\(\int \frac {1+x^8}{\sqrt [4]{-x^2+x^4} (-1+x^8)} \, dx\) [2801]

Optimal result

Integrand size = 26, antiderivative size = 272 \[ \int \frac {1+x^8}{\sqrt [4]{-x^2+x^4} \left (-1+x^8\right )} \, dx=-\frac {\left (-x^2+x^4\right )^{3/4}}{x \left (-1+x^2\right )}+\arctan \left (\frac {x}{\sqrt [4]{-x^2+x^4}}\right )-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^2+x^4}}\right )}{2 \sqrt [4]{2}}+\text {arctanh}\left (\frac {x}{\sqrt [4]{-x^2+x^4}}\right )-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^2+x^4}}\right )}{2 \sqrt [4]{2}}+\frac {5}{8} \text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x) \text {$\#$1}^3+\log \left (\sqrt [4]{-x^2+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-1+\text {$\#$1}^4}\&\right ]+\frac {1}{8} \text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {2 \log (x)-2 \log \left (\sqrt [4]{-x^2+x^4}-x \text {$\#$1}\right )+3 \log (x) \text {$\#$1}^4-3 \log \left (\sqrt [4]{-x^2+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}+\text {$\#$1}^5}\&\right ] \]

[Out]

Unintegrable

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.88 (sec) , antiderivative size = 559, normalized size of antiderivative = 2.06, number of steps used = 27, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2081, 6847, 6857, 246, 218, 212, 209, 2098, 1166, 388, 385, 1443} \[ \int \frac {1+x^8}{\sqrt [4]{-x^2+x^4} \left (-1+x^8\right )} \, dx=\frac {\sqrt {x} \sqrt [4]{x^2-1} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{\sqrt [4]{x^4-x^2}}-\frac {\sqrt {x} \sqrt [4]{x^2-1} \arctan \left (\frac {\sqrt [4]{1-i} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{2 \sqrt [4]{1-i} \sqrt [4]{x^4-x^2}}-\frac {\sqrt {x} \sqrt [4]{x^2-1} \arctan \left (\frac {\sqrt [4]{1+i} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{2 \sqrt [4]{1+i} \sqrt [4]{x^4-x^2}}-\frac {\sqrt {x} \sqrt [4]{x^2-1} \arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{2 \sqrt [4]{2} \sqrt [4]{x^4-x^2}}+\frac {\sqrt {x} \sqrt [4]{x^2-1} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{\sqrt [4]{x^4-x^2}}-\frac {\sqrt {x} \sqrt [4]{x^2-1} \text {arctanh}\left (\frac {\sqrt [4]{1-i} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{2 \sqrt [4]{1-i} \sqrt [4]{x^4-x^2}}-\frac {\sqrt {x} \sqrt [4]{x^2-1} \text {arctanh}\left (\frac {\sqrt [4]{1+i} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{2 \sqrt [4]{1+i} \sqrt [4]{x^4-x^2}}-\frac {\sqrt {x} \sqrt [4]{x^2-1} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{2 \sqrt [4]{2} \sqrt [4]{x^4-x^2}}-\frac {x (x+1) \left (\frac {1-x}{x+1}\right )^{5/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {3}{2},\frac {2 x}{x+1}\right )}{2 (1-x) \sqrt [4]{x^4-x^2}}-\frac {x \sqrt [4]{\frac {1-x}{x+1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {3}{2},\frac {2 x}{x+1}\right )}{2 \sqrt [4]{x^4-x^2}} \]

[In]

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

[Out]

(Sqrt[x]*(-1 + x^2)^(1/4)*ArcTan[Sqrt[x]/(-1 + x^2)^(1/4)])/(-x^2 + x^4)^(1/4) - (Sqrt[x]*(-1 + x^2)^(1/4)*Arc
Tan[((1 - I)^(1/4)*Sqrt[x])/(-1 + x^2)^(1/4)])/(2*(1 - I)^(1/4)*(-x^2 + x^4)^(1/4)) - (Sqrt[x]*(-1 + x^2)^(1/4
)*ArcTan[((1 + I)^(1/4)*Sqrt[x])/(-1 + x^2)^(1/4)])/(2*(1 + I)^(1/4)*(-x^2 + x^4)^(1/4)) - (Sqrt[x]*(-1 + x^2)
^(1/4)*ArcTan[(2^(1/4)*Sqrt[x])/(-1 + x^2)^(1/4)])/(2*2^(1/4)*(-x^2 + x^4)^(1/4)) + (Sqrt[x]*(-1 + x^2)^(1/4)*
ArcTanh[Sqrt[x]/(-1 + x^2)^(1/4)])/(-x^2 + x^4)^(1/4) - (Sqrt[x]*(-1 + x^2)^(1/4)*ArcTanh[((1 - I)^(1/4)*Sqrt[
x])/(-1 + x^2)^(1/4)])/(2*(1 - I)^(1/4)*(-x^2 + x^4)^(1/4)) - (Sqrt[x]*(-1 + x^2)^(1/4)*ArcTanh[((1 + I)^(1/4)
*Sqrt[x])/(-1 + x^2)^(1/4)])/(2*(1 + I)^(1/4)*(-x^2 + x^4)^(1/4)) - (Sqrt[x]*(-1 + x^2)^(1/4)*ArcTanh[(2^(1/4)
*Sqrt[x])/(-1 + x^2)^(1/4)])/(2*2^(1/4)*(-x^2 + x^4)^(1/4)) - (x*((1 - x)/(1 + x))^(1/4)*Hypergeometric2F1[1/4
, 1/2, 3/2, (2*x)/(1 + x)])/(2*(-x^2 + x^4)^(1/4)) - (x*((1 - x)/(1 + x))^(5/4)*(1 + x)*Hypergeometric2F1[1/2,
5/4, 3/2, (2*x)/(1 + x)])/(2*(1 - x)*(-x^2 + x^4)^(1/4))

Rule 209

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

Rule 212

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

Rule 218

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] &&  !Gt
Q[a/b, 0]

Rule 246

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[1/(1 - b*x^n)^(p + 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)] && IntegerQ[p
+ 1/n]

Rule 385

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

Rule 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(c*(c*((a
+ b*x^n)/(a*(c + d*x^n))))^p*(c + d*x^n)^(1/n + p)))*Hypergeometric2F1[1/n, -p, 1 + 1/n, (-(b*c - a*d))*(x^n/(
a*(c + d*x^n)))], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 1) + 1, 0]

Rule 1166

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

Rule 1443

Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{r = Rt[(-a)*c, 2]}, Dist[-c/(2
*r), Int[(d + e*x^n)^q/(r - c*x^n), x], x] - Dist[c/(2*r), Int[(d + e*x^n)^q/(r + c*x^n), x], x]] /; FreeQ[{a,
c, d, e, n, q}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[q]

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 2098

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P /. x -> Sqrt[x]]}, Int[ExpandIntegrand[(PP /. x ->
x^2)^p*Q^q, x], x] /;  !SumQ[NonfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x^2] && PolyQ[Q, x] && ILtQ[p,
0]

Rule 6847

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rule 6857

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

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \int \frac {1+x^8}{\sqrt {x} \sqrt [4]{-1+x^2} \left (-1+x^8\right )} \, dx}{\sqrt [4]{-x^2+x^4}} \\ & = \frac {\left (2 \sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1+x^{16}}{\sqrt [4]{-1+x^4} \left (-1+x^{16}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-x^2+x^4}} \\ & = \frac {\left (2 \sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{\sqrt [4]{-1+x^4}}+\frac {2}{\sqrt [4]{-1+x^4} \left (-1+x^{16}\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-x^2+x^4}} \\ & = \frac {\left (2 \sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-1+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-x^2+x^4}}+\frac {\left (4 \sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-1+x^4} \left (-1+x^{16}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-x^2+x^4}} \\ & = \frac {\left (2 \sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt [4]{-x^2+x^4}}+\frac {\left (4 \sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{8 \left (-1+x^2\right ) \sqrt [4]{-1+x^4}}-\frac {1}{8 \left (1+x^2\right ) \sqrt [4]{-1+x^4}}-\frac {1}{4 \sqrt [4]{-1+x^4} \left (1+x^4\right )}-\frac {1}{2 \sqrt [4]{-1+x^4} \left (1+x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-x^2+x^4}} \\ & = \frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+x^2\right ) \sqrt [4]{-1+x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt [4]{-x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt [4]{-1+x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt [4]{-x^2+x^4}}+\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt [4]{-x^2+x^4}}+\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt [4]{-x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-1+x^4} \left (1+x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-x^2+x^4}}-\frac {\left (2 \sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-1+x^4} \left (1+x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-x^2+x^4}} \\ & = \frac {\sqrt {x} \sqrt [4]{-1+x^2} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt [4]{-x^2+x^4}}+\frac {\sqrt {x} \sqrt [4]{-1+x^2} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt [4]{-x^2+x^4}}-\frac {\left (\sqrt [4]{-1+x} \sqrt {x} \sqrt [4]{1+x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-1+x^2} \left (1+x^2\right )^{5/4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt [4]{-x^2+x^4}}+\frac {\left (\sqrt [4]{-1+x} \sqrt {x} \sqrt [4]{1+x}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+x^2\right )^{5/4} \sqrt [4]{1+x^2}} \, dx,x,\sqrt {x}\right )}{2 \sqrt [4]{-x^2+x^4}}-\frac {\left (i \sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (i-x^4\right ) \sqrt [4]{-1+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-x^2+x^4}}-\frac {\left (i \sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-1+x^4} \left (i+x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{1-2 x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt [4]{-x^2+x^4}} \\ & = \frac {\sqrt {x} \sqrt [4]{-1+x^2} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt [4]{-x^2+x^4}}+\frac {\sqrt {x} \sqrt [4]{-1+x^2} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt [4]{-x^2+x^4}}-\frac {x \sqrt [4]{\frac {1-x}{1+x}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {3}{2},\frac {2 x}{1+x}\right )}{2 \sqrt [4]{-x^2+x^4}}-\frac {x \left (\frac {1-x}{1+x}\right )^{5/4} (1+x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {3}{2},\frac {2 x}{1+x}\right )}{2 (1-x) \sqrt [4]{-x^2+x^4}}-\frac {\left (i \sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{i-(1+i) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt [4]{-x^2+x^4}}-\frac {\left (i \sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{i+(1-i) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt [4]{-x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{2 \sqrt [4]{-x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{2 \sqrt [4]{-x^2+x^4}} \\ & = \frac {\sqrt {x} \sqrt [4]{-1+x^2} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt [4]{-x^2+x^4}}-\frac {\sqrt {x} \sqrt [4]{-1+x^2} \arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{2 \sqrt [4]{2} \sqrt [4]{-x^2+x^4}}+\frac {\sqrt {x} \sqrt [4]{-1+x^2} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt [4]{-x^2+x^4}}-\frac {\sqrt {x} \sqrt [4]{-1+x^2} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{2 \sqrt [4]{2} \sqrt [4]{-x^2+x^4}}-\frac {x \sqrt [4]{\frac {1-x}{1+x}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {3}{2},\frac {2 x}{1+x}\right )}{2 \sqrt [4]{-x^2+x^4}}-\frac {x \left (\frac {1-x}{1+x}\right )^{5/4} (1+x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {3}{2},\frac {2 x}{1+x}\right )}{2 (1-x) \sqrt [4]{-x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {1-i} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{2 \sqrt [4]{-x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {1-i} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{2 \sqrt [4]{-x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {1+i} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{2 \sqrt [4]{-x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {1+i} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{2 \sqrt [4]{-x^2+x^4}} \\ & = \frac {\sqrt {x} \sqrt [4]{-1+x^2} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt [4]{-x^2+x^4}}-\frac {\sqrt {x} \sqrt [4]{-1+x^2} \arctan \left (\frac {\sqrt [4]{1-i} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{2 \sqrt [4]{1-i} \sqrt [4]{-x^2+x^4}}-\frac {\sqrt {x} \sqrt [4]{-1+x^2} \arctan \left (\frac {\sqrt [4]{1+i} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{2 \sqrt [4]{1+i} \sqrt [4]{-x^2+x^4}}-\frac {\sqrt {x} \sqrt [4]{-1+x^2} \arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{2 \sqrt [4]{2} \sqrt [4]{-x^2+x^4}}+\frac {\sqrt {x} \sqrt [4]{-1+x^2} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt [4]{-x^2+x^4}}-\frac {\sqrt {x} \sqrt [4]{-1+x^2} \text {arctanh}\left (\frac {\sqrt [4]{1-i} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{2 \sqrt [4]{1-i} \sqrt [4]{-x^2+x^4}}-\frac {\sqrt {x} \sqrt [4]{-1+x^2} \text {arctanh}\left (\frac {\sqrt [4]{1+i} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{2 \sqrt [4]{1+i} \sqrt [4]{-x^2+x^4}}-\frac {\sqrt {x} \sqrt [4]{-1+x^2} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{2 \sqrt [4]{2} \sqrt [4]{-x^2+x^4}}-\frac {x \sqrt [4]{\frac {1-x}{1+x}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {3}{2},\frac {2 x}{1+x}\right )}{2 \sqrt [4]{-x^2+x^4}}-\frac {x \left (\frac {1-x}{1+x}\right )^{5/4} (1+x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {3}{2},\frac {2 x}{1+x}\right )}{2 (1-x) \sqrt [4]{-x^2+x^4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.79 \[ \int \frac {1+x^8}{\sqrt [4]{-x^2+x^4} \left (-1+x^8\right )} \, dx=-\frac {\sqrt {x} \left (4 \sqrt {x}-4 \sqrt [4]{-1+x^2} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )+2^{3/4} \sqrt [4]{-1+x^2} \arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )-4 \sqrt [4]{-1+x^2} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )+2^{3/4} \sqrt [4]{-1+x^2} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )-\sqrt [4]{-1+x^2} \text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right )+\log \left (\sqrt [4]{-1+x^2}-\sqrt {x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right )}{4 \sqrt [4]{x^2 \left (-1+x^2\right )}} \]

[In]

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

[Out]

-1/4*(Sqrt[x]*(4*Sqrt[x] - 4*(-1 + x^2)^(1/4)*ArcTan[Sqrt[x]/(-1 + x^2)^(1/4)] + 2^(3/4)*(-1 + x^2)^(1/4)*ArcT
an[(2^(1/4)*Sqrt[x])/(-1 + x^2)^(1/4)] - 4*(-1 + x^2)^(1/4)*ArcTanh[Sqrt[x]/(-1 + x^2)^(1/4)] + 2^(3/4)*(-1 +
x^2)^(1/4)*ArcTanh[(2^(1/4)*Sqrt[x])/(-1 + x^2)^(1/4)] - (-1 + x^2)^(1/4)*RootSum[2 - 2*#1^4 + #1^8 & , (-Log[
Sqrt[x]] + Log[(-1 + x^2)^(1/4) - Sqrt[x]*#1])/#1 & ]))/(x^2*(-1 + x^2))^(1/4)

Maple [N/A] (verified)

Time = 0.70 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.74

method	result	size
pseudoelliptic	\(-\frac {\left (\left (-\frac {\arctan \left (\frac {2^{\frac {3}{4}} \left (x^{4}-x^{2}\right )^{\frac {1}{4}}}{2 x}\right )}{2}+\frac {\ln \left (\frac {-2^{\frac {1}{4}} x -\left (x^{4}-x^{2}\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (x^{4}-x^{2}\right )^{\frac {1}{4}}}\right )}{4}\right ) 2^{\frac {3}{4}}+\ln \left (\frac {\left (x^{4}-x^{2}\right )^{\frac {1}{4}}-x}{x}\right )-\ln \left (\frac {\left (x^{4}-x^{2}\right )^{\frac {1}{4}}+x}{x}\right )-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2 \textit {\_Z}^{4}+2\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{4}-x^{2}\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}}\right )}{2}+2 \arctan \left (\frac {\left (x^{4}-x^{2}\right )^{\frac {1}{4}}}{x}\right )\right ) \left (x^{4}-x^{2}\right )^{\frac {1}{4}}+2 x}{2 \left (x^{4}-x^{2}\right )^{\frac {1}{4}}}\)	\(202\)

[In]

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

[Out]

-1/2/(x^4-x^2)^(1/4)*(((-1/2*arctan(1/2*2^(3/4)/x*(x^4-x^2)^(1/4))+1/4*ln((-2^(1/4)*x-(x^4-x^2)^(1/4))/(2^(1/4
)*x-(x^4-x^2)^(1/4))))*2^(3/4)+ln(((x^4-x^2)^(1/4)-x)/x)-ln(((x^4-x^2)^(1/4)+x)/x)-1/2*sum(ln((-_R*x+(x^4-x^2)
^(1/4))/x)/_R,_R=RootOf(_Z^8-2*_Z^4+2))+2*arctan(1/x*(x^4-x^2)^(1/4)))*(x^4-x^2)^(1/4)+2*x)

Fricas [C] (verification not implemented)

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

Time = 75.72 (sec) , antiderivative size = 1577, normalized size of antiderivative = 5.80 \[ \int \frac {1+x^8}{\sqrt [4]{-x^2+x^4} \left (-1+x^8\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/16*(sqrt(2)*(x^3 - x)*sqrt(sqrt(2)*sqrt(-I + 1))*log(-(2*sqrt(2)*sqrt(-I + 1)*((379*I + 3)*x^4 + (3*I - 379
)*x^2)*(x^4 - x^2)^(1/4) + 4*(x^4 - x^2)^(3/4)*((188*I + 191)*x^2 + 191*I - 188) + (2*sqrt(-I + 1)*sqrt(x^4 -
x^2)*((379*I + 3)*x^3 + (3*I - 379)*x) - sqrt(2)*(-(567*I + 194)*x^5 - (6*I - 758)*x^3 + (191*I - 188)*x))*sqr
t(sqrt(2)*sqrt(-I + 1)))/(x^5 + x)) - sqrt(2)*(x^3 - x)*sqrt(sqrt(2)*sqrt(-I + 1))*log(-(2*sqrt(2)*sqrt(-I + 1
)*((379*I + 3)*x^4 + (3*I - 379)*x^2)*(x^4 - x^2)^(1/4) + 4*(x^4 - x^2)^(3/4)*((188*I + 191)*x^2 + 191*I - 188
) + (2*sqrt(-I + 1)*sqrt(x^4 - x^2)*(-(379*I + 3)*x^3 - (3*I - 379)*x) - sqrt(2)*((567*I + 194)*x^5 + (6*I - 7
58)*x^3 - (191*I - 188)*x))*sqrt(sqrt(2)*sqrt(-I + 1)))/(x^5 + x)) + sqrt(2)*(x^3 - x)*sqrt(sqrt(2)*sqrt(I + 1
))*log(-(2*sqrt(2)*sqrt(I + 1)*(-(379*I - 3)*x^4 - (3*I + 379)*x^2)*(x^4 - x^2)^(1/4) + 4*(x^4 - x^2)^(3/4)*(-
(188*I - 191)*x^2 - 191*I - 188) + (2*sqrt(I + 1)*sqrt(x^4 - x^2)*(-(379*I - 3)*x^3 - (3*I + 379)*x) - sqrt(2)
*((567*I - 194)*x^5 + (6*I + 758)*x^3 - (191*I + 188)*x))*sqrt(sqrt(2)*sqrt(I + 1)))/(x^5 + x)) - sqrt(2)*(x^3
- x)*sqrt(sqrt(2)*sqrt(I + 1))*log(-(2*sqrt(2)*sqrt(I + 1)*(-(379*I - 3)*x^4 - (3*I + 379)*x^2)*(x^4 - x^2)^(
1/4) + 4*(x^4 - x^2)^(3/4)*(-(188*I - 191)*x^2 - 191*I - 188) + (2*sqrt(I + 1)*sqrt(x^4 - x^2)*((379*I - 3)*x^
3 + (3*I + 379)*x) - sqrt(2)*(-(567*I - 194)*x^5 - (6*I + 758)*x^3 + (191*I + 188)*x))*sqrt(sqrt(2)*sqrt(I + 1
)))/(x^5 + x)) - sqrt(2)*(x^3 - x)*sqrt(-sqrt(2)*sqrt(I + 1))*log(-(2*sqrt(2)*sqrt(I + 1)*(x^4 - x^2)^(1/4)*((
379*I - 3)*x^4 + (3*I + 379)*x^2) + 4*(x^4 - x^2)^(3/4)*(-(188*I - 191)*x^2 - 191*I - 188) + (2*sqrt(I + 1)*sq
rt(x^4 - x^2)*(-(379*I - 3)*x^3 - (3*I + 379)*x) - sqrt(2)*(-(567*I - 194)*x^5 - (6*I + 758)*x^3 + (191*I + 18
8)*x))*sqrt(-sqrt(2)*sqrt(I + 1)))/(x^5 + x)) + sqrt(2)*(x^3 - x)*sqrt(-sqrt(2)*sqrt(I + 1))*log(-(2*sqrt(2)*s
qrt(I + 1)*(x^4 - x^2)^(1/4)*((379*I - 3)*x^4 + (3*I + 379)*x^2) + 4*(x^4 - x^2)^(3/4)*(-(188*I - 191)*x^2 - 1
91*I - 188) + (2*sqrt(I + 1)*sqrt(x^4 - x^2)*((379*I - 3)*x^3 + (3*I + 379)*x) - sqrt(2)*((567*I - 194)*x^5 +
(6*I + 758)*x^3 - (191*I + 188)*x))*sqrt(-sqrt(2)*sqrt(I + 1)))/(x^5 + x)) - sqrt(2)*(x^3 - x)*sqrt(-sqrt(2)*s
qrt(-I + 1))*log(-(2*sqrt(2)*sqrt(-I + 1)*(x^4 - x^2)^(1/4)*(-(379*I + 3)*x^4 - (3*I - 379)*x^2) + 4*(x^4 - x^
2)^(3/4)*((188*I + 191)*x^2 + 191*I - 188) + (2*sqrt(-I + 1)*sqrt(x^4 - x^2)*((379*I + 3)*x^3 + (3*I - 379)*x)
- sqrt(2)*((567*I + 194)*x^5 + (6*I - 758)*x^3 - (191*I - 188)*x))*sqrt(-sqrt(2)*sqrt(-I + 1)))/(x^5 + x)) +
sqrt(2)*(x^3 - x)*sqrt(-sqrt(2)*sqrt(-I + 1))*log(-(2*sqrt(2)*sqrt(-I + 1)*(x^4 - x^2)^(1/4)*(-(379*I + 3)*x^4
- (3*I - 379)*x^2) + 4*(x^4 - x^2)^(3/4)*((188*I + 191)*x^2 + 191*I - 188) + (2*sqrt(-I + 1)*sqrt(x^4 - x^2)*
(-(379*I + 3)*x^3 - (3*I - 379)*x) - sqrt(2)*(-(567*I + 194)*x^5 - (6*I - 758)*x^3 + (191*I - 188)*x))*sqrt(-s
qrt(2)*sqrt(-I + 1)))/(x^5 + x)) + 2^(3/4)*(x^3 - x)*log((4*sqrt(2)*(x^4 - x^2)^(1/4)*x^2 + 2^(3/4)*(3*x^3 - x
) + 4*2^(1/4)*sqrt(x^4 - x^2)*x + 4*(x^4 - x^2)^(3/4))/(x^3 + x)) - 2^(3/4)*(x^3 - x)*log((4*sqrt(2)*(x^4 - x^
2)^(1/4)*x^2 - 2^(3/4)*(3*x^3 - x) - 4*2^(1/4)*sqrt(x^4 - x^2)*x + 4*(x^4 - x^2)^(3/4))/(x^3 + x)) - 2^(3/4)*(
-I*x^3 + I*x)*log(-(4*sqrt(2)*(x^4 - x^2)^(1/4)*x^2 - 2^(3/4)*(3*I*x^3 - I*x) + 4*I*2^(1/4)*sqrt(x^4 - x^2)*x
- 4*(x^4 - x^2)^(3/4))/(x^3 + x)) - 2^(3/4)*(I*x^3 - I*x)*log(-(4*sqrt(2)*(x^4 - x^2)^(1/4)*x^2 - 2^(3/4)*(-3*
I*x^3 + I*x) - 4*I*2^(1/4)*sqrt(x^4 - x^2)*x - 4*(x^4 - x^2)^(3/4))/(x^3 + x)) + 8*(x^3 - x)*arctan(2*((x^4 -
x^2)^(1/4)*x^2 + (x^4 - x^2)^(3/4))/x) - 8*(x^3 - x)*log((2*x^3 + 2*(x^4 - x^2)^(1/4)*x^2 + 2*sqrt(x^4 - x^2)*
x - x + 2*(x^4 - x^2)^(3/4))/x) + 16*(x^4 - x^2)^(3/4))/(x^3 - x)

Sympy [N/A]

Not integrable

Time = 2.42 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.13 \[ \int \frac {1+x^8}{\sqrt [4]{-x^2+x^4} \left (-1+x^8\right )} \, dx=\int \frac {x^{8} + 1}{\sqrt [4]{x^{2} \left (x - 1\right ) \left (x + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.10 \[ \int \frac {1+x^8}{\sqrt [4]{-x^2+x^4} \left (-1+x^8\right )} \, dx=\int { \frac {x^{8} + 1}{{\left (x^{8} - 1\right )} {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}}} \,d x } \]

[In]

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

[Out]

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

Giac [C] (verification not implemented)

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

Time = 0.34 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.05 \[ \int \frac {1+x^8}{\sqrt [4]{-x^2+x^4} \left (-1+x^8\right )} \, dx=\frac {1}{4} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{8} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {1}{4}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{8} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {1}{4}} - {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + 4 i \, \left (-\frac {1}{131072} i + \frac {1}{131072}\right )^{\frac {1}{4}} \log \left (i \, \left (-2251799813685248 i + 2251799813685248\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} - 8192\right ) - 4 i \, \left (-\frac {1}{131072} i + \frac {1}{131072}\right )^{\frac {1}{4}} \log \left (-i \, \left (-2251799813685248 i + 2251799813685248\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} - 8192\right ) + i \, \left (\frac {1}{512} i + \frac {1}{512}\right )^{\frac {1}{4}} \log \left (\left (134217728 i + 134217728\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 128 i\right ) - \left (\frac {1}{512} i + \frac {1}{512}\right )^{\frac {1}{4}} \log \left (i \, \left (134217728 i + 134217728\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 128 i\right ) + \left (\frac {1}{512} i + \frac {1}{512}\right )^{\frac {1}{4}} \log \left (-i \, \left (134217728 i + 134217728\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 128 i\right ) - i \, \left (\frac {1}{512} i + \frac {1}{512}\right )^{\frac {1}{4}} \log \left (-\left (134217728 i + 134217728\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 128 i\right ) + \left (-\frac {1}{512} i + \frac {1}{512}\right )^{\frac {1}{4}} \log \left (\left (-134217728 i + 134217728\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} - 128\right ) - \left (-\frac {1}{512} i + \frac {1}{512}\right )^{\frac {1}{4}} \log \left (-\left (-134217728 i + 134217728\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} - 128\right ) - \frac {1}{{\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}} - \arctan \left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{2} \, \log \left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {1}{2} \, \log \left (-{\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 1\right ) \]

[In]

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

[Out]

1/4*2^(3/4)*arctan(1/2*2^(3/4)*(-1/x^2 + 1)^(1/4)) - 1/8*2^(3/4)*log(2^(1/4) + (-1/x^2 + 1)^(1/4)) + 1/8*2^(3/
4)*log(2^(1/4) - (-1/x^2 + 1)^(1/4)) + 4*I*(-1/131072*I + 1/131072)^(1/4)*log(I*(-2251799813685248*I + 2251799
813685248)^(1/4)*(-1/x^2 + 1)^(1/4) - 8192) - 4*I*(-1/131072*I + 1/131072)^(1/4)*log(-I*(-2251799813685248*I +
2251799813685248)^(1/4)*(-1/x^2 + 1)^(1/4) - 8192) + I*(1/512*I + 1/512)^(1/4)*log((134217728*I + 134217728)^
(1/4)*(-1/x^2 + 1)^(1/4) + 128*I) - (1/512*I + 1/512)^(1/4)*log(I*(134217728*I + 134217728)^(1/4)*(-1/x^2 + 1)
^(1/4) + 128*I) + (1/512*I + 1/512)^(1/4)*log(-I*(134217728*I + 134217728)^(1/4)*(-1/x^2 + 1)^(1/4) + 128*I) -
I*(1/512*I + 1/512)^(1/4)*log(-(134217728*I + 134217728)^(1/4)*(-1/x^2 + 1)^(1/4) + 128*I) + (-1/512*I + 1/51
2)^(1/4)*log((-134217728*I + 134217728)^(1/4)*(-1/x^2 + 1)^(1/4) - 128) - (-1/512*I + 1/512)^(1/4)*log(-(-1342
17728*I + 134217728)^(1/4)*(-1/x^2 + 1)^(1/4) - 128) - 1/(-1/x^2 + 1)^(1/4) - arctan((-1/x^2 + 1)^(1/4)) + 1/2
*log((-1/x^2 + 1)^(1/4) + 1) - 1/2*log(-(-1/x^2 + 1)^(1/4) + 1)

Mupad [N/A]

Not integrable

Time = 7.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.10 \[ \int \frac {1+x^8}{\sqrt [4]{-x^2+x^4} \left (-1+x^8\right )} \, dx=\int \frac {x^8+1}{\left (x^8-1\right )\,{\left (x^4-x^2\right )}^{1/4}} \,d x \]

[In]

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

[Out]

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