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

Optimal. Leaf size=171 \[ -\frac {x}{8 \sqrt {x^4-1}}+\frac {1}{32} \tan ^{-1}\left (\frac {\frac {x^4}{2}-x^2-\frac {1}{2}}{x \sqrt {x^4-1}}\right )+\frac {\tan ^{-1}\left (\frac {-\frac {x^4}{2^{3/4}}+\frac {x^2}{\sqrt [4]{2}}+\frac {1}{2^{3/4}}}{x \sqrt {x^4-1}}\right )}{8\ 2^{3/4}}-\frac {1}{32} \tanh ^{-1}\left (\frac {\frac {x^4}{2}+x^2-\frac {1}{2}}{x \sqrt {x^4-1}}\right )+\frac {\tanh ^{-1}\left (\frac {2^{3/4} x \sqrt {x^4-1}}{x^4+\sqrt {2} x^2-1}\right )}{8\ 2^{3/4}} \]

________________________________________________________________________________________

Rubi [C]  time = 1.23, antiderivative size = 833, normalized size of antiderivative = 4.87, number of steps used = 57, number of rules used = 19, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.950, Rules used = {6725, 1152, 414, 423, 427, 424, 253, 222, 409, 1211, 1699, 206, 203, 1429, 1215, 1457, 540, 538, 537} \begin {gather*} -\frac {x \left (1-x^2\right )}{16 \sqrt {x^4-1}}+\frac {\sqrt {x^2+1} \Pi \left (-\sqrt [4]{-1};\left .\sin ^{-1}(x)\right |-1\right ) \sqrt {1-x^2}}{8 \sqrt {x^4-1}}+\frac {\sqrt {x^2+1} \Pi \left (\sqrt [4]{-1};\left .\sin ^{-1}(x)\right |-1\right ) \sqrt {1-x^2}}{8 \sqrt {x^4-1}}+\frac {\sqrt {x^2+1} \Pi \left (-(-1)^{3/4};\left .\sin ^{-1}(x)\right |-1\right ) \sqrt {1-x^2}}{8 \sqrt {x^4-1}}+\frac {\sqrt {x^2+1} \Pi \left ((-1)^{3/4};\left .\sin ^{-1}(x)\right |-1\right ) \sqrt {1-x^2}}{8 \sqrt {x^4-1}}-\left (\frac {1}{32}-\frac {i}{32}\right ) \tan ^{-1}\left (\frac {(1+i) x}{\sqrt {x^4-1}}\right )-\left (\frac {1}{32}-\frac {i}{32}\right ) \tanh ^{-1}\left (\frac {(1+i) x}{\sqrt {x^4-1}}\right )+\frac {\sqrt {x^2-1} \sqrt {x^2+1} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{4 \left ((2-2 i)+2 \sqrt {2}\right ) \sqrt {x^4-1}}-\frac {\sqrt {x^2-1} \sqrt {x^2+1} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{8 \left ((1+i)+\sqrt {2}\right ) \sqrt {x^4-1}}-\frac {\sqrt {x^2-1} \sqrt {x^2+1} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{8 \left ((1-i)+\sqrt {2}\right ) \sqrt {x^4-1}}-\frac {\sqrt {x^2-1} \sqrt {x^2+1} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{8 \left ((-1+i)+\sqrt {2}\right ) \sqrt {x^4-1}}-\frac {\sqrt {x^2-1} \sqrt {x^2+1} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{8 \left ((-1-i)+\sqrt {2}\right ) \sqrt {x^4-1}}+\frac {\sqrt {x^2-1} \sqrt {x^2+1} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{8 \sqrt {2} \left (1+(-1)^{3/4}\right ) \sqrt {x^4-1}}+\frac {\sqrt {x^2-1} \sqrt {x^2+1} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{8 \sqrt {2} \left (1+\sqrt [4]{-1}\right ) \sqrt {x^4-1}}+\frac {\sqrt {x^2-1} \sqrt {x^2+1} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{8 \sqrt {2} \left (1-\sqrt [4]{-1}\right ) \sqrt {x^4-1}}-\frac {\sqrt {x^2-1} \sqrt {x^2+1} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{4 \sqrt {2} \sqrt {x^4-1}}-\frac {x \left (x^2+1\right )}{16 \sqrt {x^4-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/16*(x*(1 - x^2))/Sqrt[-1 + x^4] - (x*(1 + x^2))/(16*Sqrt[-1 + x^4]) - (1/32 - I/32)*ArcTan[((1 + I)*x)/Sqrt
[-1 + x^4]] - (1/32 - I/32)*ArcTanh[((1 + I)*x)/Sqrt[-1 + x^4]] - (Sqrt[-1 + x^2]*Sqrt[1 + x^2]*EllipticF[ArcS
in[(Sqrt[2]*x)/Sqrt[-1 + x^2]], 1/2])/(4*Sqrt[2]*Sqrt[-1 + x^4]) + (Sqrt[-1 + x^2]*Sqrt[1 + x^2]*EllipticF[Arc
Sin[(Sqrt[2]*x)/Sqrt[-1 + x^2]], 1/2])/(8*Sqrt[2]*(1 - (-1)^(1/4))*Sqrt[-1 + x^4]) + (Sqrt[-1 + x^2]*Sqrt[1 +
x^2]*EllipticF[ArcSin[(Sqrt[2]*x)/Sqrt[-1 + x^2]], 1/2])/(8*Sqrt[2]*(1 + (-1)^(1/4))*Sqrt[-1 + x^4]) + (Sqrt[-
1 + x^2]*Sqrt[1 + x^2]*EllipticF[ArcSin[(Sqrt[2]*x)/Sqrt[-1 + x^2]], 1/2])/(8*Sqrt[2]*(1 + (-1)^(3/4))*Sqrt[-1
+ x^4]) - (Sqrt[-1 + x^2]*Sqrt[1 + x^2]*EllipticF[ArcSin[(Sqrt[2]*x)/Sqrt[-1 + x^2]], 1/2])/(8*((-1 - I) + Sq
rt[2])*Sqrt[-1 + x^4]) - (Sqrt[-1 + x^2]*Sqrt[1 + x^2]*EllipticF[ArcSin[(Sqrt[2]*x)/Sqrt[-1 + x^2]], 1/2])/(8*
((-1 + I) + Sqrt[2])*Sqrt[-1 + x^4]) - (Sqrt[-1 + x^2]*Sqrt[1 + x^2]*EllipticF[ArcSin[(Sqrt[2]*x)/Sqrt[-1 + x^
2]], 1/2])/(8*((1 - I) + Sqrt[2])*Sqrt[-1 + x^4]) - (Sqrt[-1 + x^2]*Sqrt[1 + x^2]*EllipticF[ArcSin[(Sqrt[2]*x)
/Sqrt[-1 + x^2]], 1/2])/(8*((1 + I) + Sqrt[2])*Sqrt[-1 + x^4]) + (Sqrt[-1 + x^2]*Sqrt[1 + x^2]*EllipticF[ArcSi
n[(Sqrt[2]*x)/Sqrt[-1 + x^2]], 1/2])/(4*((2 - 2*I) + 2*Sqrt[2])*Sqrt[-1 + x^4]) + (Sqrt[1 - x^2]*Sqrt[1 + x^2]
*EllipticPi[-(-1)^(1/4), ArcSin[x], -1])/(8*Sqrt[-1 + x^4]) + (Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[(-1)^(1/
4), ArcSin[x], -1])/(8*Sqrt[-1 + x^4]) + (Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[-(-1)^(3/4), ArcSin[x], -1])/
(8*Sqrt[-1 + x^4]) + (Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[(-1)^(3/4), ArcSin[x], -1])/(8*Sqrt[-1 + x^4])

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 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 222

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]*EllipticF[ArcSin[x/Sqrt[(a + q*x^2)/(2*q)]], 1/2])/(Sqrt[2]*Sqrt[-a]*Sqrt[a + b*x^4]), x] /; IntegerQ[q]]
/; FreeQ[{a, b}, x] && LtQ[a, 0] && GtQ[b, 0]

Rule 253

Int[((a1_.) + (b1_.)*(x_)^(n_))^(p_)*((a2_.) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[((a1 + b1*x^n)^FracPa
rt[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 409

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 414

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*
(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

Rule 423

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

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 427

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

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), 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 538

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

Rule 540

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 1152

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

Rule 1211

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

Rule 1215

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 1429

Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{r = Rt[-(a*c), 2]}, -Dist[c/(2
*r), Int[(d + e*x^n)^q/(r - c*x^n), x], x] - Dist[c/(2*r), Int[(d + e*x^n)^q/(r + c*x^n), x], x]] /; FreeQ[{a,
c, d, e, n, q}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[q]

Rule 1457

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*x^n)/e)^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 1699

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

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 {x^8}{\sqrt {-1+x^4} \left (-1+x^{16}\right )} \, dx &=\int \left (\frac {1}{8 \left (-1+x^2\right ) \sqrt {-1+x^4}}-\frac {1}{8 \left (1+x^2\right ) \sqrt {-1+x^4}}-\frac {1}{4 \sqrt {-1+x^4} \left (1+x^4\right )}+\frac {1}{2 \sqrt {-1+x^4} \left (1+x^8\right )}\right ) \, dx\\ &=\frac {1}{8} \int \frac {1}{\left (-1+x^2\right ) \sqrt {-1+x^4}} \, dx-\frac {1}{8} \int \frac {1}{\left (1+x^2\right ) \sqrt {-1+x^4}} \, dx-\frac {1}{4} \int \frac {1}{\sqrt {-1+x^4} \left (1+x^4\right )} \, dx+\frac {1}{2} \int \frac {1}{\sqrt {-1+x^4} \left (1+x^8\right )} \, dx\\ &=\frac {1}{4} i \int \frac {1}{\left (i-x^4\right ) \sqrt {-1+x^4}} \, dx+\frac {1}{4} i \int \frac {1}{\sqrt {-1+x^4} \left (i+x^4\right )} \, dx-\frac {1}{8} \int \frac {1}{\left (1-i x^2\right ) \sqrt {-1+x^4}} \, dx-\frac {1}{8} \int \frac {1}{\left (1+i x^2\right ) \sqrt {-1+x^4}} \, dx-\frac {\left (\sqrt {-1+x^2} \sqrt {1+x^2}\right ) \int \frac {1}{\sqrt {-1+x^2} \left (1+x^2\right )^{3/2}} \, dx}{8 \sqrt {-1+x^4}}+\frac {\left (\sqrt {-1+x^2} \sqrt {1+x^2}\right ) \int \frac {1}{\left (-1+x^2\right )^{3/2} \sqrt {1+x^2}} \, dx}{8 \sqrt {-1+x^4}}\\ &=-\frac {x \left (1-x^2\right )}{16 \sqrt {-1+x^4}}-\frac {x \left (1+x^2\right )}{16 \sqrt {-1+x^4}}-2 \left (\frac {1}{16} \int \frac {1}{\sqrt {-1+x^4}} \, dx\right )-\frac {1}{16} \int \frac {1-i x^2}{\left (1+i x^2\right ) \sqrt {-1+x^4}} \, dx-\frac {1}{16} \int \frac {1+i x^2}{\left (1-i x^2\right ) \sqrt {-1+x^4}} \, dx+\frac {1}{8} \int \frac {1}{\left (1-\sqrt [4]{-1} x^2\right ) \sqrt {-1+x^4}} \, dx+\frac {1}{8} \int \frac {1}{\left (1+\sqrt [4]{-1} x^2\right ) \sqrt {-1+x^4}} \, dx+\frac {1}{8} \int \frac {1}{\left (1-(-1)^{3/4} x^2\right ) \sqrt {-1+x^4}} \, dx+\frac {1}{8} \int \frac {1}{\left (1+(-1)^{3/4} x^2\right ) \sqrt {-1+x^4}} \, dx+\frac {\left (\sqrt {-1+x^2} \sqrt {1+x^2}\right ) \int \frac {\sqrt {-1+x^2}}{\sqrt {1+x^2}} \, dx}{16 \sqrt {-1+x^4}}-\frac {\left (\sqrt {-1+x^2} \sqrt {1+x^2}\right ) \int \frac {\sqrt {1+x^2}}{\sqrt {-1+x^2}} \, dx}{16 \sqrt {-1+x^4}}\\ &=-\frac {x \left (1-x^2\right )}{16 \sqrt {-1+x^4}}-\frac {x \left (1+x^2\right )}{16 \sqrt {-1+x^4}}-\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{8 \sqrt {2} \sqrt {-1+x^4}}-\frac {1}{16} \operatorname {Subst}\left (\int \frac {1}{1-2 i x^2} \, dx,x,\frac {x}{\sqrt {-1+x^4}}\right )-\frac {1}{16} \operatorname {Subst}\left (\int \frac {1}{1+2 i x^2} \, dx,x,\frac {x}{\sqrt {-1+x^4}}\right )+\frac {\int \frac {1}{\sqrt {-1+x^4}} \, dx}{8 \left (1-\sqrt [4]{-1}\right )}-\frac {\sqrt [4]{-1} \int \frac {1-x^2}{\left (1-\sqrt [4]{-1} x^2\right ) \sqrt {-1+x^4}} \, dx}{8 \left (1-\sqrt [4]{-1}\right )}+\frac {\int \frac {1}{\sqrt {-1+x^4}} \, dx}{8 \left (1+\sqrt [4]{-1}\right )}+\frac {\sqrt [4]{-1} \int \frac {1-x^2}{\left (1+\sqrt [4]{-1} x^2\right ) \sqrt {-1+x^4}} \, dx}{8 \left (1+\sqrt [4]{-1}\right )}+\frac {\int \frac {1}{\sqrt {-1+x^4}} \, dx}{8 \left (1-(-1)^{3/4}\right )}+\frac {\int \frac {1}{\sqrt {-1+x^4}} \, dx}{8 \left (1+(-1)^{3/4}\right )}+\frac {(-1)^{3/4} \int \frac {1-x^2}{\left (1+(-1)^{3/4} x^2\right ) \sqrt {-1+x^4}} \, dx}{8 \left (1+(-1)^{3/4}\right )}+\frac {\int \frac {1-x^2}{\left (1-(-1)^{3/4} x^2\right ) \sqrt {-1+x^4}} \, dx}{4 \sqrt {2} \left ((1+i)+\sqrt {2}\right )}-\frac {\left (\sqrt {1-x^2} \sqrt {1+x^2}\right ) \int \frac {\sqrt {1+x^2}}{\sqrt {1-x^2}} \, dx}{16 \sqrt {-1+x^4}}+\frac {\left (\sqrt {-1+x^2} \sqrt {1+x^2}\right ) \int \frac {\sqrt {1+x^2}}{\sqrt {-1+x^2}} \, dx}{16 \sqrt {-1+x^4}}-\frac {\left (\sqrt {-1+x^2} \sqrt {1+x^2}\right ) \int \frac {1}{\sqrt {-1+x^2} \sqrt {1+x^2}} \, dx}{8 \sqrt {-1+x^4}}\\ &=-\frac {x \left (1-x^2\right )}{16 \sqrt {-1+x^4}}-\frac {x \left (1+x^2\right )}{16 \sqrt {-1+x^4}}-\left (\frac {1}{32}-\frac {i}{32}\right ) \tan ^{-1}\left (\frac {(1+i) x}{\sqrt {-1+x^4}}\right )-\left (\frac {1}{32}-\frac {i}{32}\right ) \tanh ^{-1}\left (\frac {(1+i) x}{\sqrt {-1+x^4}}\right )-\frac {\sqrt {1-x^2} \sqrt {1+x^2} E\left (\left .\sin ^{-1}(x)\right |-1\right )}{16 \sqrt {-1+x^4}}-\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{8 \sqrt {2} \sqrt {-1+x^4}}+\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{8 \sqrt {2} \left (1-\sqrt [4]{-1}\right ) \sqrt {-1+x^4}}+\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{8 \sqrt {2} \left (1+\sqrt [4]{-1}\right ) \sqrt {-1+x^4}}+\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{8 \sqrt {2} \left (1-(-1)^{3/4}\right ) \sqrt {-1+x^4}}+\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{8 \sqrt {2} \left (1+(-1)^{3/4}\right ) \sqrt {-1+x^4}}-\frac {1}{8} \int \frac {1}{\sqrt {-1+x^4}} \, dx-\frac {\left (\sqrt [4]{-1} \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {\sqrt {1-x^2}}{\sqrt {-1-x^2} \left (1-\sqrt [4]{-1} x^2\right )} \, dx}{8 \left (1-\sqrt [4]{-1}\right ) \sqrt {-1+x^4}}+\frac {\left (\sqrt [4]{-1} \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {\sqrt {1-x^2}}{\sqrt {-1-x^2} \left (1+\sqrt [4]{-1} x^2\right )} \, dx}{8 \left (1+\sqrt [4]{-1}\right ) \sqrt {-1+x^4}}+\frac {\left ((-1)^{3/4} \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {\sqrt {1-x^2}}{\sqrt {-1-x^2} \left (1+(-1)^{3/4} x^2\right )} \, dx}{8 \left (1+(-1)^{3/4}\right ) \sqrt {-1+x^4}}+\frac {\left (\sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {\sqrt {1-x^2}}{\sqrt {-1-x^2} \left (1-(-1)^{3/4} x^2\right )} \, dx}{4 \sqrt {2} \left ((1+i)+\sqrt {2}\right ) \sqrt {-1+x^4}}+\frac {\left (\sqrt {1-x^2} \sqrt {1+x^2}\right ) \int \frac {\sqrt {1+x^2}}{\sqrt {1-x^2}} \, dx}{16 \sqrt {-1+x^4}}\\ &=-\frac {x \left (1-x^2\right )}{16 \sqrt {-1+x^4}}-\frac {x \left (1+x^2\right )}{16 \sqrt {-1+x^4}}-\left (\frac {1}{32}-\frac {i}{32}\right ) \tan ^{-1}\left (\frac {(1+i) x}{\sqrt {-1+x^4}}\right )-\left (\frac {1}{32}-\frac {i}{32}\right ) \tanh ^{-1}\left (\frac {(1+i) x}{\sqrt {-1+x^4}}\right )-\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt {2} \sqrt {-1+x^4}}+\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{8 \sqrt {2} \left (1-\sqrt [4]{-1}\right ) \sqrt {-1+x^4}}+\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{8 \sqrt {2} \left (1+\sqrt [4]{-1}\right ) \sqrt {-1+x^4}}+\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{8 \sqrt {2} \left (1-(-1)^{3/4}\right ) \sqrt {-1+x^4}}+\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{8 \sqrt {2} \left (1+(-1)^{3/4}\right ) \sqrt {-1+x^4}}-\frac {\left (\sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2}} \, dx}{8 \left (1-\sqrt [4]{-1}\right ) \sqrt {-1+x^4}}-\frac {\left (\sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2}} \, dx}{8 \left (1+\sqrt [4]{-1}\right ) \sqrt {-1+x^4}}-\frac {\left (\sqrt [4]{-1} \left (-1+(-1)^{3/4}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2} \left (1+\sqrt [4]{-1} x^2\right )} \, dx}{8 \left (1+\sqrt [4]{-1}\right ) \sqrt {-1+x^4}}-\frac {\left (\sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2}} \, dx}{8 \left (1+(-1)^{3/4}\right ) \sqrt {-1+x^4}}-\frac {\left ((-1)^{3/4} \left (-1+\sqrt [4]{-1}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2} \left (1+(-1)^{3/4} x^2\right )} \, dx}{8 \left (1+(-1)^{3/4}\right ) \sqrt {-1+x^4}}-\frac {\left (\sqrt [4]{-1} \left (1+(-1)^{3/4}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2} \left (1-\sqrt [4]{-1} x^2\right )} \, dx}{8 \left (1-\sqrt [4]{-1}\right ) \sqrt {-1+x^4}}-\frac {\left (\sqrt [4]{-1} \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2}} \, dx}{4 \sqrt {2} \left ((1+i)+\sqrt {2}\right ) \sqrt {-1+x^4}}+\frac {\left (\left (1+\sqrt [4]{-1}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2} \left (1-(-1)^{3/4} x^2\right )} \, dx}{4 \sqrt {2} \left ((1+i)+\sqrt {2}\right ) \sqrt {-1+x^4}}\\ &=-\frac {x \left (1-x^2\right )}{16 \sqrt {-1+x^4}}-\frac {x \left (1+x^2\right )}{16 \sqrt {-1+x^4}}-\left (\frac {1}{32}-\frac {i}{32}\right ) \tan ^{-1}\left (\frac {(1+i) x}{\sqrt {-1+x^4}}\right )-\left (\frac {1}{32}-\frac {i}{32}\right ) \tanh ^{-1}\left (\frac {(1+i) x}{\sqrt {-1+x^4}}\right )-\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt {2} \sqrt {-1+x^4}}+\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{8 \sqrt {2} \left (1-\sqrt [4]{-1}\right ) \sqrt {-1+x^4}}+\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{8 \sqrt {2} \left (1+\sqrt [4]{-1}\right ) \sqrt {-1+x^4}}+\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{8 \sqrt {2} \left (1-(-1)^{3/4}\right ) \sqrt {-1+x^4}}+\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{8 \sqrt {2} \left (1+(-1)^{3/4}\right ) \sqrt {-1+x^4}}-\frac {\int \frac {1}{\sqrt {-1+x^4}} \, dx}{8 \left (1-\sqrt [4]{-1}\right )}-\frac {\int \frac {1}{\sqrt {-1+x^4}} \, dx}{8 \left (1+\sqrt [4]{-1}\right )}-\frac {\int \frac {1}{\sqrt {-1+x^4}} \, dx}{8 \left (1+(-1)^{3/4}\right )}-\frac {\sqrt [4]{-1} \int \frac {1}{\sqrt {-1+x^4}} \, dx}{4 \sqrt {2} \left ((1+i)+\sqrt {2}\right )}-\frac {\left (\sqrt [4]{-1} \left (-1+(-1)^{3/4}\right ) \sqrt {1-x^2} \sqrt {1+x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (1+\sqrt [4]{-1} x^2\right )} \, dx}{8 \left (1+\sqrt [4]{-1}\right ) \sqrt {-1+x^4}}-\frac {\left ((-1)^{3/4} \left (-1+\sqrt [4]{-1}\right ) \sqrt {1-x^2} \sqrt {1+x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (1+(-1)^{3/4} x^2\right )} \, dx}{8 \left (1+(-1)^{3/4}\right ) \sqrt {-1+x^4}}-\frac {\left (\sqrt [4]{-1} \left (1+(-1)^{3/4}\right ) \sqrt {1-x^2} \sqrt {1+x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (1-\sqrt [4]{-1} x^2\right )} \, dx}{8 \left (1-\sqrt [4]{-1}\right ) \sqrt {-1+x^4}}+\frac {\left (\left (1+\sqrt [4]{-1}\right ) \sqrt {1-x^2} \sqrt {1+x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (1-(-1)^{3/4} x^2\right )} \, dx}{4 \sqrt {2} \left ((1+i)+\sqrt {2}\right ) \sqrt {-1+x^4}}\\ &=-\frac {x \left (1-x^2\right )}{16 \sqrt {-1+x^4}}-\frac {x \left (1+x^2\right )}{16 \sqrt {-1+x^4}}-\left (\frac {1}{32}-\frac {i}{32}\right ) \tan ^{-1}\left (\frac {(1+i) x}{\sqrt {-1+x^4}}\right )-\left (\frac {1}{32}-\frac {i}{32}\right ) \tanh ^{-1}\left (\frac {(1+i) x}{\sqrt {-1+x^4}}\right )-\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt {2} \sqrt {-1+x^4}}+\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{8 \sqrt {2} \left (1-(-1)^{3/4}\right ) \sqrt {-1+x^4}}-\frac {\sqrt [4]{-1} \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{8 \left ((1+i)+\sqrt {2}\right ) \sqrt {-1+x^4}}+\frac {\sqrt [4]{-1} \left (1-(-1)^{3/4}\right ) \sqrt {1-x^2} \sqrt {1+x^2} \Pi \left (-\sqrt [4]{-1};\left .\sin ^{-1}(x)\right |-1\right )}{8 \left (1+\sqrt [4]{-1}\right ) \sqrt {-1+x^4}}+\frac {\sqrt {1-x^2} \sqrt {1+x^2} \Pi \left (\sqrt [4]{-1};\left .\sin ^{-1}(x)\right |-1\right )}{8 \sqrt {-1+x^4}}+\frac {(-1)^{3/4} \left (1-\sqrt [4]{-1}\right ) \sqrt {1-x^2} \sqrt {1+x^2} \Pi \left (-(-1)^{3/4};\left .\sin ^{-1}(x)\right |-1\right )}{8 \left (1+(-1)^{3/4}\right ) \sqrt {-1+x^4}}+\frac {\sqrt {1-x^2} \sqrt {1+x^2} \Pi \left ((-1)^{3/4};\left .\sin ^{-1}(x)\right |-1\right )}{8 \sqrt {-1+x^4}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.67, size = 161, normalized size = 0.94 \begin {gather*} -\frac {\sqrt {1-x^4} F\left (\left .\sin ^{-1}(x)\right |-1\right )+\sqrt {1-x^4} \Pi \left (-i;\left .\sin ^{-1}(x)\right |-1\right )+\sqrt {1-x^4} \Pi \left (i;\left .\sin ^{-1}(x)\right |-1\right )-\sqrt {1-x^4} \Pi \left (-\sqrt [4]{-1};\left .\sin ^{-1}(x)\right |-1\right )-\sqrt {1-x^4} \Pi \left (\sqrt [4]{-1};\left .\sin ^{-1}(x)\right |-1\right )-\sqrt {1-x^4} \Pi \left (-(-1)^{3/4};\left .\sin ^{-1}(x)\right |-1\right )-\sqrt {1-x^4} \Pi \left ((-1)^{3/4};\left .\sin ^{-1}(x)\right |-1\right )+x}{8 \sqrt {x^4-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*(x + Sqrt[1 - x^4]*EllipticF[ArcSin[x], -1] + Sqrt[1 - x^4]*EllipticPi[-I, ArcSin[x], -1] + Sqrt[1 - x^4]
*EllipticPi[I, ArcSin[x], -1] - Sqrt[1 - x^4]*EllipticPi[-(-1)^(1/4), ArcSin[x], -1] - Sqrt[1 - x^4]*EllipticP
i[(-1)^(1/4), ArcSin[x], -1] - Sqrt[1 - x^4]*EllipticPi[-(-1)^(3/4), ArcSin[x], -1] - Sqrt[1 - x^4]*EllipticPi
[(-1)^(3/4), ArcSin[x], -1])/Sqrt[-1 + x^4]

________________________________________________________________________________________

IntegrateAlgebraic [C]  time = 0.79, size = 153, normalized size = 0.89 \begin {gather*} -\frac {x}{8 \sqrt {-1+x^4}}-\left (\frac {1}{32}-\frac {i}{32}\right ) \tan ^{-1}\left (\frac {(1+i) x}{\sqrt {-1+x^4}}\right )+\left (\frac {1}{32}+\frac {i}{32}\right ) \tan ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {-1+x^4}}{x}\right )-\frac {\tan ^{-1}\left (\frac {2^{3/4} x \sqrt {-1+x^4}}{1+\sqrt {2} x^2-x^4}\right )}{8\ 2^{3/4}}+\frac {\tanh ^{-1}\left (\frac {2^{3/4} x \sqrt {-1+x^4}}{-1+\sqrt {2} x^2+x^4}\right )}{8\ 2^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*x/Sqrt[-1 + x^4] - (1/32 - I/32)*ArcTan[((1 + I)*x)/Sqrt[-1 + x^4]] + (1/32 + I/32)*ArcTan[((1/2 + I/2)*S
qrt[-1 + x^4])/x] - ArcTan[(2^(3/4)*x*Sqrt[-1 + x^4])/(1 + Sqrt[2]*x^2 - x^4)]/(8*2^(3/4)) + ArcTanh[(2^(3/4)*
x*Sqrt[-1 + x^4])/(-1 + Sqrt[2]*x^2 + x^4)]/(8*2^(3/4))

________________________________________________________________________________________

fricas [B]  time = 0.75, size = 740, normalized size = 4.33 \begin {gather*} \frac {4 \cdot 2^{\frac {1}{4}} {\left (x^{4} - 1\right )} \arctan \left (\frac {2 \, x^{16} + 4 \, x^{8} + \sqrt {2} {\left (2^{\frac {3}{4}} {\left (x^{16} - 20 \, x^{12} + 34 \, x^{8} - 20 \, x^{4} + 1\right )} + 8 \, {\left (x^{11} + x^{3} + 4 \, \sqrt {2} {\left (x^{9} - x^{5}\right )}\right )} \sqrt {x^{4} - 1} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{14} - 9 \, x^{10} + 9 \, x^{6} - x^{2}\right )}\right )} \sqrt {\frac {8 \, x^{6} - 8 \, x^{2} + \sqrt {2} {\left (x^{8} + 1\right )} + 4 \, \sqrt {x^{4} - 1} {\left (2^{\frac {3}{4}} x^{3} + 2^{\frac {1}{4}} {\left (x^{5} - x\right )}\right )}}{x^{8} + 1}} + 8 \, \sqrt {2} {\left (x^{14} - x^{10} + x^{6} - x^{2}\right )} + 4 \, \sqrt {x^{4} - 1} {\left (2^{\frac {3}{4}} {\left (x^{13} - 9 \, x^{9} + 9 \, x^{5} - x\right )} + 2 \cdot 2^{\frac {1}{4}} {\left (3 \, x^{11} - 8 \, x^{7} + 3 \, x^{3}\right )}\right )} + 2}{2 \, {\left (x^{16} - 32 \, x^{12} + 66 \, x^{8} - 32 \, x^{4} + 1\right )}}\right ) - 4 \cdot 2^{\frac {1}{4}} {\left (x^{4} - 1\right )} \arctan \left (\frac {2 \, x^{16} + 4 \, x^{8} - \sqrt {2} {\left (2^{\frac {3}{4}} {\left (x^{16} - 20 \, x^{12} + 34 \, x^{8} - 20 \, x^{4} + 1\right )} - 8 \, {\left (x^{11} + x^{3} + 4 \, \sqrt {2} {\left (x^{9} - x^{5}\right )}\right )} \sqrt {x^{4} - 1} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{14} - 9 \, x^{10} + 9 \, x^{6} - x^{2}\right )}\right )} \sqrt {\frac {8 \, x^{6} - 8 \, x^{2} + \sqrt {2} {\left (x^{8} + 1\right )} - 4 \, \sqrt {x^{4} - 1} {\left (2^{\frac {3}{4}} x^{3} + 2^{\frac {1}{4}} {\left (x^{5} - x\right )}\right )}}{x^{8} + 1}} + 8 \, \sqrt {2} {\left (x^{14} - x^{10} + x^{6} - x^{2}\right )} - 4 \, \sqrt {x^{4} - 1} {\left (2^{\frac {3}{4}} {\left (x^{13} - 9 \, x^{9} + 9 \, x^{5} - x\right )} + 2 \cdot 2^{\frac {1}{4}} {\left (3 \, x^{11} - 8 \, x^{7} + 3 \, x^{3}\right )}\right )} + 2}{2 \, {\left (x^{16} - 32 \, x^{12} + 66 \, x^{8} - 32 \, x^{4} + 1\right )}}\right ) + 2^{\frac {1}{4}} {\left (x^{4} - 1\right )} \log \left (\frac {8 \, {\left (8 \, x^{6} - 8 \, x^{2} + \sqrt {2} {\left (x^{8} + 1\right )} + 4 \, \sqrt {x^{4} - 1} {\left (2^{\frac {3}{4}} x^{3} + 2^{\frac {1}{4}} {\left (x^{5} - x\right )}\right )}\right )}}{x^{8} + 1}\right ) - 2^{\frac {1}{4}} {\left (x^{4} - 1\right )} \log \left (\frac {8 \, {\left (8 \, x^{6} - 8 \, x^{2} + \sqrt {2} {\left (x^{8} + 1\right )} - 4 \, \sqrt {x^{4} - 1} {\left (2^{\frac {3}{4}} x^{3} + 2^{\frac {1}{4}} {\left (x^{5} - x\right )}\right )}\right )}}{x^{8} + 1}\right ) + 4 \, {\left (x^{4} - 1\right )} \arctan \left (\frac {\sqrt {x^{4} - 1} x}{x^{2} + 1}\right ) + 2 \, {\left (x^{4} - 1\right )} \log \left (\frac {x^{4} + 2 \, x^{2} - 2 \, \sqrt {x^{4} - 1} x - 1}{x^{4} + 1}\right ) - 8 \, \sqrt {x^{4} - 1} x}{64 \, {\left (x^{4} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(4*2^(1/4)*(x^4 - 1)*arctan(1/2*(2*x^16 + 4*x^8 + sqrt(2)*(2^(3/4)*(x^16 - 20*x^12 + 34*x^8 - 20*x^4 + 1)
+ 8*(x^11 + x^3 + 4*sqrt(2)*(x^9 - x^5))*sqrt(x^4 - 1) + 4*2^(1/4)*(x^14 - 9*x^10 + 9*x^6 - x^2))*sqrt((8*x^6
- 8*x^2 + sqrt(2)*(x^8 + 1) + 4*sqrt(x^4 - 1)*(2^(3/4)*x^3 + 2^(1/4)*(x^5 - x)))/(x^8 + 1)) + 8*sqrt(2)*(x^14
- x^10 + x^6 - x^2) + 4*sqrt(x^4 - 1)*(2^(3/4)*(x^13 - 9*x^9 + 9*x^5 - x) + 2*2^(1/4)*(3*x^11 - 8*x^7 + 3*x^3
)) + 2)/(x^16 - 32*x^12 + 66*x^8 - 32*x^4 + 1)) - 4*2^(1/4)*(x^4 - 1)*arctan(1/2*(2*x^16 + 4*x^8 - sqrt(2)*(2^
(3/4)*(x^16 - 20*x^12 + 34*x^8 - 20*x^4 + 1) - 8*(x^11 + x^3 + 4*sqrt(2)*(x^9 - x^5))*sqrt(x^4 - 1) + 4*2^(1/4
)*(x^14 - 9*x^10 + 9*x^6 - x^2))*sqrt((8*x^6 - 8*x^2 + sqrt(2)*(x^8 + 1) - 4*sqrt(x^4 - 1)*(2^(3/4)*x^3 + 2^(1
/4)*(x^5 - x)))/(x^8 + 1)) + 8*sqrt(2)*(x^14 - x^10 + x^6 - x^2) - 4*sqrt(x^4 - 1)*(2^(3/4)*(x^13 - 9*x^9 + 9*
x^5 - x) + 2*2^(1/4)*(3*x^11 - 8*x^7 + 3*x^3)) + 2)/(x^16 - 32*x^12 + 66*x^8 - 32*x^4 + 1)) + 2^(1/4)*(x^4 - 1
)*log(8*(8*x^6 - 8*x^2 + sqrt(2)*(x^8 + 1) + 4*sqrt(x^4 - 1)*(2^(3/4)*x^3 + 2^(1/4)*(x^5 - x)))/(x^8 + 1)) - 2
^(1/4)*(x^4 - 1)*log(8*(8*x^6 - 8*x^2 + sqrt(2)*(x^8 + 1) - 4*sqrt(x^4 - 1)*(2^(3/4)*x^3 + 2^(1/4)*(x^5 - x)))
/(x^8 + 1)) + 4*(x^4 - 1)*arctan(sqrt(x^4 - 1)*x/(x^2 + 1)) + 2*(x^4 - 1)*log((x^4 + 2*x^2 - 2*sqrt(x^4 - 1)*x
- 1)/(x^4 + 1)) - 8*sqrt(x^4 - 1)*x)/(x^4 - 1)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{8}}{{\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^8/(x^4-1)^(1/2)/(x^16-1),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [C]  time = 1.08, size = 214, normalized size = 1.25

method	result	size
risch	\(-\frac {x}{8 \sqrt {x^{4}-1}}+\frac {i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticF \left (i x , i\right )}{8 \sqrt {x^{4}-1}}+\frac {i \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}+1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-\sqrt {2}\, \arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{2}+x^{2}\right ) \sqrt {-2}}{2 \sqrt {x^{4}-1}}\right )-\frac {4 \underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticPi \left (i x , \underline {\hspace {1.25 ex}}\alpha ^{2}, i\right )}{\sqrt {x^{4}-1}}\right )\right )}{64}-\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{8}+1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-\frac {\arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{6}+x^{2}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{4}-1}\, \sqrt {x^{4}-1}}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{4}-1}}-\frac {2 i \underline {\hspace {1.25 ex}}\alpha ^{7} \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticPi \left (i x , \underline {\hspace {1.25 ex}}\alpha ^{6}, i\right )}{\sqrt {x^{4}-1}}\right )\right )}{32}\)	\(214\)
elliptic	\(\frac {\left (\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {x^{4}-1}}{x}+1\right )}{32}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {x^{4}-1}}{x}-1\right )}{32}+\frac {\sqrt {2}\, \ln \left (\frac {\frac {x^{4}-1}{2 x^{2}}-\frac {\sqrt {x^{4}-1}}{x}+1}{\frac {x^{4}-1}{2 x^{2}}+\frac {\sqrt {x^{4}-1}}{x}+1}\right )}{64}-\frac {\sqrt {2}\, x}{8 \sqrt {x^{4}-1}}-\frac {2^{\frac {3}{4}} \arctan \left (\frac {\sqrt {x^{4}-1}\, 2^{\frac {1}{4}}}{x}+1\right )}{16}-\frac {2^{\frac {3}{4}} \arctan \left (\frac {\sqrt {x^{4}-1}\, 2^{\frac {1}{4}}}{x}-1\right )}{16}-\frac {2^{\frac {3}{4}} \ln \left (\frac {\frac {x^{4}-1}{2 x^{2}}-\frac {2^{\frac {3}{4}} \sqrt {x^{4}-1}}{2 x}+\frac {\sqrt {2}}{2}}{\frac {x^{4}-1}{2 x^{2}}+\frac {2^{\frac {3}{4}} \sqrt {x^{4}-1}}{2 x}+\frac {\sqrt {2}}{2}}\right )}{32}\right ) \sqrt {2}}{2}\)	\(229\)
default	\(-\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{8}+1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-\frac {\arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{6}+x^{2}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{4}-1}\, \sqrt {x^{4}-1}}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{4}-1}}-\frac {2 i \underline {\hspace {1.25 ex}}\alpha ^{7} \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticPi \left (i x , \underline {\hspace {1.25 ex}}\alpha ^{6}, i\right )}{\sqrt {x^{4}-1}}\right )\right )}{32}+\frac {\left (x^{2}-1\right ) x}{16 \sqrt {\left (x^{2}+1\right ) \left (x^{2}-1\right )}}+\frac {i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticF \left (i x , i\right )}{8 \sqrt {x^{4}-1}}-\frac {x^{3}-x^{2}+x -1}{32 \sqrt {\left (1+x \right ) \left (x^{3}-x^{2}+x -1\right )}}-\frac {x^{3}+x^{2}+x +1}{32 \sqrt {\left (-1+x \right ) \left (x^{3}+x^{2}+x +1\right )}}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}+1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-i \sqrt {2}\, \arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{2}+x^{2}\right ) \sqrt {-2}}{2 \sqrt {x^{4}-1}}\right )-\frac {4 i \underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticPi \left (i x , \underline {\hspace {1.25 ex}}\alpha ^{2}, i\right )}{\sqrt {x^{4}-1}}\right )\right )}{64}\)	\(282\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*x/(x^4-1)^(1/2)+1/8*I*(x^2+1)^(1/2)*(-x^2+1)^(1/2)/(x^4-1)^(1/2)*EllipticF(I*x,I)+1/64*I*sum(_alpha*(-2^(
1/2)*arctanh(1/2*_alpha^2*(_alpha^2+x^2)*(-2)^(1/2)/(x^4-1)^(1/2))-4*_alpha^3*(x^2+1)^(1/2)*(-x^2+1)^(1/2)/(x^
4-1)^(1/2)*EllipticPi(I*x,_alpha^2,I)),_alpha=RootOf(_Z^4+1))-1/32*sum(_alpha*(-1/(_alpha^4-1)^(1/2)*arctanh(_
alpha^2*(_alpha^6+x^2)/(_alpha^4-1)^(1/2)/(x^4-1)^(1/2))-2*I*_alpha^7*(x^2+1)^(1/2)*(-x^2+1)^(1/2)/(x^4-1)^(1/
2)*EllipticPi(I*x,_alpha^6,I)),_alpha=RootOf(_Z^8+1))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{8}}{{\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^8/(x^4-1)^(1/2)/(x^16-1),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^8}{\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^8/((x^4 - 1)^(1/2)*(x^16 - 1)),x)

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________