\(\int \frac {-1+x^{16}}{\sqrt {-1+x^4} (1+x^8+x^{16})} \, dx\) [2670]

Optimal result

Integrand size = 25, antiderivative size = 239 \[ \int \frac {-1+x^{16}}{\sqrt {-1+x^4} \left (1+x^8+x^{16}\right )} \, dx=\frac {\arctan \left (\frac {-\frac {1}{\sqrt {2}}-\frac {x^2}{\sqrt {2}}+\frac {x^4}{\sqrt {2}}}{x \sqrt {-1+x^4}}\right )}{4 \sqrt {2}}+\frac {\arctan \left (\frac {-\frac {1}{\sqrt {2} \sqrt [4]{3}}-\frac {\sqrt [4]{3} x^2}{\sqrt {2}}+\frac {x^4}{\sqrt {2} \sqrt [4]{3}}}{x \sqrt {-1+x^4}}\right )}{4 \sqrt {2} \sqrt [4]{3}}-\frac {\text {arctanh}\left (\frac {-\frac {1}{\sqrt {2}}+\frac {x^2}{\sqrt {2}}+\frac {x^4}{\sqrt {2}}}{x \sqrt {-1+x^4}}\right )}{4 \sqrt {2}}-\frac {\text {arctanh}\left (\frac {-\frac {1}{\sqrt {2} \sqrt [4]{3}}+\frac {\sqrt [4]{3} x^2}{\sqrt {2}}+\frac {x^4}{\sqrt {2} \sqrt [4]{3}}}{x \sqrt {-1+x^4}}\right )}{4 \sqrt {2} \sqrt [4]{3}} \]

[Out]

1/8*arctan((-1/2*2^(1/2)-1/2*2^(1/2)*x^2+1/2*2^(1/2)*x^4)/x/(x^4-1)^(1/2))*2^(1/2)+1/24*arctan((-1/6*2^(1/2)*3
^(3/4)-1/2*3^(1/4)*x^2*2^(1/2)+1/6*x^4*2^(1/2)*3^(3/4))/x/(x^4-1)^(1/2))*2^(1/2)*3^(3/4)-1/8*arctanh((-1/2*2^(
1/2)+1/2*2^(1/2)*x^2+1/2*2^(1/2)*x^4)/x/(x^4-1)^(1/2))*2^(1/2)-1/24*arctanh((-1/6*2^(1/2)*3^(3/4)+1/2*3^(1/4)*
x^2*2^(1/2)+1/6*x^4*2^(1/2)*3^(3/4))/x/(x^4-1)^(1/2))*2^(1/2)*3^(3/4)

Rubi [C] (verified)

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

Time = 2.73 (sec) , antiderivative size = 853, normalized size of antiderivative = 3.57, number of steps used = 153, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.880, Rules used = {1600, 6860, 1743, 1223, 1202, 228, 1199, 1229, 1471, 554, 259, 552, 551, 1262, 749, 858, 223, 212, 739, 210, 415, 418} \[ \int \frac {-1+x^{16}}{\sqrt {-1+x^4} \left (1+x^8+x^{16}\right )} \, dx=-\frac {1}{32} \sqrt {\frac {1}{6} \left (3+i \sqrt {3}\right )} \left (i+\sqrt {3}\right ) \arctan \left (\frac {\left (1-i \sqrt {3}\right ) x^2+2}{\sqrt {2 \left (3+i \sqrt {3}\right )} \sqrt {x^4-1}}\right )+\frac {1}{32} \sqrt {\frac {1}{6} \left (3+i \sqrt {3}\right )} \left (i+\sqrt {3}\right ) \arctan \left (\frac {4-\left (1+i \sqrt {3}\right )^2 x^2}{2 \sqrt {2 \left (3+i \sqrt {3}\right )} \sqrt {x^4-1}}\right )+\frac {\left (i+\sqrt {3}\right ) \sqrt {x^2-1} \sqrt {x^2+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right ),\frac {1}{2}\right )}{8 \sqrt {6} \sqrt {x^4-1}}+\frac {3 \left (1+i \sqrt {3}\right ) \sqrt {x^2-1} \sqrt {x^2+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right ),\frac {1}{2}\right )}{8 \sqrt {2} \sqrt {x^4-1}}+\frac {3 \left (1-i \sqrt {3}\right ) \sqrt {x^2-1} \sqrt {x^2+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right ),\frac {1}{2}\right )}{8 \sqrt {2} \sqrt {x^4-1}}-\frac {\left (i-\sqrt {3}\right ) \sqrt {x^2-1} \sqrt {x^2+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right ),\frac {1}{2}\right )}{8 \sqrt {6} \sqrt {x^4-1}}-\frac {\sqrt {1-x^2} \sqrt {x^2+1} \operatorname {EllipticPi}\left (\frac {1}{2} \left (-i-\sqrt {3}\right ),\arcsin (x),-1\right )}{4 \sqrt {x^4-1}}-\frac {\sqrt {1-x^2} \sqrt {x^2+1} \operatorname {EllipticPi}\left (-\frac {4}{\left (i-\sqrt {3}\right )^2},\arcsin (x),-1\right )}{4 \sqrt {x^4-1}}-\frac {\sqrt {1-x^2} \sqrt {x^2+1} \operatorname {EllipticPi}\left (\frac {1}{2} \left (i-\sqrt {3}\right ),\arcsin (x),-1\right )}{4 \sqrt {x^4-1}}-\frac {\sqrt {1-x^2} \sqrt {x^2+1} \operatorname {EllipticPi}\left (\frac {1}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}},\arcsin (x),-1\right )}{4 \sqrt {x^4-1}}-\frac {\sqrt {1-x^2} \sqrt {x^2+1} \operatorname {EllipticPi}\left (\frac {2}{1-i \sqrt {3}},\arcsin (x),-1\right )}{4 \sqrt {x^4-1}}-\frac {\sqrt {1-x^2} \sqrt {x^2+1} \operatorname {EllipticPi}\left (\frac {1}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}},\arcsin (x),-1\right )}{4 \sqrt {x^4-1}}-\frac {\sqrt {1-x^2} \sqrt {x^2+1} \operatorname {EllipticPi}\left (\frac {2}{1+i \sqrt {3}},\arcsin (x),-1\right )}{4 \sqrt {x^4-1}}-\frac {\sqrt {1-x^2} \sqrt {x^2+1} \operatorname {EllipticPi}\left (-\frac {4}{\left (i+\sqrt {3}\right )^2},\arcsin (x),-1\right )}{4 \sqrt {x^4-1}} \]

[In]

Int[(-1 + x^16)/(Sqrt[-1 + x^4]*(1 + x^8 + x^16)),x]

[Out]

-1/32*(Sqrt[(3 + I*Sqrt[3])/6]*(I + Sqrt[3])*ArcTan[(2 + (1 - I*Sqrt[3])*x^2)/(Sqrt[2*(3 + I*Sqrt[3])]*Sqrt[-1
+ x^4])]) + (Sqrt[(3 + I*Sqrt[3])/6]*(I + Sqrt[3])*ArcTan[(4 - (1 + I*Sqrt[3])^2*x^2)/(2*Sqrt[2*(3 + I*Sqrt[3
])]*Sqrt[-1 + x^4])])/32 - ((I - Sqrt[3])*Sqrt[-1 + x^2]*Sqrt[1 + x^2]*EllipticF[ArcSin[(Sqrt[2]*x)/Sqrt[-1 +
x^2]], 1/2])/(8*Sqrt[6]*Sqrt[-1 + x^4]) + (3*(1 - I*Sqrt[3])*Sqrt[-1 + x^2]*Sqrt[1 + x^2]*EllipticF[ArcSin[(Sq
rt[2]*x)/Sqrt[-1 + x^2]], 1/2])/(8*Sqrt[2]*Sqrt[-1 + x^4]) + (3*(1 + I*Sqrt[3])*Sqrt[-1 + x^2]*Sqrt[1 + x^2]*E
llipticF[ArcSin[(Sqrt[2]*x)/Sqrt[-1 + x^2]], 1/2])/(8*Sqrt[2]*Sqrt[-1 + x^4]) + ((I + Sqrt[3])*Sqrt[-1 + x^2]*
Sqrt[1 + x^2]*EllipticF[ArcSin[(Sqrt[2]*x)/Sqrt[-1 + x^2]], 1/2])/(8*Sqrt[6]*Sqrt[-1 + x^4]) - (Sqrt[1 - x^2]*
Sqrt[1 + x^2]*EllipticPi[(-I - Sqrt[3])/2, ArcSin[x], -1])/(4*Sqrt[-1 + x^4]) - (Sqrt[1 - x^2]*Sqrt[1 + x^2]*E
llipticPi[-4/(I - Sqrt[3])^2, ArcSin[x], -1])/(4*Sqrt[-1 + x^4]) - (Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[(I
- Sqrt[3])/2, ArcSin[x], -1])/(4*Sqrt[-1 + x^4]) - (Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[1/Sqrt[(1 - I*Sqrt[
3])/2], ArcSin[x], -1])/(4*Sqrt[-1 + x^4]) - (Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[2/(1 - I*Sqrt[3]), ArcSin
[x], -1])/(4*Sqrt[-1 + x^4]) - (Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[1/Sqrt[(1 + I*Sqrt[3])/2], ArcSin[x], -
1])/(4*Sqrt[-1 + x^4]) - (Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[2/(1 + I*Sqrt[3]), ArcSin[x], -1])/(4*Sqrt[-1
+ x^4]) - (Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[-4/(I + Sqrt[3])^2, ArcSin[x], -1])/(4*Sqrt[-1 + x^4])

Rule 210

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

Rule 212

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

Rule 223

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

Rule 228

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

Rule 259

Int[((a1_.) + (b1_.)*(x_)^(n_))^(p_)*((a2_.) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a1 + b1*x^n)^FracPar
t[p]*((a2 + b2*x^n)^FracPart[p]/(a1*a2 + b1*b2*x^(2*n))^FracPart[p]), Int[(a1*a2 + b1*b2*x^(2*n))^p, x], x] /;
FreeQ[{a1, b1, a2, b2, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] &&  !IntegerQ[p]

Rule 415

Int[Sqrt[(a_) + (b_.)*(x_)^4]/((c_) + (d_.)*(x_)^4), x_Symbol] :> Dist[b/d, Int[1/Sqrt[a + b*x^4], x], x] - Di
st[(b*c - a*d)/d, Int[1/(Sqrt[a + b*x^4]*(c + d*x^4)), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 418

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

Rule 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b*(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-f/e, -d/c])

Rule 552

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 +
(d/c)*x^2]/Sqrt[c + d*x^2], Int[1/((a + b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[c, 0]

Rule 554

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

Rule 739

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 749

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 858

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 1199

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

Rule 1202

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

Rule 1223

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 1229

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

Rule 1262

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 1471

Int[((d_) + (e_.)*(x_)^(n_))^(q_.)*((f_) + (g_.)*(x_)^(n_))^(r_.)*((a_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :>
Dist[(a + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + (c*x^n)/e)^FracPart[p]), Int[(d + e*x^n)^(p
+ q)*(f + g*x^n)^r*(a/d + (c/e)*x^n)^p, x], x] /; FreeQ[{a, c, d, e, f, g, n, p, q, r}, x] && EqQ[n2, 2*n] &&
EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p]

Rule 1600

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 1743

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 6860

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]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {-1+x^4} \left (1+x^4+x^8+x^{12}\right )}{1+x^8+x^{16}} \, dx \\ & = \int \left (\frac {(1-x) \sqrt {-1+x^4}}{8 \left (1-x+x^2\right )}+\frac {(1+x) \sqrt {-1+x^4}}{8 \left (1+x+x^2\right )}+\frac {\sqrt {-1+x^4}}{4 \left (1-x^2+x^4\right )}+\frac {\sqrt {-1+x^4} \left (1+x^4\right )}{2 \left (1-x^4+x^8\right )}\right ) \, dx \\ & = \frac {1}{8} \int \frac {(1-x) \sqrt {-1+x^4}}{1-x+x^2} \, dx+\frac {1}{8} \int \frac {(1+x) \sqrt {-1+x^4}}{1+x+x^2} \, dx+\frac {1}{4} \int \frac {\sqrt {-1+x^4}}{1-x^2+x^4} \, dx+\frac {1}{2} \int \frac {\sqrt {-1+x^4} \left (1+x^4\right )}{1-x^4+x^8} \, dx \\ & = \frac {1}{8} \int \left (\frac {\left (-1-\frac {i}{\sqrt {3}}\right ) \sqrt {-1+x^4}}{-1-i \sqrt {3}+2 x}+\frac {\left (-1+\frac {i}{\sqrt {3}}\right ) \sqrt {-1+x^4}}{-1+i \sqrt {3}+2 x}\right ) \, dx+\frac {1}{8} \int \left (\frac {\left (1-\frac {i}{\sqrt {3}}\right ) \sqrt {-1+x^4}}{1-i \sqrt {3}+2 x}+\frac {\left (1+\frac {i}{\sqrt {3}}\right ) \sqrt {-1+x^4}}{1+i \sqrt {3}+2 x}\right ) \, dx+\frac {1}{4} \int \left (\frac {2 i \sqrt {-1+x^4}}{\sqrt {3} \left (1+i \sqrt {3}-2 x^2\right )}+\frac {2 i \sqrt {-1+x^4}}{\sqrt {3} \left (-1+i \sqrt {3}+2 x^2\right )}\right ) \, dx+\frac {1}{2} \int \left (\frac {\left (1-i \sqrt {3}\right ) \sqrt {-1+x^4}}{-1-i \sqrt {3}+2 x^4}+\frac {\left (1+i \sqrt {3}\right ) \sqrt {-1+x^4}}{-1+i \sqrt {3}+2 x^4}\right ) \, dx \\ & = \frac {i \int \frac {\sqrt {-1+x^4}}{1+i \sqrt {3}-2 x^2} \, dx}{2 \sqrt {3}}+\frac {i \int \frac {\sqrt {-1+x^4}}{-1+i \sqrt {3}+2 x^2} \, dx}{2 \sqrt {3}}+\frac {1}{2} \left (1-i \sqrt {3}\right ) \int \frac {\sqrt {-1+x^4}}{-1-i \sqrt {3}+2 x^4} \, dx+\frac {1}{24} \left (3-i \sqrt {3}\right ) \int \frac {\sqrt {-1+x^4}}{1-i \sqrt {3}+2 x} \, dx+\frac {1}{24} \left (-3+i \sqrt {3}\right ) \int \frac {\sqrt {-1+x^4}}{-1+i \sqrt {3}+2 x} \, dx+\frac {1}{2} \left (1+i \sqrt {3}\right ) \int \frac {\sqrt {-1+x^4}}{-1+i \sqrt {3}+2 x^4} \, dx-\frac {1}{24} \left (3+i \sqrt {3}\right ) \int \frac {\sqrt {-1+x^4}}{-1-i \sqrt {3}+2 x} \, dx+\frac {1}{24} \left (3+i \sqrt {3}\right ) \int \frac {\sqrt {-1+x^4}}{1+i \sqrt {3}+2 x} \, dx \\ & = -\frac {i \int \frac {-1+i \sqrt {3}-2 x^2}{\sqrt {-1+x^4}} \, dx}{8 \sqrt {3}}-\frac {i \int \frac {1+i \sqrt {3}+2 x^2}{\sqrt {-1+x^4}} \, dx}{8 \sqrt {3}}+\frac {i \int \frac {\sqrt {-1+x^4}}{\left (-1-i \sqrt {3}\right )^2-4 x^2} \, dx}{2 \sqrt {3}}-\frac {i \int \frac {\sqrt {-1+x^4}}{\left (1-i \sqrt {3}\right )^2-4 x^2} \, dx}{2 \sqrt {3}}-\frac {i \int \frac {\sqrt {-1+x^4}}{\left (-1+i \sqrt {3}\right )^2-4 x^2} \, dx}{2 \sqrt {3}}+\frac {i \int \frac {\sqrt {-1+x^4}}{\left (1+i \sqrt {3}\right )^2-4 x^2} \, dx}{2 \sqrt {3}}+\frac {1}{4} \left (-1-i \sqrt {3}\right ) \int \frac {1}{\left (1+i \sqrt {3}-2 x^2\right ) \sqrt {-1+x^4}} \, dx+\frac {1}{4} \left (1-i \sqrt {3}\right ) \int \frac {1}{\sqrt {-1+x^4}} \, dx+\frac {1}{2} \left (1-i \sqrt {3}\right ) \int \frac {1}{\sqrt {-1+x^4} \left (-1+i \sqrt {3}+2 x^4\right )} \, dx+\frac {1}{12} \left (3-i \sqrt {3}\right ) \int \frac {x \sqrt {-1+x^4}}{\left (-1+i \sqrt {3}\right )^2-4 x^2} \, dx+\frac {1}{12} \left (-3+i \sqrt {3}\right ) \int \frac {x \sqrt {-1+x^4}}{\left (1-i \sqrt {3}\right )^2-4 x^2} \, dx-\frac {1}{4} \left (-1+i \sqrt {3}\right ) \int \frac {1}{\left (-1+i \sqrt {3}+2 x^2\right ) \sqrt {-1+x^4}} \, dx+\frac {1}{4} \left (1+i \sqrt {3}\right ) \int \frac {1}{\sqrt {-1+x^4}} \, dx+\frac {1}{2} \left (1+i \sqrt {3}\right ) \int \frac {1}{\sqrt {-1+x^4} \left (-1-i \sqrt {3}+2 x^4\right )} \, dx+\frac {1}{12} \left (3+i \sqrt {3}\right ) \int \frac {x \sqrt {-1+x^4}}{\left (-1-i \sqrt {3}\right )^2-4 x^2} \, dx-\frac {1}{12} \left (3+i \sqrt {3}\right ) \int \frac {x \sqrt {-1+x^4}}{\left (1+i \sqrt {3}\right )^2-4 x^2} \, dx \\ & = \frac {\left (1-i \sqrt {3}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} \sqrt {-1+x^4}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} \sqrt {-1+x^4}}-\frac {1}{4} \int \frac {1}{\left (1-\frac {x^2}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right ) \sqrt {-1+x^4}} \, dx-\frac {1}{4} \int \frac {1}{\left (1+\frac {x^2}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right ) \sqrt {-1+x^4}} \, dx-\frac {1}{4} \int \frac {1}{\left (1-\frac {x^2}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right ) \sqrt {-1+x^4}} \, dx-\frac {1}{4} \int \frac {1}{\left (1+\frac {x^2}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right ) \sqrt {-1+x^4}} \, dx-\frac {i \int \frac {\left (-1-i \sqrt {3}\right )^2+4 x^2}{\sqrt {-1+x^4}} \, dx}{32 \sqrt {3}}+\frac {i \int \frac {\left (1-i \sqrt {3}\right )^2+4 x^2}{\sqrt {-1+x^4}} \, dx}{32 \sqrt {3}}+\frac {i \int \frac {\left (-1+i \sqrt {3}\right )^2+4 x^2}{\sqrt {-1+x^4}} \, dx}{32 \sqrt {3}}-\frac {i \int \frac {\left (1+i \sqrt {3}\right )^2+4 x^2}{\sqrt {-1+x^4}} \, dx}{32 \sqrt {3}}-\frac {1}{8} \left (-1-i \sqrt {3}\right ) \int \frac {1}{\sqrt {-1+x^4}} \, dx-\frac {1}{4} \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}{4} \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}{24} \left (3-i \sqrt {3}\right ) \text {Subst}\left (\int \frac {\sqrt {-1+x^2}}{\left (-1+i \sqrt {3}\right )^2-4 x} \, dx,x,x^2\right )+\frac {1}{24} \left (-3+i \sqrt {3}\right ) \text {Subst}\left (\int \frac {\sqrt {-1+x^2}}{\left (1-i \sqrt {3}\right )^2-4 x} \, dx,x,x^2\right )-\frac {1}{8} \left (-1+i \sqrt {3}\right ) \int \frac {1}{\sqrt {-1+x^4}} \, dx-\frac {1}{4} \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}{4} \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}{24} \left (3+i \sqrt {3}\right ) \text {Subst}\left (\int \frac {\sqrt {-1+x^2}}{\left (-1-i \sqrt {3}\right )^2-4 x} \, dx,x,x^2\right )-\frac {1}{24} \left (3+i \sqrt {3}\right ) \text {Subst}\left (\int \frac {\sqrt {-1+x^2}}{\left (1+i \sqrt {3}\right )^2-4 x} \, 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 (1+i \sqrt {3}-2 x^2\right ) \sqrt {-1+x^4}} \, dx}{2 \left (i+\sqrt {3}\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 (-1+i \sqrt {3}+2 x^2\right ) \sqrt {-1+x^4}} \, dx}{2 \left (i-\sqrt {3}\right )} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 1.76 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.79 \[ \int \frac {-1+x^{16}}{\sqrt {-1+x^4} \left (1+x^8+x^{16}\right )} \, dx=\frac {3 \arctan \left (\frac {-1-x^2+x^4}{\sqrt {2} x \sqrt {-1+x^4}}\right )+3^{3/4} \arctan \left (\frac {-1-\left (-2 i+\sqrt {-1+4 i \sqrt {3}}\right ) x^2+x^4}{\sqrt {2} \sqrt [4]{3} x \sqrt {-1+x^4}}\right )-3 \text {arctanh}\left (\frac {-1+x^2+x^4}{\sqrt {2} x \sqrt {-1+x^4}}\right )-3^{3/4} \text {arctanh}\left (\frac {-1+\left (2 i+\sqrt {-1-4 i \sqrt {3}}\right ) x^2+x^4}{\sqrt {2} \sqrt [4]{3} x \sqrt {-1+x^4}}\right )}{12 \sqrt {2}} \]

[In]

Integrate[(-1 + x^16)/(Sqrt[-1 + x^4]*(1 + x^8 + x^16)),x]

[Out]

(3*ArcTan[(-1 - x^2 + x^4)/(Sqrt[2]*x*Sqrt[-1 + x^4])] + 3^(3/4)*ArcTan[(-1 - (-2*I + Sqrt[-1 + (4*I)*Sqrt[3]]
)*x^2 + x^4)/(Sqrt[2]*3^(1/4)*x*Sqrt[-1 + x^4])] - 3*ArcTanh[(-1 + x^2 + x^4)/(Sqrt[2]*x*Sqrt[-1 + x^4])] - 3^
(3/4)*ArcTanh[(-1 + (2*I + Sqrt[-1 - (4*I)*Sqrt[3]])*x^2 + x^4)/(Sqrt[2]*3^(1/4)*x*Sqrt[-1 + x^4])])/(12*Sqrt[
2])

Maple [A] (verified)

Time = 21.34 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.95

method	result	size
elliptic	\(\frac {\left (-\frac {\ln \left (\frac {x^{4}-1}{x^{2}}+\frac {\sqrt {x^{4}-1}\, \sqrt {2}}{x}+1\right )}{8}+\frac {\arctan \left (1+\frac {\sqrt {x^{4}-1}\, \sqrt {2}}{x}\right )}{4}+\frac {3^{\frac {3}{4}} \left (\ln \left (\frac {\frac {x^{4}-1}{2 x^{2}}-\frac {3^{\frac {1}{4}} \sqrt {x^{4}-1}\, \sqrt {2}}{2 x}+\frac {\sqrt {3}}{2}}{\frac {x^{4}-1}{2 x^{2}}+\frac {3^{\frac {1}{4}} \sqrt {x^{4}-1}\, \sqrt {2}}{2 x}+\frac {\sqrt {3}}{2}}\right )+2 \arctan \left (\frac {3^{\frac {3}{4}} \sqrt {x^{4}-1}\, \sqrt {2}}{3 x}+1\right )+2 \arctan \left (\frac {3^{\frac {3}{4}} \sqrt {x^{4}-1}\, \sqrt {2}}{3 x}-1\right )\right )}{24}+\frac {\ln \left (\frac {x^{4}-1}{x^{2}}-\frac {\sqrt {x^{4}-1}\, \sqrt {2}}{x}+1\right )}{8}+\frac {\arctan \left (-1+\frac {\sqrt {x^{4}-1}\, \sqrt {2}}{x}\right )}{4}\right ) \sqrt {2}}{2}\)	\(226\)
default	\(\frac {-\frac {\sqrt {2}\, \left (\left (\left (-1+i\right ) x^{2}-1-i\right ) \sqrt {3}+\left (-1-i\right ) x^{2}-4 i x +1-i\right ) \sqrt {9 i-3 \sqrt {3}}\, \sqrt {\frac {i \left (-x^{4}+1\right )}{\left (2 i+\left (-1+i\right ) x \sqrt {3}+2 x^{2}+\left (1+i\right ) x \right )^{2}}}\, \arctan \left (\frac {3^{\frac {3}{4}} \left (\left (-1+i\right ) x \sqrt {3}-2 x^{2}+\left (-1-i\right ) x -2 i\right )}{\sqrt {\frac {i \left (-x^{4}+1\right )}{\left (2 i+\left (-1+i\right ) x \sqrt {3}+2 x^{2}+\left (1+i\right ) x \right )^{2}}}\, \left (\left (-6+6 i\right ) x \sqrt {3}+12 x^{2}+\left (6+6 i\right ) x +12 i\right )}\right )}{6}+\frac {\left (\left (\left (-1+i\right ) x^{2}-1-i\right ) \sqrt {3}+\left (-1-i\right ) x^{2}+4 i x +1-i\right ) \sqrt {2}\, \sqrt {\frac {i \left (-x^{4}+1\right )}{\left (-2 i+\left (-1+i\right ) x \sqrt {3}-2 x^{2}+\left (1+i\right ) x \right )^{2}}}\, \sqrt {9 i-3 \sqrt {3}}\, \arctan \left (\frac {3^{\frac {3}{4}} \left (\left (-1+i\right ) x \sqrt {3}+2 x^{2}+\left (-1-i\right ) x +2 i\right )}{\sqrt {\frac {i \left (-x^{4}+1\right )}{\left (-2 i+\left (-1+i\right ) x \sqrt {3}-2 x^{2}+\left (1+i\right ) x \right )^{2}}}\, \left (\left (-6+6 i\right ) x \sqrt {3}-12 x^{2}+\left (6+6 i\right ) x -12 i\right )}\right )}{6}-\frac {2 \left (\left (\left (1+i\right ) x^{2}-i x -1+i\right ) 3^{\frac {3}{4}}+3 \,3^{\frac {1}{4}} x \right ) \sqrt {\frac {i \left (-x^{4}+1\right )}{\left (-2 i+\left (-1+i\right ) x \sqrt {3}-2 x^{2}+\left (1+i\right ) x \right )^{2}}}\, \operatorname {arctanh}\left (2 \,3^{\frac {1}{4}} \sqrt {\frac {i \left (-x^{4}+1\right )}{\left (-2 i+\left (-1+i\right ) x \sqrt {3}-2 x^{2}+\left (1+i\right ) x \right )^{2}}}\right )}{3}+\frac {2 \left (\left (\left (1+i\right ) x^{2}+i x -1+i\right ) 3^{\frac {3}{4}}-3 \,3^{\frac {1}{4}} x \right ) \sqrt {\frac {i \left (-x^{4}+1\right )}{\left (2 i+\left (-1+i\right ) x \sqrt {3}+2 x^{2}+\left (1+i\right ) x \right )^{2}}}\, \operatorname {arctanh}\left (2 \,3^{\frac {1}{4}} \sqrt {\frac {i \left (-x^{4}+1\right )}{\left (2 i+\left (-1+i\right ) x \sqrt {3}+2 x^{2}+\left (1+i\right ) x \right )^{2}}}\right )}{3}+x \sqrt {2}\, \left (\left (1-i\right ) \arctan \left (\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {x^{4}-1}\, \sqrt {2}}{x}\right )+\left (1+i\right ) \arctan \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {x^{4}-1}\, \sqrt {2}}{x}\right )\right ) \sqrt {\frac {x^{4}-1}{x^{2}}}}{8 \sqrt {\frac {x^{4}-1}{x^{2}}}\, x}\)	\(615\)
pseudoelliptic	\(\frac {-\frac {\sqrt {2}\, \left (\left (\left (-1+i\right ) x^{2}-1-i\right ) \sqrt {3}+\left (-1-i\right ) x^{2}-4 i x +1-i\right ) \sqrt {9 i-3 \sqrt {3}}\, \sqrt {\frac {i \left (-x^{4}+1\right )}{\left (2 i+\left (-1+i\right ) x \sqrt {3}+2 x^{2}+\left (1+i\right ) x \right )^{2}}}\, \arctan \left (\frac {3^{\frac {3}{4}} \left (\left (-1+i\right ) x \sqrt {3}-2 x^{2}+\left (-1-i\right ) x -2 i\right )}{\sqrt {\frac {i \left (-x^{4}+1\right )}{\left (2 i+\left (-1+i\right ) x \sqrt {3}+2 x^{2}+\left (1+i\right ) x \right )^{2}}}\, \left (\left (-6+6 i\right ) x \sqrt {3}+12 x^{2}+\left (6+6 i\right ) x +12 i\right )}\right )}{6}+\frac {\left (\left (\left (-1+i\right ) x^{2}-1-i\right ) \sqrt {3}+\left (-1-i\right ) x^{2}+4 i x +1-i\right ) \sqrt {2}\, \sqrt {\frac {i \left (-x^{4}+1\right )}{\left (-2 i+\left (-1+i\right ) x \sqrt {3}-2 x^{2}+\left (1+i\right ) x \right )^{2}}}\, \sqrt {9 i-3 \sqrt {3}}\, \arctan \left (\frac {3^{\frac {3}{4}} \left (\left (-1+i\right ) x \sqrt {3}+2 x^{2}+\left (-1-i\right ) x +2 i\right )}{\sqrt {\frac {i \left (-x^{4}+1\right )}{\left (-2 i+\left (-1+i\right ) x \sqrt {3}-2 x^{2}+\left (1+i\right ) x \right )^{2}}}\, \left (\left (-6+6 i\right ) x \sqrt {3}-12 x^{2}+\left (6+6 i\right ) x -12 i\right )}\right )}{6}-\frac {2 \left (\left (\left (1+i\right ) x^{2}-i x -1+i\right ) 3^{\frac {3}{4}}+3 \,3^{\frac {1}{4}} x \right ) \sqrt {\frac {i \left (-x^{4}+1\right )}{\left (-2 i+\left (-1+i\right ) x \sqrt {3}-2 x^{2}+\left (1+i\right ) x \right )^{2}}}\, \operatorname {arctanh}\left (2 \,3^{\frac {1}{4}} \sqrt {\frac {i \left (-x^{4}+1\right )}{\left (-2 i+\left (-1+i\right ) x \sqrt {3}-2 x^{2}+\left (1+i\right ) x \right )^{2}}}\right )}{3}+\frac {2 \left (\left (\left (1+i\right ) x^{2}+i x -1+i\right ) 3^{\frac {3}{4}}-3 \,3^{\frac {1}{4}} x \right ) \sqrt {\frac {i \left (-x^{4}+1\right )}{\left (2 i+\left (-1+i\right ) x \sqrt {3}+2 x^{2}+\left (1+i\right ) x \right )^{2}}}\, \operatorname {arctanh}\left (2 \,3^{\frac {1}{4}} \sqrt {\frac {i \left (-x^{4}+1\right )}{\left (2 i+\left (-1+i\right ) x \sqrt {3}+2 x^{2}+\left (1+i\right ) x \right )^{2}}}\right )}{3}+x \sqrt {2}\, \left (\left (1-i\right ) \arctan \left (\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {x^{4}-1}\, \sqrt {2}}{x}\right )+\left (1+i\right ) \arctan \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {x^{4}-1}\, \sqrt {2}}{x}\right )\right ) \sqrt {\frac {x^{4}-1}{x^{2}}}}{8 \sqrt {\frac {x^{4}-1}{x^{2}}}\, x}\)	\(615\)
trager	\(\text {Expression too large to display}\)	\(850\)

[In]

int((x^16-1)/(x^4-1)^(1/2)/(x^16+x^8+1),x,method=_RETURNVERBOSE)

[Out]

1/2*(-1/8*ln((x^4-1)/x^2+(x^4-1)^(1/2)/x*2^(1/2)+1)+1/4*arctan(1+(x^4-1)^(1/2)/x*2^(1/2))+1/24*3^(3/4)*(ln((1/
2*(x^4-1)/x^2-1/2*3^(1/4)*(x^4-1)^(1/2)/x*2^(1/2)+1/2*3^(1/2))/(1/2*(x^4-1)/x^2+1/2*3^(1/4)*(x^4-1)^(1/2)/x*2^
(1/2)+1/2*3^(1/2)))+2*arctan(1/3*3^(3/4)*(x^4-1)^(1/2)/x*2^(1/2)+1)+2*arctan(1/3*3^(3/4)*(x^4-1)^(1/2)/x*2^(1/
2)-1))+1/8*ln((x^4-1)/x^2-(x^4-1)^(1/2)/x*2^(1/2)+1)+1/4*arctan(-1+(x^4-1)^(1/2)/x*2^(1/2)))*2^(1/2)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.37 (sec) , antiderivative size = 601, normalized size of antiderivative = 2.51 \[ \int \frac {-1+x^{16}}{\sqrt {-1+x^4} \left (1+x^8+x^{16}\right )} \, dx=\left (\frac {1}{96} i + \frac {1}{96}\right ) \cdot 3^{\frac {3}{4}} \sqrt {2} \log \left (\frac {3^{\frac {3}{4}} \sqrt {2} {\left (\left (i + 1\right ) \, x^{8} - \left (5 i + 5\right ) \, x^{4} + i + 1\right )} - 6 \cdot 3^{\frac {1}{4}} \sqrt {2} {\left (\left (i - 1\right ) \, x^{6} - \left (i - 1\right ) \, x^{2}\right )} - 12 \, {\left (x^{5} - i \, \sqrt {3} x^{3} - x\right )} \sqrt {x^{4} - 1}}{x^{8} + x^{4} + 1}\right ) - \left (\frac {1}{96} i - \frac {1}{96}\right ) \cdot 3^{\frac {3}{4}} \sqrt {2} \log \left (\frac {3^{\frac {3}{4}} \sqrt {2} {\left (-\left (i - 1\right ) \, x^{8} + \left (5 i - 5\right ) \, x^{4} - i + 1\right )} - 6 \cdot 3^{\frac {1}{4}} \sqrt {2} {\left (-\left (i + 1\right ) \, x^{6} + \left (i + 1\right ) \, x^{2}\right )} - 12 \, {\left (x^{5} + i \, \sqrt {3} x^{3} - x\right )} \sqrt {x^{4} - 1}}{x^{8} + x^{4} + 1}\right ) + \left (\frac {1}{96} i - \frac {1}{96}\right ) \cdot 3^{\frac {3}{4}} \sqrt {2} \log \left (\frac {3^{\frac {3}{4}} \sqrt {2} {\left (\left (i - 1\right ) \, x^{8} - \left (5 i - 5\right ) \, x^{4} + i - 1\right )} - 6 \cdot 3^{\frac {1}{4}} \sqrt {2} {\left (\left (i + 1\right ) \, x^{6} - \left (i + 1\right ) \, x^{2}\right )} - 12 \, {\left (x^{5} + i \, \sqrt {3} x^{3} - x\right )} \sqrt {x^{4} - 1}}{x^{8} + x^{4} + 1}\right ) - \left (\frac {1}{96} i + \frac {1}{96}\right ) \cdot 3^{\frac {3}{4}} \sqrt {2} \log \left (\frac {3^{\frac {3}{4}} \sqrt {2} {\left (-\left (i + 1\right ) \, x^{8} + \left (5 i + 5\right ) \, x^{4} - i - 1\right )} - 6 \cdot 3^{\frac {1}{4}} \sqrt {2} {\left (-\left (i - 1\right ) \, x^{6} + \left (i - 1\right ) \, x^{2}\right )} - 12 \, {\left (x^{5} - i \, \sqrt {3} x^{3} - x\right )} \sqrt {x^{4} - 1}}{x^{8} + x^{4} + 1}\right ) + \left (\frac {1}{32} i + \frac {1}{32}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (\left (i + 1\right ) \, x^{8} - \left (2 i - 2\right ) \, x^{6} - \left (3 i + 3\right ) \, x^{4} + \left (2 i - 2\right ) \, x^{2} + i + 1\right )} - 4 \, {\left (x^{5} - i \, x^{3} - x\right )} \sqrt {x^{4} - 1}}{x^{8} - x^{4} + 1}\right ) - \left (\frac {1}{32} i - \frac {1}{32}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (-\left (i - 1\right ) \, x^{8} + \left (2 i + 2\right ) \, x^{6} + \left (3 i - 3\right ) \, x^{4} - \left (2 i + 2\right ) \, x^{2} - i + 1\right )} - 4 \, {\left (x^{5} + i \, x^{3} - x\right )} \sqrt {x^{4} - 1}}{x^{8} - x^{4} + 1}\right ) + \left (\frac {1}{32} i - \frac {1}{32}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (\left (i - 1\right ) \, x^{8} - \left (2 i + 2\right ) \, x^{6} - \left (3 i - 3\right ) \, x^{4} + \left (2 i + 2\right ) \, x^{2} + i - 1\right )} - 4 \, {\left (x^{5} + i \, x^{3} - x\right )} \sqrt {x^{4} - 1}}{x^{8} - x^{4} + 1}\right ) - \left (\frac {1}{32} i + \frac {1}{32}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (-\left (i + 1\right ) \, x^{8} + \left (2 i - 2\right ) \, x^{6} + \left (3 i + 3\right ) \, x^{4} - \left (2 i - 2\right ) \, x^{2} - i - 1\right )} - 4 \, {\left (x^{5} - i \, x^{3} - x\right )} \sqrt {x^{4} - 1}}{x^{8} - x^{4} + 1}\right ) \]

[In]

integrate((x^16-1)/(x^4-1)^(1/2)/(x^16+x^8+1),x, algorithm="fricas")

[Out]

(1/96*I + 1/96)*3^(3/4)*sqrt(2)*log((3^(3/4)*sqrt(2)*((I + 1)*x^8 - (5*I + 5)*x^4 + I + 1) - 6*3^(1/4)*sqrt(2)
*((I - 1)*x^6 - (I - 1)*x^2) - 12*(x^5 - I*sqrt(3)*x^3 - x)*sqrt(x^4 - 1))/(x^8 + x^4 + 1)) - (1/96*I - 1/96)*
3^(3/4)*sqrt(2)*log((3^(3/4)*sqrt(2)*(-(I - 1)*x^8 + (5*I - 5)*x^4 - I + 1) - 6*3^(1/4)*sqrt(2)*(-(I + 1)*x^6
+ (I + 1)*x^2) - 12*(x^5 + I*sqrt(3)*x^3 - x)*sqrt(x^4 - 1))/(x^8 + x^4 + 1)) + (1/96*I - 1/96)*3^(3/4)*sqrt(2
)*log((3^(3/4)*sqrt(2)*((I - 1)*x^8 - (5*I - 5)*x^4 + I - 1) - 6*3^(1/4)*sqrt(2)*((I + 1)*x^6 - (I + 1)*x^2) -
12*(x^5 + I*sqrt(3)*x^3 - x)*sqrt(x^4 - 1))/(x^8 + x^4 + 1)) - (1/96*I + 1/96)*3^(3/4)*sqrt(2)*log((3^(3/4)*s
qrt(2)*(-(I + 1)*x^8 + (5*I + 5)*x^4 - I - 1) - 6*3^(1/4)*sqrt(2)*(-(I - 1)*x^6 + (I - 1)*x^2) - 12*(x^5 - I*s
qrt(3)*x^3 - x)*sqrt(x^4 - 1))/(x^8 + x^4 + 1)) + (1/32*I + 1/32)*sqrt(2)*log((sqrt(2)*((I + 1)*x^8 - (2*I - 2
)*x^6 - (3*I + 3)*x^4 + (2*I - 2)*x^2 + I + 1) - 4*(x^5 - I*x^3 - x)*sqrt(x^4 - 1))/(x^8 - x^4 + 1)) - (1/32*I
- 1/32)*sqrt(2)*log((sqrt(2)*(-(I - 1)*x^8 + (2*I + 2)*x^6 + (3*I - 3)*x^4 - (2*I + 2)*x^2 - I + 1) - 4*(x^5
+ I*x^3 - x)*sqrt(x^4 - 1))/(x^8 - x^4 + 1)) + (1/32*I - 1/32)*sqrt(2)*log((sqrt(2)*((I - 1)*x^8 - (2*I + 2)*x
^6 - (3*I - 3)*x^4 + (2*I + 2)*x^2 + I - 1) - 4*(x^5 + I*x^3 - x)*sqrt(x^4 - 1))/(x^8 - x^4 + 1)) - (1/32*I +
1/32)*sqrt(2)*log((sqrt(2)*(-(I + 1)*x^8 + (2*I - 2)*x^6 + (3*I + 3)*x^4 - (2*I - 2)*x^2 - I - 1) - 4*(x^5 - I
*x^3 - x)*sqrt(x^4 - 1))/(x^8 - x^4 + 1))

Sympy [F]

\[ \int \frac {-1+x^{16}}{\sqrt {-1+x^4} \left (1+x^8+x^{16}\right )} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right ) \left (x^{8} + 1\right )}{\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right ) \left (x^{4} - x^{2} + 1\right ) \left (x^{8} - x^{4} + 1\right )}\, dx \]

[In]

integrate((x**16-1)/(x**4-1)**(1/2)/(x**16+x**8+1),x)

[Out]

Integral((x - 1)*(x + 1)*(x**2 + 1)*(x**4 + 1)*(x**8 + 1)/(sqrt((x - 1)*(x + 1)*(x**2 + 1))*(x**2 - x + 1)*(x*
*2 + x + 1)*(x**4 - x**2 + 1)*(x**8 - x**4 + 1)), x)

Maxima [F]

\[ \int \frac {-1+x^{16}}{\sqrt {-1+x^4} \left (1+x^8+x^{16}\right )} \, dx=\int { \frac {x^{16} - 1}{{\left (x^{16} + x^{8} + 1\right )} \sqrt {x^{4} - 1}} \,d x } \]

[In]

integrate((x^16-1)/(x^4-1)^(1/2)/(x^16+x^8+1),x, algorithm="maxima")

[Out]

integrate((x^16 - 1)/((x^16 + x^8 + 1)*sqrt(x^4 - 1)), x)

Giac [F]

\[ \int \frac {-1+x^{16}}{\sqrt {-1+x^4} \left (1+x^8+x^{16}\right )} \, dx=\int { \frac {x^{16} - 1}{{\left (x^{16} + x^{8} + 1\right )} \sqrt {x^{4} - 1}} \,d x } \]

[In]

integrate((x^16-1)/(x^4-1)^(1/2)/(x^16+x^8+1),x, algorithm="giac")

[Out]

integrate((x^16 - 1)/((x^16 + x^8 + 1)*sqrt(x^4 - 1)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {-1+x^{16}}{\sqrt {-1+x^4} \left (1+x^8+x^{16}\right )} \, dx=\int \frac {x^{16}-1}{\sqrt {x^4-1}\,\left (x^{16}+x^8+1\right )} \,d x \]

[In]

int((x^16 - 1)/((x^4 - 1)^(1/2)*(x^8 + x^16 + 1)),x)

[Out]

int((x^16 - 1)/((x^4 - 1)^(1/2)*(x^8 + x^16 + 1)), x)