\(\int \frac {\sqrt [4]{-1+x^4} (1+x^4+x^8)}{x^6 (-1+2 x^8)} \, dx\) [1583]

Optimal result

Integrand size = 30, antiderivative size = 108 \[ \int \frac {\sqrt [4]{-1+x^4} \left (1+x^4+x^8\right )}{x^6 \left (-1+2 x^8\right )} \, dx=\frac {\sqrt [4]{-1+x^4} \left (1+4 x^4\right )}{5 x^5}+\frac {1}{8} \text {RootSum}\left [-1-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right )+2 \log (x) \text {$\#$1}^4-2 \log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+\text {$\#$1}^7}\&\right ] \]

[Out]

Unintegrable

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(793\) vs. \(2(108)=216\).

Time = 0.95 (sec) , antiderivative size = 793, normalized size of antiderivative = 7.34, number of steps used = 55, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6857, 270, 283, 338, 304, 209, 212, 1543, 525, 524, 1533, 508, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {\sqrt [4]{-1+x^4} \left (1+x^4+x^8\right )}{x^6 \left (-1+2 x^8\right )} \, dx=-\frac {\sqrt [4]{x^4-1} x^3 \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x^4,-\sqrt {2} x^4\right )}{2 \sqrt [4]{1-x^4}}-\frac {\sqrt [4]{x^4-1} x^3 \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x^4,\sqrt {2} x^4\right )}{2 \sqrt [4]{1-x^4}}+\frac {\sqrt [4]{5 \sqrt {2}-7} \arctan \left (\frac {x}{\sqrt [4]{\sqrt {2}-1} \sqrt [4]{x^4-1}}\right )}{2 \sqrt {2}}+\frac {1}{4} \sqrt [4]{5 \sqrt {2}-7} \arctan \left (\frac {x}{\sqrt [4]{\sqrt {2}-1} \sqrt [4]{x^4-1}}\right )+\frac {\sqrt [4]{7+5 \sqrt {2}} \arctan \left (1-\frac {\sqrt {2} x}{\sqrt [4]{1+\sqrt {2}} \sqrt [4]{x^4-1}}\right )}{4 \sqrt {2}}-\frac {1}{4} \sqrt [4]{7+5 \sqrt {2}} \arctan \left (1-\frac {\sqrt {2} x}{\sqrt [4]{1+\sqrt {2}} \sqrt [4]{x^4-1}}\right )-\frac {\sqrt [4]{7+5 \sqrt {2}} \arctan \left (\frac {\sqrt {2} x}{\sqrt [4]{1+\sqrt {2}} \sqrt [4]{x^4-1}}+1\right )}{4 \sqrt {2}}+\frac {1}{4} \sqrt [4]{7+5 \sqrt {2}} \arctan \left (\frac {\sqrt {2} x}{\sqrt [4]{1+\sqrt {2}} \sqrt [4]{x^4-1}}+1\right )-\frac {\sqrt [4]{5 \sqrt {2}-7} \text {arctanh}\left (\frac {x}{\sqrt [4]{\sqrt {2}-1} \sqrt [4]{x^4-1}}\right )}{2 \sqrt {2}}-\frac {1}{4} \sqrt [4]{5 \sqrt {2}-7} \text {arctanh}\left (\frac {x}{\sqrt [4]{\sqrt {2}-1} \sqrt [4]{x^4-1}}\right )+\frac {\sqrt [4]{x^4-1}}{x}-\frac {\left (x^4-1\right )^{5/4}}{5 x^5}-\frac {\sqrt [4]{7+5 \sqrt {2}} \log \left (-\frac {\sqrt {2} \sqrt [4]{1+\sqrt {2}} x}{\sqrt [4]{x^4-1}}+\frac {x^2}{\sqrt {x^4-1}}+\sqrt {1+\sqrt {2}}\right )}{8 \sqrt {2}}+\frac {1}{8} \sqrt [4]{7+5 \sqrt {2}} \log \left (-\frac {\sqrt {2} \sqrt [4]{1+\sqrt {2}} x}{\sqrt [4]{x^4-1}}+\frac {x^2}{\sqrt {x^4-1}}+\sqrt {1+\sqrt {2}}\right )+\frac {\sqrt [4]{7+5 \sqrt {2}} \log \left (\frac {\sqrt {2} \sqrt [4]{1+\sqrt {2}} x}{\sqrt [4]{x^4-1}}+\frac {x^2}{\sqrt {x^4-1}}+\sqrt {1+\sqrt {2}}\right )}{8 \sqrt {2}}-\frac {1}{8} \sqrt [4]{7+5 \sqrt {2}} \log \left (\frac {\sqrt {2} \sqrt [4]{1+\sqrt {2}} x}{\sqrt [4]{x^4-1}}+\frac {x^2}{\sqrt {x^4-1}}+\sqrt {1+\sqrt {2}}\right ) \]

[In]

Int[((-1 + x^4)^(1/4)*(1 + x^4 + x^8))/(x^6*(-1 + 2*x^8)),x]

[Out]

(-1 + x^4)^(1/4)/x - (-1 + x^4)^(5/4)/(5*x^5) - (x^3*(-1 + x^4)^(1/4)*AppellF1[3/4, -1/4, 1, 7/4, x^4, -(Sqrt[
2]*x^4)])/(2*(1 - x^4)^(1/4)) - (x^3*(-1 + x^4)^(1/4)*AppellF1[3/4, -1/4, 1, 7/4, x^4, Sqrt[2]*x^4])/(2*(1 - x
^4)^(1/4)) + ((-7 + 5*Sqrt[2])^(1/4)*ArcTan[x/((-1 + Sqrt[2])^(1/4)*(-1 + x^4)^(1/4))])/4 + ((-7 + 5*Sqrt[2])^
(1/4)*ArcTan[x/((-1 + Sqrt[2])^(1/4)*(-1 + x^4)^(1/4))])/(2*Sqrt[2]) - ((7 + 5*Sqrt[2])^(1/4)*ArcTan[1 - (Sqrt
[2]*x)/((1 + Sqrt[2])^(1/4)*(-1 + x^4)^(1/4))])/4 + ((7 + 5*Sqrt[2])^(1/4)*ArcTan[1 - (Sqrt[2]*x)/((1 + Sqrt[2
])^(1/4)*(-1 + x^4)^(1/4))])/(4*Sqrt[2]) + ((7 + 5*Sqrt[2])^(1/4)*ArcTan[1 + (Sqrt[2]*x)/((1 + Sqrt[2])^(1/4)*
(-1 + x^4)^(1/4))])/4 - ((7 + 5*Sqrt[2])^(1/4)*ArcTan[1 + (Sqrt[2]*x)/((1 + Sqrt[2])^(1/4)*(-1 + x^4)^(1/4))])
/(4*Sqrt[2]) - ((-7 + 5*Sqrt[2])^(1/4)*ArcTanh[x/((-1 + Sqrt[2])^(1/4)*(-1 + x^4)^(1/4))])/4 - ((-7 + 5*Sqrt[2
])^(1/4)*ArcTanh[x/((-1 + Sqrt[2])^(1/4)*(-1 + x^4)^(1/4))])/(2*Sqrt[2]) + ((7 + 5*Sqrt[2])^(1/4)*Log[Sqrt[1 +
Sqrt[2]] + x^2/Sqrt[-1 + x^4] - (Sqrt[2]*(1 + Sqrt[2])^(1/4)*x)/(-1 + x^4)^(1/4)])/8 - ((7 + 5*Sqrt[2])^(1/4)
*Log[Sqrt[1 + Sqrt[2]] + x^2/Sqrt[-1 + x^4] - (Sqrt[2]*(1 + Sqrt[2])^(1/4)*x)/(-1 + x^4)^(1/4)])/(8*Sqrt[2]) -
((7 + 5*Sqrt[2])^(1/4)*Log[Sqrt[1 + Sqrt[2]] + x^2/Sqrt[-1 + x^4] + (Sqrt[2]*(1 + Sqrt[2])^(1/4)*x)/(-1 + x^4
)^(1/4)])/8 + ((7 + 5*Sqrt[2])^(1/4)*Log[Sqrt[1 + Sqrt[2]] + x^2/Sqrt[-1 + x^4] + (Sqrt[2]*(1 + Sqrt[2])^(1/4)
*x)/(-1 + x^4)^(1/4)])/(8*Sqrt[2])

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 525

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPar
t[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]), Int[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x
] /; FreeQ[{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])

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 1533

Int[(((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^(n_))^(q_))/((a_) + (c_.)*(x_)^(n2_.)), 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*x^n, x]/(a + c*x^(2*n))), x], x] /; FreeQ[{a, c, d, e, f}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] &&  !Intege
rQ[q] && GtQ[q, 0] && GtQ[m, n - 1] && LeQ[m, 2*n - 1]

Rule 1543

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))^(q_))/((a_) + (c_.)*(x_)^(n2_.)), x_Symbol] :> Int[ExpandInte
grand[(d + e*x^n)^q, (f*x)^m/(a + c*x^(2*n)), x], x] /; FreeQ[{a, c, d, e, f, q, n}, x] && EqQ[n2, 2*n] && IGt
Q[n, 0] &&  !IntegerQ[q] && IntegerQ[m]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {\sqrt [4]{-1+x^4}}{x^6}-\frac {\sqrt [4]{-1+x^4}}{x^2}+\frac {x^2 \sqrt [4]{-1+x^4} \left (3+2 x^4\right )}{-1+2 x^8}\right ) \, dx \\ & = -\int \frac {\sqrt [4]{-1+x^4}}{x^6} \, dx-\int \frac {\sqrt [4]{-1+x^4}}{x^2} \, dx+\int \frac {x^2 \sqrt [4]{-1+x^4} \left (3+2 x^4\right )}{-1+2 x^8} \, dx \\ & = \frac {\sqrt [4]{-1+x^4}}{x}-\frac {\left (-1+x^4\right )^{5/4}}{5 x^5}-\int \frac {x^2}{\left (-1+x^4\right )^{3/4}} \, dx+\int \left (\frac {3 x^2 \sqrt [4]{-1+x^4}}{-1+2 x^8}+\frac {2 x^6 \sqrt [4]{-1+x^4}}{-1+2 x^8}\right ) \, dx \\ & = \frac {\sqrt [4]{-1+x^4}}{x}-\frac {\left (-1+x^4\right )^{5/4}}{5 x^5}+2 \int \frac {x^6 \sqrt [4]{-1+x^4}}{-1+2 x^8} \, dx+3 \int \frac {x^2 \sqrt [4]{-1+x^4}}{-1+2 x^8} \, dx-\text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right ) \\ & = \frac {\sqrt [4]{-1+x^4}}{x}-\frac {\left (-1+x^4\right )^{5/4}}{5 x^5}-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+3 \int \left (\frac {x^2 \sqrt [4]{-1+x^4}}{\sqrt {2} \left (-\sqrt {2}+2 x^4\right )}-\frac {x^2 \sqrt [4]{-1+x^4}}{\sqrt {2} \left (\sqrt {2}+2 x^4\right )}\right ) \, dx+\int \frac {x^2}{\left (-1+x^4\right )^{3/4}} \, dx-\int \frac {x^2 \left (-1+2 x^4\right )}{\left (-1+x^4\right )^{3/4} \left (-1+2 x^8\right )} \, dx \\ & = \frac {\sqrt [4]{-1+x^4}}{x}-\frac {\left (-1+x^4\right )^{5/4}}{5 x^5}+\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {3 \int \frac {x^2 \sqrt [4]{-1+x^4}}{-\sqrt {2}+2 x^4} \, dx}{\sqrt {2}}-\frac {3 \int \frac {x^2 \sqrt [4]{-1+x^4}}{\sqrt {2}+2 x^4} \, dx}{\sqrt {2}}-\int \left (-\frac {x^2}{\left (-1+x^4\right )^{3/4} \left (-1+2 x^8\right )}+\frac {2 x^6}{\left (-1+x^4\right )^{3/4} \left (-1+2 x^8\right )}\right ) \, dx+\text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right ) \\ & = \frac {\sqrt [4]{-1+x^4}}{x}-\frac {\left (-1+x^4\right )^{5/4}}{5 x^5}+\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-2 \int \frac {x^6}{\left (-1+x^4\right )^{3/4} \left (-1+2 x^8\right )} \, dx+\frac {\left (3 \sqrt [4]{-1+x^4}\right ) \int \frac {x^2 \sqrt [4]{1-x^4}}{-\sqrt {2}+2 x^4} \, dx}{\sqrt {2} \sqrt [4]{1-x^4}}-\frac {\left (3 \sqrt [4]{-1+x^4}\right ) \int \frac {x^2 \sqrt [4]{1-x^4}}{\sqrt {2}+2 x^4} \, dx}{\sqrt {2} \sqrt [4]{1-x^4}}+\int \frac {x^2}{\left (-1+x^4\right )^{3/4} \left (-1+2 x^8\right )} \, dx \\ & = \frac {\sqrt [4]{-1+x^4}}{x}-\frac {\left (-1+x^4\right )^{5/4}}{5 x^5}-\frac {x^3 \sqrt [4]{-1+x^4} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x^4,-\sqrt {2} x^4\right )}{2 \sqrt [4]{1-x^4}}-\frac {x^3 \sqrt [4]{-1+x^4} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x^4,\sqrt {2} x^4\right )}{2 \sqrt [4]{1-x^4}}-2 \int \left (\frac {x^2}{2 \left (-1+x^4\right )^{3/4} \left (-\sqrt {2}+2 x^4\right )}+\frac {x^2}{2 \left (-1+x^4\right )^{3/4} \left (\sqrt {2}+2 x^4\right )}\right ) \, dx+\int \left (\frac {x^2}{\sqrt {2} \left (-1+x^4\right )^{3/4} \left (-\sqrt {2}+2 x^4\right )}-\frac {x^2}{\sqrt {2} \left (-1+x^4\right )^{3/4} \left (\sqrt {2}+2 x^4\right )}\right ) \, dx \\ & = \frac {\sqrt [4]{-1+x^4}}{x}-\frac {\left (-1+x^4\right )^{5/4}}{5 x^5}-\frac {x^3 \sqrt [4]{-1+x^4} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x^4,-\sqrt {2} x^4\right )}{2 \sqrt [4]{1-x^4}}-\frac {x^3 \sqrt [4]{-1+x^4} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x^4,\sqrt {2} x^4\right )}{2 \sqrt [4]{1-x^4}}+\frac {\int \frac {x^2}{\left (-1+x^4\right )^{3/4} \left (-\sqrt {2}+2 x^4\right )} \, dx}{\sqrt {2}}-\frac {\int \frac {x^2}{\left (-1+x^4\right )^{3/4} \left (\sqrt {2}+2 x^4\right )} \, dx}{\sqrt {2}}-\int \frac {x^2}{\left (-1+x^4\right )^{3/4} \left (-\sqrt {2}+2 x^4\right )} \, dx-\int \frac {x^2}{\left (-1+x^4\right )^{3/4} \left (\sqrt {2}+2 x^4\right )} \, dx \\ & = \frac {\sqrt [4]{-1+x^4}}{x}-\frac {\left (-1+x^4\right )^{5/4}}{5 x^5}-\frac {x^3 \sqrt [4]{-1+x^4} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x^4,-\sqrt {2} x^4\right )}{2 \sqrt [4]{1-x^4}}-\frac {x^3 \sqrt [4]{-1+x^4} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x^4,\sqrt {2} x^4\right )}{2 \sqrt [4]{1-x^4}}+\frac {\text {Subst}\left (\int \frac {x^2}{-\sqrt {2}-\left (2-\sqrt {2}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt {2}}-\frac {\text {Subst}\left (\int \frac {x^2}{\sqrt {2}-\left (2+\sqrt {2}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt {2}}-\text {Subst}\left (\int \frac {x^2}{-\sqrt {2}-\left (2-\sqrt {2}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\text {Subst}\left (\int \frac {x^2}{\sqrt {2}-\left (2+\sqrt {2}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right ) \\ & = \frac {\sqrt [4]{-1+x^4}}{x}-\frac {\left (-1+x^4\right )^{5/4}}{5 x^5}-\frac {x^3 \sqrt [4]{-1+x^4} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x^4,-\sqrt {2} x^4\right )}{2 \sqrt [4]{1-x^4}}-\frac {x^3 \sqrt [4]{-1+x^4} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x^4,\sqrt {2} x^4\right )}{2 \sqrt [4]{1-x^4}}+\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {1+\sqrt {2}}-x^2}{-\sqrt {2}+\left (-2+\sqrt {2}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {1+\sqrt {2}}+x^2}{-\sqrt {2}+\left (-2+\sqrt {2}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {\text {Subst}\left (\int \frac {\sqrt {1+\sqrt {2}}-x^2}{-\sqrt {2}+\left (-2+\sqrt {2}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{2 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {\sqrt {1+\sqrt {2}}+x^2}{-\sqrt {2}+\left (-2+\sqrt {2}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{2 \sqrt {2}}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {-1+\sqrt {2}}-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{2 \left (2+\sqrt {2}\right )}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {-1+\sqrt {2}}+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{2 \left (2+\sqrt {2}\right )}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {-1+\sqrt {2}}-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{2 \sqrt {2} \left (2+\sqrt {2}\right )}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {-1+\sqrt {2}}+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{2 \sqrt {2} \left (2+\sqrt {2}\right )} \\ & = \frac {\sqrt [4]{-1+x^4}}{x}-\frac {\left (-1+x^4\right )^{5/4}}{5 x^5}-\frac {x^3 \sqrt [4]{-1+x^4} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x^4,-\sqrt {2} x^4\right )}{2 \sqrt [4]{1-x^4}}-\frac {x^3 \sqrt [4]{-1+x^4} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x^4,\sqrt {2} x^4\right )}{2 \sqrt [4]{1-x^4}}+\frac {1}{4} \sqrt [4]{-7+5 \sqrt {2}} \arctan \left (\frac {x}{\sqrt [4]{-1+\sqrt {2}} \sqrt [4]{-1+x^4}}\right )+\frac {\sqrt [4]{-7+5 \sqrt {2}} \arctan \left (\frac {x}{\sqrt [4]{-1+\sqrt {2}} \sqrt [4]{-1+x^4}}\right )}{2 \sqrt {2}}-\frac {1}{4} \sqrt [4]{-7+5 \sqrt {2}} \text {arctanh}\left (\frac {x}{\sqrt [4]{-1+\sqrt {2}} \sqrt [4]{-1+x^4}}\right )-\frac {\sqrt [4]{-7+5 \sqrt {2}} \text {arctanh}\left (\frac {x}{\sqrt [4]{-1+\sqrt {2}} \sqrt [4]{-1+x^4}}\right )}{2 \sqrt {2}}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1+\sqrt {2}}-\sqrt {2} \sqrt [4]{1+\sqrt {2}} x+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{4 \left (-2+\sqrt {2}\right )}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1+\sqrt {2}}+\sqrt {2} \sqrt [4]{1+\sqrt {2}} x+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{4 \left (-2+\sqrt {2}\right )}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1+\sqrt {2}}-\sqrt {2} \sqrt [4]{1+\sqrt {2}} x+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2} \left (-2+\sqrt {2}\right )}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1+\sqrt {2}}+\sqrt {2} \sqrt [4]{1+\sqrt {2}} x+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2} \left (-2+\sqrt {2}\right )}+\frac {1}{8} \sqrt [4]{7+5 \sqrt {2}} \text {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{1+\sqrt {2}}+2 x}{-\sqrt {1+\sqrt {2}}-\sqrt {2} \sqrt [4]{1+\sqrt {2}} x-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{8} \sqrt [4]{7+5 \sqrt {2}} \text {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{1+\sqrt {2}}-2 x}{-\sqrt {1+\sqrt {2}}+\sqrt {2} \sqrt [4]{1+\sqrt {2}} x-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {\sqrt [4]{7+5 \sqrt {2}} \text {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{1+\sqrt {2}}+2 x}{-\sqrt {1+\sqrt {2}}-\sqrt {2} \sqrt [4]{1+\sqrt {2}} x-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{8 \sqrt {2}}-\frac {\sqrt [4]{7+5 \sqrt {2}} \text {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{1+\sqrt {2}}-2 x}{-\sqrt {1+\sqrt {2}}+\sqrt {2} \sqrt [4]{1+\sqrt {2}} x-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{8 \sqrt {2}} \\ & = \frac {\sqrt [4]{-1+x^4}}{x}-\frac {\left (-1+x^4\right )^{5/4}}{5 x^5}-\frac {x^3 \sqrt [4]{-1+x^4} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x^4,-\sqrt {2} x^4\right )}{2 \sqrt [4]{1-x^4}}-\frac {x^3 \sqrt [4]{-1+x^4} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x^4,\sqrt {2} x^4\right )}{2 \sqrt [4]{1-x^4}}+\frac {1}{4} \sqrt [4]{-7+5 \sqrt {2}} \arctan \left (\frac {x}{\sqrt [4]{-1+\sqrt {2}} \sqrt [4]{-1+x^4}}\right )+\frac {\sqrt [4]{-7+5 \sqrt {2}} \arctan \left (\frac {x}{\sqrt [4]{-1+\sqrt {2}} \sqrt [4]{-1+x^4}}\right )}{2 \sqrt {2}}-\frac {1}{4} \sqrt [4]{-7+5 \sqrt {2}} \text {arctanh}\left (\frac {x}{\sqrt [4]{-1+\sqrt {2}} \sqrt [4]{-1+x^4}}\right )-\frac {\sqrt [4]{-7+5 \sqrt {2}} \text {arctanh}\left (\frac {x}{\sqrt [4]{-1+\sqrt {2}} \sqrt [4]{-1+x^4}}\right )}{2 \sqrt {2}}+\frac {1}{8} \sqrt [4]{7+5 \sqrt {2}} \log \left (\sqrt {1+\sqrt {2}}+\frac {x^2}{\sqrt {-1+x^4}}-\frac {\sqrt {2} \sqrt [4]{1+\sqrt {2}} x}{\sqrt [4]{-1+x^4}}\right )-\frac {\sqrt [4]{7+5 \sqrt {2}} \log \left (\sqrt {1+\sqrt {2}}+\frac {x^2}{\sqrt {-1+x^4}}-\frac {\sqrt {2} \sqrt [4]{1+\sqrt {2}} x}{\sqrt [4]{-1+x^4}}\right )}{8 \sqrt {2}}-\frac {1}{8} \sqrt [4]{7+5 \sqrt {2}} \log \left (\sqrt {1+\sqrt {2}}+\frac {x^2}{\sqrt {-1+x^4}}+\frac {\sqrt {2} \sqrt [4]{1+\sqrt {2}} x}{\sqrt [4]{-1+x^4}}\right )+\frac {\sqrt [4]{7+5 \sqrt {2}} \log \left (\sqrt {1+\sqrt {2}}+\frac {x^2}{\sqrt {-1+x^4}}+\frac {\sqrt {2} \sqrt [4]{1+\sqrt {2}} x}{\sqrt [4]{-1+x^4}}\right )}{8 \sqrt {2}}+\frac {1}{4} \sqrt [4]{7+5 \sqrt {2}} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} x}{\sqrt [4]{1+\sqrt {2}} \sqrt [4]{-1+x^4}}\right )-\frac {1}{4} \sqrt [4]{7+5 \sqrt {2}} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} x}{\sqrt [4]{1+\sqrt {2}} \sqrt [4]{-1+x^4}}\right )-\frac {\sqrt [4]{7+5 \sqrt {2}} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} x}{\sqrt [4]{1+\sqrt {2}} \sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2}}+\frac {\sqrt [4]{7+5 \sqrt {2}} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} x}{\sqrt [4]{1+\sqrt {2}} \sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2}} \\ & = \frac {\sqrt [4]{-1+x^4}}{x}-\frac {\left (-1+x^4\right )^{5/4}}{5 x^5}-\frac {x^3 \sqrt [4]{-1+x^4} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x^4,-\sqrt {2} x^4\right )}{2 \sqrt [4]{1-x^4}}-\frac {x^3 \sqrt [4]{-1+x^4} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x^4,\sqrt {2} x^4\right )}{2 \sqrt [4]{1-x^4}}+\frac {1}{4} \sqrt [4]{-7+5 \sqrt {2}} \arctan \left (\frac {x}{\sqrt [4]{-1+\sqrt {2}} \sqrt [4]{-1+x^4}}\right )+\frac {\sqrt [4]{-7+5 \sqrt {2}} \arctan \left (\frac {x}{\sqrt [4]{-1+\sqrt {2}} \sqrt [4]{-1+x^4}}\right )}{2 \sqrt {2}}-\frac {1}{4} \sqrt [4]{7+5 \sqrt {2}} \arctan \left (1-\frac {\sqrt {2} x}{\sqrt [4]{1+\sqrt {2}} \sqrt [4]{-1+x^4}}\right )+\frac {\sqrt [4]{7+5 \sqrt {2}} \arctan \left (1-\frac {\sqrt {2} x}{\sqrt [4]{1+\sqrt {2}} \sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2}}+\frac {1}{4} \sqrt [4]{7+5 \sqrt {2}} \arctan \left (1+\frac {\sqrt {2} x}{\sqrt [4]{1+\sqrt {2}} \sqrt [4]{-1+x^4}}\right )-\frac {\sqrt [4]{7+5 \sqrt {2}} \arctan \left (1+\frac {\sqrt {2} x}{\sqrt [4]{1+\sqrt {2}} \sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2}}-\frac {1}{4} \sqrt [4]{-7+5 \sqrt {2}} \text {arctanh}\left (\frac {x}{\sqrt [4]{-1+\sqrt {2}} \sqrt [4]{-1+x^4}}\right )-\frac {\sqrt [4]{-7+5 \sqrt {2}} \text {arctanh}\left (\frac {x}{\sqrt [4]{-1+\sqrt {2}} \sqrt [4]{-1+x^4}}\right )}{2 \sqrt {2}}+\frac {1}{8} \sqrt [4]{7+5 \sqrt {2}} \log \left (\sqrt {1+\sqrt {2}}+\frac {x^2}{\sqrt {-1+x^4}}-\frac {\sqrt {2} \sqrt [4]{1+\sqrt {2}} x}{\sqrt [4]{-1+x^4}}\right )-\frac {\sqrt [4]{7+5 \sqrt {2}} \log \left (\sqrt {1+\sqrt {2}}+\frac {x^2}{\sqrt {-1+x^4}}-\frac {\sqrt {2} \sqrt [4]{1+\sqrt {2}} x}{\sqrt [4]{-1+x^4}}\right )}{8 \sqrt {2}}-\frac {1}{8} \sqrt [4]{7+5 \sqrt {2}} \log \left (\sqrt {1+\sqrt {2}}+\frac {x^2}{\sqrt {-1+x^4}}+\frac {\sqrt {2} \sqrt [4]{1+\sqrt {2}} x}{\sqrt [4]{-1+x^4}}\right )+\frac {\sqrt [4]{7+5 \sqrt {2}} \log \left (\sqrt {1+\sqrt {2}}+\frac {x^2}{\sqrt {-1+x^4}}+\frac {\sqrt {2} \sqrt [4]{1+\sqrt {2}} x}{\sqrt [4]{-1+x^4}}\right )}{8 \sqrt {2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [4]{-1+x^4} \left (1+x^4+x^8\right )}{x^6 \left (-1+2 x^8\right )} \, dx=\frac {\sqrt [4]{-1+x^4} \left (1+4 x^4\right )}{5 x^5}+\frac {1}{8} \text {RootSum}\left [-1-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right )+2 \log (x) \text {$\#$1}^4-2 \log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+\text {$\#$1}^7}\&\right ] \]

[In]

Integrate[((-1 + x^4)^(1/4)*(1 + x^4 + x^8))/(x^6*(-1 + 2*x^8)),x]

[Out]

((-1 + x^4)^(1/4)*(1 + 4*x^4))/(5*x^5) + RootSum[-1 - 2*#1^4 + #1^8 & , (-Log[x] + Log[(-1 + x^4)^(1/4) - x*#1
] + 2*Log[x]*#1^4 - 2*Log[(-1 + x^4)^(1/4) - x*#1]*#1^4)/(-#1^3 + #1^7) & ]/8

Maple [C] (warning: unable to verify)

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

Time = 26.91 (sec) , antiderivative size = 15606, normalized size of antiderivative = 144.50

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

[In]

int((x^4-1)^(1/4)*(x^8+x^4+1)/x^6/(2*x^8-1),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 = 6.88 (sec) , antiderivative size = 1494, normalized size of antiderivative = 13.83 \[ \int \frac {\sqrt [4]{-1+x^4} \left (1+x^4+x^8\right )}{x^6 \left (-1+2 x^8\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/160*(5*sqrt(2)*x^5*sqrt(-sqrt(-61*sqrt(2) - 71))*log((686*(2*x^5 + sqrt(2)*x)*(x^4 - 1)^(3/4) - 14*(22*x^7
- 12*x^3 - sqrt(2)*(12*x^7 - 11*x^3))*(x^4 - 1)^(1/4)*sqrt(-61*sqrt(2) - 71) + (98*(6*x^6 - 2*x^2 - sqrt(2)*(2
*x^6 - 3*x^2))*sqrt(x^4 - 1) - (116*x^8 - 116*x^4 - sqrt(2)*(90*x^8 - 90*x^4 + 13) + 32)*sqrt(-61*sqrt(2) - 71
))*sqrt(-sqrt(-61*sqrt(2) - 71)))/(2*x^8 - 1)) - 5*sqrt(2)*x^5*sqrt(-sqrt(-61*sqrt(2) - 71))*log((686*(2*x^5 +
sqrt(2)*x)*(x^4 - 1)^(3/4) - 14*(22*x^7 - 12*x^3 - sqrt(2)*(12*x^7 - 11*x^3))*(x^4 - 1)^(1/4)*sqrt(-61*sqrt(2
) - 71) - (98*(6*x^6 - 2*x^2 - sqrt(2)*(2*x^6 - 3*x^2))*sqrt(x^4 - 1) - (116*x^8 - 116*x^4 - sqrt(2)*(90*x^8 -
90*x^4 + 13) + 32)*sqrt(-61*sqrt(2) - 71))*sqrt(-sqrt(-61*sqrt(2) - 71)))/(2*x^8 - 1)) - 5*sqrt(2)*x^5*sqrt(-
sqrt(61*sqrt(2) - 71))*log((686*(2*x^5 - sqrt(2)*x)*(x^4 - 1)^(3/4) - 14*(22*x^7 - 12*x^3 + sqrt(2)*(12*x^7 -
11*x^3))*(x^4 - 1)^(1/4)*sqrt(61*sqrt(2) - 71) + (98*(6*x^6 - 2*x^2 + sqrt(2)*(2*x^6 - 3*x^2))*sqrt(x^4 - 1) -
(116*x^8 - 116*x^4 + sqrt(2)*(90*x^8 - 90*x^4 + 13) + 32)*sqrt(61*sqrt(2) - 71))*sqrt(-sqrt(61*sqrt(2) - 71))
)/(2*x^8 - 1)) + 5*sqrt(2)*x^5*sqrt(-sqrt(61*sqrt(2) - 71))*log((686*(2*x^5 - sqrt(2)*x)*(x^4 - 1)^(3/4) - 14*
(22*x^7 - 12*x^3 + sqrt(2)*(12*x^7 - 11*x^3))*(x^4 - 1)^(1/4)*sqrt(61*sqrt(2) - 71) - (98*(6*x^6 - 2*x^2 + sqr
t(2)*(2*x^6 - 3*x^2))*sqrt(x^4 - 1) - (116*x^8 - 116*x^4 + sqrt(2)*(90*x^8 - 90*x^4 + 13) + 32)*sqrt(61*sqrt(2
) - 71))*sqrt(-sqrt(61*sqrt(2) - 71)))/(2*x^8 - 1)) + 5*sqrt(2)*x^5*(-61*sqrt(2) - 71)^(1/4)*log((686*(2*x^5 +
sqrt(2)*x)*(x^4 - 1)^(3/4) + 14*(22*x^7 - 12*x^3 - sqrt(2)*(12*x^7 - 11*x^3))*(x^4 - 1)^(1/4)*sqrt(-61*sqrt(2
) - 71) + (98*(6*x^6 - 2*x^2 - sqrt(2)*(2*x^6 - 3*x^2))*sqrt(x^4 - 1) + (116*x^8 - 116*x^4 - sqrt(2)*(90*x^8 -
90*x^4 + 13) + 32)*sqrt(-61*sqrt(2) - 71))*(-61*sqrt(2) - 71)^(1/4))/(2*x^8 - 1)) - 5*sqrt(2)*x^5*(-61*sqrt(2
) - 71)^(1/4)*log((686*(2*x^5 + sqrt(2)*x)*(x^4 - 1)^(3/4) + 14*(22*x^7 - 12*x^3 - sqrt(2)*(12*x^7 - 11*x^3))*
(x^4 - 1)^(1/4)*sqrt(-61*sqrt(2) - 71) - (98*(6*x^6 - 2*x^2 - sqrt(2)*(2*x^6 - 3*x^2))*sqrt(x^4 - 1) + (116*x^
8 - 116*x^4 - sqrt(2)*(90*x^8 - 90*x^4 + 13) + 32)*sqrt(-61*sqrt(2) - 71))*(-61*sqrt(2) - 71)^(1/4))/(2*x^8 -
1)) - 5*sqrt(2)*x^5*(61*sqrt(2) - 71)^(1/4)*log((686*(2*x^5 - sqrt(2)*x)*(x^4 - 1)^(3/4) + 14*(22*x^7 - 12*x^3
+ sqrt(2)*(12*x^7 - 11*x^3))*(x^4 - 1)^(1/4)*sqrt(61*sqrt(2) - 71) + (98*(6*x^6 - 2*x^2 + sqrt(2)*(2*x^6 - 3*
x^2))*sqrt(x^4 - 1) + (116*x^8 - 116*x^4 + sqrt(2)*(90*x^8 - 90*x^4 + 13) + 32)*sqrt(61*sqrt(2) - 71))*(61*sqr
t(2) - 71)^(1/4))/(2*x^8 - 1)) + 5*sqrt(2)*x^5*(61*sqrt(2) - 71)^(1/4)*log((686*(2*x^5 - sqrt(2)*x)*(x^4 - 1)^
(3/4) + 14*(22*x^7 - 12*x^3 + sqrt(2)*(12*x^7 - 11*x^3))*(x^4 - 1)^(1/4)*sqrt(61*sqrt(2) - 71) - (98*(6*x^6 -
2*x^2 + sqrt(2)*(2*x^6 - 3*x^2))*sqrt(x^4 - 1) + (116*x^8 - 116*x^4 + sqrt(2)*(90*x^8 - 90*x^4 + 13) + 32)*sqr
t(61*sqrt(2) - 71))*(61*sqrt(2) - 71)^(1/4))/(2*x^8 - 1)) - 32*(4*x^4 + 1)*(x^4 - 1)^(1/4))/x^5

Sympy [N/A]

Not integrable

Time = 35.01 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.44 \[ \int \frac {\sqrt [4]{-1+x^4} \left (1+x^4+x^8\right )}{x^6 \left (-1+2 x^8\right )} \, dx=\int \frac {\sqrt [4]{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right ) \left (x^{4} - x^{2} + 1\right )}{x^{6} \cdot \left (2 x^{8} - 1\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.28 \[ \int \frac {\sqrt [4]{-1+x^4} \left (1+x^4+x^8\right )}{x^6 \left (-1+2 x^8\right )} \, dx=\int { \frac {{\left (x^{8} + x^{4} + 1\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{{\left (2 \, x^{8} - 1\right )} x^{6}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.28 \[ \int \frac {\sqrt [4]{-1+x^4} \left (1+x^4+x^8\right )}{x^6 \left (-1+2 x^8\right )} \, dx=\int { \frac {{\left (x^{8} + x^{4} + 1\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{{\left (2 \, x^{8} - 1\right )} x^{6}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.28 \[ \int \frac {\sqrt [4]{-1+x^4} \left (1+x^4+x^8\right )}{x^6 \left (-1+2 x^8\right )} \, dx=\int \frac {{\left (x^4-1\right )}^{1/4}\,\left (x^8+x^4+1\right )}{x^6\,\left (2\,x^8-1\right )} \,d x \]

[In]

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

[Out]

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