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

Optimal. Leaf size=303 \[ -\frac {x}{8 a^8 b^8 \sqrt {-b^4+a^4 x^4}}-\frac {\text {ArcTan}\left (\frac {2^{3/4} a b x \sqrt {-b^4+a^4 x^4}}{b^4+\sqrt {2} a^2 b^2 x^2-a^4 x^4}\right )}{8\ 2^{3/4} a^9 b^9}-\frac {\text {ArcTan}\left (\frac {\frac {b^3}{2 a}+a b x^2-\frac {a^3 x^4}{2 b}}{x \sqrt {-b^4+a^4 x^4}}\right )}{32 a^9 b^9}+\frac {\tanh ^{-1}\left (\frac {\frac {b^3}{2 a}-a b x^2-\frac {a^3 x^4}{2 b}}{x \sqrt {-b^4+a^4 x^4}}\right )}{32 a^9 b^9}-\frac {\tanh ^{-1}\left (\frac {\frac {b^3}{2^{3/4} a}-\frac {a b x^2}{\sqrt [4]{2}}-\frac {a^3 x^4}{2^{3/4} b}}{x \sqrt {-b^4+a^4 x^4}}\right )}{8\ 2^{3/4} a^9 b^9} \]

[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

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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 0.94, antiderivative size = 522, normalized size of antiderivative = 1.72, number of steps used = 40, number of rules used = 19, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.528, Rules used = {6857, 1166, 425, 21, 434, 438, 437, 435, 259, 230, 227, 418, 1225, 1713, 209, 212, 1443, 1233, 1232} \begin {gather*} -\frac {\text {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 {\tanh ^{-1}\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}} F\left (\left .\text {ArcSin}\left (\frac {a x}{b}\right )\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}} \Pi \left (\frac {a^6}{\left (-a^8\right )^{3/4}};\left .\text {ArcSin}\left (\frac {a x}{b}\right )\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}} \Pi \left (\frac {\sqrt [4]{-a^8}}{a^2};\left .\text {ArcSin}\left (\frac {a x}{b}\right )\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}} \Pi \left (-\frac {\sqrt {-\sqrt {-a^8}}}{a^2};\left .\text {ArcSin}\left (\frac {a x}{b}\right )\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}} \Pi \left (\frac {\sqrt {-\sqrt {-a^8}}}{a^2};\left .\text {ArcSin}\left (\frac {a x}{b}\right )\right |-1\right )}{8 a^9 b^7 \sqrt {a^4 x^4-b^4}} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {x^8}{\sqrt {-b^4+a^4 x^4} \left (-b^{16}+a^{16} x^{16}\right )} \, dx &=\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 {\tan ^{-1}\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 {\tanh ^{-1}\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}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\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}} \Pi \left (\frac {a^6}{\left (-a^8\right )^{3/4}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\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}} \Pi \left (\frac {\sqrt [4]{-a^8}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\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}} \Pi \left (-\frac {\sqrt {-\sqrt {-a^8}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\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}} \Pi \left (\frac {\sqrt {-\sqrt {-a^8}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\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 {\tan ^{-1}\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 {\tanh ^{-1}\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}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\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}} \Pi \left (\frac {a^6}{\left (-a^8\right )^{3/4}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\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}} \Pi \left (\frac {\sqrt [4]{-a^8}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\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}} \Pi \left (-\frac {\sqrt {-\sqrt {-a^8}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\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}} \Pi \left (\frac {\sqrt {-\sqrt {-a^8}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\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 {\tan ^{-1}\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 {\tanh ^{-1}\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 .\sin ^{-1}\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}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\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}} \Pi \left (\frac {a^6}{\left (-a^8\right )^{3/4}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\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}} \Pi \left (\frac {\sqrt [4]{-a^8}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\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}} \Pi \left (-\frac {\sqrt {-\sqrt {-a^8}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\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}} \Pi \left (\frac {\sqrt {-\sqrt {-a^8}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\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 {\tan ^{-1}\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 {\tanh ^{-1}\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}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\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}} \Pi \left (\frac {a^6}{\left (-a^8\right )^{3/4}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\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}} \Pi \left (\frac {\sqrt [4]{-a^8}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\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}} \Pi \left (-\frac {\sqrt {-\sqrt {-a^8}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\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}} \Pi \left (\frac {\sqrt {-\sqrt {-a^8}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 10.85, size = 379, normalized size = 1.25 \begin {gather*} \frac {-\sqrt {-\frac {a^2}{b^2}} x+i \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )+i \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (-i;\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )+i \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (i;\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )-i \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (-\sqrt [4]{-1};\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )-i \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\sqrt [4]{-1};\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )-i \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (-(-1)^{3/4};\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )-i \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left ((-1)^{3/4};\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )}{8 a^8 \sqrt {-\frac {a^2}{b^2}} b^8 \sqrt {-b^4+a^4 x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-(Sqrt[-(a^2/b^2)]*x) + I*Sqrt[1 - (a^4*x^4)/b^4]*EllipticF[I*ArcSinh[Sqrt[-(a^2/b^2)]*x], -1] + I*Sqrt[1 - (
a^4*x^4)/b^4]*EllipticPi[-I, I*ArcSinh[Sqrt[-(a^2/b^2)]*x], -1] + I*Sqrt[1 - (a^4*x^4)/b^4]*EllipticPi[I, I*Ar
cSinh[Sqrt[-(a^2/b^2)]*x], -1] - I*Sqrt[1 - (a^4*x^4)/b^4]*EllipticPi[-(-1)^(1/4), I*ArcSinh[Sqrt[-(a^2/b^2)]*
x], -1] - I*Sqrt[1 - (a^4*x^4)/b^4]*EllipticPi[(-1)^(1/4), I*ArcSinh[Sqrt[-(a^2/b^2)]*x], -1] - I*Sqrt[1 - (a^
4*x^4)/b^4]*EllipticPi[-(-1)^(3/4), I*ArcSinh[Sqrt[-(a^2/b^2)]*x], -1] - I*Sqrt[1 - (a^4*x^4)/b^4]*EllipticPi[
(-1)^(3/4), I*ArcSinh[Sqrt[-(a^2/b^2)]*x], -1])/(8*a^8*Sqrt[-(a^2/b^2)]*b^8*Sqrt[-b^4 + a^4*x^4])

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.19, size = 1130, normalized size = 3.73

method	result	size
elliptic	\(\frac {\left (\frac {\sqrt {2}\, \ln \left (\frac {\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}-\frac {\left (a^{4} b^{4}\right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}}{x}+\sqrt {a^{4} b^{4}}}{\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}+\frac {\left (a^{4} b^{4}\right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}}{x}+\sqrt {a^{4} b^{4}}}\right )}{64 a^{8} b^{8} \left (a^{4} b^{4}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}}{\left (a^{4} b^{4}\right )^{\frac {1}{4}} x}+1\right )}{32 a^{8} b^{8} \left (a^{4} b^{4}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}}{\left (a^{4} b^{4}\right )^{\frac {1}{4}} x}-1\right )}{32 a^{8} b^{8} \left (a^{4} b^{4}\right )^{\frac {1}{4}}}-\frac {\ln \left (\frac {\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}-\frac {\sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}\, \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2}\, \sqrt {a^{4} b^{4}}}{2}}{\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}+\frac {\sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}\, \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2}\, \sqrt {a^{4} b^{4}}}{2}}\right )}{16 a^{8} b^{8} \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}-\frac {\arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{\sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}\, x}+1\right )}{8 a^{8} b^{8} \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}-\frac {\arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{\sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}\, x}-1\right )}{8 a^{8} b^{8} \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}-\frac {\sqrt {2}\, x}{8 a^{8} b^{8} \sqrt {a^{4} x^{4}-b^{4}}}\right ) \sqrt {2}}{2}\)	\(568\)
default	\(\text {Expression too large to display}\)	\(1130\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16/b^7/a^8*(1/2*(a^4*x^3-a^3*b*x^2+a^2*b^2*x-a*b^3)/a^2/b^3/((x+b/a)*(a^4*x^3-a^3*b*x^2+a^2*b^2*x-a*b^3))^(
1/2)+1/2/b/(-a^2/b^2)^(1/2)*(a^2*x^2/b^2+1)^(1/2)*(1-a^2*x^2/b^2)^(1/2)/(a^4*x^4-b^4)^(1/2)*EllipticF(x*(-a^2/
b^2)^(1/2),I)-1/2/b/(-a^2/b^2)^(1/2)*(a^2*x^2/b^2+1)^(1/2)*(1-a^2*x^2/b^2)^(1/2)/(a^4*x^4-b^4)^(1/2)*(Elliptic
F(x*(-a^2/b^2)^(1/2),I)-EllipticE(x*(-a^2/b^2)^(1/2),I)))-1/64/a^12/b^4*sum(1/_alpha^3*(-2^(1/2)/(-b^4)^(1/2)*
arctanh(_alpha^2*(_alpha^2+x^2)*a^4/(-2*b^4)^(1/2)/(a^4*x^4-b^4)^(1/2))+4/(-a^2/b^2)^(1/2)*_alpha^3*a^4/b^4*(a
^2*x^2/b^2+1)^(1/2)*(1-a^2*x^2/b^2)^(1/2)/(a^4*x^4-b^4)^(1/2)*EllipticPi(x*(-a^2/b^2)^(1/2),_alpha^2*a^2/b^2,(
a^2/b^2)^(1/2)/(-a^2/b^2)^(1/2))),_alpha=RootOf(_Z^4*a^4+b^4))+1/32/a^16*sum(1/_alpha^7*(-1/(_alpha^4*a^4-b^4)
^(1/2)*arctanh(_alpha^2/b^4*(_alpha^6*a^4+b^4*x^2)*a^4/(_alpha^4*a^4-b^4)^(1/2)/(a^4*x^4-b^4)^(1/2))+2/(-a^2/b
^2)^(1/2)*_alpha^7*a^8/b^8*(a^2*x^2/b^2+1)^(1/2)*(1-a^2*x^2/b^2)^(1/2)/(a^4*x^4-b^4)^(1/2)*EllipticPi(x*(-a^2/
b^2)^(1/2),_alpha^6*a^6/b^6,(a^2/b^2)^(1/2)/(-a^2/b^2)^(1/2))),_alpha=RootOf(_Z^8*a^8+b^8))+1/16/b^7/a^8*(-1/2
*(a^4*x^3+a^3*b*x^2+a^2*b^2*x+a*b^3)/a^2/b^3/((x-b/a)*(a^4*x^3+a^3*b*x^2+a^2*b^2*x+a*b^3))^(1/2)-1/2/b/(-a^2/b
^2)^(1/2)*(a^2*x^2/b^2+1)^(1/2)*(1-a^2*x^2/b^2)^(1/2)/(a^4*x^4-b^4)^(1/2)*EllipticF(x*(-a^2/b^2)^(1/2),I)+1/2/
b/(-a^2/b^2)^(1/2)*(a^2*x^2/b^2+1)^(1/2)*(1-a^2*x^2/b^2)^(1/2)/(a^4*x^4-b^4)^(1/2)*(EllipticF(x*(-a^2/b^2)^(1/
2),I)-EllipticE(x*(-a^2/b^2)^(1/2),I)))-1/8/a^8/b^6*(-1/2*(a^4*x^2-a^2*b^2)/b^4*x/a^2/((x^2+b^2/a^2)*(a^4*x^2-
a^2*b^2))^(1/2)+1/2/b^2/(-a^2/b^2)^(1/2)*(a^2*x^2/b^2+1)^(1/2)*(1-a^2*x^2/b^2)^(1/2)/(a^4*x^4-b^4)^(1/2)*Ellip
ticF(x*(-a^2/b^2)^(1/2),I)+1/2/b^2/(-a^2/b^2)^(1/2)*(a^2*x^2/b^2+1)^(1/2)*(1-a^2*x^2/b^2)^(1/2)/(a^4*x^4-b^4)^
(1/2)*(EllipticF(x*(-a^2/b^2)^(1/2),I)-EllipticE(x*(-a^2/b^2)^(1/2),I)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F(-1)] Timed out
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/(a^4*x^4-b^4)^(1/2)/(a^16*x^16-b^16),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)] Timed out
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/(a**4*x**4-b**4)**(1/2)/(a**16*x**16-b**16),x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\int \frac {x^8}{\sqrt {a^4\,x^4-b^4}\,\left (b^{16}-a^{16}\,x^{16}\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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