\(\int \frac {1-2 x^4+2 x^8}{\sqrt [4]{-1+x^4} (-2-x^4+2 x^8)} \, dx\) [1795]

Optimal result

Integrand size = 36, antiderivative size = 121 \[ \int \frac {1-2 x^4+2 x^8}{\sqrt [4]{-1+x^4} \left (-2-x^4+2 x^8\right )} \, dx=\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{4} \text {RootSum}\left [1-5 \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-2 \log (x)+2 \log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right )+3 \log (x) \text {$\#$1}^4-3 \log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-5 \text {$\#$1}+4 \text {$\#$1}^5}\&\right ] \]

[Out]

Unintegrable

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.91, number of steps used = 16, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6860, 246, 218, 212, 209, 385} \[ \int \frac {1-2 x^4+2 x^8}{\sqrt [4]{-1+x^4} \left (-2-x^4+2 x^8\right )} \, dx=\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4-1}}\right )-\frac {\sqrt [4]{2047-439 \sqrt {17}} \arctan \left (\frac {\sqrt [4]{\frac {2}{5+\sqrt {17}}} x}{\sqrt [4]{x^4-1}}\right )}{4 \sqrt {17}}-\frac {\sqrt [4]{2047+439 \sqrt {17}} \arctan \left (\frac {\sqrt [4]{5+\sqrt {17}} x}{\sqrt {2} \sqrt [4]{x^4-1}}\right )}{4 \sqrt {17}}+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )-\frac {\sqrt [4]{2047-439 \sqrt {17}} \text {arctanh}\left (\frac {\sqrt [4]{\frac {2}{5+\sqrt {17}}} x}{\sqrt [4]{x^4-1}}\right )}{4 \sqrt {17}}-\frac {\sqrt [4]{2047+439 \sqrt {17}} \text {arctanh}\left (\frac {\sqrt [4]{5+\sqrt {17}} x}{\sqrt {2} \sqrt [4]{x^4-1}}\right )}{4 \sqrt {17}} \]

[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 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 6860

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*} \text {integral}& = \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+\text {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{2} \text {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} \arctan \left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{17} \left (-17+11 \sqrt {17}\right ) \text {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 ) \text {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} \arctan \left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \sqrt {\frac {1}{34} \left (37-\sqrt {17}\right )} \text {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 )} \text {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 )} \text {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 )} \text {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} \arctan \left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {\sqrt [4]{2047-439 \sqrt {17}} \arctan \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}} \arctan \left (\frac {\sqrt [4]{5+\sqrt {17}} x}{\sqrt {2} \sqrt [4]{-1+x^4}}\right )}{4 \sqrt {17}}+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {\sqrt [4]{2047-439 \sqrt {17}} \text {arctanh}\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}} \text {arctanh}\left (\frac {\sqrt [4]{5+\sqrt {17}} x}{\sqrt {2} \sqrt [4]{-1+x^4}}\right )}{4 \sqrt {17}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.98 \[ \int \frac {1-2 x^4+2 x^8}{\sqrt [4]{-1+x^4} \left (-2-x^4+2 x^8\right )} \, dx=\frac {1}{4} \left (2 \left (\arctan \left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\text {arctanh}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )\right )-\text {RootSum}\left [1-5 \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-2 \log (x)+2 \log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right )+3 \log (x) \text {$\#$1}^4-3 \log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-5 \text {$\#$1}+4 \text {$\#$1}^5}\&\right ]\right ) \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Time = 7.68 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.86

method	result	size
pseudoelliptic	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{8}-5 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (3 \textit {\_R}^{4}-2\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{4}-1\right )^{\frac {1}{4}}}{x}\right )}{4 \textit {\_R}^{5}-5 \textit {\_R}}\right )}{4}+\frac {\ln \left (\frac {x +\left (x^{4}-1\right )^{\frac {1}{4}}}{x}\right )}{4}-\frac {\ln \left (\frac {\left (x^{4}-1\right )^{\frac {1}{4}}-x}{x}\right )}{4}-\frac {\arctan \left (\frac {\left (x^{4}-1\right )^{\frac {1}{4}}}{x}\right )}{2}\)	\(104\)

[In]

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

[Out]

1/4*sum((3*_R^4-2)*ln((-_R*x+(x^4-1)^(1/4))/x)/(4*_R^5-5*_R),_R=RootOf(2*_Z^8-5*_Z^4+1))+1/4*ln((x+(x^4-1)^(1/
4))/x)-1/4*ln(((x^4-1)^(1/4)-x)/x)-1/2*arctan((x^4-1)^(1/4)/x)

Fricas [C] (verification not implemented)

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

Time = 0.29 (sec) , antiderivative size = 514, normalized size of antiderivative = 4.25 \[ \int \frac {1-2 x^4+2 x^8}{\sqrt [4]{-1+x^4} \left (-2-x^4+2 x^8\right )} \, dx=\frac {1}{136} \, \sqrt {17} \sqrt {-\sqrt {439 \, \sqrt {17} + 2047}} \log \left (\frac {{\left (59 \, \sqrt {17} x - 155 \, x\right )} \sqrt {439 \, \sqrt {17} + 2047} \sqrt {-\sqrt {439 \, \sqrt {17} + 2047}} + 35152 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{136} \, \sqrt {17} \sqrt {-\sqrt {439 \, \sqrt {17} + 2047}} \log \left (-\frac {{\left (59 \, \sqrt {17} x - 155 \, x\right )} \sqrt {439 \, \sqrt {17} + 2047} \sqrt {-\sqrt {439 \, \sqrt {17} + 2047}} - 35152 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{136} \, \sqrt {17} \sqrt {-\sqrt {-439 \, \sqrt {17} + 2047}} \log \left (\frac {{\left (59 \, \sqrt {17} x + 155 \, x\right )} \sqrt {-439 \, \sqrt {17} + 2047} \sqrt {-\sqrt {-439 \, \sqrt {17} + 2047}} + 35152 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{136} \, \sqrt {17} \sqrt {-\sqrt {-439 \, \sqrt {17} + 2047}} \log \left (-\frac {{\left (59 \, \sqrt {17} x + 155 \, x\right )} \sqrt {-439 \, \sqrt {17} + 2047} \sqrt {-\sqrt {-439 \, \sqrt {17} + 2047}} - 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}{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 ) \]

[In]

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

[Out]

1/136*sqrt(17)*sqrt(-sqrt(439*sqrt(17) + 2047))*log(((59*sqrt(17)*x - 155*x)*sqrt(439*sqrt(17) + 2047)*sqrt(-s
qrt(439*sqrt(17) + 2047)) + 35152*(x^4 - 1)^(1/4))/x) - 1/136*sqrt(17)*sqrt(-sqrt(439*sqrt(17) + 2047))*log(-(
(59*sqrt(17)*x - 155*x)*sqrt(439*sqrt(17) + 2047)*sqrt(-sqrt(439*sqrt(17) + 2047)) - 35152*(x^4 - 1)^(1/4))/x)
+ 1/136*sqrt(17)*sqrt(-sqrt(-439*sqrt(17) + 2047))*log(((59*sqrt(17)*x + 155*x)*sqrt(-439*sqrt(17) + 2047)*sq
rt(-sqrt(-439*sqrt(17) + 2047)) + 35152*(x^4 - 1)^(1/4))/x) - 1/136*sqrt(17)*sqrt(-sqrt(-439*sqrt(17) + 2047))
*log(-((59*sqrt(17)*x + 155*x)*sqrt(-439*sqrt(17) + 2047)*sqrt(-sqrt(-439*sqrt(17) + 2047)) - 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*sqrt(17) + 2047)^(3/4) + 35152*(x^4 - 1)^(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) - 1/2*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)

Sympy [N/A]

Not integrable

Time = 134.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.31 \[ \int \frac {1-2 x^4+2 x^8}{\sqrt [4]{-1+x^4} \left (-2-x^4+2 x^8\right )} \, dx=\int \frac {2 x^{8} - 2 x^{4} + 1}{\sqrt [4]{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (2 x^{8} - x^{4} - 2\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.30 \[ \int \frac {1-2 x^4+2 x^8}{\sqrt [4]{-1+x^4} \left (-2-x^4+2 x^8\right )} \, dx=\int { \frac {2 \, x^{8} - 2 \, x^{4} + 1}{{\left (2 \, x^{8} - x^{4} - 2\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}} \,d x } \]

[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)

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.30 \[ \int \frac {1-2 x^4+2 x^8}{\sqrt [4]{-1+x^4} \left (-2-x^4+2 x^8\right )} \, dx=\int { \frac {2 \, x^{8} - 2 \, x^{4} + 1}{{\left (2 \, x^{8} - x^{4} - 2\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 6.11 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.29 \[ \int \frac {1-2 x^4+2 x^8}{\sqrt [4]{-1+x^4} \left (-2-x^4+2 x^8\right )} \, dx=\int -\frac {2\,x^8-2\,x^4+1}{{\left (x^4-1\right )}^{1/4}\,\left (-2\,x^8+x^4+2\right )} \,d x \]

[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)