3.6.86 \(\int \frac {-1+x^{16}}{\sqrt {1+x^4} (1+x^{16})} \, dx\)

Optimal. Leaf size=45 \[ \frac {1}{8} \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^4+2\& ,\frac {\log \left (\sqrt {x^4+1}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ] \]

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Rubi [C]  time = 7.41, antiderivative size = 2243, normalized size of antiderivative = 49.84, number of steps used = 227, number of rules used = 16, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {1586, 6725, 1729, 1209, 1198, 220, 1196, 1217, 1707, 1248, 735, 844, 215, 725, 206, 204}

result too large to display

Warning: Unable to verify antiderivative.

[In]

Int[(-1 + x^16)/(Sqrt[1 + x^4]*(1 + x^16)),x]

[Out]

-1/16*((-1)^(13/16)*Sqrt[-1 + (-1)^(1/4)]*((1 + I) + I*Sqrt[2])*ArcTan[((-1)^(3/16)*Sqrt[-1 + (-1)^(1/4)]*x)/S
qrt[1 + x^4]]) + ((-1)^(1/16)*((1 + I) + Sqrt[2])*Sqrt[-4 + (2 + 2*I)*Sqrt[2]]*ArcTan[((-1)^(3/16)*Sqrt[-1 + (
-1)^(1/4)]*x)/Sqrt[1 + x^4]])/32 - ArcTan[(Sqrt[(-1)^(3/8) - (-1)^(5/8)]*x)/Sqrt[1 + x^4]]/(8*Sqrt[(-1)^(3/8)
- (-1)^(5/8)]) - ((-1)^(5/8)*Sqrt[(-1)^(3/8) - (-1)^(5/8)]*(1 + (-1)^(5/8))*((1 - I) + Sqrt[2])*ArcTan[(Sqrt[(
-1)^(3/8) - (-1)^(5/8)]*x)/Sqrt[1 + x^4]])/(16*(I - (-1)^(1/8))) + ((-1)^(15/16)*ArcTan[((-1)^(1/16)*Sqrt[-1 +
(-1)^(3/4)]*x)/Sqrt[1 + x^4]])/(4*Sqrt[-4 - (2 - 2*I)*Sqrt[2]]) - ((-1)^(11/16)*Sqrt[-1 + (-1)^(3/4)]*((-1 +
I) + Sqrt[2])*ArcTan[((-1)^(1/16)*Sqrt[-1 + (-1)^(3/4)]*x)/Sqrt[1 + x^4]])/16 - ArcTan[(Sqrt[(-1)^(1/8) - (-1)
^(7/8)]*x)/Sqrt[1 + x^4]]/(8*Sqrt[(-1)^(1/8) - (-1)^(7/8)]) + ((-1)^(5/8)*Sqrt[(-1)^(1/8) - (-1)^(7/8)]*((-1 +
I) + Sqrt[2])*ArcTan[(Sqrt[(-1)^(1/8) - (-1)^(7/8)]*x)/Sqrt[1 + x^4]])/16 + ((-1)^(1/8)*(1 + x^2)*Sqrt[(1 + x
^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(8*(I - (-1)^(1/8))*Sqrt[1 + x^4]) + ((-1)^(1/8)*(1 + x^2)*Sqrt[
(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(8*(1 - (-1)^(1/8))*Sqrt[1 + x^4]) - ((-1)^(1/8)*(1 + x^2)
*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(8*(I + (-1)^(1/8))*Sqrt[1 + x^4]) - ((-1)^(1/8)*(1
+ x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(8*(1 + (-1)^(1/8))*Sqrt[1 + x^4]) - ((-1)^(1/
8)*(I + (-1)^(1/8))*((1 + I) + I*Sqrt[2])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(
16*Sqrt[1 + x^4]) - ((1 - (-1)^(1/8))*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(4*Sq
rt[2]*((-1 - I) + Sqrt[2])*Sqrt[1 + x^4]) - ((1 + (-1)^(1/8))*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[
2*ArcTan[x], 1/2])/(4*Sqrt[2]*((-1 - I) + Sqrt[2])*Sqrt[1 + x^4]) + ((-1)^(1/4)*(1 - (-1)^(1/8))*((-1 - I) + S
qrt[2])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(16*Sqrt[1 + x^4]) + ((-1)^(1/4)*(1
+ (-1)^(1/8))*((-1 - I) + Sqrt[2])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(16*Sqr
t[1 + x^4]) + ((-1)^(5/8)*(1 - (-1)^(1/8))*((-1 + I) + Sqrt[2])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*Elliptic
F[2*ArcTan[x], 1/2])/(16*Sqrt[1 + x^4]) - ((-1)^(5/8)*(1 + (-1)^(1/8))*((-1 + I) + Sqrt[2])*(1 + x^2)*Sqrt[(1
+ x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(16*Sqrt[1 + x^4]) + ((-1)^(5/8)*(I - (-1)^(1/8))*((1 - I) +
Sqrt[2])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(16*Sqrt[1 + x^4]) - ((1 - (-1)^(5
/8))*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(4*Sqrt[2]*((1 + I) + Sqrt[2])*Sqrt[1
+ x^4]) - ((1 + (-1)^(5/8))*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(4*Sqrt[2]*((1
+ I) + Sqrt[2])*Sqrt[1 + x^4]) + ((-1)^(1/4)*(1 - (-1)^(5/8))*((1 + I) + Sqrt[2])*(1 + x^2)*Sqrt[(1 + x^4)/(1
+ x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(16*Sqrt[1 + x^4]) + ((-1)^(1/4)*(1 + (-1)^(5/8))*((1 + I) + Sqrt[2])*(
1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(16*Sqrt[1 + x^4]) + (((-1 + I) + (2 + 2*I)*
(-1)^(1/8) + Sqrt[2])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticPi[((-1)^(3/8)*(I - (-1)^(1/8))^2)/4, 2*Ar
cTan[x], 1/2])/(16*((1 - I) + Sqrt[2])*Sqrt[1 + x^4]) - (((-1 + I) + (2 + 2*I)*(-1)^(5/8) - Sqrt[2])*(1 + x^2)
*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticPi[((-1)^(7/8)*(1 - (-1)^(1/8))^2)/4, 2*ArcTan[x], 1/2])/(16*((-1 + I) +
Sqrt[2])*Sqrt[1 + x^4]) - ((-1)^(1/8)*(1 - (-1)^(3/8))*(1 + (-1)^(5/8))*((-1 + I) + Sqrt[2])*(1 + x^2)*Sqrt[(1
+ x^4)/(1 + x^2)^2]*EllipticPi[-1/4*((-1)^(3/8)*(I + (-1)^(1/8))^2), 2*ArcTan[x], 1/2])/(32*Sqrt[1 + x^4]) +
((1/32 + I/32)*((-1 + I) + (-1)^(1/8) - (-1)^(5/8))*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticPi[-1/4*((-1
)^(7/8)*(1 + (-1)^(1/8))^2), 2*ArcTan[x], 1/2])/((1 + (-1)^(1/8))*Sqrt[1 + x^4]) - ((I - (-1)^(1/4) - 2*(-1)^(
7/8))*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticPi[(2 + (-1)^(3/8) - (-1)^(5/8))/4, 2*ArcTan[x], 1/2])/(16
*(I + (-1)^(1/4))*Sqrt[1 + x^4]) - ((I - (-1)^(1/4) + 2*(-1)^(7/8))*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*Elli
pticPi[(2 - (-1)^(3/8) + (-1)^(5/8))/4, 2*ArcTan[x], 1/2])/(16*(I + (-1)^(1/4))*Sqrt[1 + x^4]) + (((1 + I) - (
2 - 2*I)*(-1)^(3/8) + Sqrt[2])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticPi[(2 + (-1)^(1/8) - (-1)^(7/8))/
4, 2*ArcTan[x], 1/2])/(16*((-1 - I) + Sqrt[2])*Sqrt[1 + x^4]) - ((I + (-1)^(1/4) + 2*(-1)^(3/8))*(1 + x^2)*Sqr
t[(1 + x^4)/(1 + x^2)^2]*EllipticPi[(2 - (-1)^(1/8) + (-1)^(7/8))/4, 2*ArcTan[x], 1/2])/(16*(I - (-1)^(1/4))*S
qrt[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 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 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 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

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 6725

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

Rubi steps

\begin {align*} \int \frac {-1+x^{16}}{\sqrt {1+x^4} \left (1+x^{16}\right )} \, dx &=\int \frac {\sqrt {1+x^4} \left (-1+x^4-x^8+x^{12}\right )}{1+x^{16}} \, dx\\ &=\int \left (\frac {\left (-\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \sqrt {1+x^4}}{16 \left (\sqrt [16]{-1}-x\right )}+\frac {\left (-\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \sqrt {1+x^4}}{16 \left (\sqrt [16]{-1}-i x\right )}+\frac {\left (-\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \sqrt {1+x^4}}{16 \left (\sqrt [16]{-1}+i x\right )}+\frac {\left (-\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \sqrt {1+x^4}}{16 \left (\sqrt [16]{-1}+x\right )}+\frac {\left (-\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \sqrt {1+x^4}}{16 \left (\sqrt [16]{-1}-\sqrt [8]{-1} x\right )}+\frac {\left (-\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \sqrt {1+x^4}}{16 \left (\sqrt [16]{-1}+\sqrt [8]{-1} x\right )}+\frac {\left (-\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \sqrt {1+x^4}}{16 \left (\sqrt [16]{-1}-\sqrt [4]{-1} x\right )}+\frac {\left (-\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \sqrt {1+x^4}}{16 \left (\sqrt [16]{-1}+\sqrt [4]{-1} x\right )}+\frac {\left (-\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \sqrt {1+x^4}}{16 \left (\sqrt [16]{-1}-(-1)^{3/8} x\right )}+\frac {\left (-\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \sqrt {1+x^4}}{16 \left (\sqrt [16]{-1}+(-1)^{3/8} x\right )}+\frac {\left (-\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \sqrt {1+x^4}}{16 \left (\sqrt [16]{-1}-(-1)^{5/8} x\right )}+\frac {\left (-\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \sqrt {1+x^4}}{16 \left (\sqrt [16]{-1}+(-1)^{5/8} x\right )}+\frac {\left (-\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \sqrt {1+x^4}}{16 \left (\sqrt [16]{-1}-(-1)^{3/4} x\right )}+\frac {\left (-\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \sqrt {1+x^4}}{16 \left (\sqrt [16]{-1}+(-1)^{3/4} x\right )}+\frac {\left (-\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \sqrt {1+x^4}}{16 \left (\sqrt [16]{-1}-(-1)^{7/8} x\right )}+\frac {\left (-\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \sqrt {1+x^4}}{16 \left (\sqrt [16]{-1}+(-1)^{7/8} x\right )}\right ) \, dx\\ &=-\left (\frac {1}{16} \left (\sqrt [16]{-1} \left ((1+i)+i \sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [16]{-1}-\sqrt [4]{-1} x} \, dx\right )-\frac {1}{16} \left (\sqrt [16]{-1} \left ((1+i)+i \sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [16]{-1}+\sqrt [4]{-1} x} \, dx-\frac {1}{16} \left (\sqrt [16]{-1} \left ((1+i)+i \sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [16]{-1}-(-1)^{3/4} x} \, dx-\frac {1}{16} \left (\sqrt [16]{-1} \left ((1+i)+i \sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [16]{-1}+(-1)^{3/4} x} \, dx-\frac {1}{16} \left ((-1)^{9/16} \left ((-1-i)+\sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [16]{-1}-\sqrt [8]{-1} x} \, dx-\frac {1}{16} \left ((-1)^{9/16} \left ((-1-i)+\sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [16]{-1}+\sqrt [8]{-1} x} \, dx-\frac {1}{16} \left ((-1)^{9/16} \left ((-1-i)+\sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [16]{-1}-(-1)^{5/8} x} \, dx-\frac {1}{16} \left ((-1)^{9/16} \left ((-1-i)+\sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [16]{-1}+(-1)^{5/8} x} \, dx+\frac {1}{16} \left ((-1)^{9/16} \left ((-1+i)+\sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [16]{-1}-x} \, dx+\frac {1}{16} \left ((-1)^{9/16} \left ((-1+i)+\sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [16]{-1}-i x} \, dx+\frac {1}{16} \left ((-1)^{9/16} \left ((-1+i)+\sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [16]{-1}+i x} \, dx+\frac {1}{16} \left ((-1)^{9/16} \left ((-1+i)+\sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [16]{-1}+x} \, dx+\frac {1}{16} \left ((-1)^{9/16} \left ((1+i)+\sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [16]{-1}-(-1)^{3/8} x} \, dx+\frac {1}{16} \left ((-1)^{9/16} \left ((1+i)+\sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [16]{-1}+(-1)^{3/8} x} \, dx+\frac {1}{16} \left ((-1)^{9/16} \left ((1+i)+\sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [16]{-1}-(-1)^{7/8} x} \, dx+\frac {1}{16} \left ((-1)^{9/16} \left ((1+i)+\sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [16]{-1}+(-1)^{7/8} x} \, dx\\ &=-2 \left (\frac {1}{16} \left (\sqrt [8]{-1} \left ((1+i)+i \sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [8]{-1}-i x^2} \, dx\right )-2 \left (\frac {1}{16} \left (\sqrt [8]{-1} \left ((1+i)+i \sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [8]{-1}+i x^2} \, dx\right )-2 \left (\frac {1}{16} \left ((-1)^{5/8} \left ((-1-i)+\sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [8]{-1}-\sqrt [4]{-1} x^2} \, dx\right )-2 \left (\frac {1}{16} \left ((-1)^{5/8} \left ((-1-i)+\sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [8]{-1}+\sqrt [4]{-1} x^2} \, dx\right )+2 \left (\frac {1}{16} \left ((-1)^{5/8} \left ((-1+i)+\sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [8]{-1}-x^2} \, dx\right )+2 \left (\frac {1}{16} \left ((-1)^{5/8} \left ((-1+i)+\sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [8]{-1}+x^2} \, dx\right )+2 \left (\frac {1}{16} \left ((-1)^{5/8} \left ((1+i)+\sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [8]{-1}-(-1)^{3/4} x^2} \, dx\right )+2 \left (\frac {1}{16} \left ((-1)^{5/8} \left ((1+i)+\sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [8]{-1}+(-1)^{3/4} x^2} \, dx\right )\\ &=-2 \left (\frac {1}{8} \sqrt [8]{-1} \int \frac {1}{\left (\sqrt [8]{-1}+i x^2\right ) \sqrt {1+x^4}} \, dx+\frac {1}{16} \left (\sqrt [8]{-1} \left ((1+i)+i \sqrt {2}\right )\right ) \int \frac {\sqrt [8]{-1}-i x^2}{\sqrt {1+x^4}} \, dx\right )-2 \left (\frac {1}{8} \sqrt [8]{-1} \int \frac {1}{\left (\sqrt [8]{-1}-i x^2\right ) \sqrt {1+x^4}} \, dx+\frac {1}{16} \left (\sqrt [8]{-1} \left ((1+i)+i \sqrt {2}\right )\right ) \int \frac {\sqrt [8]{-1}+i x^2}{\sqrt {1+x^4}} \, dx\right )-2 \left (\frac {1}{8} \sqrt [8]{-1} \int \frac {1}{\left (\sqrt [8]{-1}+\sqrt [4]{-1} x^2\right ) \sqrt {1+x^4}} \, dx-\frac {1}{16} \left (\sqrt [8]{-1} \left ((-1-i)+\sqrt {2}\right )\right ) \int \frac {\sqrt [8]{-1}-\sqrt [4]{-1} x^2}{\sqrt {1+x^4}} \, dx\right )-2 \left (\frac {1}{8} \sqrt [8]{-1} \int \frac {1}{\left (\sqrt [8]{-1}-\sqrt [4]{-1} x^2\right ) \sqrt {1+x^4}} \, dx-\frac {1}{16} \left (\sqrt [8]{-1} \left ((-1-i)+\sqrt {2}\right )\right ) \int \frac {\sqrt [8]{-1}+\sqrt [4]{-1} x^2}{\sqrt {1+x^4}} \, dx\right )+2 \left (-\left (\frac {1}{8} \sqrt [8]{-1} \int \frac {1}{\left (\sqrt [8]{-1}+x^2\right ) \sqrt {1+x^4}} \, dx\right )-\frac {1}{16} \left ((-1)^{5/8} \left ((-1+i)+\sqrt {2}\right )\right ) \int \frac {\sqrt [8]{-1}-x^2}{\sqrt {1+x^4}} \, dx\right )+2 \left (-\left (\frac {1}{8} \sqrt [8]{-1} \int \frac {1}{\left (\sqrt [8]{-1}-x^2\right ) \sqrt {1+x^4}} \, dx\right )-\frac {1}{16} \left ((-1)^{5/8} \left ((-1+i)+\sqrt {2}\right )\right ) \int \frac {\sqrt [8]{-1}+x^2}{\sqrt {1+x^4}} \, dx\right )+2 \left (-\left (\frac {1}{8} \sqrt [8]{-1} \int \frac {1}{\left (\sqrt [8]{-1}+(-1)^{3/4} x^2\right ) \sqrt {1+x^4}} \, dx\right )+\frac {1}{16} \left (\sqrt [8]{-1} \left ((1+i)+\sqrt {2}\right )\right ) \int \frac {\sqrt [8]{-1}-(-1)^{3/4} x^2}{\sqrt {1+x^4}} \, dx\right )+2 \left (-\left (\frac {1}{8} \sqrt [8]{-1} \int \frac {1}{\left (\sqrt [8]{-1}-(-1)^{3/4} x^2\right ) \sqrt {1+x^4}} \, dx\right )+\frac {1}{16} \left (\sqrt [8]{-1} \left ((1+i)+\sqrt {2}\right )\right ) \int \frac {\sqrt [8]{-1}+(-1)^{3/4} x^2}{\sqrt {1+x^4}} \, dx\right )\\ &=-2 \left (-\frac {\sqrt [8]{-1} \int \frac {1}{\sqrt {1+x^4}} \, dx}{8 \left (i-\sqrt [8]{-1}\right )}+\frac {\sqrt [8]{-1} \int \frac {1+x^2}{\left (\sqrt [8]{-1}+i x^2\right ) \sqrt {1+x^4}} \, dx}{8 \left (1+(-1)^{5/8}\right )}+\frac {1}{16} \left ((-1)^{5/8} \left ((1+i)+i \sqrt {2}\right )\right ) \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx+\frac {1}{16} \left (\sqrt [8]{-1} \left (-i+\sqrt [8]{-1}\right ) \left ((1+i)+i \sqrt {2}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx\right )-2 \left (\frac {\sqrt [8]{-1} \int \frac {1}{\sqrt {1+x^4}} \, dx}{8 \left (i+\sqrt [8]{-1}\right )}+\frac {\sqrt [8]{-1} \int \frac {1+x^2}{\left (\sqrt [8]{-1}-i x^2\right ) \sqrt {1+x^4}} \, dx}{8 \left (1-(-1)^{5/8}\right )}-\frac {1}{16} \left ((-1)^{5/8} \left ((1+i)+i \sqrt {2}\right )\right ) \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx+\frac {1}{16} \left (\sqrt [8]{-1} \left (i+\sqrt [8]{-1}\right ) \left ((1+i)+i \sqrt {2}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx\right )-2 \left (\frac {\left (\sqrt [8]{-1} \left (\sqrt [8]{-1}+\sqrt [4]{-1}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{8 \left (-i+\sqrt [4]{-1}\right )}+\frac {\left (1+\sqrt [8]{-1}\right ) \int \frac {1+x^2}{\left (\sqrt [8]{-1}+\sqrt [4]{-1} x^2\right ) \sqrt {1+x^4}} \, dx}{8 \left (1+(-1)^{3/4}\right )}-\frac {1}{16} \left ((-1)^{3/8} \left ((-1-i)+\sqrt {2}\right )\right ) \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx-\frac {1}{16} \left (\sqrt [4]{-1} \left (1-\sqrt [8]{-1}\right ) \left ((-1-i)+\sqrt {2}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx\right )-2 \left (\frac {\left (\sqrt [8]{-1} \left (\sqrt [8]{-1}-\sqrt [4]{-1}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{8 \left (-i+\sqrt [4]{-1}\right )}-\frac {\left (i-(-1)^{5/8}\right ) \int \frac {1+x^2}{\left (\sqrt [8]{-1}-\sqrt [4]{-1} x^2\right ) \sqrt {1+x^4}} \, dx}{8 \left (i-\sqrt [4]{-1}\right )}+\frac {1}{16} \left ((-1)^{3/8} \left ((-1-i)+\sqrt {2}\right )\right ) \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx-\frac {1}{16} \left (\sqrt [4]{-1} \left (1+\sqrt [8]{-1}\right ) \left ((-1-i)+\sqrt {2}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx\right )+2 \left (\frac {\sqrt [8]{-1} \int \frac {1}{\sqrt {1+x^4}} \, dx}{8 \left (1-\sqrt [8]{-1}\right )}-\frac {\sqrt [8]{-1} \int \frac {1+x^2}{\left (\sqrt [8]{-1}+x^2\right ) \sqrt {1+x^4}} \, dx}{8 \left (1-\sqrt [8]{-1}\right )}-\frac {1}{16} \left ((-1)^{5/8} \left ((-1+i)+\sqrt {2}\right )\right ) \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx+\frac {1}{16} \left ((-1)^{5/8} \left (1-\sqrt [8]{-1}\right ) \left ((-1+i)+\sqrt {2}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx\right )+2 \left (-\frac {\sqrt [8]{-1} \int \frac {1}{\sqrt {1+x^4}} \, dx}{8 \left (1+\sqrt [8]{-1}\right )}-\frac {\sqrt [8]{-1} \int \frac {1+x^2}{\left (\sqrt [8]{-1}-x^2\right ) \sqrt {1+x^4}} \, dx}{8 \left (1+\sqrt [8]{-1}\right )}+\frac {1}{16} \left ((-1)^{5/8} \left ((-1+i)+\sqrt {2}\right )\right ) \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx-\frac {1}{16} \left ((-1)^{5/8} \left (1+\sqrt [8]{-1}\right ) \left ((-1+i)+\sqrt {2}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx\right )+2 \left (-\frac {\left (1+(-1)^{5/8}\right ) \int \frac {1+x^2}{\left (\sqrt [8]{-1}+(-1)^{3/4} x^2\right ) \sqrt {1+x^4}} \, dx}{8 \left (i+\sqrt [4]{-1}\right )}-\frac {\left (\sqrt [8]{-1} \left (\sqrt [8]{-1}+(-1)^{3/4}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{8 \left (i+\sqrt [4]{-1}\right )}+\frac {1}{16} \left ((-1)^{7/8} \left ((1+i)+\sqrt {2}\right )\right ) \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx+\frac {1}{16} \left (\sqrt [4]{-1} \left (1-(-1)^{5/8}\right ) \left ((1+i)+\sqrt {2}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx\right )+2 \left (\frac {\left (1-(-1)^{5/8}\right ) \int \frac {1+x^2}{\left (\sqrt [8]{-1}-(-1)^{3/4} x^2\right ) \sqrt {1+x^4}} \, dx}{8 \left (i+\sqrt [4]{-1}\right )}-\frac {\left (\sqrt [8]{-1} \left (\sqrt [8]{-1}-(-1)^{3/4}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{8 \left (i+\sqrt [4]{-1}\right )}-\frac {1}{16} \left ((-1)^{7/8} \left ((1+i)+\sqrt {2}\right )\right ) \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx+\frac {1}{16} \left (\sqrt [4]{-1} \left (1+(-1)^{5/8}\right ) \left ((1+i)+\sqrt {2}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx\right )\\ &=-2 \left (-\frac {(-1)^{5/8} \left ((1+i)+i \sqrt {2}\right ) x \sqrt {1+x^4}}{16 \left (1+x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {(-1)^{3/8}-(-1)^{5/8}} x}{\sqrt {1+x^4}}\right )}{16 \sqrt {(-1)^{3/8}-(-1)^{5/8}}}+\frac {(-1)^{5/8} \left ((1+i)+i \sqrt {2}\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 )}{16 \sqrt {1+x^4}}-\frac {\sqrt [8]{-1} \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 \left (i-\sqrt [8]{-1}\right ) \sqrt {1+x^4}}-\frac {\sqrt [8]{-1} \left (i-\sqrt [8]{-1}\right ) \left ((1+i)+i \sqrt {2}\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 )}{32 \sqrt {1+x^4}}+\frac {\left (1-(-1)^{5/8}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4} (-1)^{3/8} \left (i-\sqrt [8]{-1}\right )^2;2 \tan ^{-1}(x)|\frac {1}{2}\right )}{32 \left (1+(-1)^{5/8}\right ) \sqrt {1+x^4}}\right )+2 \left (\frac {(-1)^{5/8} \left ((-1+i)+\sqrt {2}\right ) x \sqrt {1+x^4}}{16 \left (1+x^2\right )}-\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt [8]{-1}-(-1)^{7/8}} x}{\sqrt {1+x^4}}\right )}{16 \sqrt {\sqrt [8]{-1}-(-1)^{7/8}}}-\frac {(-1)^{5/8} \left ((-1+i)+\sqrt {2}\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 )}{16 \sqrt {1+x^4}}+\frac {\sqrt [8]{-1} \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 \left (1-\sqrt [8]{-1}\right ) \sqrt {1+x^4}}+\frac {(-1)^{5/8} \left (1-\sqrt [8]{-1}\right ) \left ((-1+i)+\sqrt {2}\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 )}{32 \sqrt {1+x^4}}-\frac {\left (1+\sqrt [8]{-1}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4} (-1)^{7/8} \left (1-\sqrt [8]{-1}\right )^2;2 \tan ^{-1}(x)|\frac {1}{2}\right )}{32 \left (1-\sqrt [8]{-1}\right ) \sqrt {1+x^4}}\right )-2 \left (\frac {(-1)^{5/8} \left ((1+i)+i \sqrt {2}\right ) x \sqrt {1+x^4}}{16 \left (1+x^2\right )}-\frac {(-1)^{5/16} \left (i+\sqrt [8]{-1}\right ) \tan ^{-1}\left (\frac {(-1)^{3/16} \sqrt {-1+\sqrt [4]{-1}} x}{\sqrt {1+x^4}}\right )}{16 \sqrt {-1+\sqrt [4]{-1}} \left (1-(-1)^{5/8}\right )}-\frac {(-1)^{5/8} \left ((1+i)+i \sqrt {2}\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 )}{16 \sqrt {1+x^4}}+\frac {\sqrt [8]{-1} \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 \left (i+\sqrt [8]{-1}\right ) \sqrt {1+x^4}}+\frac {\sqrt [8]{-1} \left (i+\sqrt [8]{-1}\right ) \left ((1+i)+i \sqrt {2}\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 )}{32 \sqrt {1+x^4}}+\frac {\left (1+(-1)^{5/8}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (-\frac {1}{4} (-1)^{3/8} \left (i+\sqrt [8]{-1}\right )^2;2 \tan ^{-1}(x)|\frac {1}{2}\right )}{32 \left (1-(-1)^{5/8}\right ) \sqrt {1+x^4}}\right )+2 \left (-\frac {(-1)^{5/8} \left ((-1+i)+\sqrt {2}\right ) x \sqrt {1+x^4}}{16 \left (1+x^2\right )}+\frac {(-1)^{15/16} \tan ^{-1}\left (\frac {\sqrt [16]{-1} \sqrt {-1+(-1)^{3/4}} x}{\sqrt {1+x^4}}\right )}{16 \sqrt {-1+(-1)^{3/4}}}+\frac {(-1)^{5/8} \left ((-1+i)+\sqrt {2}\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 )}{16 \sqrt {1+x^4}}-\frac {\sqrt [8]{-1} \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 \left (1+\sqrt [8]{-1}\right ) \sqrt {1+x^4}}-\frac {(-1)^{5/8} \left (1+\sqrt [8]{-1}\right ) \left ((-1+i)+\sqrt {2}\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 )}{32 \sqrt {1+x^4}}-\frac {\left (1-\sqrt [8]{-1}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (-\frac {1}{4} (-1)^{7/8} \left (1+\sqrt [8]{-1}\right )^2;2 \tan ^{-1}(x)|\frac {1}{2}\right )}{32 \left (1+\sqrt [8]{-1}\right ) \sqrt {1+x^4}}\right )+2 \left (-\frac {(-1)^{7/8} \left ((1+i)+\sqrt {2}\right ) x \sqrt {1+x^4}}{16 \left (1+x^2\right )}-\frac {\sqrt [16]{-1} \left (1+\sqrt [4]{-1}\right ) \tan ^{-1}\left (\frac {(-1)^{3/16} \sqrt {-1+\sqrt [4]{-1}} x}{\sqrt {1+x^4}}\right )}{16 \sqrt {-1+\sqrt [4]{-1}} \left (i+\sqrt [4]{-1}\right )}+\frac {(-1)^{7/8} \left ((1+i)+\sqrt {2}\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 )}{16 \sqrt {1+x^4}}-\frac {\left (1+(-1)^{5/8}\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 \sqrt {2} \left ((1+i)+\sqrt {2}\right ) \sqrt {1+x^4}}+\frac {\sqrt [4]{-1} \left (1-(-1)^{5/8}\right ) \left ((1+i)+\sqrt {2}\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 )}{32 \sqrt {1+x^4}}+\frac {\sqrt [4]{-1} \left (1+(-1)^{5/8}\right )^2 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4} \left (2+(-1)^{3/8}-(-1)^{5/8}\right );2 \tan ^{-1}(x)|\frac {1}{2}\right )}{32 \left (i+\sqrt [4]{-1}\right ) \sqrt {1+x^4}}\right )+2 \left (\frac {(-1)^{7/8} \left ((1+i)+\sqrt {2}\right ) x \sqrt {1+x^4}}{16 \left (1+x^2\right )}-\frac {\tan ^{-1}\left (\frac {\sqrt {(-1)^{3/8}-(-1)^{5/8}} x}{\sqrt {1+x^4}}\right )}{16 \sqrt {(-1)^{3/8}-(-1)^{5/8}}}-\frac {(-1)^{7/8} \left ((1+i)+\sqrt {2}\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 )}{16 \sqrt {1+x^4}}-\frac {\left (1-(-1)^{5/8}\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 \sqrt {2} \left ((1+i)+\sqrt {2}\right ) \sqrt {1+x^4}}+\frac {\sqrt [4]{-1} \left (1+(-1)^{5/8}\right ) \left ((1+i)+\sqrt {2}\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 )}{32 \sqrt {1+x^4}}+\frac {\sqrt [4]{-1} \left (1-(-1)^{5/8}\right )^2 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4} \left (2-(-1)^{3/8}+(-1)^{5/8}\right );2 \tan ^{-1}(x)|\frac {1}{2}\right )}{32 \left (i+\sqrt [4]{-1}\right ) \sqrt {1+x^4}}\right )-2 \left (-\frac {(-1)^{3/8} \left ((-1-i)+\sqrt {2}\right ) x \sqrt {1+x^4}}{16 \left (1+x^2\right )}-\frac {(-1)^{3/16} \left (1-\sqrt [4]{-1}\right ) \tan ^{-1}\left (\frac {\sqrt [16]{-1} \sqrt {-1+(-1)^{3/4}} x}{\sqrt {1+x^4}}\right )}{16 \left (i-\sqrt [4]{-1}\right ) \sqrt {-1+(-1)^{3/4}}}+\frac {(-1)^{3/8} \left ((-1-i)+\sqrt {2}\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 )}{16 \sqrt {1+x^4}}+\frac {\left (1-\sqrt [8]{-1}\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 \sqrt {2} \left ((-1-i)+\sqrt {2}\right ) \sqrt {1+x^4}}-\frac {\sqrt [4]{-1} \left (1+\sqrt [8]{-1}\right ) \left ((-1-i)+\sqrt {2}\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 )}{32 \sqrt {1+x^4}}+\frac {\left (i+\sqrt [4]{-1}-2 (-1)^{3/8}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4} \left (2+\sqrt [8]{-1}-(-1)^{7/8}\right );2 \tan ^{-1}(x)|\frac {1}{2}\right )}{32 \left (i-\sqrt [4]{-1}\right ) \sqrt {1+x^4}}\right )-2 \left (\frac {(-1)^{3/8} \left ((-1-i)+\sqrt {2}\right ) x \sqrt {1+x^4}}{16 \left (1+x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt [8]{-1}-(-1)^{7/8}} x}{\sqrt {1+x^4}}\right )}{16 \sqrt {\sqrt [8]{-1}-(-1)^{7/8}}}-\frac {(-1)^{3/8} \left ((-1-i)+\sqrt {2}\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 )}{16 \sqrt {1+x^4}}+\frac {\left (1+\sqrt [8]{-1}\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 \sqrt {2} \left ((-1-i)+\sqrt {2}\right ) \sqrt {1+x^4}}-\frac {\sqrt [4]{-1} \left (1-\sqrt [8]{-1}\right ) \left ((-1-i)+\sqrt {2}\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 )}{32 \sqrt {1+x^4}}-\frac {(-1)^{3/4} \left (1+\sqrt [8]{-1}\right )^2 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4} \left (2-\sqrt [8]{-1}+(-1)^{7/8}\right );2 \tan ^{-1}(x)|\frac {1}{2}\right )}{32 \left (1+(-1)^{3/4}\right ) \sqrt {1+x^4}}\right )\\ \end {align*}

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Mathematica [C]  time = 21.34, size = 186, normalized size = 4.13 \begin {gather*} \frac {1}{4} \sqrt [4]{-1} \left (-4 F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+\Pi \left (-\sqrt [8]{-1};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+\Pi \left (\sqrt [8]{-1};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+\Pi \left (-(-1)^{3/8};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+\Pi \left ((-1)^{3/8};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+\Pi \left (-(-1)^{5/8};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+\Pi \left ((-1)^{5/8};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+\Pi \left (-(-1)^{7/8};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+\Pi \left ((-1)^{7/8};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + x^16)/(Sqrt[1 + x^4]*(1 + x^16)),x]

[Out]

((-1)^(1/4)*(-4*EllipticF[I*ArcSinh[(-1)^(1/4)*x], -1] + EllipticPi[-(-1)^(1/8), I*ArcSinh[(-1)^(1/4)*x], -1]
+ EllipticPi[(-1)^(1/8), I*ArcSinh[(-1)^(1/4)*x], -1] + EllipticPi[-(-1)^(3/8), I*ArcSinh[(-1)^(1/4)*x], -1] +
EllipticPi[(-1)^(3/8), I*ArcSinh[(-1)^(1/4)*x], -1] + EllipticPi[-(-1)^(5/8), I*ArcSinh[(-1)^(1/4)*x], -1] +
EllipticPi[(-1)^(5/8), I*ArcSinh[(-1)^(1/4)*x], -1] + EllipticPi[-(-1)^(7/8), I*ArcSinh[(-1)^(1/4)*x], -1] + E
llipticPi[(-1)^(7/8), I*ArcSinh[(-1)^(1/4)*x], -1]))/4

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IntegrateAlgebraic [A]  time = 0.30, size = 45, normalized size = 1.00 \begin {gather*} \frac {1}{8} \text {RootSum}\left [2-4 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt {1+x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-1 + x^16)/(Sqrt[1 + x^4]*(1 + x^16)),x]

[Out]

RootSum[2 - 4*#1^4 + #1^8 & , (-Log[x] + Log[Sqrt[1 + x^4] - x*#1])/#1 & ]/8

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fricas [B]  time = 1.03, size = 1459, normalized size = 32.42

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^16-1)/(x^4+1)^(1/2)/(x^16+1),x, algorithm="fricas")

[Out]

1/8*sqrt(2)*sqrt(sqrt(2)*sqrt(sqrt(2) + 2))*arctan(1/2*((4*x^10 + 4*x^6 - 2*sqrt(2)*(x^14 + x^10 + x^6 + x^2)
- (x^16 + 4*x^12 + 4*x^8 + 4*x^4 - 2*sqrt(2)*(x^12 + 2*x^8 + x^4) + 1)*sqrt(sqrt(2) + 2))*sqrt((3*sqrt(2) - 4)
*sqrt(sqrt(2) + 2))*sqrt(sqrt(2)*sqrt(sqrt(2) + 2)) - 2*((2*x^13 + 4*x^9 + 4*x^5 - sqrt(2)*(x^13 + 3*x^9 + 3*x
^5 + x) + 2*x)*sqrt(x^4 + 1)*sqrt(sqrt(2) + 2) - 2*(x^11 + 2*x^7 + x^3 - sqrt(2)*(x^11 + x^7 + x^3))*sqrt(x^4
+ 1))*sqrt(sqrt(2)*sqrt(sqrt(2) + 2)))/(x^16 + 1)) - 1/8*sqrt(2)*sqrt(sqrt(2)*sqrt(-sqrt(2) + 2))*arctan(1/2*(
2*(2*x^11 + 4*x^7 + 2*x^3 + 2*sqrt(2)*(x^11 + x^7 + x^3) + (2*x^13 + 4*x^9 + 4*x^5 + sqrt(2)*(x^13 + 3*x^9 + 3
*x^5 + x) + 2*x)*sqrt(-sqrt(2) + 2))*sqrt(x^4 + 1)*sqrt(sqrt(2)*sqrt(-sqrt(2) + 2)) + (4*x^10 + 4*x^6 + 2*sqrt
(2)*(x^14 + x^10 + x^6 + x^2) + (x^16 + 4*x^12 + 4*x^8 + 4*x^4 + 2*sqrt(2)*(x^12 + 2*x^8 + x^4) + 1)*sqrt(-sqr
t(2) + 2))*sqrt((3*sqrt(2) + 4)*sqrt(-sqrt(2) + 2))*sqrt(sqrt(2)*sqrt(-sqrt(2) + 2)))/(x^16 + 1)) - 1/32*sqrt(
2)*sqrt(sqrt(2)*sqrt(sqrt(2) + 2))*log((2*(4*x^11 + 6*x^7 + 4*x^3 - sqrt(2)*(3*x^11 + 4*x^7 + 3*x^3))*sqrt(x^4
+ 1)*sqrt(sqrt(2) + 2) - 2*(2*x^13 + 4*x^9 + 4*x^5 - sqrt(2)*(x^13 + 3*x^9 + 3*x^5 + x) + 2*x)*sqrt(x^4 + 1)
+ (x^16 + 8*x^12 + 12*x^8 + 8*x^4 - sqrt(2)*(x^16 + 6*x^12 + 8*x^8 + 6*x^4 + 1) - 2*(3*x^14 + 7*x^10 + 7*x^6 +
3*x^2 - sqrt(2)*(2*x^14 + 5*x^10 + 5*x^6 + 2*x^2))*sqrt(sqrt(2) + 2) + 1)*sqrt(sqrt(2)*sqrt(sqrt(2) + 2)))/(x
^16 + 1)) + 1/32*sqrt(2)*sqrt(sqrt(2)*sqrt(sqrt(2) + 2))*log((2*(4*x^11 + 6*x^7 + 4*x^3 - sqrt(2)*(3*x^11 + 4*
x^7 + 3*x^3))*sqrt(x^4 + 1)*sqrt(sqrt(2) + 2) - 2*(2*x^13 + 4*x^9 + 4*x^5 - sqrt(2)*(x^13 + 3*x^9 + 3*x^5 + x)
+ 2*x)*sqrt(x^4 + 1) - (x^16 + 8*x^12 + 12*x^8 + 8*x^4 - sqrt(2)*(x^16 + 6*x^12 + 8*x^8 + 6*x^4 + 1) - 2*(3*x
^14 + 7*x^10 + 7*x^6 + 3*x^2 - sqrt(2)*(2*x^14 + 5*x^10 + 5*x^6 + 2*x^2))*sqrt(sqrt(2) + 2) + 1)*sqrt(sqrt(2)*
sqrt(sqrt(2) + 2)))/(x^16 + 1)) - 1/32*sqrt(2)*sqrt(sqrt(2)*sqrt(-sqrt(2) + 2))*log(-(2*(2*x^13 + 4*x^9 + 4*x^
5 + sqrt(2)*(x^13 + 3*x^9 + 3*x^5 + x) + (4*x^11 + 6*x^7 + 4*x^3 + sqrt(2)*(3*x^11 + 4*x^7 + 3*x^3))*sqrt(-sqr
t(2) + 2) + 2*x)*sqrt(x^4 + 1) + (x^16 + 8*x^12 + 12*x^8 + 8*x^4 + sqrt(2)*(x^16 + 6*x^12 + 8*x^8 + 6*x^4 + 1)
+ 2*(3*x^14 + 7*x^10 + 7*x^6 + 3*x^2 + sqrt(2)*(2*x^14 + 5*x^10 + 5*x^6 + 2*x^2))*sqrt(-sqrt(2) + 2) + 1)*sqr
t(sqrt(2)*sqrt(-sqrt(2) + 2)))/(x^16 + 1)) + 1/32*sqrt(2)*sqrt(sqrt(2)*sqrt(-sqrt(2) + 2))*log(-(2*(2*x^13 + 4
*x^9 + 4*x^5 + sqrt(2)*(x^13 + 3*x^9 + 3*x^5 + x) + (4*x^11 + 6*x^7 + 4*x^3 + sqrt(2)*(3*x^11 + 4*x^7 + 3*x^3)
)*sqrt(-sqrt(2) + 2) + 2*x)*sqrt(x^4 + 1) - (x^16 + 8*x^12 + 12*x^8 + 8*x^4 + sqrt(2)*(x^16 + 6*x^12 + 8*x^8 +
6*x^4 + 1) + 2*(3*x^14 + 7*x^10 + 7*x^6 + 3*x^2 + sqrt(2)*(2*x^14 + 5*x^10 + 5*x^6 + 2*x^2))*sqrt(-sqrt(2) +
2) + 1)*sqrt(sqrt(2)*sqrt(-sqrt(2) + 2)))/(x^16 + 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{16} - 1}{{\left (x^{16} + 1\right )} \sqrt {x^{4} + 1}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^16-1)/(x^4+1)^(1/2)/(x^16+1),x, algorithm="giac")

[Out]

integrate((x^16 - 1)/((x^16 + 1)*sqrt(x^4 + 1)), x)

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maple [B]  time = 11.92, size = 67, normalized size = 1.49

method	result	size
elliptic	\(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (8 \textit {\_Z}^{8}-8 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (2 \textit {\_R}^{6}-\textit {\_R}^{2}\right ) \ln \left (\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}-\textit {\_R} \right )}{2 \textit {\_R}^{7}-\textit {\_R}^{3}}\right ) \sqrt {2}}{16}\)	\(67\)
default	\(\frac {\sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{16}+1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-\frac {\arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (-\underline {\hspace {1.25 ex}}\alpha ^{14}+x^{2}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{4}+1}\, \sqrt {x^{4}+1}}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{4}+1}}-\frac {2 \left (-1\right )^{\frac {3}{4}} \underline {\hspace {1.25 ex}}\alpha ^{15} \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , i \underline {\hspace {1.25 ex}}\alpha ^{14}, i\right )}{\sqrt {x^{4}+1}}\right )\right )}{16}\)	\(161\)
trager	\(\text {Expression too large to display}\)	\(2481\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^16-1)/(x^4+1)^(1/2)/(x^16+1),x,method=_RETURNVERBOSE)

[Out]

1/16*sum((2*_R^6-_R^2)/(2*_R^7-_R^3)*ln(1/2*2^(1/2)/x*(x^4+1)^(1/2)-_R),_R=RootOf(8*_Z^8-8*_Z^4+1))*2^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{16} - 1}{{\left (x^{16} + 1\right )} \sqrt {x^{4} + 1}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^16-1)/(x^4+1)^(1/2)/(x^16+1),x, algorithm="maxima")

[Out]

integrate((x^16 - 1)/((x^16 + 1)*sqrt(x^4 + 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^{16}-1}{\sqrt {x^4+1}\,\left (x^{16}+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^16 - 1)/((x^4 + 1)^(1/2)*(x^16 + 1)),x)

[Out]

int((x^16 - 1)/((x^4 + 1)^(1/2)*(x^16 + 1)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**16-1)/(x**4+1)**(1/2)/(x**16+1),x)

[Out]

Timed out

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