3.10.47 \(\int \frac {(-1+x^2) \sqrt {1+x^4}}{(1-x+x^2) (1+x+x^2)^2} \, dx\)

Optimal. Leaf size=72 \[ \frac {\sqrt {x^4+1}}{2 \left (x^2+x+1\right )}+\frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt {x^4+1}+x^2-x+1}\right )-\frac {3}{2} \tan ^{-1}\left (\frac {x}{\sqrt {x^4+1}+x^2+x+1}\right ) \]

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Rubi [C]  time = 9.86, antiderivative size = 2595, normalized size of antiderivative = 36.04, number of steps used = 248, number of rules used = 20, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.606, Rules used = {6728, 1729, 1209, 1198, 220, 1196, 1217, 1707, 1248, 735, 844, 215, 725, 206, 6742, 2153, 1227, 733, 204, 1336}

result too large to display

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((1 - I*Sqrt[3])*Sqrt[1 + x^4])/16 + ((1 + I*Sqrt[3])*Sqrt[1 + x^4])/16 - ((3 - (5*I)*Sqrt[3])*Sqrt[1 + x^4])/
48 - ((3 + (5*I)*Sqrt[3])*Sqrt[1 + x^4])/48 + ((I - Sqrt[3])*Sqrt[1 + x^4])/(3*(I - Sqrt[3] + (2*I)*x^2)) + ((
I + Sqrt[3])*Sqrt[1 + x^4])/(6*(I - Sqrt[3] + (2*I)*x^2)) - ((I/3)*x*Sqrt[1 + x^4])/(I - Sqrt[3] + (2*I)*x^2)
+ ((I + Sqrt[3])*x*Sqrt[1 + x^4])/(3*(I - Sqrt[3] + (2*I)*x^2)) + ((I - Sqrt[3])*Sqrt[1 + x^4])/(6*(I + Sqrt[3
] + (2*I)*x^2)) + ((I + Sqrt[3])*Sqrt[1 + x^4])/(3*(I + Sqrt[3] + (2*I)*x^2)) - ((I/3)*x*Sqrt[1 + x^4])/(I + S
qrt[3] + (2*I)*x^2) + ((I - Sqrt[3])*x*Sqrt[1 + x^4])/(3*(I + Sqrt[3] + (2*I)*x^2)) + (7*x*Sqrt[1 + x^4])/(6*(
1 + x^2)) - ((1 - I*Sqrt[3])*x*Sqrt[1 + x^4])/(3*(1 + x^2)) - ((1 + I*Sqrt[3])*x*Sqrt[1 + x^4])/(3*(1 + x^2))
- ((3 - (2*I)*Sqrt[3])*x*Sqrt[1 + x^4])/(12*(1 + x^2)) - ((3 + (2*I)*Sqrt[3])*x*Sqrt[1 + x^4])/(12*(1 + x^2))
- ((1 - I*Sqrt[3])*ArcSinh[x^2])/4 + ((9 - I*Sqrt[3])*ArcSinh[x^2])/48 - (3*(1 + I*Sqrt[3])*ArcSinh[x^2])/16 -
((1 + I*Sqrt[3])^2*ArcSinh[x^2])/32 + ((9 + I*Sqrt[3])*ArcSinh[x^2])/48 + (5*ArcTan[x/Sqrt[1 + x^4]])/12 - ((
1 - I*Sqrt[3])*ArcTan[x/Sqrt[1 + x^4]])/12 - ((3 - I*Sqrt[3])*ArcTan[x/Sqrt[1 + x^4]])/12 - ((1 + I*Sqrt[3])*A
rcTan[x/Sqrt[1 + x^4]])/12 - ((3 + I*Sqrt[3])*ArcTan[x/Sqrt[1 + x^4]])/12 - ((3 - (2*I)*Sqrt[3])*ArcTan[x/Sqrt
[1 + x^4]])/24 - ((3 + (2*I)*Sqrt[3])*ArcTan[x/Sqrt[1 + x^4]])/24 - ArcTan[(2*I - (I - Sqrt[3])*x^2)/(Sqrt[2*(
1 + I*Sqrt[3])]*Sqrt[1 + x^4])]/3 - ((I + Sqrt[3])*ArcTan[(2*I - (I - Sqrt[3])*x^2)/(Sqrt[2*(1 + I*Sqrt[3])]*S
qrt[1 + x^4])])/(6*(I - Sqrt[3])) + ArcTan[(2*I - (I + Sqrt[3])*x^2)/(Sqrt[2*(1 - I*Sqrt[3])]*Sqrt[1 + x^4])]/
3 + ((I - Sqrt[3])*ArcTan[(2*I - (I + Sqrt[3])*x^2)/(Sqrt[2*(1 - I*Sqrt[3])]*Sqrt[1 + x^4])])/(6*(I + Sqrt[3])
) + (I/8)*ArcTanh[(2 - (1 - I*Sqrt[3])*x^2)/(Sqrt[2*(1 - I*Sqrt[3])]*Sqrt[1 + x^4])] - (I/8)*ArcTanh[(2 - (1 +
I*Sqrt[3])*x^2)/(Sqrt[2*(1 + I*Sqrt[3])]*Sqrt[1 + x^4])] - ((3*I - 2*Sqrt[3])*ArcTanh[(2 - (1 + I*Sqrt[3])*x^
2)/(Sqrt[2*(1 + I*Sqrt[3])]*Sqrt[1 + x^4])])/24 + ((3*I + 2*Sqrt[3])*ArcTanh[(4 + (1 + I*Sqrt[3])^2*x^2)/(2*Sq
rt[2*(1 - I*Sqrt[3])]*Sqrt[1 + x^4])])/24 - (7*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticE[2*ArcTan[x], 1/
2])/(6*Sqrt[1 + x^4]) + ((1 - I*Sqrt[3])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticE[2*ArcTan[x], 1/2])/(3
*Sqrt[1 + x^4]) + ((1 + I*Sqrt[3])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticE[2*ArcTan[x], 1/2])/(3*Sqrt[
1 + x^4]) + ((3 - (2*I)*Sqrt[3])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticE[2*ArcTan[x], 1/2])/(12*Sqrt[1
+ x^4]) + ((3 + (2*I)*Sqrt[3])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticE[2*ArcTan[x], 1/2])/(12*Sqrt[1
+ x^4]) - ((1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(Sqrt[3]*(3*I - Sqrt[3])*Sqrt[1
+ x^4]) + ((1 - I*Sqrt[3])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(4*Sqrt[1 + x^4]
) - ((3 - I*Sqrt[3])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(24*Sqrt[1 + x^4]) - (
(9 - I*Sqrt[3])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(24*Sqrt[1 + x^4]) + (3*(1
+ I*Sqrt[3])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(8*Sqrt[1 + x^4]) - ((1 + I*Sq
rt[3])^2*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(16*Sqrt[1 + x^4]) - ((3 + I*Sqrt[
3])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(24*Sqrt[1 + x^4]) - ((9 + I*Sqrt[3])*(
1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(24*Sqrt[1 + x^4]) + ((1 + x^2)*Sqrt[(1 + x^
4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(Sqrt[3]*(3*I + Sqrt[3])*Sqrt[1 + x^4]) + ((3*I - Sqrt[3])*(1 + x
^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(12*(3*I + Sqrt[3])*Sqrt[1 + x^4]) + ((3*I + Sqrt
[3])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(12*(3*I - Sqrt[3])*Sqrt[1 + x^4]) - (
(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticPi[1/4, 2*ArcTan[x], 1/2])/(4*Sqrt[1 + x^4]) + ((1 - I*Sqrt[3])*
(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticPi[1/4, 2*ArcTan[x], 1/2])/(8*Sqrt[1 + x^4]) + ((2 - I*Sqrt[3])*
(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticPi[1/4, 2*ArcTan[x], 1/2])/(16*Sqrt[1 + x^4]) - ((3 - I*Sqrt[3])
*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticPi[1/4, 2*ArcTan[x], 1/2])/(24*Sqrt[1 + x^4]) + ((1 + I*Sqrt[3]
)*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticPi[1/4, 2*ArcTan[x], 1/2])/(8*Sqrt[1 + x^4]) + ((2 + I*Sqrt[3]
)*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticPi[1/4, 2*ArcTan[x], 1/2])/(16*Sqrt[1 + x^4]) - ((3 + I*Sqrt[3
])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticPi[1/4, 2*ArcTan[x], 1/2])/(24*Sqrt[1 + x^4])

Rule 204

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

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 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

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 733

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(a + c*x^2)^p)/(
e*(m + 1)), x] - Dist[(2*c*p)/(e*(m + 1)), Int[x*(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1), x], x] /; FreeQ[{a, c,
d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] &&  !ILtQ[m +
2*p + 1, 0] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 735

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(a + c*x^2)^p)/(
e*(m + 2*p + 1)), x] + Dist[(2*p)/(e*(m + 2*p + 1)), Int[(d + e*x)^m*Simp[a*e - c*d*x, x]*(a + c*x^2)^(p - 1),
x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !Ration
alQ[m] || LtQ[m, 1]) &&  !ILtQ[m + 2*p, 0] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

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 1198

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

Rule 1209

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

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 1227

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

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 1336

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(f*x)^m*(d + e*x^2)^q*(a + c*x^4)^p, x], x] /; FreeQ[{a, c, d, e, f, m, p, q}, x] && (IGtQ[p, 0] || IGtQ[q,
0] || IntegersQ[m, q])

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 1729

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

Rule 2153

Int[((c_) + (d_.)*(x_)^(n_.))^(q_)*((a_) + (b_.)*(x_)^(nn_.))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^
nn)^p, (c/(c^2 - d^2*x^(2*n)) - (d*x^n)/(c^2 - d^2*x^(2*n)))^(-q), x], x] /; FreeQ[{a, b, c, d, n, nn, p}, x]
&&  !IntegerQ[p] && ILtQ[q, 0] && IGtQ[Log[2, nn/n], 0]

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 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 {1+x^4}}{\left (1-x+x^2\right ) \left (1+x+x^2\right )^2} \, dx &=\int \left (\frac {(1+x) \sqrt {1+x^4}}{4 \left (1-x+x^2\right )}+\frac {(-1-2 x) \sqrt {1+x^4}}{2 \left (1+x+x^2\right )^2}+\frac {(-3-x) \sqrt {1+x^4}}{4 \left (1+x+x^2\right )}\right ) \, dx\\ &=\frac {1}{4} \int \frac {(1+x) \sqrt {1+x^4}}{1-x+x^2} \, dx+\frac {1}{4} \int \frac {(-3-x) \sqrt {1+x^4}}{1+x+x^2} \, dx+\frac {1}{2} \int \frac {(-1-2 x) \sqrt {1+x^4}}{\left (1+x+x^2\right )^2} \, dx\\ &=\frac {1}{4} \int \left (\frac {\left (1-i \sqrt {3}\right ) \sqrt {1+x^4}}{-1-i \sqrt {3}+2 x}+\frac {\left (1+i \sqrt {3}\right ) \sqrt {1+x^4}}{-1+i \sqrt {3}+2 x}\right ) \, dx+\frac {1}{4} \int \left (\frac {\left (-1+\frac {5 i}{\sqrt {3}}\right ) \sqrt {1+x^4}}{1-i \sqrt {3}+2 x}+\frac {\left (-1-\frac {5 i}{\sqrt {3}}\right ) \sqrt {1+x^4}}{1+i \sqrt {3}+2 x}\right ) \, dx+\frac {1}{2} \int \left (-\frac {\sqrt {1+x^4}}{\left (1+x+x^2\right )^2}-\frac {2 x \sqrt {1+x^4}}{\left (1+x+x^2\right )^2}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {\sqrt {1+x^4}}{\left (1+x+x^2\right )^2} \, dx\right )+\frac {1}{4} \left (1-i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{-1-i \sqrt {3}+2 x} \, dx+\frac {1}{4} \left (1+i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{-1+i \sqrt {3}+2 x} \, dx+\frac {1}{12} \left (-3+5 i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{1-i \sqrt {3}+2 x} \, dx-\frac {1}{12} \left (3+5 i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{1+i \sqrt {3}+2 x} \, dx-\int \frac {x \sqrt {1+x^4}}{\left (1+x+x^2\right )^2} \, dx\\ &=-\left (\frac {1}{2} \int \left (-\frac {4 \sqrt {1+x^4}}{3 \left (-1+i \sqrt {3}-2 x\right )^2}+\frac {4 i \sqrt {1+x^4}}{3 \sqrt {3} \left (-1+i \sqrt {3}-2 x\right )}-\frac {4 \sqrt {1+x^4}}{3 \left (1+i \sqrt {3}+2 x\right )^2}+\frac {4 i \sqrt {1+x^4}}{3 \sqrt {3} \left (1+i \sqrt {3}+2 x\right )}\right ) \, dx\right )+\frac {1}{2} \left (-1-i \sqrt {3}\right ) \int \frac {x \sqrt {1+x^4}}{\left (-1+i \sqrt {3}\right )^2-4 x^2} \, dx+\frac {1}{2} \left (-1+i \sqrt {3}\right ) \int \frac {x \sqrt {1+x^4}}{\left (-1-i \sqrt {3}\right )^2-4 x^2} \, dx-\frac {1}{3} \left (-3+2 i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{\left (1+i \sqrt {3}\right )^2-4 x^2} \, dx+\frac {1}{3} \left (3+2 i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{\left (1-i \sqrt {3}\right )^2-4 x^2} \, dx+\frac {1}{6} \left (3-5 i \sqrt {3}\right ) \int \frac {x \sqrt {1+x^4}}{\left (1-i \sqrt {3}\right )^2-4 x^2} \, dx+\frac {1}{6} \left (3+5 i \sqrt {3}\right ) \int \frac {x \sqrt {1+x^4}}{\left (1+i \sqrt {3}\right )^2-4 x^2} \, dx-\int \frac {\sqrt {1+x^4}}{\left (-1-i \sqrt {3}\right )^2-4 x^2} \, dx-\int \frac {\sqrt {1+x^4}}{\left (-1+i \sqrt {3}\right )^2-4 x^2} \, dx-\int \left (-\frac {2 \left (-1+i \sqrt {3}\right ) \sqrt {1+x^4}}{3 \left (-1+i \sqrt {3}-2 x\right )^2}-\frac {2 i \sqrt {1+x^4}}{3 \sqrt {3} \left (-1+i \sqrt {3}-2 x\right )}-\frac {2 \left (-1-i \sqrt {3}\right ) \sqrt {1+x^4}}{3 \left (1+i \sqrt {3}+2 x\right )^2}-\frac {2 i \sqrt {1+x^4}}{3 \sqrt {3} \left (1+i \sqrt {3}+2 x\right )}\right ) \, dx\\ &=\frac {1}{16} \int \frac {\left (-1-i \sqrt {3}\right )^2+4 x^2}{\sqrt {1+x^4}} \, dx+\frac {1}{16} \int \frac {\left (-1+i \sqrt {3}\right )^2+4 x^2}{\sqrt {1+x^4}} \, dx+\frac {2}{3} \int \frac {\sqrt {1+x^4}}{\left (-1+i \sqrt {3}-2 x\right )^2} \, dx+\frac {2}{3} \int \frac {\sqrt {1+x^4}}{\left (1+i \sqrt {3}+2 x\right )^2} \, dx+\frac {1}{4} \left (-1-i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^2}}{\left (-1+i \sqrt {3}\right )^2-4 x} \, dx,x,x^2\right )-\frac {1}{2} \left (1-i \sqrt {3}\right ) \int \frac {1}{\left (\left (-1-i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx-\frac {1}{3} \left (2 \left (1-i \sqrt {3}\right )\right ) \int \frac {\sqrt {1+x^4}}{\left (-1+i \sqrt {3}-2 x\right )^2} \, dx+\frac {1}{4} \left (-1+i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^2}}{\left (-1-i \sqrt {3}\right )^2-4 x} \, dx,x,x^2\right )-\frac {1}{2} \left (1+i \sqrt {3}\right ) \int \frac {1}{\left (\left (-1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx-\frac {1}{3} \left (2 \left (1+i \sqrt {3}\right )\right ) \int \frac {\sqrt {1+x^4}}{\left (1+i \sqrt {3}+2 x\right )^2} \, dx-\frac {1}{48} \left (3-2 i \sqrt {3}\right ) \int \frac {\left (1+i \sqrt {3}\right )^2+4 x^2}{\sqrt {1+x^4}} \, dx-\frac {1}{48} \left (3+2 i \sqrt {3}\right ) \int \frac {\left (1-i \sqrt {3}\right )^2+4 x^2}{\sqrt {1+x^4}} \, dx+\frac {1}{12} \left (3-5 i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^2}}{\left (1-i \sqrt {3}\right )^2-4 x} \, dx,x,x^2\right )+\frac {1}{6} \left (-3+5 i \sqrt {3}\right ) \int \frac {1}{\left (\left (1-i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx+\frac {1}{12} \left (3+5 i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^2}}{\left (1+i \sqrt {3}\right )^2-4 x} \, dx,x,x^2\right )-\frac {1}{6} \left (3+5 i \sqrt {3}\right ) \int \frac {1}{\left (\left (1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx\\ &=\frac {1}{16} \left (1-i \sqrt {3}\right ) \sqrt {1+x^4}+\frac {1}{16} \left (1+i \sqrt {3}\right ) \sqrt {1+x^4}-\frac {1}{48} \left (3-5 i \sqrt {3}\right ) \sqrt {1+x^4}-\frac {1}{48} \left (3+5 i \sqrt {3}\right ) \sqrt {1+x^4}-2 \left (\frac {1}{4} \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx\right )+\frac {2}{3} \int \left (\frac {\left (1+i \sqrt {3}\right ) \sqrt {1+x^4}}{2 \left (-i+\sqrt {3}-2 i x^2\right )^2}+\frac {\left (1-i \sqrt {3}\right ) x \sqrt {1+x^4}}{\left (-i+\sqrt {3}-2 i x^2\right )^2}-\frac {x^2 \sqrt {1+x^4}}{\left (-i+\sqrt {3}-2 i x^2\right )^2}\right ) \, dx+\frac {2}{3} \int \left (\frac {\left (1-i \sqrt {3}\right ) \sqrt {1+x^4}}{2 \left (i+\sqrt {3}+2 i x^2\right )^2}+\frac {\left (1+i \sqrt {3}\right ) x \sqrt {1+x^4}}{\left (i+\sqrt {3}+2 i x^2\right )^2}-\frac {x^2 \sqrt {1+x^4}}{\left (i+\sqrt {3}+2 i x^2\right )^2}\right ) \, dx-\frac {\left (3 i-5 \sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{12 \left (i-\sqrt {3}\right )}-\frac {\left (3 i-5 \sqrt {3}\right ) \int \frac {1+x^2}{\left (\left (1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx}{3 \left (i-\sqrt {3}\right )}+\frac {1}{16} \left (1-i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {-4+2 \left (1-i \sqrt {3}\right ) x}{\left (\left (-1-i \sqrt {3}\right )^2-4 x\right ) \sqrt {1+x^2}} \, dx,x,x^2\right )+\frac {1}{8} \left (1-i \sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx-\frac {1}{3} \left (2 \left (1-i \sqrt {3}\right )\right ) \int \left (\frac {\left (1+i \sqrt {3}\right ) \sqrt {1+x^4}}{2 \left (-i+\sqrt {3}-2 i x^2\right )^2}+\frac {\left (1-i \sqrt {3}\right ) x \sqrt {1+x^4}}{\left (-i+\sqrt {3}-2 i x^2\right )^2}-\frac {x^2 \sqrt {1+x^4}}{\left (-i+\sqrt {3}-2 i x^2\right )^2}\right ) \, dx-\frac {1}{24} \left (9-i \sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx+\frac {1}{16} \left (1+i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {-4+2 \left (1+i \sqrt {3}\right ) x}{\left (\left (-1+i \sqrt {3}\right )^2-4 x\right ) \sqrt {1+x^2}} \, dx,x,x^2\right )+\frac {1}{8} \left (1+i \sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx-\frac {1}{3} \left (2 \left (1+i \sqrt {3}\right )\right ) \int \left (\frac {\left (1-i \sqrt {3}\right ) \sqrt {1+x^4}}{2 \left (i+\sqrt {3}+2 i x^2\right )^2}+\frac {\left (1+i \sqrt {3}\right ) x \sqrt {1+x^4}}{\left (i+\sqrt {3}+2 i x^2\right )^2}-\frac {x^2 \sqrt {1+x^4}}{\left (i+\sqrt {3}+2 i x^2\right )^2}\right ) \, dx-\frac {1}{24} \left (9+i \sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx-\frac {1}{12} \left (-3+2 i \sqrt {3}\right ) \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx+\frac {1}{12} \left (3+2 i \sqrt {3}\right ) \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx+\frac {1}{48} \left (-3+5 i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {-4+2 \left (1+i \sqrt {3}\right ) x}{\left (\left (1-i \sqrt {3}\right )^2-4 x\right ) \sqrt {1+x^2}} \, dx,x,x^2\right )-\frac {1}{48} \left (3+5 i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {-4-\left (1+i \sqrt {3}\right )^2 x}{\left (\left (1+i \sqrt {3}\right )^2-4 x\right ) \sqrt {1+x^2}} \, dx,x,x^2\right )-\frac {\left (i-\sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{4 \left (i+\sqrt {3}\right )}-\frac {\left (i-\sqrt {3}\right ) \int \frac {1+x^2}{\left (\left (-1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx}{i+\sqrt {3}}-\frac {\left (i+\sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{4 \left (i-\sqrt {3}\right )}-\frac {\left (i+\sqrt {3}\right ) \int \frac {1+x^2}{\left (\left (-1-i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx}{i-\sqrt {3}}-\frac {\left (3 i+5 \sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{12 \left (i+\sqrt {3}\right )}-\frac {\left (3 i+5 \sqrt {3}\right ) \int \frac {1+x^2}{\left (\left (1-i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx}{3 \left (i+\sqrt {3}\right )}\\ &=\frac {1}{16} \left (1-i \sqrt {3}\right ) \sqrt {1+x^4}+\frac {1}{16} \left (1+i \sqrt {3}\right ) \sqrt {1+x^4}-\frac {1}{48} \left (3-5 i \sqrt {3}\right ) \sqrt {1+x^4}-\frac {1}{48} \left (3+5 i \sqrt {3}\right ) \sqrt {1+x^4}-\frac {\left (3-2 i \sqrt {3}\right ) x \sqrt {1+x^4}}{12 \left (1+x^2\right )}-\frac {\left (3+2 i \sqrt {3}\right ) x \sqrt {1+x^4}}{12 \left (1+x^2\right )}+\frac {1}{4} \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )+\frac {\left (3-5 i \sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{24 \left (1+i \sqrt {3}\right )}+\frac {\left (3+5 i \sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{12 \left (i-\sqrt {3}\right )^2}+\frac {\left (3-2 i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}+\frac {\left (3+2 i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}-2 \left (-\frac {x \sqrt {1+x^4}}{4 \left (1+x^2\right )}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{4 \sqrt {1+x^4}}\right )-\frac {\left (3 i-5 \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (1-i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \sqrt {1+x^4}}-\frac {\left (9-i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{48 \sqrt {1+x^4}}+\frac {\left (1+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \sqrt {1+x^4}}-\frac {\left (9+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{48 \sqrt {1+x^4}}-\frac {\left (i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (3 i+5 \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (5 i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (5 i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {2}{3} \int \frac {x^2 \sqrt {1+x^4}}{\left (-i+\sqrt {3}-2 i x^2\right )^2} \, dx-\frac {2}{3} \int \frac {x^2 \sqrt {1+x^4}}{\left (i+\sqrt {3}+2 i x^2\right )^2} \, dx-\frac {4}{3} \int \frac {\sqrt {1+x^4}}{\left (-i+\sqrt {3}-2 i x^2\right )^2} \, dx-\frac {4}{3} \int \frac {\sqrt {1+x^4}}{\left (i+\sqrt {3}+2 i x^2\right )^2} \, dx+\frac {1}{16} \left (1-i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^2\right )+\frac {1}{4} \left (1-i \sqrt {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^2\right )+\frac {1}{3} \left (1-i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{\left (i+\sqrt {3}+2 i x^2\right )^2} \, dx+\frac {1}{3} \left (2 \left (1-i \sqrt {3}\right )\right ) \int \frac {x \sqrt {1+x^4}}{\left (-i+\sqrt {3}-2 i x^2\right )^2} \, dx+\frac {1}{3} \left (2 \left (1-i \sqrt {3}\right )\right ) \int \frac {x^2 \sqrt {1+x^4}}{\left (-i+\sqrt {3}-2 i x^2\right )^2} \, dx+\frac {1}{48} \left (9-i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^2\right )+\frac {1}{12} \left (9-i \sqrt {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^2\right )+\frac {1}{16} \left (1+i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^2\right )+\frac {1}{4} \left (1+i \sqrt {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^2\right )+\frac {1}{3} \left (1+i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{\left (-i+\sqrt {3}-2 i x^2\right )^2} \, dx+\frac {1}{3} \left (2 \left (1+i \sqrt {3}\right )\right ) \int \frac {x \sqrt {1+x^4}}{\left (i+\sqrt {3}+2 i x^2\right )^2} \, dx+\frac {1}{3} \left (2 \left (1+i \sqrt {3}\right )\right ) \int \frac {x^2 \sqrt {1+x^4}}{\left (i+\sqrt {3}+2 i x^2\right )^2} \, dx+\frac {1}{3} \left (4 \left (1+i \sqrt {3}\right )\right ) \int \frac {x \sqrt {1+x^4}}{\left (-i+\sqrt {3}-2 i x^2\right )^2} \, dx-\frac {1}{3} \left (2 \left (1+i \sqrt {3}\right )^2\right ) \int \frac {x \sqrt {1+x^4}}{\left (i+\sqrt {3}+2 i x^2\right )^2} \, dx+\frac {1}{48} \left (9+i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^2\right )+\frac {1}{12} \left (9+i \sqrt {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^2\right )\\ &=\frac {1}{16} \left (1-i \sqrt {3}\right ) \sqrt {1+x^4}+\frac {1}{16} \left (1+i \sqrt {3}\right ) \sqrt {1+x^4}-\frac {1}{48} \left (3-5 i \sqrt {3}\right ) \sqrt {1+x^4}-\frac {1}{48} \left (3+5 i \sqrt {3}\right ) \sqrt {1+x^4}-\frac {i x \sqrt {1+x^4}}{6 \left (i-\sqrt {3}+2 i x^2\right )}-\frac {2 x \sqrt {1+x^4}}{3 \left (i-\sqrt {3}\right ) \left (i-\sqrt {3}+2 i x^2\right )}-\frac {i x \sqrt {1+x^4}}{6 \left (i+\sqrt {3}+2 i x^2\right )}-\frac {2 x \sqrt {1+x^4}}{3 \left (i+\sqrt {3}\right ) \left (i+\sqrt {3}+2 i x^2\right )}-\frac {\left (3-2 i \sqrt {3}\right ) x \sqrt {1+x^4}}{12 \left (1+x^2\right )}-\frac {\left (3+2 i \sqrt {3}\right ) x \sqrt {1+x^4}}{12 \left (1+x^2\right )}+\frac {1}{16} \left (1-i \sqrt {3}\right ) \sinh ^{-1}\left (x^2\right )+\frac {1}{48} \left (9-i \sqrt {3}\right ) \sinh ^{-1}\left (x^2\right )+\frac {1}{16} \left (1+i \sqrt {3}\right ) \sinh ^{-1}\left (x^2\right )+\frac {1}{48} \left (9+i \sqrt {3}\right ) \sinh ^{-1}\left (x^2\right )+\frac {1}{4} \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )+\frac {\left (3-5 i \sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{24 \left (1+i \sqrt {3}\right )}+\frac {\left (3+5 i \sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{12 \left (i-\sqrt {3}\right )^2}+\frac {\left (3-2 i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}+\frac {\left (3+2 i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}-2 \left (-\frac {x \sqrt {1+x^4}}{4 \left (1+x^2\right )}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{4 \sqrt {1+x^4}}\right )-\frac {\left (3 i-5 \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (1-i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \sqrt {1+x^4}}-\frac {\left (9-i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{48 \sqrt {1+x^4}}+\frac {\left (1+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \sqrt {1+x^4}}-\frac {\left (9+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{48 \sqrt {1+x^4}}-\frac {\left (i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (3 i+5 \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (5 i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (5 i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {1}{24} i \int \frac {i+\sqrt {3}-2 i x^2}{\sqrt {1+x^4}} \, dx-\frac {1}{24} i \int \frac {-i+\sqrt {3}+2 i x^2}{\sqrt {1+x^4}} \, dx-\frac {2}{3} \int \left (\frac {i \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}{2 \left (-i+\sqrt {3}-2 i x^2\right )^2}+\frac {i \sqrt {1+x^4}}{2 \left (-i+\sqrt {3}-2 i x^2\right )}\right ) \, dx-\frac {2}{3} \int \left (-\frac {i \left (-i-\sqrt {3}\right ) \sqrt {1+x^4}}{2 \left (i+\sqrt {3}+2 i x^2\right )^2}-\frac {i \sqrt {1+x^4}}{2 \left (i+\sqrt {3}+2 i x^2\right )}\right ) \, dx-\frac {\int \frac {-i+\sqrt {3}+2 i x^2}{\sqrt {1+x^4}} \, dx}{6 \left (i-\sqrt {3}\right )}+\frac {1}{4} \left (-1-i \sqrt {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^2}{\sqrt {1+x^4}}\right )+\frac {1}{3} \left (1-i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^2}}{\left (-i+\sqrt {3}-2 i x\right )^2} \, dx,x,x^2\right )+\frac {1}{3} \left (2 \left (1-i \sqrt {3}\right )\right ) \int \left (\frac {i \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}{2 \left (-i+\sqrt {3}-2 i x^2\right )^2}+\frac {i \sqrt {1+x^4}}{2 \left (-i+\sqrt {3}-2 i x^2\right )}\right ) \, dx+\frac {1}{12} \left (-9+i \sqrt {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^2}{\sqrt {1+x^4}}\right )+\frac {1}{4} \left (-1+i \sqrt {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^2}{\sqrt {1+x^4}}\right )+\frac {1}{3} \left (1+i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^2}}{\left (i+\sqrt {3}+2 i x\right )^2} \, dx,x,x^2\right )+\frac {1}{3} \left (2 \left (1+i \sqrt {3}\right )\right ) \int \left (-\frac {i \left (-i-\sqrt {3}\right ) \sqrt {1+x^4}}{2 \left (i+\sqrt {3}+2 i x^2\right )^2}-\frac {i \sqrt {1+x^4}}{2 \left (i+\sqrt {3}+2 i x^2\right )}\right ) \, dx+\frac {1}{3} \left (2 \left (1+i \sqrt {3}\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^2}}{\left (-i+\sqrt {3}-2 i x\right )^2} \, dx,x,x^2\right )-\frac {1}{3} \left (1+i \sqrt {3}\right )^2 \operatorname {Subst}\left (\int \frac {\sqrt {1+x^2}}{\left (i+\sqrt {3}+2 i x\right )^2} \, dx,x,x^2\right )-\frac {1}{12} \left (9+i \sqrt {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^2}{\sqrt {1+x^4}}\right )+\frac {\int \frac {i+\sqrt {3}-2 i x^2}{\sqrt {1+x^4}} \, dx}{6 \left (i+\sqrt {3}\right )}+\frac {1}{12} \left (-3 i+\sqrt {3}\right ) \int \frac {1}{\left (i+\sqrt {3}+2 i x^2\right ) \sqrt {1+x^4}} \, dx+\frac {1}{12} \left (3 i+\sqrt {3}\right ) \int \frac {1}{\left (-i+\sqrt {3}-2 i x^2\right ) \sqrt {1+x^4}} \, dx-\frac {\left (4+\left (-i+\sqrt {3}\right )^2\right ) \int \frac {1}{\left (-i+\sqrt {3}-2 i x^2\right ) \sqrt {1+x^4}} \, dx}{6 \left (-i+\sqrt {3}\right )}-\frac {\left (4+\left (i+\sqrt {3}\right )^2\right ) \int \frac {1}{\left (i+\sqrt {3}+2 i x^2\right ) \sqrt {1+x^4}} \, dx}{6 \left (i+\sqrt {3}\right )}\\ &=\frac {1}{16} \left (1-i \sqrt {3}\right ) \sqrt {1+x^4}+\frac {1}{16} \left (1+i \sqrt {3}\right ) \sqrt {1+x^4}-\frac {1}{48} \left (3-5 i \sqrt {3}\right ) \sqrt {1+x^4}-\frac {1}{48} \left (3+5 i \sqrt {3}\right ) \sqrt {1+x^4}+\frac {\left (i-\sqrt {3}\right ) \sqrt {1+x^4}}{3 \left (i-\sqrt {3}+2 i x^2\right )}+\frac {\left (i+\sqrt {3}\right ) \sqrt {1+x^4}}{6 \left (i-\sqrt {3}+2 i x^2\right )}-\frac {i x \sqrt {1+x^4}}{6 \left (i-\sqrt {3}+2 i x^2\right )}-\frac {2 x \sqrt {1+x^4}}{3 \left (i-\sqrt {3}\right ) \left (i-\sqrt {3}+2 i x^2\right )}+\frac {\left (i-\sqrt {3}\right ) \sqrt {1+x^4}}{6 \left (i+\sqrt {3}+2 i x^2\right )}+\frac {\left (i+\sqrt {3}\right ) \sqrt {1+x^4}}{3 \left (i+\sqrt {3}+2 i x^2\right )}-\frac {i x \sqrt {1+x^4}}{6 \left (i+\sqrt {3}+2 i x^2\right )}-\frac {2 x \sqrt {1+x^4}}{3 \left (i+\sqrt {3}\right ) \left (i+\sqrt {3}+2 i x^2\right )}-\frac {\left (3-2 i \sqrt {3}\right ) x \sqrt {1+x^4}}{12 \left (1+x^2\right )}-\frac {\left (3+2 i \sqrt {3}\right ) x \sqrt {1+x^4}}{12 \left (1+x^2\right )}+\frac {1}{16} \left (1-i \sqrt {3}\right ) \sinh ^{-1}\left (x^2\right )+\frac {1}{48} \left (9-i \sqrt {3}\right ) \sinh ^{-1}\left (x^2\right )+\frac {1}{16} \left (1+i \sqrt {3}\right ) \sinh ^{-1}\left (x^2\right )+\frac {1}{48} \left (9+i \sqrt {3}\right ) \sinh ^{-1}\left (x^2\right )+\frac {1}{4} \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )+\frac {\left (3-5 i \sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{24 \left (1+i \sqrt {3}\right )}+\frac {\left (3+5 i \sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{12 \left (i-\sqrt {3}\right )^2}+\frac {\left (1+i \sqrt {3}\right ) \tanh ^{-1}\left (\frac {2-\left (1-i \sqrt {3}\right ) x^2}{\sqrt {2 \left (1-i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{8 \sqrt {2 \left (1-i \sqrt {3}\right )}}+\frac {\left (1-i \sqrt {3}\right ) \tanh ^{-1}\left (\frac {2-\left (1+i \sqrt {3}\right ) x^2}{\sqrt {2 \left (1+i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{8 \sqrt {2 \left (1+i \sqrt {3}\right )}}-\frac {\left (i-3 \sqrt {3}\right ) \tanh ^{-1}\left (\frac {2-\left (1+i \sqrt {3}\right ) x^2}{\sqrt {2 \left (1+i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{8 \sqrt {6 \left (1+i \sqrt {3}\right )}}+\frac {\left (i+3 \sqrt {3}\right ) \tanh ^{-1}\left (\frac {4+\left (1+i \sqrt {3}\right )^2 x^2}{2 \sqrt {2 \left (1-i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{8 \sqrt {6 \left (1-i \sqrt {3}\right )}}+\frac {\left (3-2 i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}+\frac {\left (3+2 i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}-2 \left (-\frac {x \sqrt {1+x^4}}{4 \left (1+x^2\right )}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{4 \sqrt {1+x^4}}\right )-\frac {\left (3 i-5 \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (1-i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \sqrt {1+x^4}}-\frac {\left (9-i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{48 \sqrt {1+x^4}}+\frac {\left (1+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \sqrt {1+x^4}}-\frac {\left (9+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{48 \sqrt {1+x^4}}-\frac {\left (i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (3 i+5 \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (5 i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (5 i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {1}{3} i \int \frac {\sqrt {1+x^4}}{-i+\sqrt {3}-2 i x^2} \, dx+\frac {1}{3} i \int \frac {\sqrt {1+x^4}}{i+\sqrt {3}+2 i x^2} \, dx-2 \left (\frac {1}{12} \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx\right )-\frac {4}{3} \int \frac {\sqrt {1+x^4}}{\left (-i+\sqrt {3}-2 i x^2\right )^2} \, dx-\frac {4}{3} \int \frac {\sqrt {1+x^4}}{\left (i+\sqrt {3}+2 i x^2\right )^2} \, dx+\frac {1}{3} \left (-i-\sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (i+\sqrt {3}+2 i x\right ) \sqrt {1+x^2}} \, dx,x,x^2\right )-\frac {1}{3} \left (i-\sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{i+\sqrt {3}+2 i x^2} \, dx+\frac {1}{3} \left (i-\sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (-i+\sqrt {3}-2 i x\right ) \sqrt {1+x^2}} \, dx,x,x^2\right )+\frac {\left (i-\sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{4 \left (3 i-\sqrt {3}\right )}-\frac {\left (3 i-\sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{6 \left (i-\sqrt {3}\right )}-\frac {1}{3} \left (-1-i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{\left (-i+\sqrt {3}-2 i x^2\right )^2} \, dx+\frac {\int \frac {1-x^2}{\sqrt {1+x^4}} \, dx}{3 \left (1-i \sqrt {3}\right )}-\frac {1}{24} \left (-1+i \sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx-\frac {1}{3} \left (-1+i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{\left (i+\sqrt {3}+2 i x^2\right )^2} \, dx+\frac {\int \frac {1-x^2}{\sqrt {1+x^4}} \, dx}{3 \left (1+i \sqrt {3}\right )}+\frac {1}{24} \left (1+i \sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx-\frac {\left (1+i \sqrt {3}\right ) \int \frac {1+x^2}{\left (-i+\sqrt {3}-2 i x^2\right ) \sqrt {1+x^4}} \, dx}{2 \left (3 i-\sqrt {3}\right )}+\frac {\left (3+i \sqrt {3}\right ) \int \frac {1+x^2}{\left (-i+\sqrt {3}-2 i x^2\right ) \sqrt {1+x^4}} \, dx}{3 \left (i-\sqrt {3}\right )}+\frac {1}{6} \left (-i+\sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (i+\sqrt {3}+2 i x\right ) \sqrt {1+x^2}} \, dx,x,x^2\right )-\frac {\left (i-\sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{6 \left (i+\sqrt {3}\right )}-\frac {\left (3-i \sqrt {3}\right ) \int \frac {1+x^2}{\left (i+\sqrt {3}+2 i x^2\right ) \sqrt {1+x^4}} \, dx}{3 \left (i+\sqrt {3}\right )}+\frac {1}{6} \left (i+\sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (-i+\sqrt {3}-2 i x\right ) \sqrt {1+x^2}} \, dx,x,x^2\right )+\frac {1}{3} \left (i+\sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{-i+\sqrt {3}-2 i x^2} \, dx-\frac {\left (i+\sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{6 \left (i-\sqrt {3}\right )}+\frac {\left (1-i \sqrt {3}\right ) \int \frac {1+x^2}{\left (i+\sqrt {3}+2 i x^2\right ) \sqrt {1+x^4}} \, dx}{2 \left (3 i+\sqrt {3}\right )}+\frac {\left (i+\sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{4 \left (3 i+\sqrt {3}\right )}-\frac {\left (3 i+\sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{6 \left (i+\sqrt {3}\right )}\\ &=\frac {1}{16} \left (1-i \sqrt {3}\right ) \sqrt {1+x^4}+\frac {1}{16} \left (1+i \sqrt {3}\right ) \sqrt {1+x^4}-\frac {1}{48} \left (3-5 i \sqrt {3}\right ) \sqrt {1+x^4}-\frac {1}{48} \left (3+5 i \sqrt {3}\right ) \sqrt {1+x^4}+\frac {\left (i-\sqrt {3}\right ) \sqrt {1+x^4}}{3 \left (i-\sqrt {3}+2 i x^2\right )}+\frac {\left (i+\sqrt {3}\right ) \sqrt {1+x^4}}{6 \left (i-\sqrt {3}+2 i x^2\right )}-\frac {i x \sqrt {1+x^4}}{3 \left (i-\sqrt {3}+2 i x^2\right )}-\frac {4 x \sqrt {1+x^4}}{3 \left (i-\sqrt {3}\right ) \left (i-\sqrt {3}+2 i x^2\right )}+\frac {\left (i-\sqrt {3}\right ) \sqrt {1+x^4}}{6 \left (i+\sqrt {3}+2 i x^2\right )}+\frac {\left (i+\sqrt {3}\right ) \sqrt {1+x^4}}{3 \left (i+\sqrt {3}+2 i x^2\right )}-\frac {i x \sqrt {1+x^4}}{3 \left (i+\sqrt {3}+2 i x^2\right )}-\frac {4 x \sqrt {1+x^4}}{3 \left (i+\sqrt {3}\right ) \left (i+\sqrt {3}+2 i x^2\right )}-\frac {x \sqrt {1+x^4}}{3 \left (1-i \sqrt {3}\right ) \left (1+x^2\right )}-\frac {x \sqrt {1+x^4}}{3 \left (1+i \sqrt {3}\right ) \left (1+x^2\right )}-\frac {\left (3-2 i \sqrt {3}\right ) x \sqrt {1+x^4}}{12 \left (1+x^2\right )}-\frac {\left (3+2 i \sqrt {3}\right ) x \sqrt {1+x^4}}{12 \left (1+x^2\right )}+\frac {1}{16} \left (1-i \sqrt {3}\right ) \sinh ^{-1}\left (x^2\right )+\frac {1}{48} \left (9-i \sqrt {3}\right ) \sinh ^{-1}\left (x^2\right )+\frac {1}{16} \left (1+i \sqrt {3}\right ) \sinh ^{-1}\left (x^2\right )+\frac {1}{48} \left (9+i \sqrt {3}\right ) \sinh ^{-1}\left (x^2\right )+\frac {1}{4} \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )-\frac {\left (3 i-\sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{12 \left (i-\sqrt {3}\right )}+\frac {\left (3-5 i \sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{24 \left (1+i \sqrt {3}\right )}+\frac {\left (3+5 i \sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{12 \left (i-\sqrt {3}\right )^2}-\frac {\left (i+\sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{8 \left (3 i-\sqrt {3}\right )}-\frac {\left (i-\sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{8 \left (3 i+\sqrt {3}\right )}-\frac {\left (3 i+\sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{12 \left (i+\sqrt {3}\right )}+\frac {\left (1+i \sqrt {3}\right ) \tanh ^{-1}\left (\frac {2-\left (1-i \sqrt {3}\right ) x^2}{\sqrt {2 \left (1-i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{8 \sqrt {2 \left (1-i \sqrt {3}\right )}}+\frac {\left (1-i \sqrt {3}\right ) \tanh ^{-1}\left (\frac {2-\left (1+i \sqrt {3}\right ) x^2}{\sqrt {2 \left (1+i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{8 \sqrt {2 \left (1+i \sqrt {3}\right )}}-\frac {\left (i-3 \sqrt {3}\right ) \tanh ^{-1}\left (\frac {2-\left (1+i \sqrt {3}\right ) x^2}{\sqrt {2 \left (1+i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{8 \sqrt {6 \left (1+i \sqrt {3}\right )}}+\frac {\left (i+3 \sqrt {3}\right ) \tanh ^{-1}\left (\frac {4+\left (1+i \sqrt {3}\right )^2 x^2}{2 \sqrt {2 \left (1-i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{8 \sqrt {6 \left (1-i \sqrt {3}\right )}}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{3 \left (1-i \sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{3 \left (1+i \sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (3-2 i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}+\frac {\left (3+2 i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}-2 \left (-\frac {x \sqrt {1+x^4}}{12 \left (1+x^2\right )}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}\right )-2 \left (-\frac {x \sqrt {1+x^4}}{4 \left (1+x^2\right )}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{4 \sqrt {1+x^4}}\right )-\frac {\left (3 i-5 \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \left (3 i-\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (3 i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (1-i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}-\frac {\left (9-i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{48 \sqrt {1+x^4}}+\frac {\left (1+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}-\frac {\left (9+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{48 \sqrt {1+x^4}}-\frac {5 \left (i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {5 \left (i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \left (3 i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (3 i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (3 i+5 \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \sqrt {1+x^4}}+\frac {\left (5 i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (5 i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {1}{24} i \int \frac {i+\sqrt {3}-2 i x^2}{\sqrt {1+x^4}} \, dx-\frac {1}{24} i \int \frac {-i+\sqrt {3}+2 i x^2}{\sqrt {1+x^4}} \, dx+\frac {1}{12} i \int \frac {i+\sqrt {3}-2 i x^2}{\sqrt {1+x^4}} \, dx-\frac {1}{12} i \int \frac {-i+\sqrt {3}+2 i x^2}{\sqrt {1+x^4}} \, dx-\frac {1}{3} i \operatorname {Subst}\left (\int \frac {1}{\left (-i+\sqrt {3}-2 i x\right ) \sqrt {1+x^2}} \, dx,x,x^2\right )+\frac {1}{3} i \operatorname {Subst}\left (\int \frac {1}{\left (i+\sqrt {3}+2 i x\right ) \sqrt {1+x^2}} \, dx,x,x^2\right )+\frac {2}{3} i \int \frac {1}{\left (-i+\sqrt {3}-2 i x^2\right ) \sqrt {1+x^4}} \, dx-\frac {2}{3} i \int \frac {1}{\left (i+\sqrt {3}+2 i x^2\right ) \sqrt {1+x^4}} \, dx-\frac {\int \frac {-i+\sqrt {3}+2 i x^2}{\sqrt {1+x^4}} \, dx}{6 \left (i-\sqrt {3}\right )}-\frac {1}{12} \left (i-\sqrt {3}\right ) \int \frac {i+\sqrt {3}-2 i x^2}{\sqrt {1+x^4}} \, dx-\frac {1}{6} \left (i-\sqrt {3}\right ) \int \frac {1}{\left (-i+\sqrt {3}-2 i x^2\right ) \sqrt {1+x^4}} \, dx-\frac {1}{12} \left (3 i-\sqrt {3}\right ) \int \frac {1}{\left (i+\sqrt {3}+2 i x^2\right ) \sqrt {1+x^4}} \, dx+\frac {1}{12} \left (-1-i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^2\right )+\frac {1}{6} \left (-1-i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^2\right )+\frac {1}{12} \left (-1+i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^2\right )+\frac {1}{6} \left (-1+i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^2\right )+\frac {1}{3} \left (-i+\sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (i+\sqrt {3}+2 i x\right ) \sqrt {1+x^2}} \, dx,x,x^2\right )+\frac {\int \frac {i+\sqrt {3}-2 i x^2}{\sqrt {1+x^4}} \, dx}{6 \left (i+\sqrt {3}\right )}+\frac {1}{12} \left (i+\sqrt {3}\right ) \int \frac {-i+\sqrt {3}+2 i x^2}{\sqrt {1+x^4}} \, dx+\frac {1}{6} \left (i+\sqrt {3}\right ) \int \frac {1}{\left (i+\sqrt {3}+2 i x^2\right ) \sqrt {1+x^4}} \, dx+\frac {1}{3} \left (i+\sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-i+\sqrt {3}-2 i x\right ) \sqrt {1+x^2}} \, dx,x,x^2\right )+\frac {1}{12} \left (3 i+\sqrt {3}\right ) \int \frac {1}{\left (-i+\sqrt {3}-2 i x^2\right ) \sqrt {1+x^4}} \, dx-\frac {\left (4+\left (-i+\sqrt {3}\right )^2\right ) \int \frac {1}{\left (-i+\sqrt {3}-2 i x^2\right ) \sqrt {1+x^4}} \, dx}{6 \left (-i+\sqrt {3}\right )}-\frac {\left (4+\left (i+\sqrt {3}\right )^2\right ) \int \frac {1}{\left (i+\sqrt {3}+2 i x^2\right ) \sqrt {1+x^4}} \, dx}{6 \left (i+\sqrt {3}\right )}\\ &=\frac {1}{16} \left (1-i \sqrt {3}\right ) \sqrt {1+x^4}+\frac {1}{16} \left (1+i \sqrt {3}\right ) \sqrt {1+x^4}-\frac {1}{48} \left (3-5 i \sqrt {3}\right ) \sqrt {1+x^4}-\frac {1}{48} \left (3+5 i \sqrt {3}\right ) \sqrt {1+x^4}+\frac {\left (i-\sqrt {3}\right ) \sqrt {1+x^4}}{3 \left (i-\sqrt {3}+2 i x^2\right )}+\frac {\left (i+\sqrt {3}\right ) \sqrt {1+x^4}}{6 \left (i-\sqrt {3}+2 i x^2\right )}-\frac {i x \sqrt {1+x^4}}{3 \left (i-\sqrt {3}+2 i x^2\right )}-\frac {4 x \sqrt {1+x^4}}{3 \left (i-\sqrt {3}\right ) \left (i-\sqrt {3}+2 i x^2\right )}+\frac {\left (i-\sqrt {3}\right ) \sqrt {1+x^4}}{6 \left (i+\sqrt {3}+2 i x^2\right )}+\frac {\left (i+\sqrt {3}\right ) \sqrt {1+x^4}}{3 \left (i+\sqrt {3}+2 i x^2\right )}-\frac {i x \sqrt {1+x^4}}{3 \left (i+\sqrt {3}+2 i x^2\right )}-\frac {4 x \sqrt {1+x^4}}{3 \left (i+\sqrt {3}\right ) \left (i+\sqrt {3}+2 i x^2\right )}-\frac {x \sqrt {1+x^4}}{3 \left (1-i \sqrt {3}\right ) \left (1+x^2\right )}-\frac {x \sqrt {1+x^4}}{3 \left (1+i \sqrt {3}\right ) \left (1+x^2\right )}-\frac {\left (3-2 i \sqrt {3}\right ) x \sqrt {1+x^4}}{12 \left (1+x^2\right )}-\frac {\left (3+2 i \sqrt {3}\right ) x \sqrt {1+x^4}}{12 \left (1+x^2\right )}-\frac {3}{16} \left (1-i \sqrt {3}\right ) \sinh ^{-1}\left (x^2\right )+\frac {1}{48} \left (9-i \sqrt {3}\right ) \sinh ^{-1}\left (x^2\right )-\frac {3}{16} \left (1+i \sqrt {3}\right ) \sinh ^{-1}\left (x^2\right )+\frac {1}{48} \left (9+i \sqrt {3}\right ) \sinh ^{-1}\left (x^2\right )+\frac {1}{4} \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )-\frac {\left (3 i-\sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{12 \left (i-\sqrt {3}\right )}+\frac {\left (3-5 i \sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{24 \left (1+i \sqrt {3}\right )}+\frac {\left (3+5 i \sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{12 \left (i-\sqrt {3}\right )^2}-\frac {\left (i+\sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{8 \left (3 i-\sqrt {3}\right )}-\frac {\left (i-\sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{8 \left (3 i+\sqrt {3}\right )}-\frac {\left (3 i+\sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{12 \left (i+\sqrt {3}\right )}+\frac {\left (1+i \sqrt {3}\right ) \tanh ^{-1}\left (\frac {2-\left (1-i \sqrt {3}\right ) x^2}{\sqrt {2 \left (1-i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{8 \sqrt {2 \left (1-i \sqrt {3}\right )}}+\frac {\left (1-i \sqrt {3}\right ) \tanh ^{-1}\left (\frac {2-\left (1+i \sqrt {3}\right ) x^2}{\sqrt {2 \left (1+i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{8 \sqrt {2 \left (1+i \sqrt {3}\right )}}-\frac {\left (i-3 \sqrt {3}\right ) \tanh ^{-1}\left (\frac {2-\left (1+i \sqrt {3}\right ) x^2}{\sqrt {2 \left (1+i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{8 \sqrt {6 \left (1+i \sqrt {3}\right )}}+\frac {\left (i+3 \sqrt {3}\right ) \tanh ^{-1}\left (\frac {4+\left (1+i \sqrt {3}\right )^2 x^2}{2 \sqrt {2 \left (1-i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{8 \sqrt {6 \left (1-i \sqrt {3}\right )}}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{3 \left (1-i \sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{3 \left (1+i \sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (3-2 i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}+\frac {\left (3+2 i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}-2 \left (-\frac {x \sqrt {1+x^4}}{12 \left (1+x^2\right )}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}\right )-2 \left (-\frac {x \sqrt {1+x^4}}{4 \left (1+x^2\right )}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{4 \sqrt {1+x^4}}\right )-\frac {\left (3 i-5 \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \left (3 i-\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (3 i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (1-i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}-\frac {\left (9-i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{48 \sqrt {1+x^4}}+\frac {\left (1+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}-\frac {\left (9+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{48 \sqrt {1+x^4}}-\frac {5 \left (i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {5 \left (i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \left (3 i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (3 i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (3 i+5 \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \sqrt {1+x^4}}+\frac {\left (5 i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (5 i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {1}{3} i \operatorname {Subst}\left (\int \frac {1}{-4+\left (-i+\sqrt {3}\right )^2-x^2} \, dx,x,\frac {-2 i-\left (-i+\sqrt {3}\right ) x^2}{\sqrt {1+x^4}}\right )-\frac {1}{3} i \operatorname {Subst}\left (\int \frac {1}{-4+\left (i+\sqrt {3}\right )^2-x^2} \, dx,x,\frac {2 i-\left (i+\sqrt {3}\right ) x^2}{\sqrt {1+x^4}}\right )-2 \left (\frac {1}{12} \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx\right )-2 \left (\frac {1}{6} \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx\right )+\frac {1}{3} \left (-i-\sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{-4+\left (-i+\sqrt {3}\right )^2-x^2} \, dx,x,\frac {-2 i-\left (-i+\sqrt {3}\right ) x^2}{\sqrt {1+x^4}}\right )+\frac {1}{3} \left (i-\sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{-4+\left (i+\sqrt {3}\right )^2-x^2} \, dx,x,\frac {2 i-\left (i+\sqrt {3}\right ) x^2}{\sqrt {1+x^4}}\right )-\frac {(4 i) \int \frac {1+x^2}{\left (-i+\sqrt {3}-2 i x^2\right ) \sqrt {1+x^4}} \, dx}{\sqrt {3} \left (3 i-\sqrt {3}\right )}+\frac {\left (i-\sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{4 \left (3 i-\sqrt {3}\right )}-\frac {\left (3 i-\sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{6 \left (i-\sqrt {3}\right )}-\frac {1}{6} \left (-1-i \sqrt {3}\right ) \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx+\frac {\int \frac {1-x^2}{\sqrt {1+x^4}} \, dx}{3 \left (1-i \sqrt {3}\right )}+\frac {1}{6} \left (1-i \sqrt {3}\right ) \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx-\frac {(2 i) \int \frac {1}{\sqrt {1+x^4}} \, dx}{\sqrt {3} \left (3-i \sqrt {3}\right )}-\frac {1}{24} \left (-1+i \sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx-\frac {1}{12} \left (-1+i \sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx-\frac {1}{6} \left (-1+i \sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx+\frac {\int \frac {1-x^2}{\sqrt {1+x^4}} \, dx}{3 \left (1+i \sqrt {3}\right )}+\frac {1}{24} \left (1+i \sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx+\frac {1}{12} \left (1+i \sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx+\frac {1}{6} \left (1+i \sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx-\frac {\left (1+i \sqrt {3}\right ) \int \frac {1+x^2}{\left (-i+\sqrt {3}-2 i x^2\right ) \sqrt {1+x^4}} \, dx}{2 \left (3 i-\sqrt {3}\right )}+\frac {(2 i) \int \frac {1}{\sqrt {1+x^4}} \, dx}{\sqrt {3} \left (3+i \sqrt {3}\right )}+\frac {\left (3+i \sqrt {3}\right ) \int \frac {1+x^2}{\left (-i+\sqrt {3}-2 i x^2\right ) \sqrt {1+x^4}} \, dx}{3 \left (i-\sqrt {3}\right )}-\frac {\left (-i+\sqrt {3}\right ) \int \frac {1+x^2}{\left (-i+\sqrt {3}-2 i x^2\right ) \sqrt {1+x^4}} \, dx}{\sqrt {3} \left (3 i-\sqrt {3}\right )}-\frac {\left (i-\sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{6 \left (i+\sqrt {3}\right )}-\frac {\left (3-i \sqrt {3}\right ) \int \frac {1+x^2}{\left (i+\sqrt {3}+2 i x^2\right ) \sqrt {1+x^4}} \, dx}{3 \left (i+\sqrt {3}\right )}-\frac {\left (i+\sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{6 \left (i-\sqrt {3}\right )}-\frac {(4 i) \int \frac {1+x^2}{\left (i+\sqrt {3}+2 i x^2\right ) \sqrt {1+x^4}} \, dx}{\sqrt {3} \left (3 i+\sqrt {3}\right )}+\frac {\left (3 i-\sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{6 \left (3 i+\sqrt {3}\right )}+\frac {\left (1-i \sqrt {3}\right ) \int \frac {1+x^2}{\left (i+\sqrt {3}+2 i x^2\right ) \sqrt {1+x^4}} \, dx}{2 \left (3 i+\sqrt {3}\right )}+\frac {\left (3+i \sqrt {3}\right ) \int \frac {1+x^2}{\left (i+\sqrt {3}+2 i x^2\right ) \sqrt {1+x^4}} \, dx}{3 \left (3 i+\sqrt {3}\right )}+\frac {\left (i+\sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{4 \left (3 i+\sqrt {3}\right )}+\frac {\left (3 i+\sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{6 \left (3 i-\sqrt {3}\right )}-\frac {\left (3 i+\sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{6 \left (i+\sqrt {3}\right )}\\ &=\frac {1}{16} \left (1-i \sqrt {3}\right ) \sqrt {1+x^4}+\frac {1}{16} \left (1+i \sqrt {3}\right ) \sqrt {1+x^4}-\frac {1}{48} \left (3-5 i \sqrt {3}\right ) \sqrt {1+x^4}-\frac {1}{48} \left (3+5 i \sqrt {3}\right ) \sqrt {1+x^4}+\frac {\left (i-\sqrt {3}\right ) \sqrt {1+x^4}}{3 \left (i-\sqrt {3}+2 i x^2\right )}+\frac {\left (i+\sqrt {3}\right ) \sqrt {1+x^4}}{6 \left (i-\sqrt {3}+2 i x^2\right )}-\frac {i x \sqrt {1+x^4}}{3 \left (i-\sqrt {3}+2 i x^2\right )}-\frac {4 x \sqrt {1+x^4}}{3 \left (i-\sqrt {3}\right ) \left (i-\sqrt {3}+2 i x^2\right )}+\frac {\left (i-\sqrt {3}\right ) \sqrt {1+x^4}}{6 \left (i+\sqrt {3}+2 i x^2\right )}+\frac {\left (i+\sqrt {3}\right ) \sqrt {1+x^4}}{3 \left (i+\sqrt {3}+2 i x^2\right )}-\frac {i x \sqrt {1+x^4}}{3 \left (i+\sqrt {3}+2 i x^2\right )}-\frac {4 x \sqrt {1+x^4}}{3 \left (i+\sqrt {3}\right ) \left (i+\sqrt {3}+2 i x^2\right )}-\frac {2 x \sqrt {1+x^4}}{3 \left (1-i \sqrt {3}\right ) \left (1+x^2\right )}-\frac {\left (1-i \sqrt {3}\right ) x \sqrt {1+x^4}}{6 \left (1+x^2\right )}-\frac {2 x \sqrt {1+x^4}}{3 \left (1+i \sqrt {3}\right ) \left (1+x^2\right )}-\frac {\left (1+i \sqrt {3}\right ) x \sqrt {1+x^4}}{6 \left (1+x^2\right )}-\frac {\left (3-2 i \sqrt {3}\right ) x \sqrt {1+x^4}}{12 \left (1+x^2\right )}-\frac {\left (3+2 i \sqrt {3}\right ) x \sqrt {1+x^4}}{12 \left (1+x^2\right )}-\frac {3}{16} \left (1-i \sqrt {3}\right ) \sinh ^{-1}\left (x^2\right )+\frac {1}{48} \left (9-i \sqrt {3}\right ) \sinh ^{-1}\left (x^2\right )-\frac {3}{16} \left (1+i \sqrt {3}\right ) \sinh ^{-1}\left (x^2\right )+\frac {1}{48} \left (9+i \sqrt {3}\right ) \sinh ^{-1}\left (x^2\right )+\frac {5}{12} \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )-\frac {\left (3 i-\sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{6 \left (i-\sqrt {3}\right )}-\frac {1}{12} \left (1-i \sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )-\frac {1}{12} \left (1+i \sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )+\frac {\left (3-5 i \sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{24 \left (1+i \sqrt {3}\right )}+\frac {\left (3+5 i \sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{12 \left (i-\sqrt {3}\right )^2}-\frac {\left (i+\sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{4 \left (3 i-\sqrt {3}\right )}-\frac {\left (i-\sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{4 \left (3 i+\sqrt {3}\right )}-\frac {\left (3 i+\sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{6 \left (i+\sqrt {3}\right )}+\frac {i \tan ^{-1}\left (\frac {2 i-\left (i-\sqrt {3}\right ) x^2}{\sqrt {2 \left (1+i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{3 \sqrt {2 \left (1+i \sqrt {3}\right )}}-\frac {\left (i+\sqrt {3}\right ) \tan ^{-1}\left (\frac {2 i-\left (i-\sqrt {3}\right ) x^2}{\sqrt {2 \left (1+i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{3 \sqrt {2 \left (1+i \sqrt {3}\right )}}+\frac {i \tan ^{-1}\left (\frac {2 i-\left (i+\sqrt {3}\right ) x^2}{\sqrt {2 \left (1-i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{3 \sqrt {2 \left (1-i \sqrt {3}\right )}}-\frac {\left (i-\sqrt {3}\right ) \tan ^{-1}\left (\frac {2 i-\left (i+\sqrt {3}\right ) x^2}{\sqrt {2 \left (1-i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{3 \sqrt {2 \left (1-i \sqrt {3}\right )}}+\frac {\left (1+i \sqrt {3}\right ) \tanh ^{-1}\left (\frac {2-\left (1-i \sqrt {3}\right ) x^2}{\sqrt {2 \left (1-i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{8 \sqrt {2 \left (1-i \sqrt {3}\right )}}+\frac {\left (1-i \sqrt {3}\right ) \tanh ^{-1}\left (\frac {2-\left (1+i \sqrt {3}\right ) x^2}{\sqrt {2 \left (1+i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{8 \sqrt {2 \left (1+i \sqrt {3}\right )}}-\frac {\left (i-3 \sqrt {3}\right ) \tanh ^{-1}\left (\frac {2-\left (1+i \sqrt {3}\right ) x^2}{\sqrt {2 \left (1+i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{8 \sqrt {6 \left (1+i \sqrt {3}\right )}}+\frac {\left (i+3 \sqrt {3}\right ) \tanh ^{-1}\left (\frac {4+\left (1+i \sqrt {3}\right )^2 x^2}{2 \sqrt {2 \left (1-i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{8 \sqrt {6 \left (1-i \sqrt {3}\right )}}+\frac {2 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{3 \left (1-i \sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (1-i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \sqrt {1+x^4}}+\frac {2 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{3 \left (1+i \sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (1+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \sqrt {1+x^4}}+\frac {\left (3-2 i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}+\frac {\left (3+2 i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}-4 \left (-\frac {x \sqrt {1+x^4}}{12 \left (1+x^2\right )}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}\right )-2 \left (-\frac {x \sqrt {1+x^4}}{6 \left (1+x^2\right )}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \sqrt {1+x^4}}\right )-2 \left (-\frac {x \sqrt {1+x^4}}{4 \left (1+x^2\right )}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{4 \sqrt {1+x^4}}\right )-\frac {\left (3 i-5 \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {3} \left (3 i-\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{4 \left (3 i-\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (3 i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {11 \left (1-i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{48 \sqrt {1+x^4}}-\frac {\left (9-i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{48 \sqrt {1+x^4}}+\frac {11 \left (1+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{48 \sqrt {1+x^4}}-\frac {\left (9+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{48 \sqrt {1+x^4}}-\frac {7 \left (i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {7 \left (i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {3} \left (3 i+\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (3 i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \left (3 i+\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{4 \left (3 i+\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (3 i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \left (3 i-\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (3 i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (3 i+5 \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{4 \sqrt {1+x^4}}-\frac {i \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \sqrt {3} \sqrt {1+x^4}}+\frac {\left (5 i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (3+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \sqrt {1+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{2 \sqrt {3} \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{4 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{4 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \left (3 i-\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (5 i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}\\ \end {align*}

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Mathematica [C]  time = 1.44, size = 350, normalized size = 4.86 \begin {gather*} \frac {1}{8} \left (\frac {4 \sqrt {x^4+1}}{x^2+x+1}-i \sqrt {6+6 i \sqrt {3}} \tanh ^{-1}\left (\frac {2+\left (-1-i \sqrt {3}\right ) x^2}{\sqrt {2+2 i \sqrt {3}} \sqrt {x^4+1}}\right )-\sqrt {2+2 i \sqrt {3}} \tanh ^{-1}\left (\frac {2+\left (-1-i \sqrt {3}\right ) x^2}{\sqrt {2+2 i \sqrt {3}} \sqrt {x^4+1}}\right )+i \sqrt {6-6 i \sqrt {3}} \tanh ^{-1}\left (\frac {2+i \left (\sqrt {3}+i\right ) x^2}{\sqrt {2-2 i \sqrt {3}} \sqrt {x^4+1}}\right )-\sqrt {2-2 i \sqrt {3}} \tanh ^{-1}\left (\frac {2+i \left (\sqrt {3}+i\right ) x^2}{\sqrt {2-2 i \sqrt {3}} \sqrt {x^4+1}}\right )+4 \sqrt [4]{-1} \Pi \left (-\sqrt [6]{-1};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+4 \sqrt [4]{-1} \Pi \left (-(-1)^{5/6};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )\right )-\frac {1}{2} \sqrt [4]{-1} F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((-1)^(1/4)*EllipticF[I*ArcSinh[(-1)^(1/4)*x], -1]) + ((4*Sqrt[1 + x^4])/(1 + x + x^2) - Sqrt[2 + (2*I)*S
qrt[3]]*ArcTanh[(2 + (-1 - I*Sqrt[3])*x^2)/(Sqrt[2 + (2*I)*Sqrt[3]]*Sqrt[1 + x^4])] - I*Sqrt[6 + (6*I)*Sqrt[3]
]*ArcTanh[(2 + (-1 - I*Sqrt[3])*x^2)/(Sqrt[2 + (2*I)*Sqrt[3]]*Sqrt[1 + x^4])] - Sqrt[2 - (2*I)*Sqrt[3]]*ArcTan
h[(2 + I*(I + Sqrt[3])*x^2)/(Sqrt[2 - (2*I)*Sqrt[3]]*Sqrt[1 + x^4])] + I*Sqrt[6 - (6*I)*Sqrt[3]]*ArcTanh[(2 +
I*(I + Sqrt[3])*x^2)/(Sqrt[2 - (2*I)*Sqrt[3]]*Sqrt[1 + x^4])] + 4*(-1)^(1/4)*EllipticPi[-(-1)^(1/6), I*ArcSinh
[(-1)^(1/4)*x], -1] + 4*(-1)^(1/4)*EllipticPi[-(-1)^(5/6), I*ArcSinh[(-1)^(1/4)*x], -1])/8

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IntegrateAlgebraic [A]  time = 1.81, size = 72, normalized size = 1.00 \begin {gather*} \frac {\sqrt {1+x^4}}{2 \left (1+x+x^2\right )}+\frac {1}{2} \tan ^{-1}\left (\frac {x}{1-x+x^2+\sqrt {1+x^4}}\right )-\frac {3}{2} \tan ^{-1}\left (\frac {x}{1+x+x^2+\sqrt {1+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.52, size = 73, normalized size = 1.01 \begin {gather*} \frac {3 \, {\left (x^{2} + x + 1\right )} \arctan \left (\frac {\sqrt {x^{4} + 1}}{x^{2} + 2 \, x + 1}\right ) + {\left (x^{2} + x + 1\right )} \arctan \left (\frac {\sqrt {x^{4} + 1}}{x^{2} - 2 \, x + 1}\right ) + 2 \, \sqrt {x^{4} + 1}}{4 \, {\left (x^{2} + x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(3*(x^2 + x + 1)*arctan(sqrt(x^4 + 1)/(x^2 + 2*x + 1)) + (x^2 + x + 1)*arctan(sqrt(x^4 + 1)/(x^2 - 2*x + 1
)) + 2*sqrt(x^4 + 1))/(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^{4} + 1} {\left (x^{2} - 1\right )}}{{\left (x^{2} + x + 1\right )}^{2} {\left (x^{2} - x + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 1.05, size = 198, normalized size = 2.75

method	result	size
trager	\(\frac {\sqrt {x^{4}+1}}{2 x^{2}+2 x +2}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{8}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{7}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{6}+4 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{5}-2 \sqrt {x^{4}+1}\, x^{5}+5 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}-2 x^{4} \sqrt {x^{4}+1}+4 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}-2 \sqrt {x^{4}+1}\, x^{3}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}-2 x^{2} \sqrt {x^{4}+1}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x -2 \sqrt {x^{4}+1}\, x +\RootOf \left (\textit {\_Z}^{2}+1\right )}{\left (x^{2}-x +1\right ) \left (x^{2}+x +1\right )^{3}}\right )}{4}\)	\(198\)
risch	\(\frac {\sqrt {x^{4}+1}}{2 x^{2}+2 x +2}+\frac {\sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{2 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}+\frac {3 \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {\arctanh \left (\frac {\sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \left (x^{2}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\sqrt {x^{4}+1}}\right )}{2 \sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}}+\frac {\left (-1\right )^{\frac {3}{4}} \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , -i \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ), i\right )}{\sqrt {x^{4}+1}}\right )}{4}+\frac {3 \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (\frac {\arctanh \left (\frac {\sqrt {\frac {1}{2}-\frac {i \sqrt {3}}{2}}\, \left (x^{2}-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\sqrt {x^{4}+1}}\right )}{2 \sqrt {\frac {1}{2}-\frac {i \sqrt {3}}{2}}}+\frac {\left (-1\right )^{\frac {3}{4}} \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , -i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ), i\right )}{\sqrt {x^{4}+1}}\right )}{4}-\frac {\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {\arctanh \left (\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x^{2}-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\sqrt {\frac {1}{2}-\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+1}}\right )}{2 \sqrt {\frac {1}{2}-\frac {i \sqrt {3}}{2}}}+\frac {\left (-1\right )^{\frac {3}{4}} \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , i \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ), i\right )}{\sqrt {x^{4}+1}}\right )}{4}-\frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-\frac {\arctanh \left (\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x^{2}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+1}}\right )}{2 \sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}}+\frac {\left (-1\right )^{\frac {3}{4}} \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , i \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ), i\right )}{\sqrt {x^{4}+1}}\right )}{4}\)	\(539\)
elliptic	\(-\frac {\sqrt {2}\, \sqrt {\frac {2 \left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+2}\, \left (\sqrt {3}\, \arctanh \left (\frac {\sqrt {\frac {2 \left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+2}\, \sqrt {3}}{2}\right )-9 \arctan \left (\frac {\sqrt {\frac {2 \left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+2}\, \left (x^{2}+1\right )}{\left (\frac {\left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+1\right ) \left (-x^{2}+1\right )}\right )\right )}{12 \sqrt {\frac {\frac {\left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+1}{\left (\frac {x^{2}+1}{-x^{2}+1}+1\right )^{2}}}\, \left (\frac {x^{2}+1}{-x^{2}+1}+1\right )}+\frac {\left (\frac {3 \sqrt {3}\, \arctanh \left (\frac {\sqrt {\frac {2 \left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+2}\, \sqrt {3}}{2}\right ) \left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}-\frac {9 \arctan \left (\frac {\sqrt {\frac {2 \left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+2}\, \left (x^{2}+1\right )}{\left (\frac {\left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+1\right ) \left (-x^{2}+1\right )}\right ) \left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+\sqrt {3}\, \arctanh \left (\frac {\sqrt {\frac {2 \left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+2}\, \sqrt {3}}{2}\right )+\frac {6 \sqrt {\frac {2 \left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+2}\, \left (x^{2}+1\right )}{-x^{2}+1}-3 \arctan \left (\frac {\sqrt {\frac {2 \left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+2}\, \left (x^{2}+1\right )}{\left (\frac {\left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+1\right ) \left (-x^{2}+1\right )}\right )\right ) \sqrt {\frac {2 \left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+2}\, \sqrt {2}}{12 \left (\frac {3 \left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+1\right ) \sqrt {\frac {\frac {\left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+1}{\left (\frac {x^{2}+1}{-x^{2}+1}+1\right )^{2}}}\, \left (\frac {x^{2}+1}{-x^{2}+1}+1\right )}+\frac {\left (-\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{4 x \left (\frac {x^{4}+1}{2 x^{2}}+\frac {1}{2}\right )}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {x^{4}+1}}{x}\right )}{2}\right ) \sqrt {2}}{2}\)	\(602\)
default	\(\frac {\sqrt {x^{4}+1}}{2 x^{2}+2 x +2}+\frac {\sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{2 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}-\frac {i \sqrt {3}\, \left (\frac {\arctanh \left (\frac {\sqrt {\frac {1}{2}-\frac {i \sqrt {3}}{2}}\, \left (x^{2}-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\sqrt {x^{4}+1}}\right )}{2 \sqrt {\frac {1}{2}-\frac {i \sqrt {3}}{2}}}+\frac {\left (-1\right )^{\frac {3}{4}} \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , -i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ), i\right )}{\sqrt {x^{4}+1}}\right )}{3}+\frac {i \sqrt {3}\, \left (\frac {\arctanh \left (\frac {\sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \left (x^{2}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\sqrt {x^{4}+1}}\right )}{2 \sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}}+\frac {\left (-1\right )^{\frac {3}{4}} \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , -i \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ), i\right )}{\sqrt {x^{4}+1}}\right )}{3}-\frac {\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {\arctanh \left (\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x^{2}-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\sqrt {\frac {1}{2}-\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+1}}\right )}{2 \sqrt {\frac {1}{2}-\frac {i \sqrt {3}}{2}}}+\frac {\left (-1\right )^{\frac {3}{4}} \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , i \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ), i\right )}{\sqrt {x^{4}+1}}\right )}{4}-\frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-\frac {\arctanh \left (\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x^{2}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+1}}\right )}{2 \sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}}+\frac {\left (-1\right )^{\frac {3}{4}} \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , i \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ), i\right )}{\sqrt {x^{4}+1}}\right )}{4}-\frac {\left (\frac {3}{2}-\frac {i \sqrt {3}}{6}\right ) \left (\frac {\arctanh \left (\frac {\sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \left (x^{2}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\sqrt {x^{4}+1}}\right )}{2 \sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}}+\frac {\left (-1\right )^{\frac {3}{4}} \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , -i \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ), i\right )}{\sqrt {x^{4}+1}}\right )}{4}-\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{6}\right ) \left (\frac {\arctanh \left (\frac {\sqrt {\frac {1}{2}-\frac {i \sqrt {3}}{2}}\, \left (x^{2}-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\sqrt {x^{4}+1}}\right )}{2 \sqrt {\frac {1}{2}-\frac {i \sqrt {3}}{2}}}+\frac {\left (-1\right )^{\frac {3}{4}} \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , -i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ), i\right )}{\sqrt {x^{4}+1}}\right )}{4}\)	\(753\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(x^4+1)^(1/2)/(x^2+x+1)+1/4*RootOf(_Z^2+1)*ln(-(RootOf(_Z^2+1)*x^8+2*RootOf(_Z^2+1)*x^7+2*RootOf(_Z^2+1)*x
^6+4*RootOf(_Z^2+1)*x^5-2*(x^4+1)^(1/2)*x^5+5*RootOf(_Z^2+1)*x^4-2*x^4*(x^4+1)^(1/2)+4*RootOf(_Z^2+1)*x^3-2*(x
^4+1)^(1/2)*x^3+2*RootOf(_Z^2+1)*x^2-2*x^2*(x^4+1)^(1/2)+2*RootOf(_Z^2+1)*x-2*(x^4+1)^(1/2)*x+RootOf(_Z^2+1))/
(x^2-x+1)/(x^2+x+1)^3)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{4} + 1} {\left (x^{2} - 1\right )}}{{\left (x^{2} + x + 1\right )}^{2} {\left (x^{2} - x + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (x^2-1\right )\,\sqrt {x^4+1}}{\left (x^2-x+1\right )\,{\left (x^2+x+1\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x - 1\right ) \left (x + 1\right ) \sqrt {x^{4} + 1}}{\left (x^{2} - x + 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**4+1)**(1/2)/(x**2-x+1)/(x**2+x+1)**2,x)

[Out]

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

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