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

Optimal. Leaf size=74 \[ \frac {15}{8} \tan ^{-1}\left (\frac {x}{\sqrt [4]{2 x^4-1}}\right )-\frac {15}{8} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{2 x^4-1}}\right )+\frac {\sqrt [4]{2 x^4-1} \left (69 x^8-56 x^4-8\right )}{20 x^5 \left (x^4-1\right )} \]

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Rubi [C]  time = 2.88, antiderivative size = 198, normalized size of antiderivative = 2.68, number of steps used = 176, number of rules used = 33, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.222, Rules used = {6742, 2153, 1240, 412, 529, 237, 335, 275, 232, 407, 409, 1213, 537, 511, 510, 1248, 733, 844, 234, 220, 747, 400, 442, 444, 63, 212, 206, 203, 1336, 264, 277, 331, 298} \begin {gather*} -\frac {4 \sqrt [4]{2 x^4-1} x^3 F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};2 x^4,x^4\right )}{3 \sqrt [4]{1-2 x^4}}-\frac {\sqrt [4]{2 x^4-1} x^3 \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{3 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}+\frac {4 \sqrt [4]{2 x^4-1}}{x}+2 \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{2 x^4-1}}\right )-2 \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{2 x^4-1}}\right )-\frac {2 \left (2 x^4-1\right )^{5/4}}{5 x^5} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(4*(-1 + 2*x^4)^(1/4))/x - (2*(-1 + 2*x^4)^(5/4))/(5*x^5) - (4*x^3*(-1 + 2*x^4)^(1/4)*AppellF1[3/4, -1/4, 1, 7
/4, 2*x^4, x^4])/(3*(1 - 2*x^4)^(1/4)) + 2*2^(1/4)*ArcTan[(2^(1/4)*x)/(-1 + 2*x^4)^(1/4)] - 2*2^(1/4)*ArcTanh[
(2^(1/4)*x)/(-1 + 2*x^4)^(1/4)] - (x^3*(-1 + 2*x^4)^(1/4)*Hypergeometric2F1[-1/4, 3/4, 7/4, x^4/(1 - x^4)])/(3
*(1 - 2*x^4)^(1/4)*(1 - x^4)^(3/4))

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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 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 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

Rule 220

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 232

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2*EllipticF[(1*ArcSin[Rt[-(b/a), 2]*x])/2, 2])/(a^(3/4)*R
t[-(b/a), 2]), x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b/a]

Rule 234

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Dist[(2*Sqrt[-((b*x^2)/a)])/(b*x), Subst[Int[1/Sqrt[1 - x^4/a],
x], x, (a + b*x^2)^(1/4)], x] /; FreeQ[{a, b}, x] && NegQ[a]

Rule 237

Int[((a_) + (b_.)*(x_)^4)^(-3/4), x_Symbol] :> Dist[(x^3*(1 + a/(b*x^4))^(3/4))/(a + b*x^4)^(3/4), Int[1/(x^3*
(1 + a/(b*x^4))^(3/4)), x], x] /; FreeQ[{a, b}, x]

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 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

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 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 335

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /;
FreeQ[{a, b, p}, x] && ILtQ[n, 0] && IntegerQ[m]

Rule 400

Int[1/(((a_) + (b_.)*(x_)^2)^(3/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Dist[1/c, Int[1/(a + b*x^2)^(3/4), x],
x] - Dist[d/c, Int[x^2/((a + b*x^2)^(3/4)*(c + d*x^2)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d,
0]

Rule 407

Int[((a_) + (b_.)*(x_)^4)^(1/4)/((c_) + (d_.)*(x_)^4), x_Symbol] :> Dist[Sqrt[a + b*x^4]*Sqrt[a/(a + b*x^4)],
Subst[Int[1/(Sqrt[1 - b*x^4]*(c - (b*c - a*d)*x^4)), x], x, x/(a + b*x^4)^(1/4)], x] /; FreeQ[{a, b, c, d}, x]
&& NeQ[b*c - a*d, 0]

Rule 409

Int[1/(Sqrt[(a_) + (b_.)*(x_)^4]*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Dist[1/(2*c), Int[1/(Sqrt[a + b*x^4]*(1
- Rt[-(d/c), 2]*x^2)), x], x] + Dist[1/(2*c), Int[1/(Sqrt[a + b*x^4]*(1 + Rt[-(d/c), 2]*x^2)), x], x] /; FreeQ
[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 412

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1)*(c
+ d*x^n)^q)/(a*n*(p + 1)), x] + Dist[1/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(n*(p
+ 1) + 1) + d*(n*(p + q + 1) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p,
-1] && LtQ[0, q, 1] && IntBinomialQ[a, b, c, d, n, p, q, x]

Rule 442

Int[(x_)^2/(((a_) + (b_.)*(x_)^2)^(3/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> -Simp[(b*ArcTan[(Rt[-(b^2/a), 4]*
x)/(Sqrt[2]*(a + b*x^2)^(1/4))])/(Sqrt[2]*a*d*Rt[-(b^2/a), 4]^3), x] + Simp[(b*ArcTanh[(Rt[-(b^2/a), 4]*x)/(Sq
rt[2]*(a + b*x^2)^(1/4))])/(Sqrt[2]*a*d*Rt[-(b^2/a), 4]^3), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0
] && NegQ[b^2/a]

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 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 529

Int[((e_) + (f_.)*(x_)^4)/(((a_) + (b_.)*(x_)^4)^(3/4)*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Dist[(b*e - a*f)/(
b*c - a*d), Int[1/(a + b*x^4)^(3/4), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[(a + b*x^4)^(1/4)/(c + d*x^4),
x], x] /; FreeQ[{a, b, c, d, e, f}, x]

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-(f/e), -(d/c)
])

Rule 733

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(a + c*x^2)^p)/(
e*(m + 1)), x] - Dist[(2*c*p)/(e*(m + 1)), Int[x*(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1), x], x] /; FreeQ[{a, c,
d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] &&  !ILtQ[m +
2*p + 1, 0] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 747

Int[1/(((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(3/4)), x_Symbol] :> Dist[d, Int[1/((d^2 - e^2*x^2)*(a + c*x^
2)^(3/4)), x], x] - Dist[e, Int[x/((d^2 - e^2*x^2)*(a + c*x^2)^(3/4)), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ
[c*d^2 + a*e^2, 0]

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 1213

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[-(a*c), 2]}, Dist[Sqrt[-c],
Int[1/((d + e*x^2)*Sqrt[q + c*x^2]*Sqrt[q - c*x^2]), x], x]] /; FreeQ[{a, c, d, e}, x] && GtQ[a, 0] && LtQ[c,
0]

Rule 1240

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + c*x^4)^p, (d/
(d^2 - e^2*x^4) - (e*x^2)/(d^2 - e^2*x^4))^(-q), x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[c*d^2 + a*e^2, 0]
&&  !IntegerQ[p] && ILtQ[q, 0]

Rule 1248

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(d + e*x)^q
*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x]

Rule 1336

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(f*x)^m*(d + e*x^2)^q*(a + c*x^4)^p, x], x] /; FreeQ[{a, c, d, e, f, m, p, q}, x] && (IGtQ[p, 0] || IGtQ[q,
0] || IntegersQ[m, q])

Rule 2153

Int[((c_) + (d_.)*(x_)^(n_.))^(q_)*((a_) + (b_.)*(x_)^(nn_.))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^
nn)^p, (c/(c^2 - d^2*x^(2*n)) - (d*x^n)/(c^2 - d^2*x^(2*n)))^(-q), x], x] /; FreeQ[{a, b, c, d, n, nn, p}, x]
&&  !IntegerQ[p] && ILtQ[q, 0] && IGtQ[Log[2, nn/n], 0]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {\sqrt [4]{-1+2 x^4} \left (-2+x^8\right )}{x^6 \left (-1+x^4\right )^2} \, dx &=\int \left (-\frac {\sqrt [4]{-1+2 x^4}}{16 (-1+x)^2}-\frac {2 \sqrt [4]{-1+2 x^4}}{x^6}-\frac {4 \sqrt [4]{-1+2 x^4}}{x^2}-\frac {\sqrt [4]{-1+2 x^4}}{16 (1+x)^2}+\frac {17 \sqrt [4]{-1+2 x^4}}{8 \left (-1+x^2\right )}+\frac {\sqrt [4]{-1+2 x^4}}{4 \left (1+x^2\right )^2}+\frac {2 \sqrt [4]{-1+2 x^4}}{1+x^2}\right ) \, dx\\ &=-\left (\frac {1}{16} \int \frac {\sqrt [4]{-1+2 x^4}}{(-1+x)^2} \, dx\right )-\frac {1}{16} \int \frac {\sqrt [4]{-1+2 x^4}}{(1+x)^2} \, dx+\frac {1}{4} \int \frac {\sqrt [4]{-1+2 x^4}}{\left (1+x^2\right )^2} \, dx-2 \int \frac {\sqrt [4]{-1+2 x^4}}{x^6} \, dx+2 \int \frac {\sqrt [4]{-1+2 x^4}}{1+x^2} \, dx+\frac {17}{8} \int \frac {\sqrt [4]{-1+2 x^4}}{-1+x^2} \, dx-4 \int \frac {\sqrt [4]{-1+2 x^4}}{x^2} \, dx\\ &=\frac {4 \sqrt [4]{-1+2 x^4}}{x}-\frac {2 \left (-1+2 x^4\right )^{5/4}}{5 x^5}-\frac {1}{16} \int \left (\frac {\sqrt [4]{-1+2 x^4}}{\left (-1+x^2\right )^2}-\frac {2 x \sqrt [4]{-1+2 x^4}}{\left (-1+x^2\right )^2}+\frac {x^2 \sqrt [4]{-1+2 x^4}}{\left (-1+x^2\right )^2}\right ) \, dx-\frac {1}{16} \int \left (\frac {\sqrt [4]{-1+2 x^4}}{\left (-1+x^2\right )^2}+\frac {2 x \sqrt [4]{-1+2 x^4}}{\left (-1+x^2\right )^2}+\frac {x^2 \sqrt [4]{-1+2 x^4}}{\left (-1+x^2\right )^2}\right ) \, dx+\frac {1}{4} \int \left (\frac {\sqrt [4]{-1+2 x^4}}{\left (-1+x^4\right )^2}-\frac {2 x^2 \sqrt [4]{-1+2 x^4}}{\left (-1+x^4\right )^2}+\frac {x^4 \sqrt [4]{-1+2 x^4}}{\left (-1+x^4\right )^2}\right ) \, dx+2 \int \left (\frac {\sqrt [4]{-1+2 x^4}}{1-x^4}+\frac {x^2 \sqrt [4]{-1+2 x^4}}{-1+x^4}\right ) \, dx+\frac {17}{8} \int \left (\frac {\sqrt [4]{-1+2 x^4}}{-1+x^4}+\frac {x^2 \sqrt [4]{-1+2 x^4}}{-1+x^4}\right ) \, dx-8 \int \frac {x^2}{\left (-1+2 x^4\right )^{3/4}} \, dx\\ &=\frac {4 \sqrt [4]{-1+2 x^4}}{x}-\frac {2 \left (-1+2 x^4\right )^{5/4}}{5 x^5}-2 \left (\frac {1}{16} \int \frac {\sqrt [4]{-1+2 x^4}}{\left (-1+x^2\right )^2} \, dx\right )-2 \left (\frac {1}{16} \int \frac {x^2 \sqrt [4]{-1+2 x^4}}{\left (-1+x^2\right )^2} \, dx\right )+\frac {1}{4} \int \frac {\sqrt [4]{-1+2 x^4}}{\left (-1+x^4\right )^2} \, dx+\frac {1}{4} \int \frac {x^4 \sqrt [4]{-1+2 x^4}}{\left (-1+x^4\right )^2} \, dx-\frac {1}{2} \int \frac {x^2 \sqrt [4]{-1+2 x^4}}{\left (-1+x^4\right )^2} \, dx+2 \int \frac {\sqrt [4]{-1+2 x^4}}{1-x^4} \, dx+2 \int \frac {x^2 \sqrt [4]{-1+2 x^4}}{-1+x^4} \, dx+\frac {17}{8} \int \frac {\sqrt [4]{-1+2 x^4}}{-1+x^4} \, dx+\frac {17}{8} \int \frac {x^2 \sqrt [4]{-1+2 x^4}}{-1+x^4} \, dx-8 \operatorname {Subst}\left (\int \frac {x^2}{1-2 x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )\\ &=\frac {4 \sqrt [4]{-1+2 x^4}}{x}+\frac {x \sqrt [4]{-1+2 x^4}}{16 \left (1-x^4\right )}-\frac {2 \left (-1+2 x^4\right )^{5/4}}{5 x^5}+\frac {1}{16} \int \frac {3-4 x^4}{\left (-1+x^4\right ) \left (-1+2 x^4\right )^{3/4}} \, dx-2 \left (\frac {1}{16} \int \left (\frac {\sqrt [4]{-1+2 x^4}}{\left (-1+x^2\right )^2}+\frac {\sqrt [4]{-1+2 x^4}}{-1+x^2}\right ) \, dx\right )-2 \left (\frac {1}{16} \int \left (\frac {\sqrt [4]{-1+2 x^4}}{\left (-1+x^4\right )^2}+\frac {2 x^2 \sqrt [4]{-1+2 x^4}}{\left (-1+x^4\right )^2}+\frac {x^4 \sqrt [4]{-1+2 x^4}}{\left (-1+x^4\right )^2}\right ) \, dx\right )-\left (2 \sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )+\left (2 \sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )+\frac {\sqrt [4]{-1+2 x^4} \int \frac {x^4 \sqrt [4]{1-2 x^4}}{\left (-1+x^4\right )^2} \, dx}{4 \sqrt [4]{1-2 x^4}}-\frac {\sqrt [4]{-1+2 x^4} \int \frac {x^2 \sqrt [4]{1-2 x^4}}{\left (-1+x^4\right )^2} \, dx}{2 \sqrt [4]{1-2 x^4}}+\frac {\left (2 \sqrt [4]{-1+2 x^4}\right ) \int \frac {x^2 \sqrt [4]{1-2 x^4}}{-1+x^4} \, dx}{\sqrt [4]{1-2 x^4}}+\frac {\left (17 \sqrt [4]{-1+2 x^4}\right ) \int \frac {x^2 \sqrt [4]{1-2 x^4}}{-1+x^4} \, dx}{8 \sqrt [4]{1-2 x^4}}+\left (2 \sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-2 x^4} \left (1-x^4\right )} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )+\frac {1}{8} \left (17 \sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-2 x^4} \left (-1+x^4\right )} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )\\ &=\frac {4 \sqrt [4]{-1+2 x^4}}{x}+\frac {x \sqrt [4]{-1+2 x^4}}{16 \left (1-x^4\right )}-\frac {2 \left (-1+2 x^4\right )^{5/4}}{5 x^5}-\frac {11 x^3 \sqrt [4]{-1+2 x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};2 x^4,x^4\right )}{8 \sqrt [4]{1-2 x^4}}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{20 \sqrt [4]{1-2 x^4}}+2 \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )-2 \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )-\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{6 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}-2 \left (\frac {1}{16} \int \frac {\sqrt [4]{-1+2 x^4}}{\left (-1+x^2\right )^2} \, dx+\frac {1}{16} \int \frac {\sqrt [4]{-1+2 x^4}}{-1+x^2} \, dx\right )-\frac {1}{16} \int \frac {\sqrt [4]{-1+2 x^4}}{-1+x^4} \, dx-\frac {1}{8} \int \frac {1}{\left (-1+2 x^4\right )^{3/4}} \, dx-2 \left (\frac {1}{16} \int \frac {\sqrt [4]{-1+2 x^4}}{\left (-1+x^4\right )^2} \, dx+\frac {1}{16} \int \frac {x^4 \sqrt [4]{-1+2 x^4}}{\left (-1+x^4\right )^2} \, dx+\frac {1}{8} \int \frac {x^2 \sqrt [4]{-1+2 x^4}}{\left (-1+x^4\right )^2} \, dx\right )+\left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {1-2 x^4}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )+\left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {1-2 x^4}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )-\frac {1}{16} \left (17 \sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {1-2 x^4}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )-\frac {1}{16} \left (17 \sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {1-2 x^4}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )\\ &=\frac {4 \sqrt [4]{-1+2 x^4}}{x}+\frac {x \sqrt [4]{-1+2 x^4}}{16 \left (1-x^4\right )}-\frac {2 \left (-1+2 x^4\right )^{5/4}}{5 x^5}-\frac {11 x^3 \sqrt [4]{-1+2 x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};2 x^4,x^4\right )}{8 \sqrt [4]{1-2 x^4}}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{20 \sqrt [4]{1-2 x^4}}+2 \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )-2 \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )-\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{6 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}-2 \left (\frac {1}{16} \int \left (\frac {\sqrt [4]{-1+2 x^4}}{\left (-1+x^4\right )^2}+\frac {2 x^2 \sqrt [4]{-1+2 x^4}}{\left (-1+x^4\right )^2}+\frac {x^4 \sqrt [4]{-1+2 x^4}}{\left (-1+x^4\right )^2}\right ) \, dx+\frac {1}{16} \int \left (\frac {\sqrt [4]{-1+2 x^4}}{-1+x^4}+\frac {x^2 \sqrt [4]{-1+2 x^4}}{-1+x^4}\right ) \, dx\right )-\frac {\left (\left (1-\frac {1}{2 x^4}\right )^{3/4} x^3\right ) \int \frac {1}{\left (1-\frac {1}{2 x^4}\right )^{3/4} x^3} \, dx}{8 \left (-1+2 x^4\right )^{3/4}}-2 \left (\frac {x \sqrt [4]{-1+2 x^4}}{64 \left (1-x^4\right )}+\frac {1}{64} \int \frac {3-4 x^4}{\left (-1+x^4\right ) \left (-1+2 x^4\right )^{3/4}} \, dx+\frac {\sqrt [4]{-1+2 x^4} \int \frac {x^4 \sqrt [4]{1-2 x^4}}{\left (-1+x^4\right )^2} \, dx}{16 \sqrt [4]{1-2 x^4}}+\frac {\sqrt [4]{-1+2 x^4} \int \frac {x^2 \sqrt [4]{1-2 x^4}}{\left (-1+x^4\right )^2} \, dx}{8 \sqrt [4]{1-2 x^4}}\right )-\frac {1}{16} \left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-2 x^4} \left (-1+x^4\right )} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )-\frac {\left (17 \sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {\sqrt {2}-2 x^2} \left (1-x^2\right ) \sqrt {\sqrt {2}+2 x^2}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )}{8 \sqrt {2}}-\frac {\left (17 \sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {\sqrt {2}-2 x^2} \left (1+x^2\right ) \sqrt {\sqrt {2}+2 x^2}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )}{8 \sqrt {2}}+\left (\sqrt {2} \sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {\sqrt {2}-2 x^2} \left (1-x^2\right ) \sqrt {\sqrt {2}+2 x^2}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )+\left (\sqrt {2} \sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {\sqrt {2}-2 x^2} \left (1+x^2\right ) \sqrt {\sqrt {2}+2 x^2}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )\\ &=\frac {4 \sqrt [4]{-1+2 x^4}}{x}+\frac {x \sqrt [4]{-1+2 x^4}}{16 \left (1-x^4\right )}-\frac {2 \left (-1+2 x^4\right )^{5/4}}{5 x^5}-\frac {11 x^3 \sqrt [4]{-1+2 x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};2 x^4,x^4\right )}{8 \sqrt [4]{1-2 x^4}}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{20 \sqrt [4]{1-2 x^4}}+2 \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )-2 \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )-\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (-\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{16 \sqrt [4]{2}}-\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{16 \sqrt [4]{2}}-\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{6 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}-2 \left (\frac {x \sqrt [4]{-1+2 x^4}}{64 \left (1-x^4\right )}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{80 \sqrt [4]{1-2 x^4}}+\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{24 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}-\frac {1}{64} \int \frac {\sqrt [4]{-1+2 x^4}}{-1+x^4} \, dx-\frac {1}{32} \int \frac {1}{\left (-1+2 x^4\right )^{3/4}} \, dx\right )-2 \left (\frac {1}{16} \int \frac {\sqrt [4]{-1+2 x^4}}{\left (-1+x^4\right )^2} \, dx+\frac {1}{16} \int \frac {x^4 \sqrt [4]{-1+2 x^4}}{\left (-1+x^4\right )^2} \, dx+\frac {1}{16} \int \frac {\sqrt [4]{-1+2 x^4}}{-1+x^4} \, dx+\frac {1}{16} \int \frac {x^2 \sqrt [4]{-1+2 x^4}}{-1+x^4} \, dx+\frac {1}{8} \int \frac {x^2 \sqrt [4]{-1+2 x^4}}{\left (-1+x^4\right )^2} \, dx\right )+\frac {\left (\left (1-\frac {1}{2 x^4}\right )^{3/4} x^3\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1-\frac {x^4}{2}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )}{8 \left (-1+2 x^4\right )^{3/4}}+\frac {1}{32} \left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {1-2 x^4}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )+\frac {1}{32} \left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {1-2 x^4}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )\\ &=\frac {4 \sqrt [4]{-1+2 x^4}}{x}+\frac {x \sqrt [4]{-1+2 x^4}}{16 \left (1-x^4\right )}-\frac {2 \left (-1+2 x^4\right )^{5/4}}{5 x^5}-\frac {11 x^3 \sqrt [4]{-1+2 x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};2 x^4,x^4\right )}{8 \sqrt [4]{1-2 x^4}}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{20 \sqrt [4]{1-2 x^4}}+2 \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )-2 \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )-\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (-\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{16 \sqrt [4]{2}}-\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{16 \sqrt [4]{2}}-\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{6 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}+\frac {\left (\left (1-\frac {1}{2 x^4}\right )^{3/4} x^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {x^2}{2}\right )^{3/4}} \, dx,x,\frac {1}{x^2}\right )}{16 \left (-1+2 x^4\right )^{3/4}}-2 \left (\frac {x \sqrt [4]{-1+2 x^4}}{64 \left (1-x^4\right )}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{80 \sqrt [4]{1-2 x^4}}+\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{24 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}-\frac {\left (\left (1-\frac {1}{2 x^4}\right )^{3/4} x^3\right ) \int \frac {1}{\left (1-\frac {1}{2 x^4}\right )^{3/4} x^3} \, dx}{32 \left (-1+2 x^4\right )^{3/4}}-\frac {1}{64} \left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-2 x^4} \left (-1+x^4\right )} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )\right )-2 \left (\frac {x \sqrt [4]{-1+2 x^4}}{64 \left (1-x^4\right )}+\frac {1}{64} \int \frac {3-4 x^4}{\left (-1+x^4\right ) \left (-1+2 x^4\right )^{3/4}} \, dx+\frac {\sqrt [4]{-1+2 x^4} \int \frac {x^4 \sqrt [4]{1-2 x^4}}{\left (-1+x^4\right )^2} \, dx}{16 \sqrt [4]{1-2 x^4}}+\frac {\sqrt [4]{-1+2 x^4} \int \frac {x^2 \sqrt [4]{1-2 x^4}}{-1+x^4} \, dx}{16 \sqrt [4]{1-2 x^4}}+\frac {\sqrt [4]{-1+2 x^4} \int \frac {x^2 \sqrt [4]{1-2 x^4}}{\left (-1+x^4\right )^2} \, dx}{8 \sqrt [4]{1-2 x^4}}+\frac {1}{16} \left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-2 x^4} \left (-1+x^4\right )} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )\right )+\frac {\left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {\sqrt {2}-2 x^2} \left (1-x^2\right ) \sqrt {\sqrt {2}+2 x^2}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )}{16 \sqrt {2}}+\frac {\left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {\sqrt {2}-2 x^2} \left (1+x^2\right ) \sqrt {\sqrt {2}+2 x^2}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )}{16 \sqrt {2}}\\ &=\frac {4 \sqrt [4]{-1+2 x^4}}{x}+\frac {x \sqrt [4]{-1+2 x^4}}{16 \left (1-x^4\right )}-\frac {2 \left (-1+2 x^4\right )^{5/4}}{5 x^5}-\frac {11 x^3 \sqrt [4]{-1+2 x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};2 x^4,x^4\right )}{8 \sqrt [4]{1-2 x^4}}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{20 \sqrt [4]{1-2 x^4}}+2 \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )-2 \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )+\frac {\left (2-\frac {1}{x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \csc ^{-1}\left (\sqrt {2} x^2\right )\right |2\right )}{8 \sqrt [4]{2} \left (-1+2 x^4\right )^{3/4}}-\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (-\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{32 \sqrt [4]{2}}-\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{32 \sqrt [4]{2}}-\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{6 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}-2 \left (\frac {x \sqrt [4]{-1+2 x^4}}{64 \left (1-x^4\right )}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{80 \sqrt [4]{1-2 x^4}}+\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{24 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}+\frac {\left (\left (1-\frac {1}{2 x^4}\right )^{3/4} x^3\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1-\frac {x^4}{2}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )}{32 \left (-1+2 x^4\right )^{3/4}}+\frac {1}{128} \left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {1-2 x^4}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )+\frac {1}{128} \left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {1-2 x^4}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )\right )-2 \left (\frac {x \sqrt [4]{-1+2 x^4}}{64 \left (1-x^4\right )}-\frac {x^3 \sqrt [4]{-1+2 x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};2 x^4,x^4\right )}{48 \sqrt [4]{1-2 x^4}}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{80 \sqrt [4]{1-2 x^4}}+\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{24 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}-\frac {1}{64} \int \frac {\sqrt [4]{-1+2 x^4}}{-1+x^4} \, dx-\frac {1}{32} \int \frac {1}{\left (-1+2 x^4\right )^{3/4}} \, dx-\frac {1}{32} \left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {1-2 x^4}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )-\frac {1}{32} \left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {1-2 x^4}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )\right )\\ &=\frac {4 \sqrt [4]{-1+2 x^4}}{x}+\frac {x \sqrt [4]{-1+2 x^4}}{16 \left (1-x^4\right )}-\frac {2 \left (-1+2 x^4\right )^{5/4}}{5 x^5}-\frac {11 x^3 \sqrt [4]{-1+2 x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};2 x^4,x^4\right )}{8 \sqrt [4]{1-2 x^4}}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{20 \sqrt [4]{1-2 x^4}}+2 \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )-2 \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )+\frac {\left (2-\frac {1}{x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \csc ^{-1}\left (\sqrt {2} x^2\right )\right |2\right )}{8 \sqrt [4]{2} \left (-1+2 x^4\right )^{3/4}}-\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (-\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{32 \sqrt [4]{2}}-\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{32 \sqrt [4]{2}}-\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{6 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}-2 \left (\frac {x \sqrt [4]{-1+2 x^4}}{64 \left (1-x^4\right )}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{80 \sqrt [4]{1-2 x^4}}+\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{24 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}+\frac {\left (\left (1-\frac {1}{2 x^4}\right )^{3/4} x^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {x^2}{2}\right )^{3/4}} \, dx,x,\frac {1}{x^2}\right )}{64 \left (-1+2 x^4\right )^{3/4}}+\frac {\left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {\sqrt {2}-2 x^2} \left (1-x^2\right ) \sqrt {\sqrt {2}+2 x^2}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )}{64 \sqrt {2}}+\frac {\left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {\sqrt {2}-2 x^2} \left (1+x^2\right ) \sqrt {\sqrt {2}+2 x^2}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )}{64 \sqrt {2}}\right )-2 \left (\frac {x \sqrt [4]{-1+2 x^4}}{64 \left (1-x^4\right )}-\frac {x^3 \sqrt [4]{-1+2 x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};2 x^4,x^4\right )}{48 \sqrt [4]{1-2 x^4}}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{80 \sqrt [4]{1-2 x^4}}+\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{24 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}-\frac {\left (\left (1-\frac {1}{2 x^4}\right )^{3/4} x^3\right ) \int \frac {1}{\left (1-\frac {1}{2 x^4}\right )^{3/4} x^3} \, dx}{32 \left (-1+2 x^4\right )^{3/4}}-\frac {1}{64} \left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-2 x^4} \left (-1+x^4\right )} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )-\frac {\left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {\sqrt {2}-2 x^2} \left (1-x^2\right ) \sqrt {\sqrt {2}+2 x^2}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )}{16 \sqrt {2}}-\frac {\left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {\sqrt {2}-2 x^2} \left (1+x^2\right ) \sqrt {\sqrt {2}+2 x^2}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )}{16 \sqrt {2}}\right )\\ &=\frac {4 \sqrt [4]{-1+2 x^4}}{x}+\frac {x \sqrt [4]{-1+2 x^4}}{16 \left (1-x^4\right )}-\frac {2 \left (-1+2 x^4\right )^{5/4}}{5 x^5}-\frac {11 x^3 \sqrt [4]{-1+2 x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};2 x^4,x^4\right )}{8 \sqrt [4]{1-2 x^4}}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{20 \sqrt [4]{1-2 x^4}}+2 \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )-2 \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )+\frac {\left (2-\frac {1}{x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \csc ^{-1}\left (\sqrt {2} x^2\right )\right |2\right )}{8 \sqrt [4]{2} \left (-1+2 x^4\right )^{3/4}}-\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (-\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{32 \sqrt [4]{2}}-\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{32 \sqrt [4]{2}}-\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{6 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}-2 \left (\frac {x \sqrt [4]{-1+2 x^4}}{64 \left (1-x^4\right )}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{80 \sqrt [4]{1-2 x^4}}+\frac {\left (2-\frac {1}{x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \csc ^{-1}\left (\sqrt {2} x^2\right )\right |2\right )}{32 \sqrt [4]{2} \left (-1+2 x^4\right )^{3/4}}+\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (-\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{128 \sqrt [4]{2}}+\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{128 \sqrt [4]{2}}+\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{24 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}\right )-2 \left (\frac {x \sqrt [4]{-1+2 x^4}}{64 \left (1-x^4\right )}-\frac {x^3 \sqrt [4]{-1+2 x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};2 x^4,x^4\right )}{48 \sqrt [4]{1-2 x^4}}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{80 \sqrt [4]{1-2 x^4}}-\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (-\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{32 \sqrt [4]{2}}-\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{32 \sqrt [4]{2}}+\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{24 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}+\frac {\left (\left (1-\frac {1}{2 x^4}\right )^{3/4} x^3\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1-\frac {x^4}{2}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )}{32 \left (-1+2 x^4\right )^{3/4}}+\frac {1}{128} \left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {1-2 x^4}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )+\frac {1}{128} \left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {1-2 x^4}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )\right )\\ &=\frac {4 \sqrt [4]{-1+2 x^4}}{x}+\frac {x \sqrt [4]{-1+2 x^4}}{16 \left (1-x^4\right )}-\frac {2 \left (-1+2 x^4\right )^{5/4}}{5 x^5}-\frac {11 x^3 \sqrt [4]{-1+2 x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};2 x^4,x^4\right )}{8 \sqrt [4]{1-2 x^4}}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{20 \sqrt [4]{1-2 x^4}}+2 \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )-2 \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )+\frac {\left (2-\frac {1}{x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \csc ^{-1}\left (\sqrt {2} x^2\right )\right |2\right )}{8 \sqrt [4]{2} \left (-1+2 x^4\right )^{3/4}}-\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (-\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{32 \sqrt [4]{2}}-\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{32 \sqrt [4]{2}}-\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{6 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}-2 \left (\frac {x \sqrt [4]{-1+2 x^4}}{64 \left (1-x^4\right )}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{80 \sqrt [4]{1-2 x^4}}+\frac {\left (2-\frac {1}{x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \csc ^{-1}\left (\sqrt {2} x^2\right )\right |2\right )}{32 \sqrt [4]{2} \left (-1+2 x^4\right )^{3/4}}+\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (-\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{128 \sqrt [4]{2}}+\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{128 \sqrt [4]{2}}+\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{24 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}\right )-2 \left (\frac {x \sqrt [4]{-1+2 x^4}}{64 \left (1-x^4\right )}-\frac {x^3 \sqrt [4]{-1+2 x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};2 x^4,x^4\right )}{48 \sqrt [4]{1-2 x^4}}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{80 \sqrt [4]{1-2 x^4}}-\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (-\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{32 \sqrt [4]{2}}-\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{32 \sqrt [4]{2}}+\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{24 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}+\frac {\left (\left (1-\frac {1}{2 x^4}\right )^{3/4} x^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {x^2}{2}\right )^{3/4}} \, dx,x,\frac {1}{x^2}\right )}{64 \left (-1+2 x^4\right )^{3/4}}+\frac {\left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {\sqrt {2}-2 x^2} \left (1-x^2\right ) \sqrt {\sqrt {2}+2 x^2}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )}{64 \sqrt {2}}+\frac {\left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {\sqrt {2}-2 x^2} \left (1+x^2\right ) \sqrt {\sqrt {2}+2 x^2}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )}{64 \sqrt {2}}\right )\\ &=\frac {4 \sqrt [4]{-1+2 x^4}}{x}+\frac {x \sqrt [4]{-1+2 x^4}}{16 \left (1-x^4\right )}-\frac {2 \left (-1+2 x^4\right )^{5/4}}{5 x^5}-\frac {11 x^3 \sqrt [4]{-1+2 x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};2 x^4,x^4\right )}{8 \sqrt [4]{1-2 x^4}}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{20 \sqrt [4]{1-2 x^4}}+2 \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )-2 \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )+\frac {\left (2-\frac {1}{x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \csc ^{-1}\left (\sqrt {2} x^2\right )\right |2\right )}{8 \sqrt [4]{2} \left (-1+2 x^4\right )^{3/4}}-\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (-\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{32 \sqrt [4]{2}}-\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{32 \sqrt [4]{2}}-\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{6 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}-2 \left (\frac {x \sqrt [4]{-1+2 x^4}}{64 \left (1-x^4\right )}-\frac {x^3 \sqrt [4]{-1+2 x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};2 x^4,x^4\right )}{48 \sqrt [4]{1-2 x^4}}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{80 \sqrt [4]{1-2 x^4}}+\frac {\left (2-\frac {1}{x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \csc ^{-1}\left (\sqrt {2} x^2\right )\right |2\right )}{32 \sqrt [4]{2} \left (-1+2 x^4\right )^{3/4}}-\frac {3 \sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (-\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{128 \sqrt [4]{2}}-\frac {3 \sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{128 \sqrt [4]{2}}+\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{24 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}\right )-2 \left (\frac {x \sqrt [4]{-1+2 x^4}}{64 \left (1-x^4\right )}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{80 \sqrt [4]{1-2 x^4}}+\frac {\left (2-\frac {1}{x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \csc ^{-1}\left (\sqrt {2} x^2\right )\right |2\right )}{32 \sqrt [4]{2} \left (-1+2 x^4\right )^{3/4}}+\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (-\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{128 \sqrt [4]{2}}+\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{128 \sqrt [4]{2}}+\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{24 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}\right )\\ \end {align*}

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Mathematica [C]  time = 0.23, size = 92, normalized size = 1.24 \begin {gather*} \frac {25 \left (1-2 x^4\right )^{3/4} \sqrt [4]{1-x^4} x^8 \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )+138 x^{12}-181 x^8+40 x^4+8}{20 x^5 \left (x^4-1\right ) \left (2 x^4-1\right )^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(8 + 40*x^4 - 181*x^8 + 138*x^12 + 25*x^8*(1 - 2*x^4)^(3/4)*(1 - x^4)^(1/4)*Hypergeometric2F1[3/4, 3/4, 7/4, x
^4/(1 - x^4)])/(20*x^5*(-1 + x^4)*(-1 + 2*x^4)^(3/4))

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IntegrateAlgebraic [A]  time = 0.34, size = 74, normalized size = 1.00 \begin {gather*} \frac {\sqrt [4]{-1+2 x^4} \left (-8-56 x^4+69 x^8\right )}{20 x^5 \left (-1+x^4\right )}+\frac {15}{8} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-1+2 x^4}}\right )-\frac {15}{8} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+2 x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-1 + 2*x^4)^(1/4)*(-8 - 56*x^4 + 69*x^8))/(20*x^5*(-1 + x^4)) + (15*ArcTan[x/(-1 + 2*x^4)^(1/4)])/8 - (15*Ar
cTanh[x/(-1 + 2*x^4)^(1/4)])/8

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fricas [B]  time = 3.95, size = 151, normalized size = 2.04 \begin {gather*} \frac {75 \, {\left (x^{9} - x^{5}\right )} \arctan \left (\frac {2 \, {\left ({\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + {\left (2 \, x^{4} - 1\right )}^{\frac {3}{4}} x\right )}}{x^{4} - 1}\right ) + 75 \, {\left (x^{9} - x^{5}\right )} \log \left (-\frac {3 \, x^{4} - 2 \, {\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 2 \, \sqrt {2 \, x^{4} - 1} x^{2} - 2 \, {\left (2 \, x^{4} - 1\right )}^{\frac {3}{4}} x - 1}{x^{4} - 1}\right ) + 4 \, {\left (69 \, x^{8} - 56 \, x^{4} - 8\right )} {\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}}}{80 \, {\left (x^{9} - x^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/80*(75*(x^9 - x^5)*arctan(2*((2*x^4 - 1)^(1/4)*x^3 + (2*x^4 - 1)^(3/4)*x)/(x^4 - 1)) + 75*(x^9 - x^5)*log(-(
3*x^4 - 2*(2*x^4 - 1)^(1/4)*x^3 + 2*sqrt(2*x^4 - 1)*x^2 - 2*(2*x^4 - 1)^(3/4)*x - 1)/(x^4 - 1)) + 4*(69*x^8 -
56*x^4 - 8)*(2*x^4 - 1)^(1/4))/(x^9 - x^5)

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giac [A]  time = 0.44, size = 108, normalized size = 1.46 \begin {gather*} -\frac {2 \, {\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}} {\left (\frac {1}{x^{4}} - 2\right )}}{5 \, x} - \frac {4 \, {\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}}}{x} + \frac {{\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}}}{4 \, x {\left (\frac {1}{x^{4}} - 1\right )}} + \frac {15}{8} \, \arctan \left (\frac {{\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {15}{16} \, \log \left (\frac {{\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}}}{x} + 1\right ) - \frac {15}{16} \, \log \left ({\left | \frac {{\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}}}{x} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*(2*x^4 - 1)^(1/4)*(1/x^4 - 2)/x - 4*(2*x^4 - 1)^(1/4)/x + 1/4*(2*x^4 - 1)^(1/4)/(x*(1/x^4 - 1)) + 15/8*ar
ctan((2*x^4 - 1)^(1/4)/x) + 15/16*log((2*x^4 - 1)^(1/4)/x + 1) - 15/16*log(abs((2*x^4 - 1)^(1/4)/x - 1))

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maple [C]  time = 4.37, size = 196, normalized size = 2.65

method	result	size
trager	\(\frac {\left (2 x^{4}-1\right )^{\frac {1}{4}} \left (69 x^{8}-56 x^{4}-8\right )}{20 x^{5} \left (x^{4}-1\right )}+\frac {15 \ln \left (\frac {2 \left (2 x^{4}-1\right )^{\frac {3}{4}} x -2 \sqrt {2 x^{4}-1}\, x^{2}+2 \left (2 x^{4}-1\right )^{\frac {1}{4}} x^{3}-3 x^{4}+1}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )}{16}+\frac {15 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-2 \sqrt {2 x^{4}-1}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+3 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \left (2 x^{4}-1\right )^{\frac {3}{4}} x -2 \left (2 x^{4}-1\right )^{\frac {1}{4}} x^{3}-\RootOf \left (\textit {\_Z}^{2}+1\right )}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )}{16}\)	\(196\)
risch	\(\frac {138 x^{12}-181 x^{8}+40 x^{4}+8}{20 x^{5} \left (x^{4}-1\right ) \left (2 x^{4}-1\right )^{\frac {3}{4}}}+\frac {\left (-\frac {15 \ln \left (\frac {12 x^{12}+8 \left (8 x^{12}-12 x^{8}+6 x^{4}-1\right )^{\frac {1}{4}} x^{9}+4 \sqrt {8 x^{12}-12 x^{8}+6 x^{4}-1}\, x^{6}-16 x^{8}+2 \left (8 x^{12}-12 x^{8}+6 x^{4}-1\right )^{\frac {3}{4}} x^{3}-8 \left (8 x^{12}-12 x^{8}+6 x^{4}-1\right )^{\frac {1}{4}} x^{5}-2 \sqrt {8 x^{12}-12 x^{8}+6 x^{4}-1}\, x^{2}+7 x^{4}+2 \left (8 x^{12}-12 x^{8}+6 x^{4}-1\right )^{\frac {1}{4}} x -1}{\left (2 x^{4}-1\right )^{2} \left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )}{16}+\frac {15 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {12 x^{12}-8 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (8 x^{12}-12 x^{8}+6 x^{4}-1\right )^{\frac {1}{4}} x^{9}-4 \sqrt {8 x^{12}-12 x^{8}+6 x^{4}-1}\, x^{6}-16 x^{8}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (8 x^{12}-12 x^{8}+6 x^{4}-1\right )^{\frac {3}{4}} x^{3}+8 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (8 x^{12}-12 x^{8}+6 x^{4}-1\right )^{\frac {1}{4}} x^{5}+2 \sqrt {8 x^{12}-12 x^{8}+6 x^{4}-1}\, x^{2}+7 x^{4}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (8 x^{12}-12 x^{8}+6 x^{4}-1\right )^{\frac {1}{4}} x -1}{\left (2 x^{4}-1\right )^{2} \left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )}{16}\right ) \left (\left (2 x^{4}-1\right )^{3}\right )^{\frac {1}{4}}}{\left (2 x^{4}-1\right )^{\frac {3}{4}}}\)	\(471\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^4-1)^(1/4)*(x^8-2)/x^6/(x^4-1)^2,x,method=_RETURNVERBOSE)

[Out]

1/20*(2*x^4-1)^(1/4)*(69*x^8-56*x^4-8)/x^5/(x^4-1)+15/16*ln((2*(2*x^4-1)^(3/4)*x-2*(2*x^4-1)^(1/2)*x^2+2*(2*x^
4-1)^(1/4)*x^3-3*x^4+1)/(-1+x)/(1+x)/(x^2+1))+15/16*RootOf(_Z^2+1)*ln((-2*(2*x^4-1)^(1/2)*RootOf(_Z^2+1)*x^2+3
*RootOf(_Z^2+1)*x^4+2*(2*x^4-1)^(3/4)*x-2*(2*x^4-1)^(1/4)*x^3-RootOf(_Z^2+1))/(-1+x)/(1+x)/(x^2+1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (x^8-2\right )\,{\left (2\,x^4-1\right )}^{1/4}}{x^6\,{\left (x^4-1\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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