3.29.2 \(\int \frac {1+x^8}{\sqrt [4]{-x^2+x^4} (-1+x^8)} \, dx\) [2802]

Optimal. Leaf size=272 \[ -\frac {\left (-x^2+x^4\right )^{3/4}}{x \left (-1+x^2\right )}+\text {ArcTan}\left (\frac {x}{\sqrt [4]{-x^2+x^4}}\right )-\frac {\text {ArcTan}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^2+x^4}}\right )}{2 \sqrt [4]{2}}+\tanh ^{-1}\left (\frac {x}{\sqrt [4]{-x^2+x^4}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^2+x^4}}\right )}{2 \sqrt [4]{2}}+\frac {5}{8} \text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\& ,\frac {-\log (x) \text {$\#$1}^3+\log \left (\sqrt [4]{-x^2+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-1+\text {$\#$1}^4}\& \right ]+\frac {1}{8} \text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\& ,\frac {2 \log (x)-2 \log \left (\sqrt [4]{-x^2+x^4}-x \text {$\#$1}\right )+3 \log (x) \text {$\#$1}^4-3 \log \left (\sqrt [4]{-x^2+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}+\text {$\#$1}^5}\& \right ] \]

[Out]

Unintegrable

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Rubi [C] Result contains complex when optimal does not.
time = 0.46, antiderivative size = 559, normalized size of antiderivative = 2.06, number of steps used = 27, number of rules used = 12, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2081, 6847, 6857, 246, 218, 212, 209, 2098, 1166, 388, 385, 1443} \begin {gather*} \frac {\sqrt {x} \sqrt [4]{x^2-1} \text {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} \text {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} \text {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} \text {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 {x (x+1) \left (\frac {1-x}{x+1}\right )^{5/4} \, _2F_1\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}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {3}{2};\frac {2 x}{x+1}\right )}{2 \sqrt [4]{x^4-x^2}}+\frac {\sqrt {x} \sqrt [4]{x^2-1} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{\sqrt [4]{x^4-x^2}}-\frac {\sqrt {x} \sqrt [4]{x^2-1} \tanh ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{2 \sqrt [4]{2} \sqrt [4]{x^4-x^2}} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {1+x^8}{\sqrt [4]{-x^2+x^4} \left (-1+x^8\right )} \, dx &=\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} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt [4]{-x^2+x^4}}+\frac {\sqrt {x} \sqrt [4]{-1+x^2} \tanh ^{-1}\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} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt [4]{-x^2+x^4}}+\frac {\sqrt {x} \sqrt [4]{-1+x^2} \tanh ^{-1}\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}} \, _2F_1\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) \, _2F_1\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} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt [4]{-x^2+x^4}}-\frac {\sqrt {x} \sqrt [4]{-1+x^2} \tan ^{-1}\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} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt [4]{-x^2+x^4}}-\frac {\sqrt {x} \sqrt [4]{-1+x^2} \tanh ^{-1}\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}} \, _2F_1\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) \, _2F_1\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} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt [4]{-x^2+x^4}}-\frac {\sqrt {x} \sqrt [4]{-1+x^2} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt [4]{-x^2+x^4}}-\frac {\sqrt {x} \sqrt [4]{-1+x^2} \tanh ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\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}} \, _2F_1\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) \, _2F_1\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*}

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Mathematica [A]
time = 0.00, size = 216, normalized size = 0.79 \begin {gather*} -\frac {\sqrt {x} \left (4 \sqrt {x}-4 \sqrt [4]{-1+x^2} \text {ArcTan}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )+2^{3/4} \sqrt [4]{-1+x^2} \text {ArcTan}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )-4 \sqrt [4]{-1+x^2} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )+2^{3/4} \sqrt [4]{-1+x^2} \tanh ^{-1}\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 )}} \end {gather*}

Antiderivative was successfully verified.

[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)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 1.
time = 74.34, size = 2034, normalized size = 7.48

method	result	size
risch	\(\text {Expression too large to display}\)	\(2034\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/(x^2*(x^2-1))^(1/4)+1/2*ln((2*(x^4-x^2)^(3/4)+2*(x^4-x^2)^(1/2)*x+2*x^2*(x^4-x^2)^(1/4)+2*x^3-x)/x)+1/2*Roo
tOf(_Z^2+1)*ln((-2*(x^4-x^2)^(1/2)*RootOf(_Z^2+1)*x+2*RootOf(_Z^2+1)*x^3+2*(x^4-x^2)^(3/4)-2*x^2*(x^4-x^2)^(1/
4)-RootOf(_Z^2+1)*x)/x)-1/8*RootOf(_Z^2+1)*RootOf(_Z^4-8-8*RootOf(_Z^2+1))*ln((-367*(x^4-x^2)^(1/2)*RootOf(_Z^
2+1)*RootOf(_Z^4-8-8*RootOf(_Z^2+1))^3*x+1519*(x^4-x^2)^(1/2)*RootOf(_Z^4-8-8*RootOf(_Z^2+1))^3*x-3038*RootOf(
_Z^4-8-8*RootOf(_Z^2+1))^2*RootOf(_Z^2+1)*(x^4-x^2)^(1/4)*x^2-8*RootOf(_Z^4-8-8*RootOf(_Z^2+1))*RootOf(_Z^2+1)
^2*x^3-734*RootOf(_Z^4-8-8*RootOf(_Z^2+1))^2*(x^4-x^2)^(1/4)*x^2-418*RootOf(_Z^4-8-8*RootOf(_Z^2+1))*RootOf(_Z
^2+1)*x^3+7544*RootOf(_Z^2+1)*(x^4-x^2)^(3/4)+32*RootOf(_Z^4-8-8*RootOf(_Z^2+1))*RootOf(_Z^2+1)^2*x-4932*x^3*R
ootOf(_Z^4-8-8*RootOf(_Z^2+1))-4608*(x^4-x^2)^(3/4)+1152*RootOf(_Z^4-8-8*RootOf(_Z^2+1))*RootOf(_Z^2+1)*x+1918
*x*RootOf(_Z^4-8-8*RootOf(_Z^2+1)))/(RootOf(_Z^2+1)*x^2-4*x^2-4*RootOf(_Z^2+1)-1)/x)-1/8*RootOf(_Z^4-8-8*RootO
f(_Z^2+1))*ln((1519*(x^4-x^2)^(1/2)*RootOf(_Z^2+1)*RootOf(_Z^4-8-8*RootOf(_Z^2+1))^3*x+367*(x^4-x^2)^(1/2)*Roo
tOf(_Z^4-8-8*RootOf(_Z^2+1))^3*x+3038*RootOf(_Z^4-8-8*RootOf(_Z^2+1))^2*RootOf(_Z^2+1)*(x^4-x^2)^(1/4)*x^2+274
*RootOf(_Z^4-8-8*RootOf(_Z^2+1))*RootOf(_Z^2+1)^2*x^3+734*RootOf(_Z^4-8-8*RootOf(_Z^2+1))^2*(x^4-x^2)^(1/4)*x^
2+4924*RootOf(_Z^4-8-8*RootOf(_Z^2+1))*RootOf(_Z^2+1)*x^3+7544*RootOf(_Z^2+1)*(x^4-x^2)^(3/4)-1096*RootOf(_Z^4
-8-8*RootOf(_Z^2+1))*RootOf(_Z^2+1)^2*x-144*x^3*RootOf(_Z^4-8-8*RootOf(_Z^2+1))-4608*(x^4-x^2)^(3/4)-1886*Root
Of(_Z^4-8-8*RootOf(_Z^2+1))*RootOf(_Z^2+1)*x+56*x*RootOf(_Z^4-8-8*RootOf(_Z^2+1)))/(RootOf(_Z^2+1)*x^2-4*x^2-4
*RootOf(_Z^2+1)-1)/x)+1/8*RootOf(_Z^2+1)*RootOf(_Z^4-8+8*RootOf(_Z^2+1))*ln(-(-367*(x^4-x^2)^(1/2)*RootOf(_Z^2
+1)*RootOf(_Z^4-8+8*RootOf(_Z^2+1))^3*x-1519*(x^4-x^2)^(1/2)*RootOf(_Z^4-8+8*RootOf(_Z^2+1))^3*x-3038*RootOf(_
Z^2+1)*(x^4-x^2)^(1/4)*RootOf(_Z^4-8+8*RootOf(_Z^2+1))^2*x^2+8*RootOf(_Z^4-8+8*RootOf(_Z^2+1))*RootOf(_Z^2+1)^
2*x^3+734*(x^4-x^2)^(1/4)*RootOf(_Z^4-8+8*RootOf(_Z^2+1))^2*x^2-418*RootOf(_Z^4-8+8*RootOf(_Z^2+1))*RootOf(_Z^
2+1)*x^3+7544*RootOf(_Z^2+1)*(x^4-x^2)^(3/4)-32*RootOf(_Z^4-8+8*RootOf(_Z^2+1))*RootOf(_Z^2+1)^2*x+4932*RootOf
(_Z^4-8+8*RootOf(_Z^2+1))*x^3+4608*(x^4-x^2)^(3/4)+1152*RootOf(_Z^4-8+8*RootOf(_Z^2+1))*RootOf(_Z^2+1)*x-1918*
RootOf(_Z^4-8+8*RootOf(_Z^2+1))*x)/(RootOf(_Z^2+1)*x^2+4*x^2-4*RootOf(_Z^2+1)+1)/x)-1/8*RootOf(_Z^4-8+8*RootOf
(_Z^2+1))*ln((-1519*(x^4-x^2)^(1/2)*RootOf(_Z^2+1)*RootOf(_Z^4-8+8*RootOf(_Z^2+1))^3*x+274*RootOf(_Z^4-8+8*Roo
tOf(_Z^2+1))*RootOf(_Z^2+1)^2*x^3-3038*RootOf(_Z^2+1)*(x^4-x^2)^(1/4)*RootOf(_Z^4-8+8*RootOf(_Z^2+1))^2*x^2+36
7*(x^4-x^2)^(1/2)*RootOf(_Z^4-8+8*RootOf(_Z^2+1))^3*x-4924*RootOf(_Z^4-8+8*RootOf(_Z^2+1))*RootOf(_Z^2+1)*x^3+
734*(x^4-x^2)^(1/4)*RootOf(_Z^4-8+8*RootOf(_Z^2+1))^2*x^2-1096*RootOf(_Z^4-8+8*RootOf(_Z^2+1))*RootOf(_Z^2+1)^
2*x-7544*RootOf(_Z^2+1)*(x^4-x^2)^(3/4)-144*RootOf(_Z^4-8+8*RootOf(_Z^2+1))*x^3+1886*RootOf(_Z^4-8+8*RootOf(_Z
^2+1))*RootOf(_Z^2+1)*x-4608*(x^4-x^2)^(3/4)+56*RootOf(_Z^4-8+8*RootOf(_Z^2+1))*x)/(RootOf(_Z^2+1)*x^2+4*x^2-4
*RootOf(_Z^2+1)+1)/x)+1/16*RootOf(_Z^4-8-8*RootOf(_Z^2+1))*RootOf(_Z^4-8+8*RootOf(_Z^2+1))*ln((-(x^4-x^2)^(1/2
)*RootOf(_Z^4-8-8*RootOf(_Z^2+1))^3*RootOf(_Z^4-8+8*RootOf(_Z^2+1))^3*x+4*RootOf(_Z^4-8-8*RootOf(_Z^2+1))^2*(x
^4-x^2)^(1/4)*RootOf(_Z^4-8+8*RootOf(_Z^2+1))^2*x^2-12*RootOf(_Z^4-8-8*RootOf(_Z^2+1))*RootOf(_Z^4-8+8*RootOf(
_Z^2+1))*x^3+32*(x^4-x^2)^(3/4)+4*RootOf(_Z^4-8-8*RootOf(_Z^2+1))*RootOf(_Z^4-8+8*RootOf(_Z^2+1))*x)/x/(x^2+1)
)+1/16*RootOf(_Z^2+1)*RootOf(_Z^4-8-8*RootOf(_Z^2+1))*RootOf(_Z^4-8+8*RootOf(_Z^2+1))*ln(((x^4-x^2)^(1/2)*Root
Of(_Z^4-8-8*RootOf(_Z^2+1))^3*RootOf(_Z^4-8+8*RootOf(_Z^2+1))^3*RootOf(_Z^2+1)*x-4*RootOf(_Z^4-8-8*RootOf(_Z^2
+1))^2*(x^4-x^2)^(1/4)*RootOf(_Z^4-8+8*RootOf(_Z^2+1))^2*x^2-12*RootOf(_Z^4-8-8*RootOf(_Z^2+1))*RootOf(_Z^4-8+
8*RootOf(_Z^2+1))*RootOf(_Z^2+1)*x^3+4*RootOf(_Z^4-8-8*RootOf(_Z^2+1))*RootOf(_Z^4-8+8*RootOf(_Z^2+1))*RootOf(
_Z^2+1)*x+32*(x^4-x^2)^(3/4))/x/(x^2+1))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 0.46, size = 282, normalized size = 1.04 \begin {gather*} -\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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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 + 225179
9813685248)^(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 + 1/5
12)^(1/4)*log((-134217728*I + 134217728)^(1/4)*(-1/x^2 + 1)^(1/4) + 128) - (-1/512*I + 1/512)^(1/4)*log(-(-134
217728*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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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