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

Optimal. Leaf size=108 \[ \frac {1}{8} \text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^4-1\& ,\frac {-2 \text {$\#$1}^4 \log \left (\sqrt [4]{x^4-1}-\text {$\#$1} x\right )+2 \text {$\#$1}^4 \log (x)+\log \left (\sqrt [4]{x^4-1}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}^7-\text {$\#$1}^3}\& \right ]+\frac {\sqrt [4]{x^4-1} \left (4 x^4+1\right )}{5 x^5} \]

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Rubi [B]  time = 1.87, antiderivative size = 793, normalized size of antiderivative = 7.34, number of steps used = 55, number of rules used = 18, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6725, 264, 277, 331, 298, 203, 206, 1529, 511, 510, 1519, 494, 297, 1162, 617, 204, 1165, 628} \begin {gather*} -\frac {\sqrt [4]{x^4-1} x^3 F_1\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 F_1\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}+\frac {\sqrt [4]{5 \sqrt {2}-7} \tan ^{-1}\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} \tan ^{-1}\left (\frac {x}{\sqrt [4]{\sqrt {2}-1} \sqrt [4]{x^4-1}}\right )+\frac {\sqrt [4]{7+5 \sqrt {2}} \tan ^{-1}\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}} \tan ^{-1}\left (1-\frac {\sqrt {2} x}{\sqrt [4]{1+\sqrt {2}} \sqrt [4]{x^4-1}}\right )-\frac {\sqrt [4]{7+5 \sqrt {2}} \tan ^{-1}\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}} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{1+\sqrt {2}} \sqrt [4]{x^4-1}}+1\right )-\frac {\sqrt [4]{5 \sqrt {2}-7} \tanh ^{-1}\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} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{\sqrt {2}-1} \sqrt [4]{x^4-1}}\right )-\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 ) \end {gather*}

Warning: Unable to verify antiderivative.

[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 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 511

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPa
rt[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 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 1519

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 1529

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 6725

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

Rubi steps

\begin {align*} \int \frac {\sqrt [4]{-1+x^4} \left (1+x^4+x^8\right )}{x^6 \left (-1+2 x^8\right )} \, dx &=\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-\operatorname {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} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{2} \operatorname {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} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \tanh ^{-1}\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+\operatorname {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} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \operatorname {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} F_1\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} F_1\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} F_1\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} F_1\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} F_1\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} F_1\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 {\operatorname {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 {\operatorname {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}}-\operatorname {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 )-\operatorname {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} F_1\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} F_1\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} \operatorname {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} \operatorname {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 {\operatorname {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 {\operatorname {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 {\operatorname {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 {\operatorname {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 {\operatorname {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 {\operatorname {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} F_1\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} F_1\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}} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-1+\sqrt {2}} \sqrt [4]{-1+x^4}}\right )+\frac {\sqrt [4]{-7+5 \sqrt {2}} \tan ^{-1}\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}} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+\sqrt {2}} \sqrt [4]{-1+x^4}}\right )-\frac {\sqrt [4]{-7+5 \sqrt {2}} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+\sqrt {2}} \sqrt [4]{-1+x^4}}\right )}{2 \sqrt {2}}-\frac {\operatorname {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 {\operatorname {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 {\operatorname {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 {\operatorname {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}} \operatorname {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}} \operatorname {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}} \operatorname {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}} \operatorname {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} F_1\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} F_1\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}} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-1+\sqrt {2}} \sqrt [4]{-1+x^4}}\right )+\frac {\sqrt [4]{-7+5 \sqrt {2}} \tan ^{-1}\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}} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+\sqrt {2}} \sqrt [4]{-1+x^4}}\right )-\frac {\sqrt [4]{-7+5 \sqrt {2}} \tanh ^{-1}\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}} \operatorname {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}} \operatorname {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}} \operatorname {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}} \operatorname {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} F_1\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} F_1\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}} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-1+\sqrt {2}} \sqrt [4]{-1+x^4}}\right )+\frac {\sqrt [4]{-7+5 \sqrt {2}} \tan ^{-1}\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}} \tan ^{-1}\left (1-\frac {\sqrt {2} x}{\sqrt [4]{1+\sqrt {2}} \sqrt [4]{-1+x^4}}\right )+\frac {\sqrt [4]{7+5 \sqrt {2}} \tan ^{-1}\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}} \tan ^{-1}\left (1+\frac {\sqrt {2} x}{\sqrt [4]{1+\sqrt {2}} \sqrt [4]{-1+x^4}}\right )-\frac {\sqrt [4]{7+5 \sqrt {2}} \tan ^{-1}\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}} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+\sqrt {2}} \sqrt [4]{-1+x^4}}\right )-\frac {\sqrt [4]{-7+5 \sqrt {2}} \tanh ^{-1}\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*}

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Mathematica [F]  time = 0.26, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{-1+x^4} \left (1+x^4+x^8\right )}{x^6 \left (-1+2 x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.26, size = 108, normalized size = 1.00 \begin {gather*} \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 ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-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

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fricas [B]  time = 16.81, size = 5743, normalized size = 53.18

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/160*(20*sqrt(2)*x^5*(61*sqrt(2) - 71)^(1/4)*arctan(1/4802*(sqrt(2)*(784*x^8 - 784*x^4 + 2*(90*x^6 - 58*x^2 +
sqrt(2)*(58*x^6 - 45*x^2))*sqrt(x^4 - 1)*sqrt(61*sqrt(2) - 71) + 49*sqrt(2)*(10*x^8 - 10*x^4 + 3) + 98)*sqrt(
sqrt(61*sqrt(2) - 71)*(5*sqrt(2) + 1))*(61*sqrt(2) - 71)^(1/4) + 28*(49*(2*x^5 + sqrt(2)*(3*x^5 - x) - 3*x)*(x
^4 - 1)^(3/4) + (58*x^7 - 45*x^3 + sqrt(2)*(45*x^7 - 29*x^3))*(x^4 - 1)^(1/4)*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*sqrt(2) - 71)^(1/4))/(2*x^8 - 1)) - 20*x^5*(61*sqrt(2) + 71)^
(1/4)*arctan(-1/823543*(1070843080384*x^48 - 5913486747392*x^44 + 13489291746112*x^40 - 16041721625216*x^36 +
10177133566992*x^32 - 2984994192768*x^28 + 169531267808*x^24 + 263533760*x^20 + 41516449716*x^16 - 9342271792*
x^12 + 1011310804*x^8 - 46118408*x^4 - 67228*(810688*x^46 - 3300704*x^42 + 5566048*x^38 - 4690832*x^34 + 16850
88*x^30 - 100080*x^26 + 179824*x^22 - 109448*x^18 - 52228*x^14 + 12210*x^10 - 578*x^6 + 11*x^2 - sqrt(2)*(5646
40*x^46 - 5779968*x^42 + 16974192*x^38 - 21532576*x^34 + 12159632*x^30 - 2257344*x^26 + 37112*x^22 - 174448*x^
18 + 3362*x^14 + 5680*x^10 - 293*x^6 + 6*x^2))*sqrt(x^4 - 1)*sqrt(61*sqrt(2) + 71) + sqrt(14)*(5488*(391136*x^
45 - 2069248*x^41 + 3437184*x^37 - 1877760*x^33 - 154768*x^29 + 199680*x^25 + 65728*x^21 + 11968*x^17 - 3850*x
^13 - 80*x^9 - sqrt(2)*(332800*x^45 - 2089936*x^41 + 4083712*x^37 - 3148160*x^33 + 816896*x^29 - 68232*x^25 +
68352*x^21 + 7312*x^17 - 2688*x^13 - 57*x^9))*(x^4 - 1)^(3/4)*sqrt(61*sqrt(2) + 71) - (98*(5163328*x^46 - 3427
7664*x^42 + 72312672*x^38 - 63645680*x^34 + 23071008*x^30 - 4224976*x^26 + 1737648*x^22 - 75000*x^18 - 93468*x
^14 + 34214*x^10 - 2114*x^6 + 45*x^2 - sqrt(2)*(5769760*x^46 - 34301216*x^42 + 70731504*x^38 - 63826672*x^34 +
24244048*x^30 - 3234000*x^26 + 836088*x^22 - 238584*x^18 + 1002*x^14 + 19366*x^10 - 1309*x^6 + 29*x^2))*sqrt(
x^4 - 1) - (68566464*x^48 - 584166592*x^44 + 1803246976*x^40 - 2593305760*x^36 + 1779401648*x^32 - 497618400*x
^28 + 46058368*x^24 - 27645008*x^20 + 3558004*x^16 + 2170628*x^12 - 282040*x^8 + 16382*x^4 - 2*sqrt(2)*(342506
56*x^48 - 258415904*x^44 + 740587408*x^40 - 1014652720*x^36 + 674825568*x^32 - 188789840*x^28 + 21702920*x^24
- 10965592*x^20 + 586570*x^16 + 979526*x^12 - 114699*x^8 + 6361*x^4 - 127) - 335)*sqrt(61*sqrt(2) + 71))*(61*s
qrt(2) + 71)^(3/4) + 14*(196*(394240*x^45 - 2873408*x^41 + 6497632*x^37 - 6047616*x^33 + 2140416*x^29 - 144480
*x^25 + 40048*x^21 + 3616*x^17 - 11552*x^13 + 1148*x^9 - 34*x^5 - sqrt(2)*(567840*x^45 - 3201312*x^41 + 612616
0*x^37 - 4823936*x^33 + 1294864*x^29 - 59920*x^25 + 126152*x^21 - 24256*x^17 - 6310*x^13 + 742*x^9 - 23*x^5))*
(x^4 - 1)^(3/4) + (10498496*x^47 - 52397824*x^43 + 73747104*x^39 - 17336960*x^35 - 25311648*x^31 + 8428160*x^2
7 + 492624*x^23 + 2002112*x^19 - 103924*x^15 - 24880*x^11 + 6882*x^7 - 264*x^3 - sqrt(2)*(4728064*x^47 - 26310
880*x^43 + 33288576*x^39 + 7460272*x^35 - 30588800*x^31 + 9467600*x^27 + 1411776*x^23 + 453208*x^19 + 114832*x
^15 - 29830*x^11 + 5304*x^7 - 193*x^3))*(x^4 - 1)^(1/4)*sqrt(61*sqrt(2) + 71))*sqrt(61*sqrt(2) + 71) - 2151296
*(5248*x^47 - 28864*x^43 + 60400*x^39 - 62224*x^35 + 34296*x^31 - 10760*x^27 + 1940*x^23 - 12*x^19 - 14*x^15 -
10*x^11 + sqrt(2)*(2576*x^47 - 14368*x^43 + 34160*x^39 - 43048*x^35 + 29120*x^31 - 9444*x^27 + 1124*x^23 - 14
2*x^19 + 15*x^15 + 7*x^11))*(x^4 - 1)^(1/4) - 392*(4349632*x^48 - 39505760*x^44 + 109692576*x^40 - 129159296*x
^36 + 59250016*x^32 + 1415120*x^28 - 5926256*x^24 + 53312*x^20 - 235396*x^16 + 69874*x^12 - 3822*x^8 + (264668
8*x^46 - 15237728*x^42 + 28884656*x^38 - 21695616*x^34 + 4499664*x^30 + 798800*x^26 + 129144*x^22 + 13888*x^18
- 40254*x^14 + 642*x^10 + 103*x^6 - 2*sqrt(2)*(637936*x^46 - 4066368*x^42 + 8185104*x^38 - 6611040*x^34 + 188
2712*x^30 - 208784*x^26 + 210552*x^22 - 17416*x^18 - 12925*x^14 + 184*x^10 + 37*x^6))*sqrt(x^4 - 1)*sqrt(61*sq
rt(2) + 71) - 98*sqrt(2)*(46192*x^48 - 269184*x^44 + 638992*x^40 - 767552*x^36 + 483864*x^32 - 169440*x^28 + 5
4584*x^24 - 17328*x^20 - 565*x^16 + 464*x^12 - 27*x^8))*(61*sqrt(2) + 71)^(1/4))*sqrt((1372*(sqrt(2)*x^6 + x^2
)*sqrt(x^4 - 1) - 7*(2*x^8 - 5*sqrt(2)*(2*x^8 - 1) - 1)*sqrt(61*sqrt(2) + 71) + 2*((26*x^5 - sqrt(2)*(32*x^5 -
13*x) - 32*x)*(x^4 - 1)^(3/4)*sqrt(61*sqrt(2) + 71) - 49*(6*x^7 - 2*x^3 - sqrt(2)*(2*x^7 - 3*x^3))*(x^4 - 1)^
(1/4))*(61*sqrt(2) + 71)^(1/4))/(2*x^8 - 1)) + 98*((84770240*x^45 - 506339456*x^41 + 1062618144*x^37 - 9791178
88*x^33 + 374087776*x^29 - 44624960*x^25 + 13075472*x^21 - 2727456*x^17 - 1971636*x^13 + 249112*x^9 - 19046*x^
5 - sqrt(2)*(38490112*x^45 - 271881568*x^41 + 620203136*x^37 - 595757904*x^33 + 224354048*x^29 - 19751280*x^25
+ 9549504*x^21 - 4580904*x^17 - 744768*x^13 + 131986*x^9 - 11832*x^5 + 335*x) + 508*x)*(x^4 - 1)^(3/4)*sqrt(6
1*sqrt(2) + 71) - 49*(10656960*x^47 - 62174656*x^43 + 133850272*x^39 - 134582176*x^35 + 63040480*x^31 - 116299
20*x^27 + 2046800*x^23 - 1532240*x^19 + 270460*x^15 + 59892*x^11 - 6078*x^7 + 174*x^3 - sqrt(2)*(1409088*x^47
- 14855264*x^43 + 41389664*x^39 - 50887632*x^35 + 31535520*x^31 - 10006512*x^27 + 1266608*x^23 + 218392*x^19 -
129068*x^15 + 64162*x^11 - 5106*x^7 + 135*x^3))*(x^4 - 1)^(1/4))*(61*sqrt(2) + 71)^(3/4) + 6588344*sqrt(2)*(1
21248*x^48 - 664768*x^44 + 1475824*x^40 - 1696656*x^36 + 1056912*x^32 - 324992*x^28 + 28600*x^24 + 648*x^20 +
4450*x^16 - 1316*x^12 + 51*x^8 - x^4) + 2744*(49*(1050528*x^46 - 5554048*x^42 + 10575248*x^38 - 9024896*x^34 +
3514320*x^30 - 922752*x^26 + 492648*x^22 - 123808*x^18 - 9582*x^14 + 2456*x^10 - 115*x^6 - sqrt(2)*(253856*x^
46 - 2382752*x^42 + 6088688*x^38 - 6303936*x^34 + 2421584*x^30 + 103568*x^26 - 189032*x^22 + 19920*x^18 - 1380
6*x^14 + 1990*x^10 - 85*x^6))*sqrt(x^4 - 1) - (1653472*x^48 - 7958880*x^44 + 16108208*x^40 - 15601200*x^36 + 4
931952*x^32 + 2562384*x^28 - 2103240*x^24 + 613640*x^20 - 188682*x^16 - 15742*x^12 - 2105*x^8 + 193*x^4 + 2*sq
rt(2)*(698112*x^48 - 3230800*x^44 + 5655024*x^40 - 4718720*x^36 + 1737824*x^32 + 362312*x^28 - 831032*x^24 + 3
29272*x^20 - 12856*x^16 + 10347*x^12 + 583*x^8 - 66*x^4))*sqrt(61*sqrt(2) + 71))*sqrt(61*sqrt(2) + 71) - 38416
*(49*(196960*x^45 - 980384*x^41 + 1882624*x^37 - 1780608*x^33 + 906480*x^29 - 283216*x^25 + 60128*x^21 - 1600*
x^17 - 514*x^13 + 126*x^9 + sqrt(2)*(17824*x^45 + 23312*x^41 - 19456*x^37 - 259008*x^33 + 409488*x^29 - 189944
*x^25 + 20128*x^21 - 2752*x^17 + 498*x^13 - 91*x^9))*(x^4 - 1)^(3/4) + (826400*x^47 - 6554208*x^43 + 16724800*
x^39 - 19063552*x^35 + 10249296*x^31 - 2596336*x^27 + 594304*x^23 - 155296*x^19 - 29478*x^15 + 3922*x^11 + 148
*x^7 - sqrt(2)*(853312*x^47 - 4698352*x^43 + 9017776*x^39 - 6885696*x^35 + 842336*x^31 + 1198920*x^27 - 273064
*x^23 - 33248*x^19 - 24988*x^15 + 2901*x^11 + 103*x^7))*(x^4 - 1)^(1/4)*sqrt(61*sqrt(2) + 71))*(61*sqrt(2) + 7
1)^(1/4) + 823543)/(1783744*x^48 - 12228608*x^44 + 29945024*x^40 - 33926144*x^36 + 17890064*x^32 - 3611648*x^2
8 + 433824*x^24 - 284160*x^20 - 21180*x^16 + 20608*x^12 - 1588*x^8 + 64*x^4 - 1)) - 20*x^5*(61*sqrt(2) + 71)^(
1/4)*arctan(1/823543*(1070843080384*x^48 - 5913486747392*x^44 + 13489291746112*x^40 - 16041721625216*x^36 + 10
177133566992*x^32 - 2984994192768*x^28 + 169531267808*x^24 + 263533760*x^20 + 41516449716*x^16 - 9342271792*x^
12 + 1011310804*x^8 - 46118408*x^4 - 67228*(810688*x^46 - 3300704*x^42 + 5566048*x^38 - 4690832*x^34 + 1685088
*x^30 - 100080*x^26 + 179824*x^22 - 109448*x^18 - 52228*x^14 + 12210*x^10 - 578*x^6 + 11*x^2 - sqrt(2)*(564640
*x^46 - 5779968*x^42 + 16974192*x^38 - 21532576*x^34 + 12159632*x^30 - 2257344*x^26 + 37112*x^22 - 174448*x^18
+ 3362*x^14 + 5680*x^10 - 293*x^6 + 6*x^2))*sqrt(x^4 - 1)*sqrt(61*sqrt(2) + 71) + sqrt(14)*(5488*(391136*x^45
- 2069248*x^41 + 3437184*x^37 - 1877760*x^33 - 154768*x^29 + 199680*x^25 + 65728*x^21 + 11968*x^17 - 3850*x^1
3 - 80*x^9 - sqrt(2)*(332800*x^45 - 2089936*x^41 + 4083712*x^37 - 3148160*x^33 + 816896*x^29 - 68232*x^25 + 68
352*x^21 + 7312*x^17 - 2688*x^13 - 57*x^9))*(x^4 - 1)^(3/4)*sqrt(61*sqrt(2) + 71) + (98*(5163328*x^46 - 342776
64*x^42 + 72312672*x^38 - 63645680*x^34 + 23071008*x^30 - 4224976*x^26 + 1737648*x^22 - 75000*x^18 - 93468*x^1
4 + 34214*x^10 - 2114*x^6 + 45*x^2 - sqrt(2)*(5769760*x^46 - 34301216*x^42 + 70731504*x^38 - 63826672*x^34 + 2
4244048*x^30 - 3234000*x^26 + 836088*x^22 - 238584*x^18 + 1002*x^14 + 19366*x^10 - 1309*x^6 + 29*x^2))*sqrt(x^
4 - 1) - (68566464*x^48 - 584166592*x^44 + 1803246976*x^40 - 2593305760*x^36 + 1779401648*x^32 - 497618400*x^2
8 + 46058368*x^24 - 27645008*x^20 + 3558004*x^16 + 2170628*x^12 - 282040*x^8 + 16382*x^4 - 2*sqrt(2)*(34250656
*x^48 - 258415904*x^44 + 740587408*x^40 - 1014652720*x^36 + 674825568*x^32 - 188789840*x^28 + 21702920*x^24 -
10965592*x^20 + 586570*x^16 + 979526*x^12 - 114699*x^8 + 6361*x^4 - 127) - 335)*sqrt(61*sqrt(2) + 71))*(61*sqr
t(2) + 71)^(3/4) + 14*(196*(394240*x^45 - 2873408*x^41 + 6497632*x^37 - 6047616*x^33 + 2140416*x^29 - 144480*x
^25 + 40048*x^21 + 3616*x^17 - 11552*x^13 + 1148*x^9 - 34*x^5 - sqrt(2)*(567840*x^45 - 3201312*x^41 + 6126160*
x^37 - 4823936*x^33 + 1294864*x^29 - 59920*x^25 + 126152*x^21 - 24256*x^17 - 6310*x^13 + 742*x^9 - 23*x^5))*(x
^4 - 1)^(3/4) + (10498496*x^47 - 52397824*x^43 + 73747104*x^39 - 17336960*x^35 - 25311648*x^31 + 8428160*x^27
+ 492624*x^23 + 2002112*x^19 - 103924*x^15 - 24880*x^11 + 6882*x^7 - 264*x^3 - sqrt(2)*(4728064*x^47 - 2631088
0*x^43 + 33288576*x^39 + 7460272*x^35 - 30588800*x^31 + 9467600*x^27 + 1411776*x^23 + 453208*x^19 + 114832*x^1
5 - 29830*x^11 + 5304*x^7 - 193*x^3))*(x^4 - 1)^(1/4)*sqrt(61*sqrt(2) + 71))*sqrt(61*sqrt(2) + 71) - 2151296*(
5248*x^47 - 28864*x^43 + 60400*x^39 - 62224*x^35 + 34296*x^31 - 10760*x^27 + 1940*x^23 - 12*x^19 - 14*x^15 - 1
0*x^11 + sqrt(2)*(2576*x^47 - 14368*x^43 + 34160*x^39 - 43048*x^35 + 29120*x^31 - 9444*x^27 + 1124*x^23 - 142*
x^19 + 15*x^15 + 7*x^11))*(x^4 - 1)^(1/4) + 392*(4349632*x^48 - 39505760*x^44 + 109692576*x^40 - 129159296*x^3
6 + 59250016*x^32 + 1415120*x^28 - 5926256*x^24 + 53312*x^20 - 235396*x^16 + 69874*x^12 - 3822*x^8 + (2646688*
x^46 - 15237728*x^42 + 28884656*x^38 - 21695616*x^34 + 4499664*x^30 + 798800*x^26 + 129144*x^22 + 13888*x^18 -
40254*x^14 + 642*x^10 + 103*x^6 - 2*sqrt(2)*(637936*x^46 - 4066368*x^42 + 8185104*x^38 - 6611040*x^34 + 18827
12*x^30 - 208784*x^26 + 210552*x^22 - 17416*x^18 - 12925*x^14 + 184*x^10 + 37*x^6))*sqrt(x^4 - 1)*sqrt(61*sqrt
(2) + 71) - 98*sqrt(2)*(46192*x^48 - 269184*x^44 + 638992*x^40 - 767552*x^36 + 483864*x^32 - 169440*x^28 + 545
84*x^24 - 17328*x^20 - 565*x^16 + 464*x^12 - 27*x^8))*(61*sqrt(2) + 71)^(1/4))*sqrt((1372*(sqrt(2)*x^6 + x^2)*
sqrt(x^4 - 1) - 7*(2*x^8 - 5*sqrt(2)*(2*x^8 - 1) - 1)*sqrt(61*sqrt(2) + 71) - 2*((26*x^5 - sqrt(2)*(32*x^5 - 1
3*x) - 32*x)*(x^4 - 1)^(3/4)*sqrt(61*sqrt(2) + 71) - 49*(6*x^7 - 2*x^3 - sqrt(2)*(2*x^7 - 3*x^3))*(x^4 - 1)^(1
/4))*(61*sqrt(2) + 71)^(1/4))/(2*x^8 - 1)) - 98*((84770240*x^45 - 506339456*x^41 + 1062618144*x^37 - 979117888
*x^33 + 374087776*x^29 - 44624960*x^25 + 13075472*x^21 - 2727456*x^17 - 1971636*x^13 + 249112*x^9 - 19046*x^5
- sqrt(2)*(38490112*x^45 - 271881568*x^41 + 620203136*x^37 - 595757904*x^33 + 224354048*x^29 - 19751280*x^25 +
9549504*x^21 - 4580904*x^17 - 744768*x^13 + 131986*x^9 - 11832*x^5 + 335*x) + 508*x)*(x^4 - 1)^(3/4)*sqrt(61*
sqrt(2) + 71) - 49*(10656960*x^47 - 62174656*x^43 + 133850272*x^39 - 134582176*x^35 + 63040480*x^31 - 11629920
*x^27 + 2046800*x^23 - 1532240*x^19 + 270460*x^15 + 59892*x^11 - 6078*x^7 + 174*x^3 - sqrt(2)*(1409088*x^47 -
14855264*x^43 + 41389664*x^39 - 50887632*x^35 + 31535520*x^31 - 10006512*x^27 + 1266608*x^23 + 218392*x^19 - 1
29068*x^15 + 64162*x^11 - 5106*x^7 + 135*x^3))*(x^4 - 1)^(1/4))*(61*sqrt(2) + 71)^(3/4) + 6588344*sqrt(2)*(121
248*x^48 - 664768*x^44 + 1475824*x^40 - 1696656*x^36 + 1056912*x^32 - 324992*x^28 + 28600*x^24 + 648*x^20 + 44
50*x^16 - 1316*x^12 + 51*x^8 - x^4) + 2744*(49*(1050528*x^46 - 5554048*x^42 + 10575248*x^38 - 9024896*x^34 + 3
514320*x^30 - 922752*x^26 + 492648*x^22 - 123808*x^18 - 9582*x^14 + 2456*x^10 - 115*x^6 - sqrt(2)*(253856*x^46
- 2382752*x^42 + 6088688*x^38 - 6303936*x^34 + 2421584*x^30 + 103568*x^26 - 189032*x^22 + 19920*x^18 - 13806*
x^14 + 1990*x^10 - 85*x^6))*sqrt(x^4 - 1) - (1653472*x^48 - 7958880*x^44 + 16108208*x^40 - 15601200*x^36 + 493
1952*x^32 + 2562384*x^28 - 2103240*x^24 + 613640*x^20 - 188682*x^16 - 15742*x^12 - 2105*x^8 + 193*x^4 + 2*sqrt
(2)*(698112*x^48 - 3230800*x^44 + 5655024*x^40 - 4718720*x^36 + 1737824*x^32 + 362312*x^28 - 831032*x^24 + 329
272*x^20 - 12856*x^16 + 10347*x^12 + 583*x^8 - 66*x^4))*sqrt(61*sqrt(2) + 71))*sqrt(61*sqrt(2) + 71) + 38416*(
49*(196960*x^45 - 980384*x^41 + 1882624*x^37 - 1780608*x^33 + 906480*x^29 - 283216*x^25 + 60128*x^21 - 1600*x^
17 - 514*x^13 + 126*x^9 + sqrt(2)*(17824*x^45 + 23312*x^41 - 19456*x^37 - 259008*x^33 + 409488*x^29 - 189944*x
^25 + 20128*x^21 - 2752*x^17 + 498*x^13 - 91*x^9))*(x^4 - 1)^(3/4) + (826400*x^47 - 6554208*x^43 + 16724800*x^
39 - 19063552*x^35 + 10249296*x^31 - 2596336*x^27 + 594304*x^23 - 155296*x^19 - 29478*x^15 + 3922*x^11 + 148*x
^7 - sqrt(2)*(853312*x^47 - 4698352*x^43 + 9017776*x^39 - 6885696*x^35 + 842336*x^31 + 1198920*x^27 - 273064*x
^23 - 33248*x^19 - 24988*x^15 + 2901*x^11 + 103*x^7))*(x^4 - 1)^(1/4)*sqrt(61*sqrt(2) + 71))*(61*sqrt(2) + 71)
^(1/4) + 823543)/(1783744*x^48 - 12228608*x^44 + 29945024*x^40 - 33926144*x^36 + 17890064*x^32 - 3611648*x^28
+ 433824*x^24 - 284160*x^20 - 21180*x^16 + 20608*x^12 - 1588*x^8 + 64*x^4 - 1)) + 5*x^5*(61*sqrt(2) + 71)^(1/4
)*log(686*(1372*(sqrt(2)*x^6 + x^2)*sqrt(x^4 - 1) - 7*(2*x^8 - 5*sqrt(2)*(2*x^8 - 1) - 1)*sqrt(61*sqrt(2) + 71
) + 2*((26*x^5 - sqrt(2)*(32*x^5 - 13*x) - 32*x)*(x^4 - 1)^(3/4)*sqrt(61*sqrt(2) + 71) - 49*(6*x^7 - 2*x^3 - s
qrt(2)*(2*x^7 - 3*x^3))*(x^4 - 1)^(1/4))*(61*sqrt(2) + 71)^(1/4))/(2*x^8 - 1)) - 5*x^5*(61*sqrt(2) + 71)^(1/4)
*log(686*(1372*(sqrt(2)*x^6 + x^2)*sqrt(x^4 - 1) - 7*(2*x^8 - 5*sqrt(2)*(2*x^8 - 1) - 1)*sqrt(61*sqrt(2) + 71)
- 2*((26*x^5 - sqrt(2)*(32*x^5 - 13*x) - 32*x)*(x^4 - 1)^(3/4)*sqrt(61*sqrt(2) + 71) - 49*(6*x^7 - 2*x^3 - sq
rt(2)*(2*x^7 - 3*x^3))*(x^4 - 1)^(1/4))*(61*sqrt(2) + 71)^(1/4))/(2*x^8 - 1)) + 32*(4*x^4 + 1)*(x^4 - 1)^(1/4)
)/x^5

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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^4-1)^(1/4)*(x^8+x^4+1)/x^6/(2*x^8-1),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 = 32.46, size = 15605, normalized size = 144.49 \[\text {output too large to display}\]

Verification of antiderivative is not currently implemented for this CAS.

[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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \left (2 x^{8} - 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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