\(\int \frac {\sqrt [4]{2+x^4} (-4+x^8)}{x^6 (-4-2 x^4+x^8)} \, dx\) [1067]

Optimal result

Integrand size = 30, antiderivative size = 80 \[ \int \frac {\sqrt [4]{2+x^4} \left (-4+x^8\right )}{x^6 \left (-4-2 x^4+x^8\right )} \, dx=\frac {\sqrt [4]{2+x^4} \left (-1+2 x^4\right )}{5 x^5}+\frac {1}{8} \text {RootSum}\left [-1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{2+x^4}-x \text {$\#$1}\right )}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \]

[Out]

Unintegrable

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(877\) vs. \(2(80)=160\).

Time = 2.24 (sec) , antiderivative size = 877, normalized size of antiderivative = 10.96, number of steps used = 53, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.567, Rules used = {6860, 270, 283, 338, 304, 209, 212, 1542, 524, 1532, 508, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {\sqrt [4]{2+x^4} \left (-4+x^8\right )}{x^6 \left (-4-2 x^4+x^8\right )} \, dx=-\frac {\operatorname {AppellF1}\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 {\operatorname {AppellF1}\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 )} \arctan \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}} \arctan \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}} \arctan \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}} \arctan \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}} \arctan \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}} \arctan \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 )} \text {arctanh}\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}} \text {arctanh}\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} \]

[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 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 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[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 270

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 283

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 303

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 304

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 338

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 508

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 524

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)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], 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 631

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 642

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 1176

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 1179

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 1532

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 1542

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 6860

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*} \text {integral}& = \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} \text {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} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )+\frac {1}{4} \text {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} \arctan \left (\frac {x}{\sqrt [4]{2+x^4}}\right )-\frac {1}{4} \text {arctanh}\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} \text {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 \operatorname {AppellF1}\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 \operatorname {AppellF1}\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} \arctan \left (\frac {x}{\sqrt [4]{2+x^4}}\right )-\frac {1}{4} \text {arctanh}\left (\frac {x}{\sqrt [4]{2+x^4}}\right )+\frac {1}{4} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )-\frac {1}{4} \text {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 \operatorname {AppellF1}\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 \operatorname {AppellF1}\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 \operatorname {AppellF1}\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 \operatorname {AppellF1}\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 \operatorname {AppellF1}\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 \operatorname {AppellF1}\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 \text {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 \text {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 ) \text {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 ) \text {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 \operatorname {AppellF1}\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 \operatorname {AppellF1}\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 {\text {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 {\text {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 ) \text {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 ) \text {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 {\text {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 {\text {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 ) \text {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 ) \text {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 \operatorname {AppellF1}\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 \operatorname {AppellF1}\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}} \arctan \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 )} \arctan \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}} \text {arctanh}\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 )} \text {arctanh}\left (\frac {\sqrt [4]{\frac {2}{-1+\sqrt {5}}} x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {5}}-\frac {\text {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 {\text {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 ) \text {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 ) \text {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}} \text {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}} \text {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}} \text {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}} \text {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 \operatorname {AppellF1}\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 \operatorname {AppellF1}\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}} \arctan \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 )} \arctan \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}} \text {arctanh}\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 )} \text {arctanh}\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}} \text {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}} \text {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}} \text {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}} \text {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 \operatorname {AppellF1}\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 \operatorname {AppellF1}\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}} \arctan \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 )} \arctan \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}} \arctan \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}} \arctan \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}} \arctan \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}} \arctan \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}} \text {arctanh}\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 )} \text {arctanh}\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*}

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [4]{2+x^4} \left (-4+x^8\right )}{x^6 \left (-4-2 x^4+x^8\right )} \, dx=\frac {\sqrt [4]{2+x^4} \left (-1+2 x^4\right )}{5 x^5}+\frac {1}{8} \text {RootSum}\left [-1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{2+x^4}-x \text {$\#$1}\right )}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \]

[In]

Integrate[((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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 1.

Time = 32.28 (sec) , antiderivative size = 6505, normalized size of antiderivative = 81.31

\[\text {output too large to display}\]

[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

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 4.41 (sec) , antiderivative size = 1398, normalized size of antiderivative = 17.48 \[ \int \frac {\sqrt [4]{2+x^4} \left (-4+x^8\right )}{x^6 \left (-4-2 x^4+x^8\right )} \, dx=\text {Too large to display} \]

[In]

integrate((x^4+2)^(1/4)*(x^8-4)/x^6/(x^8-2*x^4-4),x, algorithm="fricas")

[Out]

-1/80*(sqrt(5)*x^5*sqrt(-sqrt(sqrt(5) - 2))*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(-sqrt(sqrt(5) - 2)))/(x^8 - 2*x^4 - 4))
- sqrt(5)*x^5*sqrt(-sqrt(sqrt(5) - 2))*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(-sqrt(sqrt(5) - 2)))/(x^8 - 2*x^4 - 4)) + s
qrt(5)*x^5*sqrt(-sqrt(-sqrt(5) - 2))*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 + 1
4*x^4 - sqrt(5)*(3*x^8 + 6*x^4 + 4) + 4)*sqrt(-sqrt(5) - 2))*sqrt(-sqrt(-sqrt(5) - 2)))/(x^8 - 2*x^4 - 4)) - s
qrt(5)*x^5*sqrt(-sqrt(-sqrt(5) - 2))*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 + 1
4*x^4 - sqrt(5)*(3*x^8 + 6*x^4 + 4) + 4)*sqrt(-sqrt(5) - 2))*sqrt(-sqrt(-sqrt(5) - 2)))/(x^8 - 2*x^4 - 4)) + s
qrt(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)) - sqrt(5)*x^5*(sq
rt(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)) + 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)*s
qrt(-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)) - 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)*sq
rt(-sqrt(5) - 2))*(-sqrt(5) - 2)^(1/4))/(x^8 - 2*x^4 - 4)) - 16*(2*x^4 - 1)*(x^4 + 2)^(1/4))/x^5

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt [4]{2+x^4} \left (-4+x^8\right )}{x^6 \left (-4-2 x^4+x^8\right )} \, dx=\text {Timed out} \]

[In]

integrate((x**4+2)**(1/4)*(x**8-4)/x**6/(x**8-2*x**4-4),x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.38 \[ \int \frac {\sqrt [4]{2+x^4} \left (-4+x^8\right )}{x^6 \left (-4-2 x^4+x^8\right )} \, dx=\int { \frac {{\left (x^{8} - 4\right )} {\left (x^{4} + 2\right )}^{\frac {1}{4}}}{{\left (x^{8} - 2 \, x^{4} - 4\right )} x^{6}} \,d x } \]

[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)

Giac [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.38 \[ \int \frac {\sqrt [4]{2+x^4} \left (-4+x^8\right )}{x^6 \left (-4-2 x^4+x^8\right )} \, dx=\int { \frac {{\left (x^{8} - 4\right )} {\left (x^{4} + 2\right )}^{\frac {1}{4}}}{{\left (x^{8} - 2 \, x^{4} - 4\right )} x^{6}} \,d x } \]

[In]

integrate((x^4+2)^(1/4)*(x^8-4)/x^6/(x^8-2*x^4-4),x, algorithm="giac")

[Out]

integrate((x^8 - 4)*(x^4 + 2)^(1/4)/((x^8 - 2*x^4 - 4)*x^6), x)

Mupad [N/A]

Not integrable

Time = 5.98 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.42 \[ \int \frac {\sqrt [4]{2+x^4} \left (-4+x^8\right )}{x^6 \left (-4-2 x^4+x^8\right )} \, dx=-\int \frac {{\left (x^4+2\right )}^{1/4}\,\left (x^8-4\right )}{x^6\,\left (-x^8+2\,x^4+4\right )} \,d x \]

[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)