3.14.9 \(\int \frac {-1+x^8}{\sqrt [4]{-x^2+x^4} (1+x^8)} \, dx\)
Optimal. Leaf size=94 \[ \frac {1}{4} \text {RootSum}\left [\text {$\#$1}^{16}-4 \text {$\#$1}^{12}+6 \text {$\#$1}^8-4 \text {$\#$1}^4+2\& ,\frac {\log \left (\sqrt [4]{x^4-x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]+\tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^2}}\right )+\tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^2}}\right ) \]
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Rubi [C] time = 30.69, antiderivative size = 4085, normalized size of antiderivative = 43.46,
number of steps used = 581, number of rules used = 31, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.192,
Rules used = {2056, 1586, 6715, 6725, 2153, 1240, 408, 240, 212, 206, 203, 377, 208,
205, 511, 510, 1248, 735, 844, 230, 305, 220, 1196, 746, 399, 490, 1217, 1707, 444, 63,
298}
result too large to display
Warning: Unable to verify antiderivative.
[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 63
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 203
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])
Rule 205
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]
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 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rule 212
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] &&
!GtQ[a/b, 0]
Rule 220
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Rule 230
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 240
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 298
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 305
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 377
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[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 408
Int[((a_) + (b_.)*(x_)^4)^(p_)/((c_) + (d_.)*(x_)^4), x_Symbol] :> Dist[b/d, Int[(a + b*x^4)^(p - 1), x], x] -
Dist[(b*c - a*d)/d, Int[(a + b*x^4)^(p - 1)/(c + d*x^4), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0
] && (EqQ[p, 3/4] || EqQ[p, 5/4])
Rule 444
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 490
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), In
t[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 510
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)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), 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 511
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPa
rt[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 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 746
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 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 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 1240
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 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 2056
Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] && !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] && !PolyQ[P, x, 2]
Rule 2153
Int[((c_) + (d_.)*(x_)^(n_.))^(q_)*((a_) + (b_.)*(x_)^(nn_.))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^
nn)^p, (c/(c^2 - d^2*x^(2*n)) - (d*x^n)/(c^2 - d^2*x^(2*n)))^(-q), x], x] /; FreeQ[{a, b, c, d, n, nn, p}, x]
&& !IntegerQ[p] && ILtQ[q, 0] && IGtQ[Log[2, nn/n], 0]
Rule 6715
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 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^8}{\sqrt [4]{-x^2+x^4} \left (1+x^8\right )} \, dx &=\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 ) \operatorname {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}} \end {align*}
rest of steps removed due to Latex formating problem.
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Mathematica [F] time = 1.25, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-1+x^8}{\sqrt [4]{-x^2+x^4} \left (1+x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Integrate[(-1 + x^8)/((-x^2 + x^4)^(1/4)*(1 + x^8)),x]
[Out]
Integrate[(-1 + x^8)/((-x^2 + x^4)^(1/4)*(1 + x^8)), x]
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IntegrateAlgebraic [A] time = 0.44, size = 94, normalized size = 1.00 \begin {gather*} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-x^2+x^4}}\right )+\tanh ^{-1}\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 ] \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(-1 + x^8)/((-x^2 + x^4)^(1/4)*(1 + x^8)),x]
[Out]
ArcTan[x/(-x^2 + x^4)^(1/4)] + ArcTanh[x/(-x^2 + x^4)^(1/4)] + RootSum[2 - 4*#1^4 + 6*#1^8 - 4*#1^12 + #1^16 &
, (-Log[x] + Log[(-x^2 + x^4)^(1/4) - x*#1])/#1 & ]/4
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fricas [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^8-1)/(x^4-x^2)^(1/4)/(x^8+1),x, algorithm="fricas")
[Out]
Timed out
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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,x):;OUTPUT:Inva
lid _EXT in replace_ext Error: Bad Argument ValueInvalid _EXT in replace_ext Error: Bad Argument Value
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maple [F] time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {x^{8}-1}{\left (x^{4}-x^{2}\right )^{\frac {1}{4}} \left (x^{8}+1\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[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)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{8} - 1}{{\left (x^{8} + 1\right )} {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^8-1}{\left (x^8+1\right )\,{\left (x^4-x^2\right )}^{1/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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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**8-1)/(x**4-x**2)**(1/4)/(x**8+1),x)
[Out]
Timed out
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