln \left (x^2-x+1+\sqrt {x^3+x}\,2{}\mathrm {i}\right )\,1{}\mathrm {i}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log((x + x^3)^(1/2)*2i - x + x^2 + 1)*1i)/2 - (log(x + x^2 + 1)*1i)/2 + (x + x^3)^(1/2)/(x + x^2 + 1)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x \left (x^{2} + 1\right )} \left (x - 1\right ) \left (x + 1\right )}{\left (x^{2} + 1\right ) \left (x^{2} + x + 1\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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