3.11.68 \(\int \frac {\sqrt [4]{2+x^4} (-4+x^8)}{x^6 (-4-2 x^4+x^8)} \, dx\)
Optimal. Leaf size=80 \[ \frac {1}{8} \text {RootSum}\left [\text {$\#$1}^8-\text {$\#$1}^4-1\& ,\frac {\log \left (\sqrt [4]{x^4+2}-\text {$\#$1} x\right )-\log (x)}{2 \text {$\#$1}^7-\text {$\#$1}^3}\& \right ]+\frac {\sqrt [4]{x^4+2} \left (2 x^4-1\right )}{5 x^5} \]
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Rubi [B] time = 2.10, antiderivative size = 877, normalized size of antiderivative = 10.96,
number of steps used = 53, number of rules used = 17, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.567, Rules used
= {6728, 264, 277, 331, 298, 203, 206, 1528, 510, 1518, 494, 297, 1162, 617, 204, 1165, 628} \begin {gather*} -\frac {F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};-\frac {x^4}{2},\frac {x^4}{1-\sqrt {5}}\right ) x^3}{3\ 2^{3/4} \sqrt {5} \left (1-\sqrt {5}\right )}+\frac {F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {x^4}{1+\sqrt {5}},-\frac {x^4}{2}\right ) x^3}{3\ 2^{3/4} \sqrt {5} \left (1+\sqrt {5}\right )}+\frac {\sqrt [4]{\frac {1}{2} \left (-29+13 \sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {2}{-1+\sqrt {5}}} x}{\sqrt [4]{x^4+2}}\right )}{4 \sqrt {5}}+\frac {\sqrt [4]{-2+\sqrt {5}} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {2}{-1+\sqrt {5}}} x}{\sqrt [4]{x^4+2}}\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{29+13 \sqrt {5}} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{1+\sqrt {5}} \sqrt [4]{x^4+2}}\right )}{4\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{2+\sqrt {5}} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{1+\sqrt {5}} \sqrt [4]{x^4+2}}\right )}{2 \sqrt {10}}-\frac {\sqrt [4]{29+13 \sqrt {5}} \tan ^{-1}\left (\frac {2^{3/4} x}{\sqrt [4]{1+\sqrt {5}} \sqrt [4]{x^4+2}}+1\right )}{4\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{2+\sqrt {5}} \tan ^{-1}\left (\frac {2^{3/4} x}{\sqrt [4]{1+\sqrt {5}} \sqrt [4]{x^4+2}}+1\right )}{2 \sqrt {10}}-\frac {\sqrt [4]{\frac {1}{2} \left (-29+13 \sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {2}{-1+\sqrt {5}}} x}{\sqrt [4]{x^4+2}}\right )}{4 \sqrt {5}}-\frac {\sqrt [4]{-2+\sqrt {5}} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {2}{-1+\sqrt {5}}} x}{\sqrt [4]{x^4+2}}\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{29+13 \sqrt {5}} \log \left (\frac {2 x^2}{\sqrt {x^4+2}}-\frac {2 \sqrt [4]{2 \left (1+\sqrt {5}\right )} x}{\sqrt [4]{x^4+2}}+\sqrt {2 \left (1+\sqrt {5}\right )}\right )}{8\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{2+\sqrt {5}} \log \left (\frac {2 x^2}{\sqrt {x^4+2}}-\frac {2 \sqrt [4]{2 \left (1+\sqrt {5}\right )} x}{\sqrt [4]{x^4+2}}+\sqrt {2 \left (1+\sqrt {5}\right )}\right )}{4 \sqrt {10}}+\frac {\sqrt [4]{29+13 \sqrt {5}} \log \left (\frac {2 x^2}{\sqrt {x^4+2}}+\frac {2 \sqrt [4]{2 \left (1+\sqrt {5}\right )} x}{\sqrt [4]{x^4+2}}+\sqrt {2 \left (1+\sqrt {5}\right )}\right )}{8\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{2+\sqrt {5}} \log \left (\frac {2 x^2}{\sqrt {x^4+2}}+\frac {2 \sqrt [4]{2 \left (1+\sqrt {5}\right )} x}{\sqrt [4]{x^4+2}}+\sqrt {2 \left (1+\sqrt {5}\right )}\right )}{4 \sqrt {10}}+\frac {\sqrt [4]{x^4+2}}{2 x}-\frac {\left (x^4+2\right )^{5/4}}{10 x^5} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Int[((2 + x^4)^(1/4)*(-4 + x^8))/(x^6*(-4 - 2*x^4 + x^8)),x]
[Out]
(2 + x^4)^(1/4)/(2*x) - (2 + x^4)^(5/4)/(10*x^5) - (x^3*AppellF1[3/4, -1/4, 1, 7/4, -1/2*x^4, x^4/(1 - Sqrt[5]
)])/(3*2^(3/4)*Sqrt[5]*(1 - Sqrt[5])) + (x^3*AppellF1[3/4, 1, -1/4, 7/4, x^4/(1 + Sqrt[5]), -1/2*x^4])/(3*2^(3
/4)*Sqrt[5]*(1 + Sqrt[5])) + ((-2 + Sqrt[5])^(1/4)*ArcTan[((2/(-1 + Sqrt[5]))^(1/4)*x)/(2 + x^4)^(1/4)])/(2*Sq
rt[5]) + (((-29 + 13*Sqrt[5])/2)^(1/4)*ArcTan[((2/(-1 + Sqrt[5]))^(1/4)*x)/(2 + x^4)^(1/4)])/(4*Sqrt[5]) - ((2
+ Sqrt[5])^(1/4)*ArcTan[1 - (2^(3/4)*x)/((1 + Sqrt[5])^(1/4)*(2 + x^4)^(1/4))])/(2*Sqrt[10]) + ((29 + 13*Sqrt
[5])^(1/4)*ArcTan[1 - (2^(3/4)*x)/((1 + Sqrt[5])^(1/4)*(2 + x^4)^(1/4))])/(4*2^(3/4)*Sqrt[5]) + ((2 + Sqrt[5])
^(1/4)*ArcTan[1 + (2^(3/4)*x)/((1 + Sqrt[5])^(1/4)*(2 + x^4)^(1/4))])/(2*Sqrt[10]) - ((29 + 13*Sqrt[5])^(1/4)*
ArcTan[1 + (2^(3/4)*x)/((1 + Sqrt[5])^(1/4)*(2 + x^4)^(1/4))])/(4*2^(3/4)*Sqrt[5]) - ((-2 + Sqrt[5])^(1/4)*Arc
Tanh[((2/(-1 + Sqrt[5]))^(1/4)*x)/(2 + x^4)^(1/4)])/(2*Sqrt[5]) - (((-29 + 13*Sqrt[5])/2)^(1/4)*ArcTanh[((2/(-
1 + Sqrt[5]))^(1/4)*x)/(2 + x^4)^(1/4)])/(4*Sqrt[5]) + ((2 + Sqrt[5])^(1/4)*Log[Sqrt[2*(1 + Sqrt[5])] + (2*x^2
)/Sqrt[2 + x^4] - (2*(2*(1 + Sqrt[5]))^(1/4)*x)/(2 + x^4)^(1/4)])/(4*Sqrt[10]) - ((29 + 13*Sqrt[5])^(1/4)*Log[
Sqrt[2*(1 + Sqrt[5])] + (2*x^2)/Sqrt[2 + x^4] - (2*(2*(1 + Sqrt[5]))^(1/4)*x)/(2 + x^4)^(1/4)])/(8*2^(3/4)*Sqr
t[5]) - ((2 + Sqrt[5])^(1/4)*Log[Sqrt[2*(1 + Sqrt[5])] + (2*x^2)/Sqrt[2 + x^4] + (2*(2*(1 + Sqrt[5]))^(1/4)*x)
/(2 + x^4)^(1/4)])/(4*Sqrt[10]) + ((29 + 13*Sqrt[5])^(1/4)*Log[Sqrt[2*(1 + Sqrt[5])] + (2*x^2)/Sqrt[2 + x^4] +
(2*(2*(1 + Sqrt[5]))^(1/4)*x)/(2 + x^4)^(1/4)])/(8*2^(3/4)*Sqrt[5])
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 204
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[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 264
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]
Rule 277
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Rule 297
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))
Rule 298
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b),
2]]}, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &
& !GtQ[a/b, 0]
Rule 331
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(
p + (m + 1)/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[
p, -2^(-1)] && IntegersQ[m, p + (m + 1)/n]
Rule 494
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{k = Denominato
r[p]}, Dist[(k*a^(p + (m + 1)/n))/n, Subst[Int[(x^((k*(m + 1))/n - 1)*(c - (b*c - a*d)*x^k)^q)/(1 - b*x^k)^(p
+ q + (m + 1)/n + 1), x], x, x^(n/k)/(a + b*x^n)^(1/k)], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && Ration
alQ[m, p] && IntegersQ[p + (m + 1)/n, q] && LtQ[-1, p, 0]
Rule 510
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a^p*c^q
*(e*x)^(m + 1)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])
Rule 617
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 628
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Rule 1162
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Rule 1165
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Rule 1518
Int[(((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^(n_))^(q_))/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symbol
] :> Dist[(e*f^n)/c, Int[(f*x)^(m - n)*(d + e*x^n)^(q - 1), x], x] - Dist[f^n/c, Int[((f*x)^(m - n)*(d + e*x^n
)^(q - 1)*Simp[a*e - (c*d - b*e)*x^n, x])/(a + b*x^n + c*x^(2*n)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && E
qQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && !IntegerQ[q] && GtQ[q, 0] && GtQ[m, n - 1] && LeQ[m, 2*n
- 1]
Rule 1528
Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))^(q_))/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symbol]
:> Int[ExpandIntegrand[(d + e*x^n)^q, (f*x)^m/(a + b*x^n + c*x^(2*n)), x], x] /; FreeQ[{a, b, c, d, e, f, q,
n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && !IntegerQ[q] && IntegerQ[m]
Rule 6728
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{2+x^4} \left (-4+x^8\right )}{x^6 \left (-4-2 x^4+x^8\right )} \, dx &=\int \left (\frac {\sqrt [4]{2+x^4}}{x^6}-\frac {\sqrt [4]{2+x^4}}{2 x^2}+\frac {x^2 \left (-2+x^4\right ) \sqrt [4]{2+x^4}}{2 \left (-4-2 x^4+x^8\right )}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {\sqrt [4]{2+x^4}}{x^2} \, dx\right )+\frac {1}{2} \int \frac {x^2 \left (-2+x^4\right ) \sqrt [4]{2+x^4}}{-4-2 x^4+x^8} \, dx+\int \frac {\sqrt [4]{2+x^4}}{x^6} \, dx\\ &=\frac {\sqrt [4]{2+x^4}}{2 x}-\frac {\left (2+x^4\right )^{5/4}}{10 x^5}-\frac {1}{2} \int \frac {x^2}{\left (2+x^4\right )^{3/4}} \, dx+\frac {1}{2} \int \left (-\frac {2 x^2 \sqrt [4]{2+x^4}}{-4-2 x^4+x^8}+\frac {x^6 \sqrt [4]{2+x^4}}{-4-2 x^4+x^8}\right ) \, dx\\ &=\frac {\sqrt [4]{2+x^4}}{2 x}-\frac {\left (2+x^4\right )^{5/4}}{10 x^5}+\frac {1}{2} \int \frac {x^6 \sqrt [4]{2+x^4}}{-4-2 x^4+x^8} \, dx-\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )-\int \frac {x^2 \sqrt [4]{2+x^4}}{-4-2 x^4+x^8} \, dx\\ &=\frac {\sqrt [4]{2+x^4}}{2 x}-\frac {\left (2+x^4\right )^{5/4}}{10 x^5}-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )+\frac {1}{2} \int \frac {x^2}{\left (2+x^4\right )^{3/4}} \, dx-\frac {1}{2} \int \frac {x^2 \left (-4-4 x^4\right )}{\left (2+x^4\right )^{3/4} \left (-4-2 x^4+x^8\right )} \, dx-\int \left (-\frac {x^2 \sqrt [4]{2+x^4}}{\sqrt {5} \left (2+2 \sqrt {5}-2 x^4\right )}-\frac {x^2 \sqrt [4]{2+x^4}}{\sqrt {5} \left (-2+2 \sqrt {5}+2 x^4\right )}\right ) \, dx\\ &=\frac {\sqrt [4]{2+x^4}}{2 x}-\frac {\left (2+x^4\right )^{5/4}}{10 x^5}+\frac {1}{4} \tan ^{-1}\left (\frac {x}{\sqrt [4]{2+x^4}}\right )-\frac {1}{4} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{2+x^4}}\right )-\frac {1}{2} \int \left (-\frac {4 x^2}{\left (2+x^4\right )^{3/4} \left (-4-2 x^4+x^8\right )}-\frac {4 x^6}{\left (2+x^4\right )^{3/4} \left (-4-2 x^4+x^8\right )}\right ) \, dx+\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )+\frac {\int \frac {x^2 \sqrt [4]{2+x^4}}{2+2 \sqrt {5}-2 x^4} \, dx}{\sqrt {5}}+\frac {\int \frac {x^2 \sqrt [4]{2+x^4}}{-2+2 \sqrt {5}+2 x^4} \, dx}{\sqrt {5}}\\ &=\frac {\sqrt [4]{2+x^4}}{2 x}-\frac {\left (2+x^4\right )^{5/4}}{10 x^5}-\frac {x^3 F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};-\frac {x^4}{2},\frac {x^4}{1-\sqrt {5}}\right )}{3\ 2^{3/4} \sqrt {5} \left (1-\sqrt {5}\right )}+\frac {x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {x^4}{1+\sqrt {5}},-\frac {x^4}{2}\right )}{3\ 2^{3/4} \sqrt {5} \left (1+\sqrt {5}\right )}+\frac {1}{4} \tan ^{-1}\left (\frac {x}{\sqrt [4]{2+x^4}}\right )-\frac {1}{4} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{2+x^4}}\right )+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )+2 \int \frac {x^2}{\left (2+x^4\right )^{3/4} \left (-4-2 x^4+x^8\right )} \, dx+2 \int \frac {x^6}{\left (2+x^4\right )^{3/4} \left (-4-2 x^4+x^8\right )} \, dx\\ &=\frac {\sqrt [4]{2+x^4}}{2 x}-\frac {\left (2+x^4\right )^{5/4}}{10 x^5}-\frac {x^3 F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};-\frac {x^4}{2},\frac {x^4}{1-\sqrt {5}}\right )}{3\ 2^{3/4} \sqrt {5} \left (1-\sqrt {5}\right )}+\frac {x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {x^4}{1+\sqrt {5}},-\frac {x^4}{2}\right )}{3\ 2^{3/4} \sqrt {5} \left (1+\sqrt {5}\right )}+2 \int \left (-\frac {x^2}{\sqrt {5} \left (2+2 \sqrt {5}-2 x^4\right ) \left (2+x^4\right )^{3/4}}-\frac {x^2}{\sqrt {5} \left (2+x^4\right )^{3/4} \left (-2+2 \sqrt {5}+2 x^4\right )}\right ) \, dx+2 \int \left (-\frac {\left (2+2 \sqrt {5}\right ) x^2}{2 \sqrt {5} \left (2+2 \sqrt {5}-2 x^4\right ) \left (2+x^4\right )^{3/4}}+\frac {\left (-2+2 \sqrt {5}\right ) x^2}{2 \sqrt {5} \left (2+x^4\right )^{3/4} \left (-2+2 \sqrt {5}+2 x^4\right )}\right ) \, dx\\ &=\frac {\sqrt [4]{2+x^4}}{2 x}-\frac {\left (2+x^4\right )^{5/4}}{10 x^5}-\frac {x^3 F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};-\frac {x^4}{2},\frac {x^4}{1-\sqrt {5}}\right )}{3\ 2^{3/4} \sqrt {5} \left (1-\sqrt {5}\right )}+\frac {x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {x^4}{1+\sqrt {5}},-\frac {x^4}{2}\right )}{3\ 2^{3/4} \sqrt {5} \left (1+\sqrt {5}\right )}-\frac {2 \int \frac {x^2}{\left (2+2 \sqrt {5}-2 x^4\right ) \left (2+x^4\right )^{3/4}} \, dx}{\sqrt {5}}-\frac {2 \int \frac {x^2}{\left (2+x^4\right )^{3/4} \left (-2+2 \sqrt {5}+2 x^4\right )} \, dx}{\sqrt {5}}+\frac {1}{5} \left (2 \left (5-\sqrt {5}\right )\right ) \int \frac {x^2}{\left (2+x^4\right )^{3/4} \left (-2+2 \sqrt {5}+2 x^4\right )} \, dx-\frac {1}{5} \left (2 \left (5+\sqrt {5}\right )\right ) \int \frac {x^2}{\left (2+2 \sqrt {5}-2 x^4\right ) \left (2+x^4\right )^{3/4}} \, dx\\ &=\frac {\sqrt [4]{2+x^4}}{2 x}-\frac {\left (2+x^4\right )^{5/4}}{10 x^5}-\frac {x^3 F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};-\frac {x^4}{2},\frac {x^4}{1-\sqrt {5}}\right )}{3\ 2^{3/4} \sqrt {5} \left (1-\sqrt {5}\right )}+\frac {x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {x^4}{1+\sqrt {5}},-\frac {x^4}{2}\right )}{3\ 2^{3/4} \sqrt {5} \left (1+\sqrt {5}\right )}-\frac {2 \operatorname {Subst}\left (\int \frac {x^2}{-2+2 \sqrt {5}-\left (-6+2 \sqrt {5}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )}{\sqrt {5}}-\frac {2 \operatorname {Subst}\left (\int \frac {x^2}{2+2 \sqrt {5}-\left (6+2 \sqrt {5}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )}{\sqrt {5}}+\frac {1}{5} \left (2 \left (5-\sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {x^2}{-2+2 \sqrt {5}-\left (-6+2 \sqrt {5}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )-\frac {1}{5} \left (2 \left (5+\sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {x^2}{2+2 \sqrt {5}-\left (6+2 \sqrt {5}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )\\ &=\frac {\sqrt [4]{2+x^4}}{2 x}-\frac {\left (2+x^4\right )^{5/4}}{10 x^5}-\frac {x^3 F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};-\frac {x^4}{2},\frac {x^4}{1-\sqrt {5}}\right )}{3\ 2^{3/4} \sqrt {5} \left (1-\sqrt {5}\right )}+\frac {x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {x^4}{1+\sqrt {5}},-\frac {x^4}{2}\right )}{3\ 2^{3/4} \sqrt {5} \left (1+\sqrt {5}\right )}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {1+\sqrt {5}}-\sqrt {2} x^2}{-2+2 \sqrt {5}+\left (6-2 \sqrt {5}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )}{\sqrt {10}}-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {1+\sqrt {5}}+\sqrt {2} x^2}{-2+2 \sqrt {5}+\left (6-2 \sqrt {5}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )}{\sqrt {10}}+\frac {\left (1-\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\sqrt {5}}-\sqrt {2} x^2}{-2+2 \sqrt {5}+\left (6-2 \sqrt {5}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )}{\sqrt {10}}-\frac {\left (1-\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\sqrt {5}}+\sqrt {2} x^2}{-2+2 \sqrt {5}+\left (6-2 \sqrt {5}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )}{\sqrt {10}}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+\sqrt {5}}-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )}{\sqrt {10} \left (3+\sqrt {5}\right )}+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+\sqrt {5}}+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )}{\sqrt {10} \left (3+\sqrt {5}\right )}-\frac {\left (1+\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+\sqrt {5}}-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )}{\sqrt {10} \left (3+\sqrt {5}\right )}+\frac {\left (1+\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+\sqrt {5}}+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )}{\sqrt {10} \left (3+\sqrt {5}\right )}\\ &=\frac {\sqrt [4]{2+x^4}}{2 x}-\frac {\left (2+x^4\right )^{5/4}}{10 x^5}-\frac {x^3 F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};-\frac {x^4}{2},\frac {x^4}{1-\sqrt {5}}\right )}{3\ 2^{3/4} \sqrt {5} \left (1-\sqrt {5}\right )}+\frac {x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {x^4}{1+\sqrt {5}},-\frac {x^4}{2}\right )}{3\ 2^{3/4} \sqrt {5} \left (1+\sqrt {5}\right )}+\frac {\sqrt [4]{-2+\sqrt {5}} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {2}{-1+\sqrt {5}}} x}{\sqrt [4]{2+x^4}}\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (-29+13 \sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {2}{-1+\sqrt {5}}} x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {5}}-\frac {\sqrt [4]{-2+\sqrt {5}} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {2}{-1+\sqrt {5}}} x}{\sqrt [4]{2+x^4}}\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (-29+13 \sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {2}{-1+\sqrt {5}}} x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {5}}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )}-\sqrt [4]{2 \left (1+\sqrt {5}\right )} x+x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {5} \left (3-\sqrt {5}\right )}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )}+\sqrt [4]{2 \left (1+\sqrt {5}\right )} x+x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {5} \left (3-\sqrt {5}\right )}-\frac {\left (1-\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )}-\sqrt [4]{2 \left (1+\sqrt {5}\right )} x+x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {5} \left (3-\sqrt {5}\right )}-\frac {\left (1-\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )}+\sqrt [4]{2 \left (1+\sqrt {5}\right )} x+x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {5} \left (3-\sqrt {5}\right )}+\frac {\sqrt [4]{2+\sqrt {5}} \operatorname {Subst}\left (\int \frac {\sqrt [4]{2 \left (1+\sqrt {5}\right )}+2 x}{-\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )}-\sqrt [4]{2 \left (1+\sqrt {5}\right )} x-x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {10}}+\frac {\sqrt [4]{2+\sqrt {5}} \operatorname {Subst}\left (\int \frac {\sqrt [4]{2 \left (1+\sqrt {5}\right )}-2 x}{-\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )}+\sqrt [4]{2 \left (1+\sqrt {5}\right )} x-x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {10}}-\frac {\sqrt [4]{29+13 \sqrt {5}} \operatorname {Subst}\left (\int \frac {\sqrt [4]{2 \left (1+\sqrt {5}\right )}+2 x}{-\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )}-\sqrt [4]{2 \left (1+\sqrt {5}\right )} x-x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )}{8\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{29+13 \sqrt {5}} \operatorname {Subst}\left (\int \frac {\sqrt [4]{2 \left (1+\sqrt {5}\right )}-2 x}{-\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )}+\sqrt [4]{2 \left (1+\sqrt {5}\right )} x-x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )}{8\ 2^{3/4} \sqrt {5}}\\ &=\frac {\sqrt [4]{2+x^4}}{2 x}-\frac {\left (2+x^4\right )^{5/4}}{10 x^5}-\frac {x^3 F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};-\frac {x^4}{2},\frac {x^4}{1-\sqrt {5}}\right )}{3\ 2^{3/4} \sqrt {5} \left (1-\sqrt {5}\right )}+\frac {x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {x^4}{1+\sqrt {5}},-\frac {x^4}{2}\right )}{3\ 2^{3/4} \sqrt {5} \left (1+\sqrt {5}\right )}+\frac {\sqrt [4]{-2+\sqrt {5}} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {2}{-1+\sqrt {5}}} x}{\sqrt [4]{2+x^4}}\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (-29+13 \sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {2}{-1+\sqrt {5}}} x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {5}}-\frac {\sqrt [4]{-2+\sqrt {5}} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {2}{-1+\sqrt {5}}} x}{\sqrt [4]{2+x^4}}\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (-29+13 \sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {2}{-1+\sqrt {5}}} x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {5}}+\frac {\sqrt [4]{2+\sqrt {5}} \log \left (\sqrt {2 \left (1+\sqrt {5}\right )}+\frac {2 x^2}{\sqrt {2+x^4}}-\frac {2 \sqrt [4]{2 \left (1+\sqrt {5}\right )} x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {10}}-\frac {\sqrt [4]{29+13 \sqrt {5}} \log \left (\sqrt {2 \left (1+\sqrt {5}\right )}+\frac {2 x^2}{\sqrt {2+x^4}}-\frac {2 \sqrt [4]{2 \left (1+\sqrt {5}\right )} x}{\sqrt [4]{2+x^4}}\right )}{8\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{2+\sqrt {5}} \log \left (\sqrt {2 \left (1+\sqrt {5}\right )}+\frac {2 x^2}{\sqrt {2+x^4}}+\frac {2 \sqrt [4]{2 \left (1+\sqrt {5}\right )} x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {10}}+\frac {\sqrt [4]{29+13 \sqrt {5}} \log \left (\sqrt {2 \left (1+\sqrt {5}\right )}+\frac {2 x^2}{\sqrt {2+x^4}}+\frac {2 \sqrt [4]{2 \left (1+\sqrt {5}\right )} x}{\sqrt [4]{2+x^4}}\right )}{8\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{2+\sqrt {5}} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {2^{3/4} x}{\sqrt [4]{1+\sqrt {5}} \sqrt [4]{2+x^4}}\right )}{2 \sqrt {10}}-\frac {\sqrt [4]{2+\sqrt {5}} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {2^{3/4} x}{\sqrt [4]{1+\sqrt {5}} \sqrt [4]{2+x^4}}\right )}{2 \sqrt {10}}-\frac {\sqrt [4]{29+13 \sqrt {5}} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {2^{3/4} x}{\sqrt [4]{1+\sqrt {5}} \sqrt [4]{2+x^4}}\right )}{4\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{29+13 \sqrt {5}} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {2^{3/4} x}{\sqrt [4]{1+\sqrt {5}} \sqrt [4]{2+x^4}}\right )}{4\ 2^{3/4} \sqrt {5}}\\ &=\frac {\sqrt [4]{2+x^4}}{2 x}-\frac {\left (2+x^4\right )^{5/4}}{10 x^5}-\frac {x^3 F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};-\frac {x^4}{2},\frac {x^4}{1-\sqrt {5}}\right )}{3\ 2^{3/4} \sqrt {5} \left (1-\sqrt {5}\right )}+\frac {x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {x^4}{1+\sqrt {5}},-\frac {x^4}{2}\right )}{3\ 2^{3/4} \sqrt {5} \left (1+\sqrt {5}\right )}+\frac {\sqrt [4]{-2+\sqrt {5}} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {2}{-1+\sqrt {5}}} x}{\sqrt [4]{2+x^4}}\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (-29+13 \sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {2}{-1+\sqrt {5}}} x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {5}}-\frac {\sqrt [4]{2+\sqrt {5}} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{1+\sqrt {5}} \sqrt [4]{2+x^4}}\right )}{2 \sqrt {10}}+\frac {\sqrt [4]{29+13 \sqrt {5}} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{1+\sqrt {5}} \sqrt [4]{2+x^4}}\right )}{4\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{2+\sqrt {5}} \tan ^{-1}\left (1+\frac {2^{3/4} x}{\sqrt [4]{1+\sqrt {5}} \sqrt [4]{2+x^4}}\right )}{2 \sqrt {10}}-\frac {\sqrt [4]{29+13 \sqrt {5}} \tan ^{-1}\left (1+\frac {2^{3/4} x}{\sqrt [4]{1+\sqrt {5}} \sqrt [4]{2+x^4}}\right )}{4\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{-2+\sqrt {5}} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {2}{-1+\sqrt {5}}} x}{\sqrt [4]{2+x^4}}\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (-29+13 \sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {2}{-1+\sqrt {5}}} x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {5}}+\frac {\sqrt [4]{2+\sqrt {5}} \log \left (\sqrt {2 \left (1+\sqrt {5}\right )}+\frac {2 x^2}{\sqrt {2+x^4}}-\frac {2 \sqrt [4]{2 \left (1+\sqrt {5}\right )} x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {10}}-\frac {\sqrt [4]{29+13 \sqrt {5}} \log \left (\sqrt {2 \left (1+\sqrt {5}\right )}+\frac {2 x^2}{\sqrt {2+x^4}}-\frac {2 \sqrt [4]{2 \left (1+\sqrt {5}\right )} x}{\sqrt [4]{2+x^4}}\right )}{8\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{2+\sqrt {5}} \log \left (\sqrt {2 \left (1+\sqrt {5}\right )}+\frac {2 x^2}{\sqrt {2+x^4}}+\frac {2 \sqrt [4]{2 \left (1+\sqrt {5}\right )} x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {10}}+\frac {\sqrt [4]{29+13 \sqrt {5}} \log \left (\sqrt {2 \left (1+\sqrt {5}\right )}+\frac {2 x^2}{\sqrt {2+x^4}}+\frac {2 \sqrt [4]{2 \left (1+\sqrt {5}\right )} x}{\sqrt [4]{2+x^4}}\right )}{8\ 2^{3/4} \sqrt {5}}\\ \end {align*}
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Mathematica [F] time = 6.58, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{2+x^4} \left (-4+x^8\right )}{x^6 \left (-4-2 x^4+x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Integrate[((2 + x^4)^(1/4)*(-4 + x^8))/(x^6*(-4 - 2*x^4 + x^8)),x]
[Out]
Integrate[((2 + x^4)^(1/4)*(-4 + x^8))/(x^6*(-4 - 2*x^4 + x^8)), x]
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IntegrateAlgebraic [A] time = 0.00, size = 80, normalized size = 1.00 \begin {gather*} \frac {\sqrt [4]{2+x^4} \left (-1+2 x^4\right )}{5 x^5}+\frac {1}{8} \text {RootSum}\left [-1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{2+x^4}-x \text {$\#$1}\right )}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[((2 + x^4)^(1/4)*(-4 + x^8))/(x^6*(-4 - 2*x^4 + x^8)),x]
[Out]
((2 + x^4)^(1/4)*(-1 + 2*x^4))/(5*x^5) + RootSum[-1 - #1^4 + #1^8 & , (-Log[x] + Log[(2 + x^4)^(1/4) - x*#1])/
(-#1^3 + 2*#1^7) & ]/8
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fricas [B] time = 14.54, size = 5805, normalized size = 72.56
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^4+2)^(1/4)*(x^8-4)/x^6/(x^8-2*x^4-4),x, algorithm="fricas")
[Out]
-1/160*(4*sqrt(5)*sqrt(2)*x^5*(sqrt(5) + 2)^(1/4)*arctan(-1/4*(3205700*x^48 + 37345168*x^44 + 180959952*x^40 +
465140928*x^36 + 664687232*x^32 + 494124288*x^28 + 141907200*x^24 - 6450176*x^20 + 1075200*x^16 + 2797568*x^1
2 + 1691648*x^8 + 278528*x^4 - 16*(95823*x^46 + 1251203*x^42 + 6523442*x^38 + 16526620*x^34 + 19670784*x^30 +
7602736*x^26 - 1179424*x^22 + 1387328*x^18 + 445952*x^14 - 243456*x^10 - 46592*x^6 - 3072*x^2 + sqrt(5)*(48196
*x^46 - 196187*x^42 - 3617546*x^38 - 13241892*x^34 - 19800240*x^30 - 11105648*x^26 - 526432*x^22 - 398016*x^18
- 373504*x^14 + 58112*x^10 + 13824*x^6 + 1024*x^2))*sqrt(x^4 + 2)*sqrt(sqrt(5) + 2) + sqrt(2)*(32*(4449*x^45
+ 634636*x^41 + 3555644*x^37 + 6358496*x^33 + 2708704*x^29 - 1547296*x^25 + 190272*x^21 + 181376*x^17 - 108800
*x^13 + 5632*x^9 + sqrt(5)*(5925*x^45 - 282704*x^41 - 1807476*x^37 - 3570256*x^33 - 2134720*x^29 + 190240*x^25
- 209984*x^21 - 98176*x^17 + 47360*x^13 - 2560*x^9))*(x^4 + 2)^(3/4)*sqrt(sqrt(5) + 2) - (2*sqrt(x^4 + 2)*(sq
rt(5)*sqrt(2)*(10653*x^46 + 2956548*x^42 + 20110664*x^38 + 50652360*x^34 + 56023600*x^30 + 25198208*x^26 + 372
5952*x^22 + 757632*x^18 + 1234176*x^14 - 217088*x^10 - 77824*x^6 - 6144*x^2) + sqrt(2)*(108077*x^46 - 5431312*
x^42 - 40263280*x^38 - 102843464*x^34 - 112864528*x^30 - 50417856*x^26 - 9076224*x^22 - 2229120*x^18 - 2436864
*x^14 + 578560*x^10 + 186368*x^6 + 14336*x^2)) + (sqrt(5)*sqrt(2)*(28915*x^48 - 8366472*x^44 - 80621076*x^40 -
293448080*x^36 - 499554944*x^32 - 380351744*x^28 - 90915392*x^24 - 11139840*x^20 - 17656832*x^16 + 94208*x^12
+ 1137664*x^8 + 290816*x^4 + 20480) - sqrt(2)*(190537*x^48 - 17103640*x^44 - 172683372*x^40 - 639099216*x^36
- 1098574144*x^32 - 842361344*x^28 - 201443776*x^24 - 22353664*x^20 - 38754304*x^16 - 204800*x^12 + 2425856*x^
8 + 634880*x^4 + 45056))*sqrt(sqrt(5) + 2))*(sqrt(5) + 2)^(3/4) - 8*(4*(11786*x^45 - 418982*x^41 - 3153502*x^3
7 - 7711008*x^33 - 7243648*x^29 - 1668480*x^25 + 72544*x^21 - 338048*x^17 + 23296*x^13 + 30720*x^9 + 3584*x^5
+ sqrt(5)*(2097*x^45 + 266982*x^41 + 1696566*x^37 + 3848848*x^33 + 3350352*x^29 + 602560*x^25 - 43104*x^21 + 2
00064*x^17 + 1536*x^13 - 12288*x^9 - 1536*x^5))*(x^4 + 2)^(3/4) - (12471*x^47 + 710367*x^43 + 2116698*x^39 - 3
578860*x^35 - 18142976*x^31 - 16588112*x^27 - 1714848*x^23 - 1272384*x^19 - 446976*x^15 - 139008*x^11 + 70144*
x^7 + 11264*x^3 + sqrt(5)*(9172*x^47 - 210451*x^43 - 628290*x^39 + 2143588*x^35 + 8786384*x^31 + 8031184*x^27
+ 1024032*x^23 + 498368*x^19 + 118016*x^15 + 44800*x^11 - 33280*x^7 - 5120*x^3))*(x^4 + 2)^(1/4)*sqrt(sqrt(5)
+ 2))*sqrt(sqrt(5) + 2) - 256*(2073*x^47 + 31642*x^43 + 149420*x^39 + 309672*x^35 + 314176*x^31 + 198976*x^27
+ 124608*x^23 + 27520*x^19 - 10496*x^15 + 3584*x^11 + sqrt(5)*(953*x^47 + 10346*x^43 + 54828*x^39 + 165000*x^3
5 + 269856*x^31 + 208768*x^27 + 52032*x^23 + 5760*x^19 + 6400*x^15 - 1536*x^11))*(x^4 + 2)^(1/4) - 8*(2*sqrt(5
)*sqrt(2)*(6759*x^48 + 192756*x^44 + 1338912*x^40 + 3951568*x^36 + 5386016*x^32 + 2908960*x^28 + 366208*x^24 +
417664*x^20 + 157952*x^16 - 25088*x^12 - 9216*x^8) - sqrt(x^4 + 2)*(sqrt(5)*sqrt(2)*(12951*x^46 - 734356*x^42
- 4933932*x^38 - 10666960*x^34 - 7733808*x^30 + 208192*x^26 + 222272*x^22 - 739840*x^18 + 166656*x^14 + 22528
*x^10 - 5120*x^6) + sqrt(2)*(3493*x^46 + 2013180*x^42 + 12514388*x^38 + 26406800*x^34 + 18862832*x^30 - 954432
*x^26 - 1374912*x^22 + 1328128*x^18 - 426752*x^14 - 55296*x^10 + 11264*x^6))*sqrt(sqrt(5) + 2) + 2*sqrt(2)*(19
945*x^48 - 252392*x^44 - 2918376*x^40 - 9824736*x^36 - 13811360*x^32 - 6643360*x^28 + 793216*x^24 + 89984*x^20
- 152320*x^16 + 77312*x^12 + 21504*x^8))*(sqrt(5) + 2)^(1/4))*sqrt((4*(x^6 - 6*x^2 + sqrt(5)*(x^6 - 2*x^2))*s
qrt(x^4 + 2) + (x^8 - 2*x^4 + sqrt(5)*(x^8 - 2*x^4 - 4) - 4)*sqrt(sqrt(5) + 2) + 2*((sqrt(5)*sqrt(2)*x^5 - sqr
t(2)*(x^5 + 4*x))*(x^4 + 2)^(3/4)*sqrt(sqrt(5) + 2) - 2*(sqrt(5)*sqrt(2)*x^3 - sqrt(2)*(x^7 - x^3))*(x^4 + 2)^
(1/4))*(sqrt(5) + 2)^(1/4))/(x^8 - 2*x^4 - 4)) - 4*((x^4 + 2)^(3/4)*(sqrt(5)*sqrt(2)*(60399*x^45 - 2243940*x^4
1 - 17911864*x^37 - 47674696*x^33 - 51732592*x^29 - 17641088*x^25 - 228224*x^21 - 2926464*x^17 - 241920*x^13 +
77824*x^9 + 57344*x^5 + 6144*x) + sqrt(2)*(53127*x^45 + 6673680*x^41 + 45520576*x^37 + 114960072*x^33 + 12177
0576*x^29 + 42292928*x^25 + 2253056*x^21 + 6572928*x^17 + 80640*x^13 - 283648*x^9 - 141312*x^5 - 14336*x))*sqr
t(sqrt(5) + 2) - (x^4 + 2)^(1/4)*(sqrt(5)*sqrt(2)*(95519*x^47 - 417922*x^43 - 6811916*x^39 - 24252224*x^35 - 3
8254288*x^31 - 27904608*x^27 - 7052352*x^23 - 93696*x^19 - 1097472*x^15 + 267776*x^11 + 125952*x^7 + 12288*x^3
) + sqrt(2)*(172357*x^47 + 4920726*x^43 + 31853380*x^39 + 88748912*x^35 + 119801296*x^31 + 71833760*x^27 + 101
58272*x^23 + 179968*x^19 + 4671232*x^15 - 68096*x^11 - 218112*x^7 - 24576*x^3)))*(sqrt(5) + 2)^(3/4) + 32*sqrt
(5)*(44757*x^48 + 528500*x^44 + 2531006*x^40 + 6388884*x^36 + 9172544*x^32 + 7394272*x^28 + 2846624*x^24 + 106
688*x^20 - 75776*x^16 + 115200*x^12 + 15872*x^8 + 1024*x^4) + 16*((91357*x^46 + 1895268*x^42 + 10212936*x^38 +
22875904*x^34 + 22078448*x^30 + 6990272*x^26 + 1117568*x^22 + 2320896*x^18 + 413952*x^14 - 74752*x^10 - 20480
*x^6 + sqrt(5)*(47877*x^46 - 154580*x^42 - 2723168*x^38 - 8670208*x^34 - 10594640*x^30 - 3899328*x^26 + 805376
*x^22 + 61952*x^18 + 42240*x^14 + 58368*x^10 + 10240*x^6))*sqrt(x^4 + 2) - 2*(23987*x^48 + 227560*x^44 + 93835
0*x^40 + 1981652*x^36 + 1955168*x^32 + 712608*x^28 + 496288*x^24 + 274624*x^20 - 612864*x^16 + 116224*x^12 - 8
704*x^8 - 7168*x^4 + 2*sqrt(5)*(5352*x^48 + 52677*x^44 + 202941*x^40 + 418518*x^36 + 526744*x^32 + 300896*x^28
- 223888*x^24 - 296544*x^20 + 36992*x^16 - 43520*x^12 + 256*x^8 + 1536*x^4))*sqrt(sqrt(5) + 2))*sqrt(sqrt(5)
+ 2) + 32*(2*(x^4 + 2)^(3/4)*(sqrt(5)*sqrt(2)*(11608*x^45 + 30607*x^41 - 58200*x^37 - 138724*x^33 + 244200*x^2
9 + 571344*x^25 + 243360*x^21 + 38272*x^17 + 128*x^13 + 3584*x^9) + sqrt(2)*(24419*x^45 + 289341*x^41 + 125450
8*x^37 + 2575396*x^33 + 2690040*x^29 + 1562960*x^25 + 649760*x^21 + 125952*x^17 + 20608*x^13 - 7168*x^9)) - (x
^4 + 2)^(1/4)*(sqrt(5)*sqrt(2)*(9667*x^47 + 271524*x^43 + 1604384*x^39 + 3720976*x^35 + 3252512*x^31 - 143712*
x^27 - 1076608*x^23 + 108928*x^19 - 15104*x^15 - 39424*x^11 + 3072*x^7) + sqrt(2)*(27805*x^47 - 275016*x^43 -
3020872*x^39 - 9166816*x^35 - 11334816*x^31 - 4883488*x^27 - 483712*x^23 - 1009280*x^19 - 75520*x^15 + 79360*x
^11 - 7168*x^7))*sqrt(sqrt(5) + 2))*(sqrt(5) + 2)^(1/4) + 16384)/(60929*x^48 - 8635020*x^44 - 72157788*x^40 -
223468208*x^36 - 314189280*x^32 - 181711040*x^28 - 22351296*x^24 - 8686848*x^20 - 7921152*x^16 + 1174528*x^12
+ 500736*x^8 + 77824*x^4 + 4096)) + 4*sqrt(5)*sqrt(2)*x^5*(sqrt(5) + 2)^(1/4)*arctan(1/4*(3205700*x^48 + 37345
168*x^44 + 180959952*x^40 + 465140928*x^36 + 664687232*x^32 + 494124288*x^28 + 141907200*x^24 - 6450176*x^20 +
1075200*x^16 + 2797568*x^12 + 1691648*x^8 + 278528*x^4 - 16*(95823*x^46 + 1251203*x^42 + 6523442*x^38 + 16526
620*x^34 + 19670784*x^30 + 7602736*x^26 - 1179424*x^22 + 1387328*x^18 + 445952*x^14 - 243456*x^10 - 46592*x^6
- 3072*x^2 + sqrt(5)*(48196*x^46 - 196187*x^42 - 3617546*x^38 - 13241892*x^34 - 19800240*x^30 - 11105648*x^26
- 526432*x^22 - 398016*x^18 - 373504*x^14 + 58112*x^10 + 13824*x^6 + 1024*x^2))*sqrt(x^4 + 2)*sqrt(sqrt(5) + 2
) + sqrt(2)*(32*(4449*x^45 + 634636*x^41 + 3555644*x^37 + 6358496*x^33 + 2708704*x^29 - 1547296*x^25 + 190272*
x^21 + 181376*x^17 - 108800*x^13 + 5632*x^9 + sqrt(5)*(5925*x^45 - 282704*x^41 - 1807476*x^37 - 3570256*x^33 -
2134720*x^29 + 190240*x^25 - 209984*x^21 - 98176*x^17 + 47360*x^13 - 2560*x^9))*(x^4 + 2)^(3/4)*sqrt(sqrt(5)
+ 2) + (2*sqrt(x^4 + 2)*(sqrt(5)*sqrt(2)*(10653*x^46 + 2956548*x^42 + 20110664*x^38 + 50652360*x^34 + 56023600
*x^30 + 25198208*x^26 + 3725952*x^22 + 757632*x^18 + 1234176*x^14 - 217088*x^10 - 77824*x^6 - 6144*x^2) + sqrt
(2)*(108077*x^46 - 5431312*x^42 - 40263280*x^38 - 102843464*x^34 - 112864528*x^30 - 50417856*x^26 - 9076224*x^
22 - 2229120*x^18 - 2436864*x^14 + 578560*x^10 + 186368*x^6 + 14336*x^2)) + (sqrt(5)*sqrt(2)*(28915*x^48 - 836
6472*x^44 - 80621076*x^40 - 293448080*x^36 - 499554944*x^32 - 380351744*x^28 - 90915392*x^24 - 11139840*x^20 -
17656832*x^16 + 94208*x^12 + 1137664*x^8 + 290816*x^4 + 20480) - sqrt(2)*(190537*x^48 - 17103640*x^44 - 17268
3372*x^40 - 639099216*x^36 - 1098574144*x^32 - 842361344*x^28 - 201443776*x^24 - 22353664*x^20 - 38754304*x^16
- 204800*x^12 + 2425856*x^8 + 634880*x^4 + 45056))*sqrt(sqrt(5) + 2))*(sqrt(5) + 2)^(3/4) - 8*(4*(11786*x^45
- 418982*x^41 - 3153502*x^37 - 7711008*x^33 - 7243648*x^29 - 1668480*x^25 + 72544*x^21 - 338048*x^17 + 23296*x
^13 + 30720*x^9 + 3584*x^5 + sqrt(5)*(2097*x^45 + 266982*x^41 + 1696566*x^37 + 3848848*x^33 + 3350352*x^29 + 6
02560*x^25 - 43104*x^21 + 200064*x^17 + 1536*x^13 - 12288*x^9 - 1536*x^5))*(x^4 + 2)^(3/4) - (12471*x^47 + 710
367*x^43 + 2116698*x^39 - 3578860*x^35 - 18142976*x^31 - 16588112*x^27 - 1714848*x^23 - 1272384*x^19 - 446976*
x^15 - 139008*x^11 + 70144*x^7 + 11264*x^3 + sqrt(5)*(9172*x^47 - 210451*x^43 - 628290*x^39 + 2143588*x^35 + 8
786384*x^31 + 8031184*x^27 + 1024032*x^23 + 498368*x^19 + 118016*x^15 + 44800*x^11 - 33280*x^7 - 5120*x^3))*(x
^4 + 2)^(1/4)*sqrt(sqrt(5) + 2))*sqrt(sqrt(5) + 2) - 256*(2073*x^47 + 31642*x^43 + 149420*x^39 + 309672*x^35 +
314176*x^31 + 198976*x^27 + 124608*x^23 + 27520*x^19 - 10496*x^15 + 3584*x^11 + sqrt(5)*(953*x^47 + 10346*x^4
3 + 54828*x^39 + 165000*x^35 + 269856*x^31 + 208768*x^27 + 52032*x^23 + 5760*x^19 + 6400*x^15 - 1536*x^11))*(x
^4 + 2)^(1/4) + 8*(2*sqrt(5)*sqrt(2)*(6759*x^48 + 192756*x^44 + 1338912*x^40 + 3951568*x^36 + 5386016*x^32 + 2
908960*x^28 + 366208*x^24 + 417664*x^20 + 157952*x^16 - 25088*x^12 - 9216*x^8) - sqrt(x^4 + 2)*(sqrt(5)*sqrt(2
)*(12951*x^46 - 734356*x^42 - 4933932*x^38 - 10666960*x^34 - 7733808*x^30 + 208192*x^26 + 222272*x^22 - 739840
*x^18 + 166656*x^14 + 22528*x^10 - 5120*x^6) + sqrt(2)*(3493*x^46 + 2013180*x^42 + 12514388*x^38 + 26406800*x^
34 + 18862832*x^30 - 954432*x^26 - 1374912*x^22 + 1328128*x^18 - 426752*x^14 - 55296*x^10 + 11264*x^6))*sqrt(s
qrt(5) + 2) + 2*sqrt(2)*(19945*x^48 - 252392*x^44 - 2918376*x^40 - 9824736*x^36 - 13811360*x^32 - 6643360*x^28
+ 793216*x^24 + 89984*x^20 - 152320*x^16 + 77312*x^12 + 21504*x^8))*(sqrt(5) + 2)^(1/4))*sqrt((4*(x^6 - 6*x^2
+ sqrt(5)*(x^6 - 2*x^2))*sqrt(x^4 + 2) + (x^8 - 2*x^4 + sqrt(5)*(x^8 - 2*x^4 - 4) - 4)*sqrt(sqrt(5) + 2) - 2*
((sqrt(5)*sqrt(2)*x^5 - sqrt(2)*(x^5 + 4*x))*(x^4 + 2)^(3/4)*sqrt(sqrt(5) + 2) - 2*(sqrt(5)*sqrt(2)*x^3 - sqrt
(2)*(x^7 - x^3))*(x^4 + 2)^(1/4))*(sqrt(5) + 2)^(1/4))/(x^8 - 2*x^4 - 4)) + 4*((x^4 + 2)^(3/4)*(sqrt(5)*sqrt(2
)*(60399*x^45 - 2243940*x^41 - 17911864*x^37 - 47674696*x^33 - 51732592*x^29 - 17641088*x^25 - 228224*x^21 - 2
926464*x^17 - 241920*x^13 + 77824*x^9 + 57344*x^5 + 6144*x) + sqrt(2)*(53127*x^45 + 6673680*x^41 + 45520576*x^
37 + 114960072*x^33 + 121770576*x^29 + 42292928*x^25 + 2253056*x^21 + 6572928*x^17 + 80640*x^13 - 283648*x^9 -
141312*x^5 - 14336*x))*sqrt(sqrt(5) + 2) - (x^4 + 2)^(1/4)*(sqrt(5)*sqrt(2)*(95519*x^47 - 417922*x^43 - 68119
16*x^39 - 24252224*x^35 - 38254288*x^31 - 27904608*x^27 - 7052352*x^23 - 93696*x^19 - 1097472*x^15 + 267776*x^
11 + 125952*x^7 + 12288*x^3) + sqrt(2)*(172357*x^47 + 4920726*x^43 + 31853380*x^39 + 88748912*x^35 + 119801296
*x^31 + 71833760*x^27 + 10158272*x^23 + 179968*x^19 + 4671232*x^15 - 68096*x^11 - 218112*x^7 - 24576*x^3)))*(s
qrt(5) + 2)^(3/4) + 32*sqrt(5)*(44757*x^48 + 528500*x^44 + 2531006*x^40 + 6388884*x^36 + 9172544*x^32 + 739427
2*x^28 + 2846624*x^24 + 106688*x^20 - 75776*x^16 + 115200*x^12 + 15872*x^8 + 1024*x^4) + 16*((91357*x^46 + 189
5268*x^42 + 10212936*x^38 + 22875904*x^34 + 22078448*x^30 + 6990272*x^26 + 1117568*x^22 + 2320896*x^18 + 41395
2*x^14 - 74752*x^10 - 20480*x^6 + sqrt(5)*(47877*x^46 - 154580*x^42 - 2723168*x^38 - 8670208*x^34 - 10594640*x
^30 - 3899328*x^26 + 805376*x^22 + 61952*x^18 + 42240*x^14 + 58368*x^10 + 10240*x^6))*sqrt(x^4 + 2) - 2*(23987
*x^48 + 227560*x^44 + 938350*x^40 + 1981652*x^36 + 1955168*x^32 + 712608*x^28 + 496288*x^24 + 274624*x^20 - 61
2864*x^16 + 116224*x^12 - 8704*x^8 - 7168*x^4 + 2*sqrt(5)*(5352*x^48 + 52677*x^44 + 202941*x^40 + 418518*x^36
+ 526744*x^32 + 300896*x^28 - 223888*x^24 - 296544*x^20 + 36992*x^16 - 43520*x^12 + 256*x^8 + 1536*x^4))*sqrt(
sqrt(5) + 2))*sqrt(sqrt(5) + 2) - 32*(2*(x^4 + 2)^(3/4)*(sqrt(5)*sqrt(2)*(11608*x^45 + 30607*x^41 - 58200*x^37
- 138724*x^33 + 244200*x^29 + 571344*x^25 + 243360*x^21 + 38272*x^17 + 128*x^13 + 3584*x^9) + sqrt(2)*(24419*
x^45 + 289341*x^41 + 1254508*x^37 + 2575396*x^33 + 2690040*x^29 + 1562960*x^25 + 649760*x^21 + 125952*x^17 + 2
0608*x^13 - 7168*x^9)) - (x^4 + 2)^(1/4)*(sqrt(5)*sqrt(2)*(9667*x^47 + 271524*x^43 + 1604384*x^39 + 3720976*x^
35 + 3252512*x^31 - 143712*x^27 - 1076608*x^23 + 108928*x^19 - 15104*x^15 - 39424*x^11 + 3072*x^7) + sqrt(2)*(
27805*x^47 - 275016*x^43 - 3020872*x^39 - 9166816*x^35 - 11334816*x^31 - 4883488*x^27 - 483712*x^23 - 1009280*
x^19 - 75520*x^15 + 79360*x^11 - 7168*x^7))*sqrt(sqrt(5) + 2))*(sqrt(5) + 2)^(1/4) + 16384)/(60929*x^48 - 8635
020*x^44 - 72157788*x^40 - 223468208*x^36 - 314189280*x^32 - 181711040*x^28 - 22351296*x^24 - 8686848*x^20 - 7
921152*x^16 + 1174528*x^12 + 500736*x^8 + 77824*x^4 + 4096)) + sqrt(5)*sqrt(2)*x^5*(sqrt(5) + 2)^(1/4)*log(2*(
4*(x^6 - 6*x^2 + sqrt(5)*(x^6 - 2*x^2))*sqrt(x^4 + 2) + (x^8 - 2*x^4 + sqrt(5)*(x^8 - 2*x^4 - 4) - 4)*sqrt(sqr
t(5) + 2) + 2*((sqrt(5)*sqrt(2)*x^5 - sqrt(2)*(x^5 + 4*x))*(x^4 + 2)^(3/4)*sqrt(sqrt(5) + 2) - 2*(sqrt(5)*sqrt
(2)*x^3 - sqrt(2)*(x^7 - x^3))*(x^4 + 2)^(1/4))*(sqrt(5) + 2)^(1/4))/(x^8 - 2*x^4 - 4)) - sqrt(5)*sqrt(2)*x^5*
(sqrt(5) + 2)^(1/4)*log(2*(4*(x^6 - 6*x^2 + sqrt(5)*(x^6 - 2*x^2))*sqrt(x^4 + 2) + (x^8 - 2*x^4 + sqrt(5)*(x^8
- 2*x^4 - 4) - 4)*sqrt(sqrt(5) + 2) - 2*((sqrt(5)*sqrt(2)*x^5 - sqrt(2)*(x^5 + 4*x))*(x^4 + 2)^(3/4)*sqrt(sqr
t(5) + 2) - 2*(sqrt(5)*sqrt(2)*x^3 - sqrt(2)*(x^7 - x^3))*(x^4 + 2)^(1/4))*(sqrt(5) + 2)^(1/4))/(x^8 - 2*x^4 -
4)) + 8*sqrt(5)*x^5*(sqrt(5) - 2)^(1/4)*arctan(1/4*(sqrt(2)*(11*x^8 + 22*x^4 + 2*(11*x^6 + 14*x^2 + sqrt(5)*(
5*x^6 + 6*x^2))*sqrt(x^4 + 2)*sqrt(sqrt(5) - 2) + sqrt(5)*(5*x^8 + 10*x^4 + 4) + 12)*sqrt((sqrt(5) - 1)*sqrt(s
qrt(5) - 2))*(sqrt(5) - 2)^(1/4) + 4*((3*x^5 + sqrt(5)*(x^5 + 2*x) + 2*x)*(x^4 + 2)^(3/4) + (7*x^7 + 8*x^3 + s
qrt(5)*(3*x^7 + 4*x^3))*(x^4 + 2)^(1/4)*sqrt(sqrt(5) - 2))*(sqrt(5) - 2)^(1/4))/(x^8 - 2*x^4 - 4)) + 2*sqrt(5)
*x^5*(sqrt(5) - 2)^(1/4)*log((4*(x^5 + sqrt(5)*x - x)*(x^4 + 2)^(3/4) + 2*(3*x^7 + 2*x^3 + sqrt(5)*(x^7 + 2*x^
3))*(x^4 + 2)^(1/4)*sqrt(sqrt(5) - 2) + (2*(sqrt(5)*x^6 + x^6 + 4*x^2)*sqrt(x^4 + 2) + (7*x^8 + 14*x^4 + sqrt(
5)*(3*x^8 + 6*x^4 + 4) + 4)*sqrt(sqrt(5) - 2))*(sqrt(5) - 2)^(1/4))/(x^8 - 2*x^4 - 4)) - 2*sqrt(5)*x^5*(sqrt(5
) - 2)^(1/4)*log((4*(x^5 + sqrt(5)*x - x)*(x^4 + 2)^(3/4) + 2*(3*x^7 + 2*x^3 + sqrt(5)*(x^7 + 2*x^3))*(x^4 + 2
)^(1/4)*sqrt(sqrt(5) - 2) - (2*(sqrt(5)*x^6 + x^6 + 4*x^2)*sqrt(x^4 + 2) + (7*x^8 + 14*x^4 + sqrt(5)*(3*x^8 +
6*x^4 + 4) + 4)*sqrt(sqrt(5) - 2))*(sqrt(5) - 2)^(1/4))/(x^8 - 2*x^4 - 4)) - 32*(2*x^4 - 1)*(x^4 + 2)^(1/4))/x
^5
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^4+2)^(1/4)*(x^8-4)/x^6/(x^8-2*x^4-4),x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to convert to real 1/4 Error: Bad Argument ValueUnable to convert to real 1/4 Error: Bad Argument Value
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maple [B] time = 21.35, size = 6507, normalized size = 81.34 \[\text {output too large to display}\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^4+2)^(1/4)*(x^8-4)/x^6/(x^8-2*x^4-4),x)
[Out]
result too large to display
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{8} - 4\right )} {\left (x^{4} + 2\right )}^{\frac {1}{4}}}{{\left (x^{8} - 2 \, x^{4} - 4\right )} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^4+2)^(1/4)*(x^8-4)/x^6/(x^8-2*x^4-4),x, algorithm="maxima")
[Out]
integrate((x^8 - 4)*(x^4 + 2)^(1/4)/((x^8 - 2*x^4 - 4)*x^6), x)
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {{\left (x^4+2\right )}^{1/4}\,\left (x^8-4\right )}{x^6\,\left (-x^8+2\,x^4+4\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-((x^4 + 2)^(1/4)*(x^8 - 4))/(x^6*(2*x^4 - x^8 + 4)),x)
[Out]
-int(((x^4 + 2)^(1/4)*(x^8 - 4))/(x^6*(2*x^4 - x^8 + 4)), 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**4+2)**(1/4)*(x**8-4)/x**6/(x**8-2*x**4-4),x)
[Out]
Timed out
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