\(\int \frac {x^8}{\sqrt {-b^4+a^4 x^4} (-b^{16}+a^{16} x^{16})} \, dx\) [2864]

Optimal result

Integrand size = 36, antiderivative size = 303 \[ \int \frac {x^8}{\sqrt {-b^4+a^4 x^4} \left (-b^{16}+a^{16} x^{16}\right )} \, dx=-\frac {x}{8 a^8 b^8 \sqrt {-b^4+a^4 x^4}}-\frac {\arctan \left (\frac {2^{3/4} a b x \sqrt {-b^4+a^4 x^4}}{b^4+\sqrt {2} a^2 b^2 x^2-a^4 x^4}\right )}{8\ 2^{3/4} a^9 b^9}-\frac {\arctan \left (\frac {\frac {b^3}{2 a}+a b x^2-\frac {a^3 x^4}{2 b}}{x \sqrt {-b^4+a^4 x^4}}\right )}{32 a^9 b^9}+\frac {\text {arctanh}\left (\frac {\frac {b^3}{2 a}-a b x^2-\frac {a^3 x^4}{2 b}}{x \sqrt {-b^4+a^4 x^4}}\right )}{32 a^9 b^9}-\frac {\text {arctanh}\left (\frac {\frac {b^3}{2^{3/4} a}-\frac {a b x^2}{\sqrt [4]{2}}-\frac {a^3 x^4}{2^{3/4} b}}{x \sqrt {-b^4+a^4 x^4}}\right )}{8\ 2^{3/4} a^9 b^9} \]

[Out]

-1/8*x/a^8/b^8/(a^4*x^4-b^4)^(1/2)-1/16*arctan(2^(3/4)*a*b*x*(a^4*x^4-b^4)^(1/2)/(b^4+2^(1/2)*a^2*b^2*x^2-a^4*
x^4))*2^(1/4)/a^9/b^9-1/32*arctan((1/2*b^3/a+a*b*x^2-1/2*a^3*x^4/b)/x/(a^4*x^4-b^4)^(1/2))/a^9/b^9+1/32*arctan
h((1/2*b^3/a-a*b*x^2-1/2*a^3*x^4/b)/x/(a^4*x^4-b^4)^(1/2))/a^9/b^9-1/16*arctanh((1/2*b^3*2^(1/4)/a-1/2*a*b*x^2
*2^(3/4)-1/2*a^3*x^4*2^(1/4)/b)/x/(a^4*x^4-b^4)^(1/2))*2^(1/4)/a^9/b^9

Rubi [C] (verified)

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

Time = 0.82 (sec) , antiderivative size = 522, normalized size of antiderivative = 1.72, number of steps used = 40, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.528, Rules used = {6857, 1166, 425, 21, 434, 438, 437, 435, 259, 230, 227, 418, 1225, 1713, 209, 212, 1443, 1233, 1232} \[ \int \frac {x^8}{\sqrt {-b^4+a^4 x^4} \left (-b^{16}+a^{16} x^{16}\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {a^4 x^4-b^4}}\right )}{16 \sqrt {2} \left (-a^4\right )^{9/4} b^9}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {a^4 x^4-b^4}}\right )}{16 \sqrt {2} \left (-a^4\right )^{9/4} b^9}-\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {a x}{b}\right ),-1\right )}{4 a^9 b^7 \sqrt {a^4 x^4-b^4}}-\frac {x \left (b^2-a^2 x^2\right )}{16 a^8 b^{10} \sqrt {a^4 x^4-b^4}}-\frac {x \left (a^2 x^2+b^2\right )}{16 a^8 b^{10} \sqrt {a^4 x^4-b^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {a^6}{\left (-a^8\right )^{3/4}},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{8 a^9 b^7 \sqrt {a^4 x^4-b^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {\sqrt [4]{-a^8}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{8 a^9 b^7 \sqrt {a^4 x^4-b^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (-\frac {\sqrt {-\sqrt {-a^8}}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{8 a^9 b^7 \sqrt {a^4 x^4-b^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {\sqrt {-\sqrt {-a^8}}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{8 a^9 b^7 \sqrt {a^4 x^4-b^4}} \]

[In]

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

[Out]

-1/16*(x*(b^2 - a^2*x^2))/(a^8*b^10*Sqrt[-b^4 + a^4*x^4]) - (x*(b^2 + a^2*x^2))/(16*a^8*b^10*Sqrt[-b^4 + a^4*x
^4]) - ArcTan[(Sqrt[2]*(-a^4)^(1/4)*b*x)/Sqrt[-b^4 + a^4*x^4]]/(16*Sqrt[2]*(-a^4)^(9/4)*b^9) - ArcTanh[(Sqrt[2
]*(-a^4)^(1/4)*b*x)/Sqrt[-b^4 + a^4*x^4]]/(16*Sqrt[2]*(-a^4)^(9/4)*b^9) - (Sqrt[1 - (a^4*x^4)/b^4]*EllipticF[A
rcSin[(a*x)/b], -1])/(4*a^9*b^7*Sqrt[-b^4 + a^4*x^4]) + (Sqrt[1 - (a^4*x^4)/b^4]*EllipticPi[a^6/(-a^8)^(3/4),
ArcSin[(a*x)/b], -1])/(8*a^9*b^7*Sqrt[-b^4 + a^4*x^4]) + (Sqrt[1 - (a^4*x^4)/b^4]*EllipticPi[(-a^8)^(1/4)/a^2,
ArcSin[(a*x)/b], -1])/(8*a^9*b^7*Sqrt[-b^4 + a^4*x^4]) + (Sqrt[1 - (a^4*x^4)/b^4]*EllipticPi[-(Sqrt[-Sqrt[-a^
8]]/a^2), ArcSin[(a*x)/b], -1])/(8*a^9*b^7*Sqrt[-b^4 + a^4*x^4]) + (Sqrt[1 - (a^4*x^4)/b^4]*EllipticPi[Sqrt[-S
qrt[-a^8]]/a^2, ArcSin[(a*x)/b], -1])/(8*a^9*b^7*Sqrt[-b^4 + a^4*x^4])

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*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 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 227

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

Rule 230

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Dist[Sqrt[1 + b*(x^4/a)]/Sqrt[a + b*x^4], Int[1/Sqrt[1 + b*(x^4/
a)], x], x] /; FreeQ[{a, b}, x] && NegQ[b/a] &&  !GtQ[a, 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 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 425

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]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]

Rule 434

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 435

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

Rule 437

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

Rule 438

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

Rule 1166

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/e)*x^2)^p, x], x] /; FreeQ[{
a, c, d, e, p, q}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p]

Rule 1225

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 1232

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

Rule 1233

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

Rule 1443

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 1713

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 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}& = \int \left (\frac {1}{8 a^8 b^6 \left (-b^2+a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}}-\frac {1}{8 a^8 b^6 \left (b^2+a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}}-\frac {1}{4 a^8 b^4 \sqrt {-b^4+a^4 x^4} \left (b^4+a^4 x^4\right )}+\frac {1}{2 a^8 \sqrt {-b^4+a^4 x^4} \left (b^8+a^8 x^8\right )}\right ) \, dx \\ & = \frac {\int \frac {1}{\sqrt {-b^4+a^4 x^4} \left (b^8+a^8 x^8\right )} \, dx}{2 a^8}+\frac {\int \frac {1}{\left (-b^2+a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}} \, dx}{8 a^8 b^6}-\frac {\int \frac {1}{\left (b^2+a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}} \, dx}{8 a^8 b^6}-\frac {\int \frac {1}{\sqrt {-b^4+a^4 x^4} \left (b^4+a^4 x^4\right )} \, dx}{4 a^8 b^4} \\ & = -\frac {\int \frac {1}{\left (1-\frac {\sqrt {-a^4} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx}{8 a^8 b^8}-\frac {\int \frac {1}{\left (1+\frac {\sqrt {-a^4} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx}{8 a^8 b^8}-\frac {\int \frac {1}{\sqrt {-b^4+a^4 x^4} \left (\sqrt {-a^8} b^4-a^8 x^4\right )} \, dx}{4 \sqrt {-a^8} b^4}-\frac {\int \frac {1}{\sqrt {-b^4+a^4 x^4} \left (\sqrt {-a^8} b^4+a^8 x^4\right )} \, dx}{4 \sqrt {-a^8} b^4}-\frac {\left (\sqrt {-b^2+a^2 x^2} \sqrt {b^2+a^2 x^2}\right ) \int \frac {1}{\sqrt {-b^2+a^2 x^2} \left (b^2+a^2 x^2\right )^{3/2}} \, dx}{8 a^8 b^6 \sqrt {-b^4+a^4 x^4}}+\frac {\left (\sqrt {-b^2+a^2 x^2} \sqrt {b^2+a^2 x^2}\right ) \int \frac {1}{\left (-b^2+a^2 x^2\right )^{3/2} \sqrt {b^2+a^2 x^2}} \, dx}{8 a^8 b^6 \sqrt {-b^4+a^4 x^4}} \\ & = -\frac {x \left (b^2-a^2 x^2\right )}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}-\frac {x \left (b^2+a^2 x^2\right )}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}-2 \frac {\int \frac {1}{\sqrt {-b^4+a^4 x^4}} \, dx}{16 a^8 b^8}-\frac {\int \frac {1-\frac {\sqrt {-a^4} x^2}{b^2}}{\left (1+\frac {\sqrt {-a^4} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx}{16 a^8 b^8}-\frac {\int \frac {1+\frac {\sqrt {-a^4} x^2}{b^2}}{\left (1-\frac {\sqrt {-a^4} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx}{16 a^8 b^8}+\frac {\int \frac {1}{\left (1-\frac {\sqrt [4]{-a^8} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx}{8 a^8 b^8}+\frac {\int \frac {1}{\left (1+\frac {\sqrt [4]{-a^8} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx}{8 a^8 b^8}+\frac {\int \frac {1}{\left (1-\frac {\sqrt {-\sqrt {-a^8}} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx}{8 a^8 b^8}+\frac {\int \frac {1}{\left (1+\frac {\sqrt {-\sqrt {-a^8}} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx}{8 a^8 b^8}+\frac {\left (\sqrt {-b^2+a^2 x^2} \sqrt {b^2+a^2 x^2}\right ) \int \frac {-a^2 b^2+a^4 x^2}{\sqrt {-b^2+a^2 x^2} \sqrt {b^2+a^2 x^2}} \, dx}{16 a^{10} b^{10} \sqrt {-b^4+a^4 x^4}}-\frac {\left (\sqrt {-b^2+a^2 x^2} \sqrt {b^2+a^2 x^2}\right ) \int \frac {a^2 b^2+a^4 x^2}{\sqrt {-b^2+a^2 x^2} \sqrt {b^2+a^2 x^2}} \, dx}{16 a^{10} b^{10} \sqrt {-b^4+a^4 x^4}} \\ & = -\frac {x \left (b^2-a^2 x^2\right )}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}-\frac {x \left (b^2+a^2 x^2\right )}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}-\frac {\text {Subst}\left (\int \frac {1}{1-2 \sqrt {-a^4} b^2 x^2} \, dx,x,\frac {x}{\sqrt {-b^4+a^4 x^4}}\right )}{16 a^8 b^8}-\frac {\text {Subst}\left (\int \frac {1}{1+2 \sqrt {-a^4} b^2 x^2} \, dx,x,\frac {x}{\sqrt {-b^4+a^4 x^4}}\right )}{16 a^8 b^8}+\frac {\left (\sqrt {-b^2+a^2 x^2} \sqrt {b^2+a^2 x^2}\right ) \int \frac {\sqrt {-b^2+a^2 x^2}}{\sqrt {b^2+a^2 x^2}} \, dx}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}-\frac {\left (\sqrt {-b^2+a^2 x^2} \sqrt {b^2+a^2 x^2}\right ) \int \frac {\sqrt {b^2+a^2 x^2}}{\sqrt {-b^2+a^2 x^2}} \, dx}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}-2 \frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{16 a^8 b^8 \sqrt {-b^4+a^4 x^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\left (1-\frac {\sqrt [4]{-a^8} x^2}{b^2}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{8 a^8 b^8 \sqrt {-b^4+a^4 x^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\left (1+\frac {\sqrt [4]{-a^8} x^2}{b^2}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{8 a^8 b^8 \sqrt {-b^4+a^4 x^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\left (1-\frac {\sqrt {-\sqrt {-a^8}} x^2}{b^2}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{8 a^8 b^8 \sqrt {-b^4+a^4 x^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\left (1+\frac {\sqrt {-\sqrt {-a^8}} x^2}{b^2}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{8 a^8 b^8 \sqrt {-b^4+a^4 x^4}} \\ & = -\frac {x \left (b^2-a^2 x^2\right )}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}-\frac {x \left (b^2+a^2 x^2\right )}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{16 \sqrt {2} \left (-a^4\right )^{9/4} b^9}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{16 \sqrt {2} \left (-a^4\right )^{9/4} b^9}-\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {a x}{b}\right ),-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {a^6}{\left (-a^8\right )^{3/4}},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {\sqrt [4]{-a^8}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (-\frac {\sqrt {-\sqrt {-a^8}}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {\sqrt {-\sqrt {-a^8}}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\left (\sqrt {-b^2+a^2 x^2} \sqrt {b^2+a^2 x^2}\right ) \int \frac {\sqrt {b^2+a^2 x^2}}{\sqrt {-b^2+a^2 x^2}} \, dx}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}-\frac {\left (\sqrt {-b^2+a^2 x^2} \sqrt {b^2+a^2 x^2}\right ) \int \frac {1}{\sqrt {-b^2+a^2 x^2} \sqrt {b^2+a^2 x^2}} \, dx}{8 a^8 b^8 \sqrt {-b^4+a^4 x^4}}-\frac {\left (\sqrt {b^2+a^2 x^2} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \int \frac {\sqrt {b^2+a^2 x^2}}{\sqrt {1-\frac {a^2 x^2}{b^2}}} \, dx}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}} \\ & = -\frac {x \left (b^2-a^2 x^2\right )}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}-\frac {x \left (b^2+a^2 x^2\right )}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{16 \sqrt {2} \left (-a^4\right )^{9/4} b^9}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{16 \sqrt {2} \left (-a^4\right )^{9/4} b^9}-\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {a x}{b}\right ),-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {a^6}{\left (-a^8\right )^{3/4}},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {\sqrt [4]{-a^8}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (-\frac {\sqrt {-\sqrt {-a^8}}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {\sqrt {-\sqrt {-a^8}}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}-\frac {\int \frac {1}{\sqrt {-b^4+a^4 x^4}} \, dx}{8 a^8 b^8}+\frac {\left (\sqrt {b^2+a^2 x^2} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \int \frac {\sqrt {b^2+a^2 x^2}}{\sqrt {1-\frac {a^2 x^2}{b^2}}} \, dx}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}-\frac {\left (\left (b^2+a^2 x^2\right ) \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \int \frac {\sqrt {1+\frac {a^2 x^2}{b^2}}}{\sqrt {1-\frac {a^2 x^2}{b^2}}} \, dx}{16 a^8 b^{10} \sqrt {1+\frac {a^2 x^2}{b^2}} \sqrt {-b^4+a^4 x^4}} \\ & = -\frac {x \left (b^2-a^2 x^2\right )}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}-\frac {x \left (b^2+a^2 x^2\right )}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{16 \sqrt {2} \left (-a^4\right )^{9/4} b^9}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{16 \sqrt {2} \left (-a^4\right )^{9/4} b^9}-\frac {\left (b^2+a^2 x^2\right ) \sqrt {1-\frac {a^2 x^2}{b^2}} E\left (\left .\arcsin \left (\frac {a x}{b}\right )\right |-1\right )}{16 a^9 b^9 \sqrt {1+\frac {a^2 x^2}{b^2}} \sqrt {-b^4+a^4 x^4}}-\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {a x}{b}\right ),-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {a^6}{\left (-a^8\right )^{3/4}},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {\sqrt [4]{-a^8}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (-\frac {\sqrt {-\sqrt {-a^8}}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {\sqrt {-\sqrt {-a^8}}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\left (\left (b^2+a^2 x^2\right ) \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \int \frac {\sqrt {1+\frac {a^2 x^2}{b^2}}}{\sqrt {1-\frac {a^2 x^2}{b^2}}} \, dx}{16 a^8 b^{10} \sqrt {1+\frac {a^2 x^2}{b^2}} \sqrt {-b^4+a^4 x^4}}-\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{8 a^8 b^8 \sqrt {-b^4+a^4 x^4}} \\ & = -\frac {x \left (b^2-a^2 x^2\right )}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}-\frac {x \left (b^2+a^2 x^2\right )}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{16 \sqrt {2} \left (-a^4\right )^{9/4} b^9}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{16 \sqrt {2} \left (-a^4\right )^{9/4} b^9}-\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {a x}{b}\right ),-1\right )}{4 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {a^6}{\left (-a^8\right )^{3/4}},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {\sqrt [4]{-a^8}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (-\frac {\sqrt {-\sqrt {-a^8}}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \operatorname {EllipticPi}\left (\frac {\sqrt {-\sqrt {-a^8}}}{a^2},\arcsin \left (\frac {a x}{b}\right ),-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 2.55 (sec) , antiderivative size = 594, normalized size of antiderivative = 1.96 \[ \int \frac {x^8}{\sqrt {-b^4+a^4 x^4} \left (-b^{16}+a^{16} x^{16}\right )} \, dx=-\frac {\frac {4 a b x}{\sqrt {-b^4+a^4 x^4}}+(2-2 i) \arctan \left (\frac {(1+i) a b x}{i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}}\right )+(1+i) \arctan \left (\frac {i b^4+(1-i) a b^3 x-(1+i) a^3 b x^3-i a^4 x^4+\left (b^2-(1+i) a b x-i a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}}{i b^4-(1-i) a b^3 x+(1+i) a^3 b x^3-i a^4 x^4+\left (b^2+(1+i) a b x-i a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}}\right )-2 a b \text {RootSum}\left [16 a^8 b^8+32 i a^6 b^6 \text {$\#$1}^2+8 a^4 b^4 \text {$\#$1}^4-8 i a^2 b^2 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-8 a^6 b^6 \log (x)+8 a^6 b^6 \log \left (i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}-x \text {$\#$1}\right )-4 i a^4 b^4 \log (x) \text {$\#$1}^2+4 i a^4 b^4 \log \left (i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2-2 a^2 b^2 \log (x) \text {$\#$1}^4+2 a^2 b^2 \log \left (i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4-i \log (x) \text {$\#$1}^6+i \log \left (i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^6}{8 a^6 b^6 \text {$\#$1}-4 i a^4 b^4 \text {$\#$1}^3-6 a^2 b^2 \text {$\#$1}^5-i \text {$\#$1}^7}\&\right ]}{32 a^9 b^9} \]

[In]

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

[Out]

-1/32*((4*a*b*x)/Sqrt[-b^4 + a^4*x^4] + (2 - 2*I)*ArcTan[((1 + I)*a*b*x)/(I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^
4])] + (1 + I)*ArcTan[(I*b^4 + (1 - I)*a*b^3*x - (1 + I)*a^3*b*x^3 - I*a^4*x^4 + (b^2 - (1 + I)*a*b*x - I*a^2*
x^2)*Sqrt[-b^4 + a^4*x^4])/(I*b^4 - (1 - I)*a*b^3*x + (1 + I)*a^3*b*x^3 - I*a^4*x^4 + (b^2 + (1 + I)*a*b*x - I
*a^2*x^2)*Sqrt[-b^4 + a^4*x^4])] - 2*a*b*RootSum[16*a^8*b^8 + (32*I)*a^6*b^6*#1^2 + 8*a^4*b^4*#1^4 - (8*I)*a^2
*b^2*#1^6 + #1^8 & , (-8*a^6*b^6*Log[x] + 8*a^6*b^6*Log[I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4] - x*#1] - (4*I)
*a^4*b^4*Log[x]*#1^2 + (4*I)*a^4*b^4*Log[I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4] - x*#1]*#1^2 - 2*a^2*b^2*Log[x
]*#1^4 + 2*a^2*b^2*Log[I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4] - x*#1]*#1^4 - I*Log[x]*#1^6 + I*Log[I*b^2 + a^2
*x^2 + Sqrt[-b^4 + a^4*x^4] - x*#1]*#1^6)/(8*a^6*b^6*#1 - (4*I)*a^4*b^4*#1^3 - 6*a^2*b^2*#1^5 - I*#1^7) & ])/(
a^9*b^9)

Maple [A] (verified)

Time = 5.38 (sec) , antiderivative size = 492, normalized size of antiderivative = 1.62

method	result	size
elliptic	\(\frac {\left (-\frac {\ln \left (\frac {\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}-\frac {\sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}\, \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2}\, \sqrt {a^{4} b^{4}}}{2}}{\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}+\frac {\sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}\, \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2}\, \sqrt {a^{4} b^{4}}}{2}}\right )+2 \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{\sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}\, x}+1\right )+2 \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{\sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}\, x}-1\right )}{16 a^{8} b^{8} \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}-\frac {\sqrt {2}\, x}{8 a^{8} b^{8} \sqrt {a^{4} x^{4}-b^{4}}}+\frac {\sqrt {2}\, \left (\ln \left (\frac {\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}-\frac {\left (a^{4} b^{4}\right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}}{x}+\sqrt {a^{4} b^{4}}}{\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}+\frac {\left (a^{4} b^{4}\right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}}{x}+\sqrt {a^{4} b^{4}}}\right )+2 \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}}{\left (a^{4} b^{4}\right )^{\frac {1}{4}} x}+1\right )+2 \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}}{\left (a^{4} b^{4}\right )^{\frac {1}{4}} x}-1\right )\right )}{64 a^{8} b^{8} \left (a^{4} b^{4}\right )^{\frac {1}{4}}}\right ) \sqrt {2}}{2}\)	\(492\)
pseudoelliptic	\(-\frac {i \left (2 \left (a^{4} x^{4}-b^{4}\right ) \sqrt {i a^{2} b^{2}}\, \sqrt {-i a^{2} b^{2}}\, \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (16 a^{8} b^{8}+32 i a^{6} b^{6} \textit {\_Z}^{2}+8 b^{4} \textit {\_Z}^{4} a^{4}-8 i a^{2} b^{2} \textit {\_Z}^{6}+\textit {\_Z}^{8}\right )}{\sum }\frac {\left (8 i a^{6} b^{6}-4 a^{4} b^{4} \textit {\_R}^{2}+2 i a^{2} b^{2} \textit {\_R}^{4}-\textit {\_R}^{6}\right ) \ln \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}+\left (-a^{2} x^{2}-i b^{2}\right ) \operatorname {csgn}\left (a^{2}\right )-\textit {\_R} x}{x}\right )}{\textit {\_R} \left (-8 i a^{6} b^{6}-4 a^{4} b^{4} \textit {\_R}^{2}+6 i a^{2} b^{2} \textit {\_R}^{4}-\textit {\_R}^{6}\right )}\right )+\frac {\sqrt {2}\, \sqrt {-i a^{2} b^{2}}\, \left (a^{4} x^{4}-b^{4}\right ) \ln \left (\frac {a^{2} \left (-2 i a^{2} b^{2} x +2 \sqrt {i a^{2} b^{2}}\, a^{2} x^{2}+2 i \sqrt {i a^{2} b^{2}}\, b^{2}+\sqrt {2}\, \sqrt {i a^{2} b^{2}}\, \sqrt {a^{4} x^{4}-b^{4}}\right )}{a^{2} x^{2}+i b^{2}-2 \sqrt {i a^{2} b^{2}}\, x}\right )}{2}+\frac {\sqrt {2}\, \sqrt {-i a^{2} b^{2}}\, \left (a^{4} x^{4}-b^{4}\right ) \ln \left (-\frac {2 a^{2} \left (-\frac {\sqrt {2}\, \sqrt {i a^{2} b^{2}}\, \sqrt {a^{4} x^{4}-b^{4}}}{2}+\left (a^{2} x^{2}+i b^{2}\right ) \sqrt {i a^{2} b^{2}}+i a^{2} b^{2} x \right )}{a^{2} x^{2}+i b^{2}+2 \sqrt {i a^{2} b^{2}}\, x}\right )}{2}+\sqrt {i a^{2} b^{2}}\, \sqrt {2}\, \left (a^{4} x^{4}-b^{4}\right ) \ln \left (\frac {\left (-2 i a^{2} b^{2} x +\sqrt {2}\, \sqrt {-i a^{2} b^{2}}\, \sqrt {a^{4} x^{4}-b^{4}}\right ) a^{2}}{a^{2} x^{2}+i b^{2}}\right )+4 \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {i a^{2} b^{2}}\, \sqrt {-i a^{2} b^{2}}\, x +\ln \left (2\right ) \sqrt {2}\, \left (a x -b \right ) \left (a x +b \right ) \left (a^{2} x^{2}+b^{2}\right ) \left (\sqrt {i a^{2} b^{2}}+\sqrt {-i a^{2} b^{2}}\right )\right )}{16 \sqrt {i a^{2} b^{2}}\, \sqrt {-i a^{2} b^{2}}\, \left (-2 \sqrt {i a^{2} b^{2}}+\left (1+i\right ) a b \right ) a^{6} \left (-a x +i b \right ) \left (2 \sqrt {i a^{2} b^{2}}+\left (1+i\right ) a b \right ) \left (a x -b \right ) \left (a x +b \right ) b^{6} \left (a x +i b \right )}\)	\(785\)
default	\(\text {Expression too large to display}\)	\(1130\)

[In]

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

[Out]

1/2*(-1/16/a^8/b^8/(2^(1/2)*(a^4*b^4)^(1/2))^(1/2)*(ln((1/2*(a^4*x^4-b^4)/x^2-1/2*(2^(1/2)*(a^4*b^4)^(1/2))^(1
/2)*(a^4*x^4-b^4)^(1/2)*2^(1/2)/x+1/2*2^(1/2)*(a^4*b^4)^(1/2))/(1/2*(a^4*x^4-b^4)/x^2+1/2*(2^(1/2)*(a^4*b^4)^(
1/2))^(1/2)*(a^4*x^4-b^4)^(1/2)*2^(1/2)/x+1/2*2^(1/2)*(a^4*b^4)^(1/2)))+2*arctan(1/(2^(1/2)*(a^4*b^4)^(1/2))^(
1/2)*(a^4*x^4-b^4)^(1/2)*2^(1/2)/x+1)+2*arctan(1/(2^(1/2)*(a^4*b^4)^(1/2))^(1/2)*(a^4*x^4-b^4)^(1/2)*2^(1/2)/x
-1))-1/8/a^8/b^8/(a^4*x^4-b^4)^(1/2)*2^(1/2)*x+1/64/a^8/b^8/(a^4*b^4)^(1/4)*2^(1/2)*(ln((1/2*(a^4*x^4-b^4)/x^2
-(a^4*b^4)^(1/4)*(a^4*x^4-b^4)^(1/2)/x+(a^4*b^4)^(1/2))/(1/2*(a^4*x^4-b^4)/x^2+(a^4*b^4)^(1/4)*(a^4*x^4-b^4)^(
1/2)/x+(a^4*b^4)^(1/2)))+2*arctan(1/(a^4*b^4)^(1/4)*(a^4*x^4-b^4)^(1/2)/x+1)+2*arctan(1/(a^4*b^4)^(1/4)*(a^4*x
^4-b^4)^(1/2)/x-1)))*2^(1/2)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 39.32 (sec) , antiderivative size = 917, normalized size of antiderivative = 3.03 \[ \int \frac {x^8}{\sqrt {-b^4+a^4 x^4} \left (-b^{16}+a^{16} x^{16}\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/32*(4*sqrt(a^4*x^4 - b^4)*a*b*x - (1/2)^(1/4)*(a^13*b^9*x^4 - a^9*b^13)*(-1/(a^36*b^36))^(1/4)*log(1/2*(4*(
1/2)^(3/4)*(a^32*b^28*x^6 - a^28*b^32*x^2)*(-1/(a^36*b^36))^(3/4) + 2*(2*sqrt(1/2)*a^20*b^20*x^3*sqrt(-1/(a^36
*b^36)) - a^4*x^5 + b^4*x)*sqrt(a^4*x^4 - b^4) - (1/2)^(1/4)*(a^16*b^8*x^8 - 4*a^12*b^12*x^4 + a^8*b^16)*(-1/(
a^36*b^36))^(1/4))/(a^8*x^8 + b^8)) + (1/2)^(1/4)*(a^13*b^9*x^4 - a^9*b^13)*(-1/(a^36*b^36))^(1/4)*log(-1/2*(4
*(1/2)^(3/4)*(a^32*b^28*x^6 - a^28*b^32*x^2)*(-1/(a^36*b^36))^(3/4) - 2*(2*sqrt(1/2)*a^20*b^20*x^3*sqrt(-1/(a^
36*b^36)) - a^4*x^5 + b^4*x)*sqrt(a^4*x^4 - b^4) - (1/2)^(1/4)*(a^16*b^8*x^8 - 4*a^12*b^12*x^4 + a^8*b^16)*(-1
/(a^36*b^36))^(1/4))/(a^8*x^8 + b^8)) + (1/2)^(1/4)*(-I*a^13*b^9*x^4 + I*a^9*b^13)*(-1/(a^36*b^36))^(1/4)*log(
-1/2*(4*(1/2)^(3/4)*(I*a^32*b^28*x^6 - I*a^28*b^32*x^2)*(-1/(a^36*b^36))^(3/4) + 2*(2*sqrt(1/2)*a^20*b^20*x^3*
sqrt(-1/(a^36*b^36)) + a^4*x^5 - b^4*x)*sqrt(a^4*x^4 - b^4) + (1/2)^(1/4)*(I*a^16*b^8*x^8 - 4*I*a^12*b^12*x^4
+ I*a^8*b^16)*(-1/(a^36*b^36))^(1/4))/(a^8*x^8 + b^8)) + (1/2)^(1/4)*(I*a^13*b^9*x^4 - I*a^9*b^13)*(-1/(a^36*b
^36))^(1/4)*log(-1/2*(4*(1/2)^(3/4)*(-I*a^32*b^28*x^6 + I*a^28*b^32*x^2)*(-1/(a^36*b^36))^(3/4) + 2*(2*sqrt(1/
2)*a^20*b^20*x^3*sqrt(-1/(a^36*b^36)) + a^4*x^5 - b^4*x)*sqrt(a^4*x^4 - b^4) + (1/2)^(1/4)*(-I*a^16*b^8*x^8 +
4*I*a^12*b^12*x^4 - I*a^8*b^16)*(-1/(a^36*b^36))^(1/4))/(a^8*x^8 + b^8)) - 2*(a^4*x^4 - b^4)*arctan(sqrt(a^4*x
^4 - b^4)*a*x/(a^2*b*x^2 + b^3)) - (a^4*x^4 - b^4)*log((a^4*x^4 + 2*a^2*b^2*x^2 - b^4 - 2*sqrt(a^4*x^4 - b^4)*
a*b*x)/(a^4*x^4 + b^4)))/(a^13*b^9*x^4 - a^9*b^13)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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