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

Optimal. Leaf size=137 \[ \frac {1}{8} \text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^4+2\& ,\frac {\log \left (\sqrt [4]{x^4-x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]-\frac {\left (x^4-x^2\right )^{3/4}}{2 x \left (x^2-1\right )}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4-x^2}}\right )}{4 \sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4-x^2}}\right )}{4 \sqrt [4]{2}} \]

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Rubi [C]  time = 0.56, antiderivative size = 471, normalized size of antiderivative = 3.44, number of steps used = 21, number of rules used = 10, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {2056, 6715, 2073, 1152, 380, 377, 212, 206, 203, 1429} \begin {gather*} -\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 )}{4 (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 )}{4 \sqrt [4]{x^4-x^2}}-\frac {\sqrt {x} \sqrt [4]{x^2-1} \tan ^{-1}\left (\frac {\sqrt [4]{1-i} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{4 \sqrt [4]{1-i} \sqrt [4]{x^4-x^2}}-\frac {\sqrt {x} \sqrt [4]{x^2-1} \tan ^{-1}\left (\frac {\sqrt [4]{1+i} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{4 \sqrt [4]{1+i} \sqrt [4]{x^4-x^2}}-\frac {\sqrt {x} \sqrt [4]{x^2-1} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{4 \sqrt [4]{2} \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 )}{4 \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 )}{4 \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 )}{4 \sqrt [4]{2} \sqrt [4]{x^4-x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(Sqrt[x]*(-1 + x^2)^(1/4)*ArcTan[((1 - I)^(1/4)*Sqrt[x])/(-1 + x^2)^(1/4)])/((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)])/(4*(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)])/(4*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)])/(4*(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)])/(4*(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)])/(4*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)])/(4*(-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)])/(4*(1 - x)*(-x^2 + x^4)
^(1/4))

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 377

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 380

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(x*(a + b*x^n)^p*Hypergeome
tric2F1[1/n, -p, 1 + 1/n, -(((b*c - a*d)*x^n)/(a*(c + d*x^n)))])/(c*((c*(a + b*x^n))/(a*(c + d*x^n)))^p*(c + d
*x^n)^(1/n + p)), x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 1) + 1, 0]

Rule 1152

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*x^2)/e)^p, x], x] /; FreeQ[{
a, c, d, e, p, q}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p]

Rule 1429

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 2056

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 2073

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 6715

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]

Rubi steps

\begin {align*} \int \frac {1}{\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}{\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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+x^2\right ) \sqrt [4]{-1+x^4}} \, dx,x,\sqrt {x}\right )}{4 \sqrt [4]{-x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt [4]{-1+x^4}} \, dx,x,\sqrt {x}\right )}{4 \sqrt [4]{-x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1+x^4} \left (1+x^4\right )} \, dx,x,\sqrt {x}\right )}{2 \sqrt [4]{-x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \operatorname {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 {\left (\sqrt [4]{-1+x} \sqrt {x} \sqrt [4]{1+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1+x^2} \left (1+x^2\right )^{5/4}} \, dx,x,\sqrt {x}\right )}{4 \sqrt [4]{-x^2+x^4}}+\frac {\left (\sqrt [4]{-1+x} \sqrt {x} \sqrt [4]{1+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+x^2\right )^{5/4} \sqrt [4]{1+x^2}} \, dx,x,\sqrt {x}\right )}{4 \sqrt [4]{-x^2+x^4}}-\frac {\left (i \sqrt {x} \sqrt [4]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (i-x^4\right ) \sqrt [4]{-1+x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt [4]{-x^2+x^4}}-\frac {\left (i \sqrt {x} \sqrt [4]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1+x^4} \left (i+x^4\right )} \, dx,x,\sqrt {x}\right )}{2 \sqrt [4]{-x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-2 x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{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 )}{4 \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 )}{4 (1-x) \sqrt [4]{-x^2+x^4}}-\frac {\left (i \sqrt {x} \sqrt [4]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{i-(1+i) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{2 \sqrt [4]{-x^2+x^4}}-\frac {\left (i \sqrt {x} \sqrt [4]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{i+(1-i) x^4} \, 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 ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{4 \sqrt [4]{-x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{4 \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 )}{4 \sqrt [4]{2} \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 )}{4 \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 )}{4 \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 )}{4 (1-x) \sqrt [4]{-x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {1-i} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{4 \sqrt [4]{-x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {1-i} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{4 \sqrt [4]{-x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {1+i} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{4 \sqrt [4]{-x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {1+i} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{4 \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 )}{4 \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 )}{4 \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 )}{4 \sqrt [4]{2} \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 )}{4 \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 )}{4 \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 )}{4 \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 )}{4 \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 )}{4 (1-x) \sqrt [4]{-x^2+x^4}}\\ \end {align*}

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Mathematica [C]  time = 16.10, size = 272, normalized size = 1.99 \begin {gather*} \frac {x \left (-8+\sqrt [4]{\frac {1}{x^2}-1} \left (-(-2)^{3/4} \log \left (\sqrt [4]{-2}-\sqrt [4]{\frac {1}{x^2}-1}\right )+\frac {2 \log \left (-\sqrt [4]{\frac {1}{x^2}-1}+\sqrt [4]{-1-i}\right )}{\sqrt [4]{-1-i}}+\frac {2 \log \left (-\sqrt [4]{\frac {1}{x^2}-1}+\sqrt [4]{-1+i}\right )}{\sqrt [4]{-1+i}}+(-2)^{3/4} \log \left (\sqrt [4]{\frac {1}{x^2}-1}+\sqrt [4]{-2}\right )-\frac {2 \log \left (\sqrt [4]{\frac {1}{x^2}-1}+\sqrt [4]{-1-i}\right )}{\sqrt [4]{-1-i}}-\frac {2 \log \left (\sqrt [4]{\frac {1}{x^2}-1}+\sqrt [4]{-1+i}\right )}{\sqrt [4]{-1+i}}+\frac {4 \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{x^2}-1}}{\sqrt [4]{-1-i}}\right )}{\sqrt [4]{-1-i}}+\frac {4 \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{x^2}-1}}{\sqrt [4]{-1+i}}\right )}{\sqrt [4]{-1+i}}-(2+2 i) \sqrt [4]{2} \tanh ^{-1}\left (\frac {(1+i) \sqrt [4]{\frac {1}{x^2}-1}}{2^{3/4}}\right )\right )\right )}{16 \sqrt [4]{x^2 \left (x^2-1\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(-8 + (-1 + x^(-2))^(1/4)*((4*ArcTan[(-1 + x^(-2))^(1/4)/(-1 - I)^(1/4)])/(-1 - I)^(1/4) + (4*ArcTan[(-1 +
x^(-2))^(1/4)/(-1 + I)^(1/4)])/(-1 + I)^(1/4) - (2 + 2*I)*2^(1/4)*ArcTanh[((1 + I)*(-1 + x^(-2))^(1/4))/2^(3/4
)] - (-2)^(3/4)*Log[(-2)^(1/4) - (-1 + x^(-2))^(1/4)] + (2*Log[(-1 - I)^(1/4) - (-1 + x^(-2))^(1/4)])/(-1 - I)
^(1/4) + (2*Log[(-1 + I)^(1/4) - (-1 + x^(-2))^(1/4)])/(-1 + I)^(1/4) + (-2)^(3/4)*Log[(-2)^(1/4) + (-1 + x^(-
2))^(1/4)] - (2*Log[(-1 - I)^(1/4) + (-1 + x^(-2))^(1/4)])/(-1 - I)^(1/4) - (2*Log[(-1 + I)^(1/4) + (-1 + x^(-
2))^(1/4)])/(-1 + I)^(1/4))))/(16*(x^2*(-1 + x^2))^(1/4))

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IntegrateAlgebraic [A]  time = 0.33, size = 137, normalized size = 1.00 \begin {gather*} -\frac {\left (-x^2+x^4\right )^{3/4}}{2 x \left (-1+x^2\right )}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^2+x^4}}\right )}{4 \sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^2+x^4}}\right )}{4 \sqrt [4]{2}}+\frac {1}{8} \text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{-x^2+x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F(-1)]  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(1/(x^4-x^2)^(1/4)/(x^8-1),x, algorithm="fricas")

[Out]

Timed out

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giac [B]  time = 0.25, size = 244, normalized size = 1.78 \begin {gather*} -\frac {1}{8} \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}{16} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {1}{4}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{16} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {1}{4}} - {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - 8 i \, \left (-\frac {1}{33554432} i + \frac {1}{33554432}\right )^{\frac {1}{4}} \log \left (i \, \left (-37778931862957161709568 i + 37778931862957161709568\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} - 524288\right ) + 8 i \, \left (-\frac {1}{33554432} i + \frac {1}{33554432}\right )^{\frac {1}{4}} \log \left (-i \, \left (-37778931862957161709568 i + 37778931862957161709568\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} - 524288\right ) - i \, \left (\frac {1}{8192} i + \frac {1}{8192}\right )^{\frac {1}{4}} \log \left (\left (549755813888 i + 549755813888\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 1024 i\right ) + \left (\frac {1}{8192} i + \frac {1}{8192}\right )^{\frac {1}{4}} \log \left (i \, \left (549755813888 i + 549755813888\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 1024 i\right ) - \left (\frac {1}{8192} i + \frac {1}{8192}\right )^{\frac {1}{4}} \log \left (-i \, \left (549755813888 i + 549755813888\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 1024 i\right ) + i \, \left (\frac {1}{8192} i + \frac {1}{8192}\right )^{\frac {1}{4}} \log \left (-\left (549755813888 i + 549755813888\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 1024 i\right ) - \left (-\frac {1}{8192} i + \frac {1}{8192}\right )^{\frac {1}{4}} \log \left (\left (-549755813888 i + 549755813888\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} - 1024\right ) + \left (-\frac {1}{8192} i + \frac {1}{8192}\right )^{\frac {1}{4}} \log \left (-\left (-549755813888 i + 549755813888\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} - 1024\right ) + \frac {1}{2 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*2^(3/4)*arctan(1/2*2^(3/4)*(-1/x^2 + 1)^(1/4)) + 1/16*2^(3/4)*log(2^(1/4) + (-1/x^2 + 1)^(1/4)) - 1/16*2^
(3/4)*log(2^(1/4) - (-1/x^2 + 1)^(1/4)) - 8*I*(-1/33554432*I + 1/33554432)^(1/4)*log(I*(-377789318629571617095
68*I + 37778931862957161709568)^(1/4)*(-1/x^2 + 1)^(1/4) - 524288) + 8*I*(-1/33554432*I + 1/33554432)^(1/4)*lo
g(-I*(-37778931862957161709568*I + 37778931862957161709568)^(1/4)*(-1/x^2 + 1)^(1/4) - 524288) - I*(1/8192*I +
1/8192)^(1/4)*log((549755813888*I + 549755813888)^(1/4)*(-1/x^2 + 1)^(1/4) + 1024*I) + (1/8192*I + 1/8192)^(1
/4)*log(I*(549755813888*I + 549755813888)^(1/4)*(-1/x^2 + 1)^(1/4) + 1024*I) - (1/8192*I + 1/8192)^(1/4)*log(-
I*(549755813888*I + 549755813888)^(1/4)*(-1/x^2 + 1)^(1/4) + 1024*I) + I*(1/8192*I + 1/8192)^(1/4)*log(-(54975
5813888*I + 549755813888)^(1/4)*(-1/x^2 + 1)^(1/4) + 1024*I) - (-1/8192*I + 1/8192)^(1/4)*log((-549755813888*I
+ 549755813888)^(1/4)*(-1/x^2 + 1)^(1/4) - 1024) + (-1/8192*I + 1/8192)^(1/4)*log(-(-549755813888*I + 5497558
13888)^(1/4)*(-1/x^2 + 1)^(1/4) - 1024) + 1/2/(-1/x^2 + 1)^(1/4)

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maple [B]  time = 131.01, size = 3571, normalized size = 26.07

method	result	size
risch	\(\text {Expression too large to display}\)	\(3571\)
trager	\(\text {Expression too large to display}\)	\(3683\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*x/(x^2*(x^2-1))^(1/4)+1/8*RootOf(RootOf(2*_Z^8-2*_Z^4+1)^4+_Z^4-1)*ln(-(548*RootOf(RootOf(2*_Z^8-2*_Z^4+1
)^4+_Z^4-1)*RootOf(2*_Z^8-2*_Z^4+1)^8*x^3-6076*(x^4-x^2)^(1/2)*RootOf(2*_Z^8-2*_Z^4+1)^4*RootOf(RootOf(2*_Z^8-
2*_Z^4+1)^4+_Z^4-1)^3*x-2192*RootOf(RootOf(2*_Z^8-2*_Z^4+1)^4+_Z^4-1)*RootOf(2*_Z^8-2*_Z^4+1)^8*x+6076*(x^4-x^
2)^(1/4)*RootOf(RootOf(2*_Z^8-2*_Z^4+1)^4+_Z^4-1)^2*RootOf(2*_Z^8-2*_Z^4+1)^4*x^2-5472*RootOf(RootOf(2*_Z^8-2*
_Z^4+1)^4+_Z^4-1)*RootOf(2*_Z^8-2*_Z^4+1)^4*x^3+3772*(x^4-x^2)^(3/4)*RootOf(2*_Z^8-2*_Z^4+1)^4+3772*(x^4-x^2)^
(1/2)*RootOf(RootOf(2*_Z^8-2*_Z^4+1)^4+_Z^4-1)^3*x+4078*RootOf(RootOf(2*_Z^8-2*_Z^4+1)^4+_Z^4-1)*RootOf(2*_Z^8
-2*_Z^4+1)^4*x-3772*(x^4-x^2)^(1/4)*RootOf(RootOf(2*_Z^8-2*_Z^4+1)^4+_Z^4-1)^2*x^2+2527*RootOf(RootOf(2*_Z^8-2
*_Z^4+1)^4+_Z^4-1)*x^3-734*(x^4-x^2)^(3/4)-1463*RootOf(RootOf(2*_Z^8-2*_Z^4+1)^4+_Z^4-1)*x)/x/(2*x^2*RootOf(2*
_Z^8-2*_Z^4+1)^4-8*RootOf(2*_Z^8-2*_Z^4+1)^4+3*x^2+5))-1/4*ln((-16*RootOf(2*_Z^8-2*_Z^4+1)^9*x^3-1468*(x^4-x^2
)^(1/2)*RootOf(2*_Z^8-2*_Z^4+1)^7*x+64*RootOf(2*_Z^8-2*_Z^4+1)^9*x-6076*(x^4-x^2)^(1/4)*RootOf(2*_Z^8-2*_Z^4+1
)^6*x^2-402*RootOf(2*_Z^8-2*_Z^4+1)^5*x^3+3772*(x^4-x^2)^(3/4)*RootOf(2*_Z^8-2*_Z^4+1)^4+3772*(x^4-x^2)^(1/2)*
RootOf(2*_Z^8-2*_Z^4+1)^3*x+1088*RootOf(2*_Z^8-2*_Z^4+1)^5*x+2304*(x^4-x^2)^(1/4)*RootOf(2*_Z^8-2*_Z^4+1)^2*x^
2-2261*RootOf(2*_Z^8-2*_Z^4+1)*x^3-3038*(x^4-x^2)^(3/4)+399*RootOf(2*_Z^8-2*_Z^4+1)*x)/x/(2*x^2*RootOf(2*_Z^8-
2*_Z^4+1)^4-8*RootOf(2*_Z^8-2*_Z^4+1)^4-5*x^2+3))*RootOf(2*_Z^8-2*_Z^4+1)^5+1/8*ln((-16*RootOf(2*_Z^8-2*_Z^4+1
)^9*x^3-1468*(x^4-x^2)^(1/2)*RootOf(2*_Z^8-2*_Z^4+1)^7*x+64*RootOf(2*_Z^8-2*_Z^4+1)^9*x-6076*(x^4-x^2)^(1/4)*R
ootOf(2*_Z^8-2*_Z^4+1)^6*x^2-402*RootOf(2*_Z^8-2*_Z^4+1)^5*x^3+3772*(x^4-x^2)^(3/4)*RootOf(2*_Z^8-2*_Z^4+1)^4+
3772*(x^4-x^2)^(1/2)*RootOf(2*_Z^8-2*_Z^4+1)^3*x+1088*RootOf(2*_Z^8-2*_Z^4+1)^5*x+2304*(x^4-x^2)^(1/4)*RootOf(
2*_Z^8-2*_Z^4+1)^2*x^2-2261*RootOf(2*_Z^8-2*_Z^4+1)*x^3-3038*(x^4-x^2)^(3/4)+399*RootOf(2*_Z^8-2*_Z^4+1)*x)/x/
(2*x^2*RootOf(2*_Z^8-2*_Z^4+1)^4-8*RootOf(2*_Z^8-2*_Z^4+1)^4-5*x^2+3))*RootOf(2*_Z^8-2*_Z^4+1)-1/4*RootOf(Root
Of(2*_Z^8-2*_Z^4+1)^4+_Z^4-1)*ln(-(-16*RootOf(RootOf(2*_Z^8-2*_Z^4+1)^4+_Z^4-1)*RootOf(2*_Z^8-2*_Z^4+1)^8*x^3+
1468*(x^4-x^2)^(1/2)*RootOf(2*_Z^8-2*_Z^4+1)^4*RootOf(RootOf(2*_Z^8-2*_Z^4+1)^4+_Z^4-1)^3*x+64*RootOf(RootOf(2
*_Z^8-2*_Z^4+1)^4+_Z^4-1)*RootOf(2*_Z^8-2*_Z^4+1)^8*x-6076*(x^4-x^2)^(1/4)*RootOf(RootOf(2*_Z^8-2*_Z^4+1)^4+_Z
^4-1)^2*RootOf(2*_Z^8-2*_Z^4+1)^4*x^2+434*RootOf(RootOf(2*_Z^8-2*_Z^4+1)^4+_Z^4-1)*RootOf(2*_Z^8-2*_Z^4+1)^4*x
^3+3772*(x^4-x^2)^(3/4)*RootOf(2*_Z^8-2*_Z^4+1)^4+2304*(x^4-x^2)^(1/2)*RootOf(RootOf(2*_Z^8-2*_Z^4+1)^4+_Z^4-1
)^3*x-1216*RootOf(RootOf(2*_Z^8-2*_Z^4+1)^4+_Z^4-1)*RootOf(2*_Z^8-2*_Z^4+1)^4*x+3772*(x^4-x^2)^(1/4)*RootOf(Ro
otOf(2*_Z^8-2*_Z^4+1)^4+_Z^4-1)^2*x^2-2679*RootOf(RootOf(2*_Z^8-2*_Z^4+1)^4+_Z^4-1)*x^3-734*(x^4-x^2)^(3/4)+15
51*RootOf(RootOf(2*_Z^8-2*_Z^4+1)^4+_Z^4-1)*x)/x/(2*x^2*RootOf(2*_Z^8-2*_Z^4+1)^4-8*RootOf(2*_Z^8-2*_Z^4+1)^4+
3*x^2+5))*RootOf(2*_Z^8-2*_Z^4+1)^4+1/8*RootOf(RootOf(2*_Z^8-2*_Z^4+1)^4+_Z^4-1)*ln(-(-16*RootOf(RootOf(2*_Z^8
-2*_Z^4+1)^4+_Z^4-1)*RootOf(2*_Z^8-2*_Z^4+1)^8*x^3+1468*(x^4-x^2)^(1/2)*RootOf(2*_Z^8-2*_Z^4+1)^4*RootOf(RootO
f(2*_Z^8-2*_Z^4+1)^4+_Z^4-1)^3*x+64*RootOf(RootOf(2*_Z^8-2*_Z^4+1)^4+_Z^4-1)*RootOf(2*_Z^8-2*_Z^4+1)^8*x-6076*
(x^4-x^2)^(1/4)*RootOf(RootOf(2*_Z^8-2*_Z^4+1)^4+_Z^4-1)^2*RootOf(2*_Z^8-2*_Z^4+1)^4*x^2+434*RootOf(RootOf(2*_
Z^8-2*_Z^4+1)^4+_Z^4-1)*RootOf(2*_Z^8-2*_Z^4+1)^4*x^3+3772*(x^4-x^2)^(3/4)*RootOf(2*_Z^8-2*_Z^4+1)^4+2304*(x^4
-x^2)^(1/2)*RootOf(RootOf(2*_Z^8-2*_Z^4+1)^4+_Z^4-1)^3*x-1216*RootOf(RootOf(2*_Z^8-2*_Z^4+1)^4+_Z^4-1)*RootOf(
2*_Z^8-2*_Z^4+1)^4*x+3772*(x^4-x^2)^(1/4)*RootOf(RootOf(2*_Z^8-2*_Z^4+1)^4+_Z^4-1)^2*x^2-2679*RootOf(RootOf(2*
_Z^8-2*_Z^4+1)^4+_Z^4-1)*x^3-734*(x^4-x^2)^(3/4)+1551*RootOf(RootOf(2*_Z^8-2*_Z^4+1)^4+_Z^4-1)*x)/x/(2*x^2*Roo
tOf(2*_Z^8-2*_Z^4+1)^4-8*RootOf(2*_Z^8-2*_Z^4+1)^4+3*x^2+5))-1/8*RootOf(2*_Z^8-2*_Z^4+1)*ln((548*RootOf(2*_Z^8
-2*_Z^4+1)^9*x^3+6076*(x^4-x^2)^(1/2)*RootOf(2*_Z^8-2*_Z^4+1)^7*x-2192*RootOf(2*_Z^8-2*_Z^4+1)^9*x+6076*(x^4-x
^2)^(1/4)*RootOf(2*_Z^8-2*_Z^4+1)^6*x^2+4376*RootOf(2*_Z^8-2*_Z^4+1)^5*x^3+3772*(x^4-x^2)^(3/4)*RootOf(2*_Z^8-
2*_Z^4+1)^4-2304*(x^4-x^2)^(1/2)*RootOf(2*_Z^8-2*_Z^4+1)^3*x+306*RootOf(2*_Z^8-2*_Z^4+1)^5*x-2304*(x^4-x^2)^(1
/4)*RootOf(2*_Z^8-2*_Z^4+1)^2*x^2-2397*RootOf(2*_Z^8-2*_Z^4+1)*x^3-3038*(x^4-x^2)^(3/4)+423*RootOf(2*_Z^8-2*_Z
^4+1)*x)/x/(2*x^2*RootOf(2*_Z^8-2*_Z^4+1)^4-8*RootOf(2*_Z^8-2*_Z^4+1)^4-5*x^2+3))+1/8*RootOf(2*_Z^8-2*_Z^4+1)*
RootOf(RootOf(2*_Z^8-2*_Z^4+1)^4+_Z^4-1)*ln((-4*(x^4-x^2)^(1/2)*RootOf(RootOf(2*_Z^8-2*_Z^4+1)^4+_Z^4-1)^3*Roo
tOf(2*_Z^8-2*_Z^4+1)^3*x+4*(x^4-x^2)^(1/4)*RootOf(RootOf(2*_Z^8-2*_Z^4+1)^4+_Z^4-1)^2*RootOf(2*_Z^8-2*_Z^4+1)^
2*x^2-3*RootOf(RootOf(2*_Z^8-2*_Z^4+1)^4+_Z^4-1)*RootOf(2*_Z^8-2*_Z^4+1)*x^3+2*(x^4-x^2)^(3/4)+RootOf(RootOf(2
*_Z^8-2*_Z^4+1)^4+_Z^4-1)*RootOf(2*_Z^8-2*_Z^4+1)*x)/x/(x^2+1))+1/4*RootOf(RootOf(2*_Z^8-2*_Z^4+1)^4+_Z^4-1)*l
n((8*(x^4-x^2)^(1/2)*RootOf(RootOf(2*_Z^8-2*_Z^4+1)^4+_Z^4-1)^3*RootOf(2*_Z^8-2*_Z^4+1)^7*x-4*(x^4-x^2)^(1/2)*
RootOf(RootOf(2*_Z^8-2*_Z^4+1)^4+_Z^4-1)^3*RootOf(2*_Z^8-2*_Z^4+1)^3*x-6*RootOf(RootOf(2*_Z^8-2*_Z^4+1)^4+_Z^4
-1)*RootOf(2*_Z^8-2*_Z^4+1)^5*x^3-4*(x^4-x^2)^(1/4)*RootOf(RootOf(2*_Z^8-2*_Z^4+1)^4+_Z^4-1)^2*RootOf(2*_Z^8-2
*_Z^4+1)^2*x^2+2*RootOf(RootOf(2*_Z^8-2*_Z^4+1)^4+_Z^4-1)*RootOf(2*_Z^8-2*_Z^4+1)^5*x+3*RootOf(RootOf(2*_Z^8-2
*_Z^4+1)^4+_Z^4-1)*RootOf(2*_Z^8-2*_Z^4+1)*x^3+2*(x^4-x^2)^(3/4)-RootOf(RootOf(2*_Z^8-2*_Z^4+1)^4+_Z^4-1)*Root
Of(2*_Z^8-2*_Z^4+1)*x)/x/(x^2+1))*RootOf(2*_Z^8-2*_Z^4+1)^5-1/8*RootOf(RootOf(2*_Z^8-2*_Z^4+1)^4+_Z^4-1)*ln((8
*(x^4-x^2)^(1/2)*RootOf(RootOf(2*_Z^8-2*_Z^4+1)^4+_Z^4-1)^3*RootOf(2*_Z^8-2*_Z^4+1)^7*x-4*(x^4-x^2)^(1/2)*Root
Of(RootOf(2*_Z^8-2*_Z^4+1)^4+_Z^4-1)^3*RootOf(2*_Z^8-2*_Z^4+1)^3*x-6*RootOf(RootOf(2*_Z^8-2*_Z^4+1)^4+_Z^4-1)*
RootOf(2*_Z^8-2*_Z^4+1)^5*x^3-4*(x^4-x^2)^(1/4)*RootOf(RootOf(2*_Z^8-2*_Z^4+1)^4+_Z^4-1)^2*RootOf(2*_Z^8-2*_Z^
4+1)^2*x^2+2*RootOf(RootOf(2*_Z^8-2*_Z^4+1)^4+_Z^4-1)*RootOf(2*_Z^8-2*_Z^4+1)^5*x+3*RootOf(RootOf(2*_Z^8-2*_Z^
4+1)^4+_Z^4-1)*RootOf(2*_Z^8-2*_Z^4+1)*x^3+2*(x^4-x^2)^(3/4)-RootOf(RootOf(2*_Z^8-2*_Z^4+1)^4+_Z^4-1)*RootOf(2
*_Z^8-2*_Z^4+1)*x)/x/(x^2+1))*RootOf(2*_Z^8-2*_Z^4+1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {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(1/(x**4-x**2)**(1/4)/(x**8-1),x)

[Out]

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

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