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

Optimal. Leaf size=121 \[ -\frac {1}{4} \text {RootSum}\left [2 \text {$\#$1}^8-5 \text {$\#$1}^4+1\& ,\frac {-3 \text {$\#$1}^4 \log \left (\sqrt [4]{x^4-1}-\text {$\#$1} x\right )+3 \text {$\#$1}^4 \log (x)+2 \log \left (\sqrt [4]{x^4-1}-\text {$\#$1} x\right )-2 \log (x)}{4 \text {$\#$1}^5-5 \text {$\#$1}}\& \right ]+\frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )+\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4-1}}\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.60, antiderivative size = 231, normalized size of antiderivative = 1.91, number of steps used = 16, number of rules used = 6, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6728, 240, 212, 206, 203, 377} \begin {gather*} \frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )-\frac {\sqrt [4]{2047-439 \sqrt {17}} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {2}{5+\sqrt {17}}} x}{\sqrt [4]{x^4-1}}\right )}{4 \sqrt {17}}-\frac {\sqrt [4]{2047+439 \sqrt {17}} \tan ^{-1}\left (\frac {\sqrt [4]{5+\sqrt {17}} x}{\sqrt {2} \sqrt [4]{x^4-1}}\right )}{4 \sqrt {17}}+\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )-\frac {\sqrt [4]{2047-439 \sqrt {17}} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {2}{5+\sqrt {17}}} x}{\sqrt [4]{x^4-1}}\right )}{4 \sqrt {17}}-\frac {\sqrt [4]{2047+439 \sqrt {17}} \tanh ^{-1}\left (\frac {\sqrt [4]{5+\sqrt {17}} x}{\sqrt {2} \sqrt [4]{x^4-1}}\right )}{4 \sqrt {17}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

ArcTan[x/(-1 + x^4)^(1/4)]/2 - ((2047 - 439*Sqrt[17])^(1/4)*ArcTan[((2/(5 + Sqrt[17]))^(1/4)*x)/(-1 + x^4)^(1/
4)])/(4*Sqrt[17]) - ((2047 + 439*Sqrt[17])^(1/4)*ArcTan[((5 + Sqrt[17])^(1/4)*x)/(Sqrt[2]*(-1 + x^4)^(1/4))])/
(4*Sqrt[17]) + ArcTanh[x/(-1 + x^4)^(1/4)]/2 - ((2047 - 439*Sqrt[17])^(1/4)*ArcTanh[((2/(5 + Sqrt[17]))^(1/4)*
x)/(-1 + x^4)^(1/4)])/(4*Sqrt[17]) - ((2047 + 439*Sqrt[17])^(1/4)*ArcTanh[((5 + Sqrt[17])^(1/4)*x)/(Sqrt[2]*(-
1 + x^4)^(1/4))])/(4*Sqrt[17])

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 240

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 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 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {1-2 x^4+2 x^8}{\sqrt [4]{-1+x^4} \left (-2-x^4+2 x^8\right )} \, dx &=\int \left (\frac {1}{\sqrt [4]{-1+x^4}}+\frac {3-x^4}{\sqrt [4]{-1+x^4} \left (-2-x^4+2 x^8\right )}\right ) \, dx\\ &=\int \frac {1}{\sqrt [4]{-1+x^4}} \, dx+\int \frac {3-x^4}{\sqrt [4]{-1+x^4} \left (-2-x^4+2 x^8\right )} \, dx\\ &=\int \left (\frac {-1+\frac {11}{\sqrt {17}}}{\sqrt [4]{-1+x^4} \left (-1-\sqrt {17}+4 x^4\right )}+\frac {-1-\frac {11}{\sqrt {17}}}{\sqrt [4]{-1+x^4} \left (-1+\sqrt {17}+4 x^4\right )}\right ) \, dx+\operatorname {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{17} \left (-17+11 \sqrt {17}\right ) \int \frac {1}{\sqrt [4]{-1+x^4} \left (-1-\sqrt {17}+4 x^4\right )} \, dx-\frac {1}{17} \left (17+11 \sqrt {17}\right ) \int \frac {1}{\sqrt [4]{-1+x^4} \left (-1+\sqrt {17}+4 x^4\right )} \, dx\\ &=\frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{17} \left (-17+11 \sqrt {17}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-\sqrt {17}-\left (3-\sqrt {17}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{17} \left (17+11 \sqrt {17}\right ) \operatorname {Subst}\left (\int \frac {1}{-1+\sqrt {17}-\left (3+\sqrt {17}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )\\ &=\frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \sqrt {\frac {1}{34} \left (37-\sqrt {17}\right )} \operatorname {Subst}\left (\int \frac {1}{\sqrt {5+\sqrt {17}}-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \sqrt {\frac {1}{34} \left (37-\sqrt {17}\right )} \operatorname {Subst}\left (\int \frac {1}{\sqrt {5+\sqrt {17}}+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \sqrt {\frac {1}{34} \left (37+\sqrt {17}\right )} \operatorname {Subst}\left (\int \frac {1}{\sqrt {5-\sqrt {17}}-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \sqrt {\frac {1}{34} \left (37+\sqrt {17}\right )} \operatorname {Subst}\left (\int \frac {1}{\sqrt {5-\sqrt {17}}+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )\\ &=\frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {\sqrt [4]{2047-439 \sqrt {17}} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {2}{5+\sqrt {17}}} x}{\sqrt [4]{-1+x^4}}\right )}{4 \sqrt {17}}-\frac {\sqrt [4]{2047+439 \sqrt {17}} \tan ^{-1}\left (\frac {\sqrt [4]{5+\sqrt {17}} x}{\sqrt {2} \sqrt [4]{-1+x^4}}\right )}{4 \sqrt {17}}+\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {\sqrt [4]{2047-439 \sqrt {17}} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {2}{5+\sqrt {17}}} x}{\sqrt [4]{-1+x^4}}\right )}{4 \sqrt {17}}-\frac {\sqrt [4]{2047+439 \sqrt {17}} \tanh ^{-1}\left (\frac {\sqrt [4]{5+\sqrt {17}} x}{\sqrt {2} \sqrt [4]{-1+x^4}}\right )}{4 \sqrt {17}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.30, size = 223, normalized size = 1.84 \begin {gather*} \frac {1}{68} \left (34 \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )-\sqrt {17} \sqrt [4]{2047-439 \sqrt {17}} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {2}{5+\sqrt {17}}} x}{\sqrt [4]{x^4-1}}\right )-\sqrt {17} \sqrt [4]{2047+439 \sqrt {17}} \tan ^{-1}\left (\frac {\sqrt [4]{5+\sqrt {17}} x}{\sqrt {2} \sqrt [4]{x^4-1}}\right )+34 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )-\sqrt {17} \sqrt [4]{2047-439 \sqrt {17}} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {2}{5+\sqrt {17}}} x}{\sqrt [4]{x^4-1}}\right )-\sqrt {17} \sqrt [4]{2047+439 \sqrt {17}} \tanh ^{-1}\left (\frac {\sqrt [4]{5+\sqrt {17}} x}{\sqrt {2} \sqrt [4]{x^4-1}}\right )\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(34*ArcTan[x/(-1 + x^4)^(1/4)] - Sqrt[17]*(2047 - 439*Sqrt[17])^(1/4)*ArcTan[((2/(5 + Sqrt[17]))^(1/4)*x)/(-1
+ x^4)^(1/4)] - Sqrt[17]*(2047 + 439*Sqrt[17])^(1/4)*ArcTan[((5 + Sqrt[17])^(1/4)*x)/(Sqrt[2]*(-1 + x^4)^(1/4)
)] + 34*ArcTanh[x/(-1 + x^4)^(1/4)] - Sqrt[17]*(2047 - 439*Sqrt[17])^(1/4)*ArcTanh[((2/(5 + Sqrt[17]))^(1/4)*x
)/(-1 + x^4)^(1/4)] - Sqrt[17]*(2047 + 439*Sqrt[17])^(1/4)*ArcTanh[((5 + Sqrt[17])^(1/4)*x)/(Sqrt[2]*(-1 + x^4
)^(1/4))])/68

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.35, size = 121, normalized size = 1.00 \begin {gather*} \frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{4} \text {RootSum}\left [1-5 \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-2 \log (x)+2 \log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right )+3 \log (x) \text {$\#$1}^4-3 \log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-5 \text {$\#$1}+4 \text {$\#$1}^5}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[x/(-1 + x^4)^(1/4)]/2 + ArcTanh[x/(-1 + x^4)^(1/4)]/2 - RootSum[1 - 5*#1^4 + 2*#1^8 & , (-2*Log[x] + 2*
Log[(-1 + x^4)^(1/4) - x*#1] + 3*Log[x]*#1^4 - 3*Log[(-1 + x^4)^(1/4) - x*#1]*#1^4)/(-5*#1 + 4*#1^5) & ]/4

________________________________________________________________________________________

fricas [B]  time = 0.65, size = 456, normalized size = 3.77 \begin {gather*} -\frac {1}{34} \, \sqrt {17} {\left (439 \, \sqrt {17} + 2047\right )}^{\frac {1}{4}} \arctan \left (\frac {{\left (\sqrt {2} {\left (3 \, \sqrt {17} x - 7 \, x\right )} \sqrt {-\frac {{\left (\sqrt {17} x^{2} - 37 \, x^{2}\right )} \sqrt {439 \, \sqrt {17} + 2047} - 1352 \, \sqrt {x^{4} - 1}}{x^{2}}} - 52 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}} {\left (3 \, \sqrt {17} - 7\right )}\right )} {\left (439 \, \sqrt {17} + 2047\right )}^{\frac {1}{4}}}{2704 \, x}\right ) - \frac {1}{34} \, \sqrt {17} {\left (-439 \, \sqrt {17} + 2047\right )}^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (3 \, \sqrt {17} x + 7 \, x\right )} {\left (-439 \, \sqrt {17} + 2047\right )}^{\frac {1}{4}} \sqrt {\frac {{\left (\sqrt {17} x^{2} + 37 \, x^{2}\right )} \sqrt {-439 \, \sqrt {17} + 2047} + 1352 \, \sqrt {x^{4} - 1}}{x^{2}}} - 52 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}} {\left (3 \, \sqrt {17} + 7\right )} {\left (-439 \, \sqrt {17} + 2047\right )}^{\frac {1}{4}}}{2704 \, x}\right ) - \frac {1}{136} \, \sqrt {17} {\left (439 \, \sqrt {17} + 2047\right )}^{\frac {1}{4}} \log \left (\frac {{\left (59 \, \sqrt {17} x - 155 \, x\right )} {\left (439 \, \sqrt {17} + 2047\right )}^{\frac {3}{4}} + 35152 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{136} \, \sqrt {17} {\left (439 \, \sqrt {17} + 2047\right )}^{\frac {1}{4}} \log \left (-\frac {{\left (59 \, \sqrt {17} x - 155 \, x\right )} {\left (439 \, \sqrt {17} + 2047\right )}^{\frac {3}{4}} - 35152 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{136} \, \sqrt {17} {\left (-439 \, \sqrt {17} + 2047\right )}^{\frac {1}{4}} \log \left (\frac {{\left (59 \, \sqrt {17} x + 155 \, x\right )} {\left (-439 \, \sqrt {17} + 2047\right )}^{\frac {3}{4}} + 35152 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{136} \, \sqrt {17} {\left (-439 \, \sqrt {17} + 2047\right )}^{\frac {1}{4}} \log \left (-\frac {{\left (59 \, \sqrt {17} x + 155 \, x\right )} {\left (-439 \, \sqrt {17} + 2047\right )}^{\frac {3}{4}} - 35152 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \arctan \left (\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{4} \, \log \left (\frac {x + {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{4} \, \log \left (-\frac {x - {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/34*sqrt(17)*(439*sqrt(17) + 2047)^(1/4)*arctan(1/2704*(sqrt(2)*(3*sqrt(17)*x - 7*x)*sqrt(-((sqrt(17)*x^2 -
37*x^2)*sqrt(439*sqrt(17) + 2047) - 1352*sqrt(x^4 - 1))/x^2) - 52*(x^4 - 1)^(1/4)*(3*sqrt(17) - 7))*(439*sqrt(
17) + 2047)^(1/4)/x) - 1/34*sqrt(17)*(-439*sqrt(17) + 2047)^(1/4)*arctan(1/2704*(sqrt(2)*(3*sqrt(17)*x + 7*x)*
(-439*sqrt(17) + 2047)^(1/4)*sqrt(((sqrt(17)*x^2 + 37*x^2)*sqrt(-439*sqrt(17) + 2047) + 1352*sqrt(x^4 - 1))/x^
2) - 52*(x^4 - 1)^(1/4)*(3*sqrt(17) + 7)*(-439*sqrt(17) + 2047)^(1/4))/x) - 1/136*sqrt(17)*(439*sqrt(17) + 204
7)^(1/4)*log(((59*sqrt(17)*x - 155*x)*(439*sqrt(17) + 2047)^(3/4) + 35152*(x^4 - 1)^(1/4))/x) + 1/136*sqrt(17)
*(439*sqrt(17) + 2047)^(1/4)*log(-((59*sqrt(17)*x - 155*x)*(439*sqrt(17) + 2047)^(3/4) - 35152*(x^4 - 1)^(1/4)
)/x) - 1/136*sqrt(17)*(-439*sqrt(17) + 2047)^(1/4)*log(((59*sqrt(17)*x + 155*x)*(-439*sqrt(17) + 2047)^(3/4) +
35152*(x^4 - 1)^(1/4))/x) + 1/136*sqrt(17)*(-439*sqrt(17) + 2047)^(1/4)*log(-((59*sqrt(17)*x + 155*x)*(-439*s
qrt(17) + 2047)^(3/4) - 35152*(x^4 - 1)^(1/4))/x) - 1/2*arctan((x^4 - 1)^(1/4)/x) + 1/4*log((x + (x^4 - 1)^(1/
4))/x) - 1/4*log(-(x - (x^4 - 1)^(1/4))/x)

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to convert to real 1/4 Error: Bad Argument ValueUnable to convert to real 1/4 Error: Bad Argument Value

________________________________________________________________________________________

maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {2 x^{8}-2 x^{4}+1}{\left (x^{4}-1\right )^{\frac {1}{4}} \left (2 x^{8}-x^{4}-2\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________