\(\int \frac {-1+x^8}{\sqrt [4]{-x^2+x^4} (1+x^8)} \, dx\) [1310]

Optimal result

Integrand size = 26, antiderivative size = 94 \[ \int \frac {-1+x^8}{\sqrt [4]{-x^2+x^4} \left (1+x^8\right )} \, dx=\arctan \left (\frac {x}{\sqrt [4]{-x^2+x^4}}\right )+\text {arctanh}\left (\frac {x}{\sqrt [4]{-x^2+x^4}}\right )+\frac {1}{4} \text {RootSum}\left [2-4 \text {$\#$1}^4+6 \text {$\#$1}^8-4 \text {$\#$1}^{12}+\text {$\#$1}^{16}\&,\frac {-\log (x)+\log \left (\sqrt [4]{-x^2+x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]

[Out]

Unintegrable

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 14.23 (sec) , antiderivative size = 4085, normalized size of antiderivative = 43.46, number of steps used = 581, number of rules used = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.192, Rules used = {2081, 1600, 6847, 6857, 2184, 1254, 399, 246, 218, 212, 209, 385, 214, 211, 525, 524, 1262, 749, 858, 236, 311, 226, 1210, 760, 408, 504, 1231, 1721, 455, 65, 304} \[ \text {Too Large to display} \]

[In]

Int[(-1 + x^8)/((-x^2 + x^4)^(1/4)*(1 + x^8)),x]

[Out]

((1/24 - I/24)*(-1)^(1/16)*(1 - (1 - I)/Sqrt[2])*Sqrt[x]*(1 - x^2))/(-x^2 + x^4)^(1/4) + ((-1)^(13/16)*((-1 +
I) + Sqrt[2])*Sqrt[x]*(1 - x^2))/(24*(-x^2 + x^4)^(1/4)) - ((1/8 - I/8)*(-1)^(11/16)*(1 - (1 - I)/Sqrt[2])*x^(
3/2)*Sqrt[-1 + x^2])/((-x^2 + x^4)^(1/4)*(1 + Sqrt[-1 + x^2])) + ((-1)^(7/16)*((-1 + I) + Sqrt[2])*x^(3/2)*Sqr
t[-1 + x^2])/(8*(-x^2 + x^4)^(1/4)*(1 + Sqrt[-1 + x^2])) + ((1/4 + I/4)*(1 + (-1)^(1/4))*Sqrt[x]*(-1 + x^2)^(1
/4)*ArcTan[Sqrt[x]/(-1 + x^2)^(1/4)])/(-x^2 + x^4)^(1/4) + (((1 + I) - I*Sqrt[2])*Sqrt[x]*(-1 + x^2)^(1/4)*Arc
Tan[Sqrt[x]/(-1 + x^2)^(1/4)])/(4*(-x^2 + x^4)^(1/4)) + ((I/4)*((-1 - I) + Sqrt[2])*Sqrt[x]*(-1 + x^2)^(1/4)*A
rcTan[Sqrt[x]/(-1 + x^2)^(1/4)])/(-x^2 + x^4)^(1/4) - ((I/4)*((1 + I) + Sqrt[2])*Sqrt[x]*(-1 + x^2)^(1/4)*ArcT
an[Sqrt[x]/(-1 + x^2)^(1/4)])/(-x^2 + x^4)^(1/4) + ((I/4)*(-1 + (-1)^(1/4))^(3/4)*((1 + I) + Sqrt[2])*Sqrt[x]*
(-1 + x^2)^(1/4)*ArcTan[((1 - (-1)^(1/4))^(1/4)*Sqrt[x])/(-1 + x^2)^(1/4)])/(-x^2 + x^4)^(1/4) - ((-1)^(1/16)*
(-1 + (-1)^(1/4))^(3/4)*((1 + I) + I*Sqrt[2])*Sqrt[x]*(-1 + x^2)^(1/4)*ArcTan[((-1)^(15/16)*(-1 + (-1)^(1/4))^
(1/4)*Sqrt[x])/(-1 + x^2)^(1/4)])/(4*(-x^2 + x^4)^(1/4)) - ((-1)^(1/4)*(1 + (-1)^(1/4))^(3/4)*((-1 - I) + Sqrt
[2])*Sqrt[x]*(-1 + x^2)^(1/4)*ArcTan[((1 + (-1)^(1/4))^(1/4)*Sqrt[x])/(-1 + x^2)^(1/4)])/(4*(-x^2 + x^4)^(1/4)
) - ((-1)^(9/16)*(1 + (-1)^(1/4))^(3/4)*((-1 + I) + Sqrt[2])*Sqrt[x]*(-1 + x^2)^(1/4)*ArcTan[((-1)^(15/16)*(1
+ (-1)^(1/4))^(1/4)*Sqrt[x])/(-1 + x^2)^(1/4)])/(4*(-x^2 + x^4)^(1/4)) - ((1/16 - I/16)*(-1)^(1/16)*(-1 - (-1)
^(1/4))^(3/4)*(1 - (1 - I)/Sqrt[2])*Sqrt[x]*(-1 + x^2)^(1/4)*ArcTan[(-1 + x^2)^(1/4)/(-1 - (-1)^(1/4))^(1/4)])
/(-x^2 + x^4)^(1/4) - ((-1)^(13/16)*(-1 - (-1)^(1/4))^(3/4)*((-1 + I) + Sqrt[2])*Sqrt[x]*(-1 + x^2)^(1/4)*ArcT
an[(-1 + x^2)^(1/4)/(-1 - (-1)^(1/4))^(1/4)])/(16*(-x^2 + x^4)^(1/4)) + ((1/32 + I/32)*(-1)^(1/16)*(-1 - (-1)^
(1/4))^(3/4)*(1 - (1 - I)/Sqrt[2])*Sqrt[x^2]*(-1 + x^2)^(1/4)*ArcTan[((-1)^(1/8)*(-1 + x^2)^(1/4))/((-1 - (-1)
^(1/4))^(1/4)*Sqrt[x^2])])/(Sqrt[x]*(-x^2 + x^4)^(1/4)) + ((-1)^(9/16)*(1 + (-1)^(1/4))^(3/4)*((-1 + I) + Sqrt
[2])*Sqrt[x^2]*(-1 + x^2)^(1/4)*ArcTan[((-1)^(1/8)*(-1 + x^2)^(1/4))/((-1 - (-1)^(1/4))^(1/4)*Sqrt[x^2])])/(32
*Sqrt[x]*(-x^2 + x^4)^(1/4)) - ((1/32 - I/32)*(-1)^(1/16)*(-1 - (-1)^(1/4))^(3/4)*(1 - (1 - I)/Sqrt[2])*Sqrt[x
^2]*(-1 + x^2)^(1/4)*ArcTan[((-1)^(5/8)*(-1 + x^2)^(1/4))/((-1 - (-1)^(1/4))^(1/4)*Sqrt[x^2])])/(Sqrt[x]*(-x^2
+ x^4)^(1/4)) - ((-1)^(13/16)*(-1 - (-1)^(1/4))^(3/4)*((-1 + I) + Sqrt[2])*Sqrt[x^2]*(-1 + x^2)^(1/4)*ArcTan[
((-1)^(5/8)*(-1 + x^2)^(1/4))/((-1 - (-1)^(1/4))^(1/4)*Sqrt[x^2])])/(32*Sqrt[x]*(-x^2 + x^4)^(1/4)) + ((1/4 +
I/4)*(1 + (-1)^(1/4))*Sqrt[x]*(-1 + x^2)^(1/4)*ArcTanh[Sqrt[x]/(-1 + x^2)^(1/4)])/(-x^2 + x^4)^(1/4) + (((1 +
I) - I*Sqrt[2])*Sqrt[x]*(-1 + x^2)^(1/4)*ArcTanh[Sqrt[x]/(-1 + x^2)^(1/4)])/(4*(-x^2 + x^4)^(1/4)) + ((I/4)*((
-1 - I) + Sqrt[2])*Sqrt[x]*(-1 + x^2)^(1/4)*ArcTanh[Sqrt[x]/(-1 + x^2)^(1/4)])/(-x^2 + x^4)^(1/4) - ((I/4)*((1
+ I) + Sqrt[2])*Sqrt[x]*(-1 + x^2)^(1/4)*ArcTanh[Sqrt[x]/(-1 + x^2)^(1/4)])/(-x^2 + x^4)^(1/4) + ((I/4)*(-1 +
(-1)^(1/4))^(3/4)*((1 + I) + Sqrt[2])*Sqrt[x]*(-1 + x^2)^(1/4)*ArcTanh[((1 - (-1)^(1/4))^(1/4)*Sqrt[x])/(-1 +
x^2)^(1/4)])/(-x^2 + x^4)^(1/4) - ((-1)^(1/16)*(-1 + (-1)^(1/4))^(3/4)*((1 + I) + I*Sqrt[2])*Sqrt[x]*(-1 + x^
2)^(1/4)*ArcTanh[((-1)^(15/16)*(-1 + (-1)^(1/4))^(1/4)*Sqrt[x])/(-1 + x^2)^(1/4)])/(4*(-x^2 + x^4)^(1/4)) - ((
-1)^(1/4)*(1 + (-1)^(1/4))^(3/4)*((-1 - I) + Sqrt[2])*Sqrt[x]*(-1 + x^2)^(1/4)*ArcTanh[((1 + (-1)^(1/4))^(1/4)
*Sqrt[x])/(-1 + x^2)^(1/4)])/(4*(-x^2 + x^4)^(1/4)) - ((-1)^(9/16)*(1 + (-1)^(1/4))^(3/4)*((-1 + I) + Sqrt[2])
*Sqrt[x]*(-1 + x^2)^(1/4)*ArcTanh[((-1)^(15/16)*(1 + (-1)^(1/4))^(1/4)*Sqrt[x])/(-1 + x^2)^(1/4)])/(4*(-x^2 +
x^4)^(1/4)) + ((1/16 - I/16)*(-1)^(1/16)*(-1 - (-1)^(1/4))^(3/4)*(1 - (1 - I)/Sqrt[2])*Sqrt[x]*(-1 + x^2)^(1/4
)*ArcTanh[(-1 + x^2)^(1/4)/(-1 - (-1)^(1/4))^(1/4)])/(-x^2 + x^4)^(1/4) + ((-1)^(13/16)*(-1 - (-1)^(1/4))^(3/4
)*((-1 + I) + Sqrt[2])*Sqrt[x]*(-1 + x^2)^(1/4)*ArcTanh[(-1 + x^2)^(1/4)/(-1 - (-1)^(1/4))^(1/4)])/(16*(-x^2 +
x^4)^(1/4)) + ((1/8 - I/8)*(-1)^(11/16)*(1 - (1 - I)/Sqrt[2])*(-1 + x^2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1 + x^2])^
2]*(1 + Sqrt[-1 + x^2])*EllipticE[2*ArcTan[(-1 + x^2)^(1/4)], 1/2])/(Sqrt[x]*(-x^2 + x^4)^(1/4)) - ((-1)^(7/16
)*((-1 + I) + Sqrt[2])*(-1 + x^2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1 + x^2])*EllipticE[2*ArcT
an[(-1 + x^2)^(1/4)], 1/2])/(8*Sqrt[x]*(-x^2 + x^4)^(1/4)) - ((1/16 - I/16)*(-1)^(11/16)*(1 - (1 - I)/Sqrt[2])
*(-1 + x^2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1 + x^2])*EllipticF[2*ArcTan[(-1 + x^2)^(1/4)],
1/2])/(Sqrt[x]*(-x^2 + x^4)^(1/4)) + ((-1)^(7/16)*((-1 + I) + Sqrt[2])*(-1 + x^2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1
+ x^2])^2]*(1 + Sqrt[-1 + x^2])*EllipticF[2*ArcTan[(-1 + x^2)^(1/4)], 1/2])/(16*Sqrt[x]*(-x^2 + x^4)^(1/4)) +
((-1)^(3/16)*(1 - Sqrt[-1 - (1 + I)/Sqrt[2]])*(-1 + x^2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1 +
x^2])*EllipticF[2*ArcTan[(-1 + x^2)^(1/4)], 1/2])/(4*(4*I + (4 + 4*I)*Sqrt[2])*Sqrt[x]*(-x^2 + x^4)^(1/4)) +
((-1)^(15/16)*(1 + Sqrt[-1 - (1 + I)/Sqrt[2]])*(-1 + x^2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1
+ x^2])*EllipticF[2*ArcTan[(-1 + x^2)^(1/4)], 1/2])/(16*(2 + (-1)^(1/4))*Sqrt[x]*(-x^2 + x^4)^(1/4)) + ((-1)^(
11/16)*(1 + Sqrt[-1 - (1 + I)/Sqrt[2]])*(-1 + x^2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1 + x^2])
*EllipticF[2*ArcTan[(-1 + x^2)^(1/4)], 1/2])/(16*(I + (1 + I)*Sqrt[2])*Sqrt[x]*(-x^2 + x^4)^(1/4)) - ((-1)^(11
/16)*(1 + Sqrt[-1 - (1 + I)/Sqrt[2]])*(-1 + x^2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1 + x^2])*E
llipticF[2*ArcTan[(-1 + x^2)^(1/4)], 1/2])/(8*(2*I + (2 + 2*I)*Sqrt[2])*Sqrt[x]*(-x^2 + x^4)^(1/4)) + ((-1)^(1
5/16)*(2 - Sqrt[-4 - (2 + 2*I)*Sqrt[2]])*(-1 + x^2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1 + x^2]
)*EllipticF[2*ArcTan[(-1 + x^2)^(1/4)], 1/2])/(32*(2 + (-1)^(1/4))*Sqrt[x]*(-x^2 + x^4)^(1/4)) - ((-1)^(11/16)
*(2 - Sqrt[-4 - (2 + 2*I)*Sqrt[2]])*(-1 + x^2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1 + x^2])*Ell
ipticF[2*ArcTan[(-1 + x^2)^(1/4)], 1/2])/(16*(2*I + (2 + 2*I)*Sqrt[2])*Sqrt[x]*(-x^2 + x^4)^(1/4)) + ((-1)^(11
/16)*(2 - Sqrt[-4 - (2 + 2*I)*Sqrt[2]])*(-1 + x^2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1 + x^2])
*EllipticF[2*ArcTan[(-1 + x^2)^(1/4)], 1/2])/(8*(4*I + (4 + 4*I)*Sqrt[2])*Sqrt[x]*(-x^2 + x^4)^(1/4)) + ((-1)^
(3/16)*(2 + Sqrt[-4 - (2 + 2*I)*Sqrt[2]])*(-1 + x^2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1 + x^2
])*EllipticF[2*ArcTan[(-1 + x^2)^(1/4)], 1/2])/(16*(2*I + (2 + 2*I)*Sqrt[2])*Sqrt[x]*(-x^2 + x^4)^(1/4)) + ((1
/64 - I/64)*(-1)^(11/16)*Sqrt[(-1 - (-1)^(1/4))/2]*((-1 + I) + Sqrt[2])*(1 - Sqrt[-1 - (-1)^(1/4)])^2*(-1 + x^
2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1 + x^2])*EllipticPi[(2 - (-1)^(1/4)/Sqrt[-1 - (-1)^(1/4)
])/4, 2*ArcTan[(-1 + x^2)^(1/4)], 1/2])/((2 + (-1)^(1/4))*Sqrt[x]*(-x^2 + x^4)^(1/4)) - ((-1)^(7/16)*Sqrt[-1 -
(-1)^(1/4)]*((-1 + I) + Sqrt[2])*(1 - Sqrt[-1 - (-1)^(1/4)])^2*(-1 + x^2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1 + x^2])
^2]*(1 + Sqrt[-1 + x^2])*EllipticPi[(2 - (-1)^(1/4)/Sqrt[-1 - (-1)^(1/4)])/4, 2*ArcTan[(-1 + x^2)^(1/4)], 1/2]
)/(64*(2 + (-1)^(1/4))*Sqrt[x]*(-x^2 + x^4)^(1/4)) - ((1/64 - I/64)*(-1)^(11/16)*Sqrt[(-1 - (-1)^(1/4))/2]*((-
1 + I) + Sqrt[2])*(1 + Sqrt[-1 - (-1)^(1/4)])^2*(-1 + x^2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1
+ x^2])*EllipticPi[(2 + (-1)^(1/4)/Sqrt[-1 - (-1)^(1/4)])/4, 2*ArcTan[(-1 + x^2)^(1/4)], 1/2])/((2 + (-1)^(1/
4))*Sqrt[x]*(-x^2 + x^4)^(1/4)) + ((-1)^(7/16)*Sqrt[-1 - (-1)^(1/4)]*((-1 + I) + Sqrt[2])*(1 + Sqrt[-1 - (-1)^
(1/4)])^2*(-1 + x^2)^(1/4)*Sqrt[x^2/(1 + Sqrt[-1 + x^2])^2]*(1 + Sqrt[-1 + x^2])*EllipticPi[(2 + (-1)^(1/4)/Sq
rt[-1 - (-1)^(1/4)])/4, 2*ArcTan[(-1 + x^2)^(1/4)], 1/2])/(64*(2 + (-1)^(1/4))*Sqrt[x]*(-x^2 + x^4)^(1/4))

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 209

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

Rule 211

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

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 214

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

Rule 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[a/b, 0]

Rule 226

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)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 236

Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Dist[2*(Sqrt[(-b)*(x^2/a)]/(b*x)), Subst[Int[x^2/Sqrt[1 - x^4/a
], x], x, (a + b*x^2)^(1/4)], x] /; FreeQ[{a, b}, x] && NegQ[a]

Rule 246

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[1/(1 - b*x^n)^(p + 1/n + 1), x], x
, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p
+ 1/n]

Rule 304

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}
, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !
GtQ[a/b, 0]

Rule 311

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

Rule 385

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

Rule 399

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

Rule 408

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

Rule 455

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 504

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

Rule 524

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*
((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 525

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPar
t[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]), Int[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x
] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] &&  !(IntegerQ[
p] || GtQ[a, 0])

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 760

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

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 1210

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)]/(q*Sqrt[a + c*x^4
]))*EllipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

Rule 1231

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 1254

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + c*x^4)^p, (d/
(d^2 - e^2*x^4) - e*(x^2/(d^2 - e^2*x^4)))^(-q), x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[c*d^2 + a*e^2, 0]
&&  !IntegerQ[p] && ILtQ[q, 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 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 1721

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))]/(4*d*e*A*q*Sqrt[a + c*x^4]))*El
lipticPi[Cancel[-(B*d - A*e)^2/(4*d*e*A*B)], 2*ArcTan[q*x], 1/2], 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 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 2184

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 6847

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rule 6857

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

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \int \frac {-1+x^8}{\sqrt {x} \sqrt [4]{-1+x^2} \left (1+x^8\right )} \, dx}{\sqrt [4]{-x^2+x^4}} \\ & = \frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \int \frac {\left (-1+x^2\right )^{3/4} \left (1+x^2+x^4+x^6\right )}{\sqrt {x} \left (1+x^8\right )} \, dx}{\sqrt [4]{-x^2+x^4}} \\ & = \frac {\left (2 \sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {\left (-1+x^4\right )^{3/4} \left (1+x^4+x^8+x^{12}\right )}{1+x^{16}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-x^2+x^4}} \\ & = \frac {\left (2 \sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \left (\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (-1+x^4\right )^{3/4}}{16 \left (\sqrt [16]{-1}-x\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (-1+x^4\right )^{3/4}}{16 \left (\sqrt [16]{-1}-i x\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (-1+x^4\right )^{3/4}}{16 \left (\sqrt [16]{-1}+i x\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (-1+x^4\right )^{3/4}}{16 \left (\sqrt [16]{-1}+x\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (-1+x^4\right )^{3/4}}{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 ) \left (-1+x^4\right )^{3/4}}{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 ) \left (-1+x^4\right )^{3/4}}{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 ) \left (-1+x^4\right )^{3/4}}{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 ) \left (-1+x^4\right )^{3/4}}{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 ) \left (-1+x^4\right )^{3/4}}{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 ) \left (-1+x^4\right )^{3/4}}{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 ) \left (-1+x^4\right )^{3/4}}{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 ) \left (-1+x^4\right )^{3/4}}{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 ) \left (-1+x^4\right )^{3/4}}{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 ) \left (-1+x^4\right )^{3/4}}{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 ) \left (-1+x^4\right )^{3/4}}{16 \left (\sqrt [16]{-1}+(-1)^{7/8} x\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-x^2+x^4}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.36 \[ \int \frac {-1+x^8}{\sqrt [4]{-x^2+x^4} \left (1+x^8\right )} \, dx=\frac {\sqrt {x} \sqrt [4]{-1+x^2} \left (4 \left (\arctan \left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )+\text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )\right )+\text {RootSum}\left [2-4 \text {$\#$1}^4+6 \text {$\#$1}^8-4 \text {$\#$1}^{12}+\text {$\#$1}^{16}\&,\frac {-\log \left (\sqrt {x}\right )+\log \left (\sqrt [4]{-1+x^2}-\sqrt {x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right )}{4 \sqrt [4]{x^2 \left (-1+x^2\right )}} \]

[In]

Integrate[(-1 + x^8)/((-x^2 + x^4)^(1/4)*(1 + x^8)),x]

[Out]

(Sqrt[x]*(-1 + x^2)^(1/4)*(4*(ArcTan[Sqrt[x]/(-1 + x^2)^(1/4)] + ArcTanh[Sqrt[x]/(-1 + x^2)^(1/4)]) + RootSum[
2 - 4*#1^4 + 6*#1^8 - 4*#1^12 + #1^16 & , (-Log[Sqrt[x]] + Log[(-1 + x^2)^(1/4) - Sqrt[x]*#1])/#1 & ]))/(4*(x^
2*(-1 + x^2))^(1/4))

Maple [F(-1)]

Timed out.

\[\int \frac {x^{8}-1}{\left (x^{4}-x^{2}\right )^{\frac {1}{4}} \left (x^{8}+1\right )}d x\]

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [N/A]

Not integrable

Time = 108.87 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.38 \[ \int \frac {-1+x^8}{\sqrt [4]{-x^2+x^4} \left (1+x^8\right )} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right )}{\sqrt [4]{x^{2} \left (x - 1\right ) \left (x + 1\right )} \left (x^{8} + 1\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.28 \[ \int \frac {-1+x^8}{\sqrt [4]{-x^2+x^4} \left (1+x^8\right )} \, dx=\int { \frac {x^{8} - 1}{{\left (x^{8} + 1\right )} {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}}} \,d x } \]

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {-1+x^8}{\sqrt [4]{-x^2+x^4} \left (1+x^8\right )} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Invalid _EXT in replace_ext Error: Bad Argument ValueInvalid _EXT in replace_ext Error: Bad Argument ValueD
one

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.28 \[ \int \frac {-1+x^8}{\sqrt [4]{-x^2+x^4} \left (1+x^8\right )} \, dx=\int \frac {x^8-1}{\left (x^8+1\right )\,{\left (x^4-x^2\right )}^{1/4}} \,d x \]

[In]

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

[Out]

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