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

Optimal. Leaf size=217 \[ \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )-\frac {1}{2} \sqrt {\frac {1}{10} \left (\sqrt {5}-1\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}} x}{\sqrt [4]{x^4-1}}\right )-\frac {1}{2} \sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{x^4-1}}\right )+\tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )-\frac {1}{2} \sqrt {\frac {1}{10} \left (\sqrt {5}-1\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}} x}{\sqrt [4]{x^4-1}}\right )-\frac {1}{2} \sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{x^4-1}}\right ) \]

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Rubi [A]  time = 0.38, antiderivative size = 233, normalized size of antiderivative = 1.07, number of steps used = 15, number of rules used = 7, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.206, Rules used = {6728, 240, 212, 206, 203, 1428, 377} \begin {gather*} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )-\frac {\sqrt [4]{\frac {1}{2} \left (3-\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} x}{\sqrt [4]{x^4-1}}\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x}{\sqrt [4]{x^4-1}}\right )}{2 \sqrt {5}}+\tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )-\frac {\sqrt [4]{\frac {1}{2} \left (3-\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} x}{\sqrt [4]{x^4-1}}\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x}{\sqrt [4]{x^4-1}}\right )}{2 \sqrt {5}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

ArcTan[x/(-1 + x^4)^(1/4)] - (((3 - Sqrt[5])/2)^(1/4)*ArcTan[((2/(3 + Sqrt[5]))^(1/4)*x)/(-1 + x^4)^(1/4)])/(2
*Sqrt[5]) - (((3 + Sqrt[5])/2)^(1/4)*ArcTan[(((3 + Sqrt[5])/2)^(1/4)*x)/(-1 + x^4)^(1/4)])/(2*Sqrt[5]) + ArcTa
nh[x/(-1 + x^4)^(1/4)] - (((3 - Sqrt[5])/2)^(1/4)*ArcTanh[((2/(3 + Sqrt[5]))^(1/4)*x)/(-1 + x^4)^(1/4)])/(2*Sq
rt[5]) - (((3 + Sqrt[5])/2)^(1/4)*ArcTanh[(((3 + Sqrt[5])/2)^(1/4)*x)/(-1 + x^4)^(1/4)])/(2*Sqrt[5])

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 1428

Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{r = Rt[b^2 -
4*a*c, 2]}, Dist[(2*c)/r, Int[(d + e*x^n)^q/(b - r + 2*c*x^n), x], x] - Dist[(2*c)/r, Int[(d + e*x^n)^q/(b +
r + 2*c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] &&  !IntegerQ[q]

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 (-1-x^4+x^8\right )} \, dx &=\int \left (\frac {2}{\sqrt [4]{-1+x^4}}+\frac {1}{\sqrt [4]{-1+x^4} \left (-1-x^4+x^8\right )}\right ) \, dx\\ &=2 \int \frac {1}{\sqrt [4]{-1+x^4}} \, dx+\int \frac {1}{\sqrt [4]{-1+x^4} \left (-1-x^4+x^8\right )} \, dx\\ &=2 \operatorname {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {2 \int \frac {1}{\sqrt [4]{-1+x^4} \left (-1-\sqrt {5}+2 x^4\right )} \, dx}{\sqrt {5}}-\frac {2 \int \frac {1}{\sqrt [4]{-1+x^4} \left (-1+\sqrt {5}+2 x^4\right )} \, dx}{\sqrt {5}}\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {1}{-1-\sqrt {5}-\left (1-\sqrt {5}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt {5}}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{-1+\sqrt {5}-\left (1+\sqrt {5}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt {5}}+\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )\\ &=\tan ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {3-\sqrt {5}}-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt {10}}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {3+\sqrt {5}}-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt {10}}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {3-\sqrt {5}}+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt {10}}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {3+\sqrt {5}}+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt {10}}\\ &=\tan ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} x}{\sqrt [4]{-1+x^4}}\right )}{2^{3/4} \sqrt {5} \sqrt [4]{3+\sqrt {5}}}-\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x}{\sqrt [4]{-1+x^4}}\right )}{2 \sqrt {5}}+\tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} x}{\sqrt [4]{-1+x^4}}\right )}{2^{3/4} \sqrt {5} \sqrt [4]{3+\sqrt {5}}}-\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x}{\sqrt [4]{-1+x^4}}\right )}{2 \sqrt {5}}\\ \end {align*}

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Mathematica [A]  time = 0.55, size = 239, normalized size = 1.10 \begin {gather*} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )-\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} x}{\sqrt [4]{x^4-1}}\right )}{5+\sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (3-\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x}{\sqrt [4]{x^4-1}}\right )}{\sqrt {5}-5}+\tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )-\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} x}{\sqrt [4]{x^4-1}}\right )}{5+\sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (3-\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x}{\sqrt [4]{x^4-1}}\right )}{\sqrt {5}-5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[x/(-1 + x^4)^(1/4)] - (((3 + Sqrt[5])/2)^(1/4)*ArcTan[((2/(3 + Sqrt[5]))^(1/4)*x)/(-1 + x^4)^(1/4)])/(5
+ Sqrt[5]) + (((3 - Sqrt[5])/2)^(1/4)*ArcTan[(((3 + Sqrt[5])/2)^(1/4)*x)/(-1 + x^4)^(1/4)])/(-5 + Sqrt[5]) +
ArcTanh[x/(-1 + x^4)^(1/4)] - (((3 + Sqrt[5])/2)^(1/4)*ArcTanh[((2/(3 + Sqrt[5]))^(1/4)*x)/(-1 + x^4)^(1/4)])/
(5 + Sqrt[5]) + (((3 - Sqrt[5])/2)^(1/4)*ArcTanh[(((3 + Sqrt[5])/2)^(1/4)*x)/(-1 + x^4)^(1/4)])/(-5 + Sqrt[5])

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IntegrateAlgebraic [A]  time = 0.71, size = 217, normalized size = 1.00 \begin {gather*} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \sqrt {\frac {1}{10} \left (-1+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-1+x^4}}\right )+\tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \sqrt {\frac {1}{10} \left (-1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-1+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[x/(-1 + x^4)^(1/4)] - (Sqrt[(-1 + Sqrt[5])/10]*ArcTan[(Sqrt[-1/2 + Sqrt[5]/2]*x)/(-1 + x^4)^(1/4)])/2 -
(Sqrt[(1 + Sqrt[5])/10]*ArcTan[(Sqrt[1/2 + Sqrt[5]/2]*x)/(-1 + x^4)^(1/4)])/2 + ArcTanh[x/(-1 + x^4)^(1/4)] -
(Sqrt[(-1 + Sqrt[5])/10]*ArcTanh[(Sqrt[-1/2 + Sqrt[5]/2]*x)/(-1 + x^4)^(1/4)])/2 - (Sqrt[(1 + Sqrt[5])/10]*Ar
cTanh[(Sqrt[1/2 + Sqrt[5]/2]*x)/(-1 + x^4)^(1/4)])/2

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fricas [B]  time = 0.52, size = 406, normalized size = 1.87 \begin {gather*} \frac {1}{10} \, \sqrt {10} \sqrt {\sqrt {5} + 1} \arctan \left (\frac {\sqrt {10} \sqrt {2} {\left (\sqrt {5} x - 5 \, x\right )} \sqrt {\sqrt {5} + 1} \sqrt {\frac {\sqrt {5} x^{2} + x^{2} + 2 \, \sqrt {x^{4} - 1}}{x^{2}}} - 2 \, \sqrt {10} {\left (x^{4} - 1\right )}^{\frac {1}{4}} \sqrt {\sqrt {5} + 1} {\left (\sqrt {5} - 5\right )}}{40 \, x}\right ) - \frac {1}{10} \, \sqrt {10} \sqrt {\sqrt {5} - 1} \arctan \left (\frac {\sqrt {10} \sqrt {2} {\left (\sqrt {5} x + 5 \, x\right )} \sqrt {\sqrt {5} - 1} \sqrt {\frac {\sqrt {5} x^{2} - x^{2} + 2 \, \sqrt {x^{4} - 1}}{x^{2}}} - 2 \, \sqrt {10} {\left (x^{4} - 1\right )}^{\frac {1}{4}} {\left (\sqrt {5} + 5\right )} \sqrt {\sqrt {5} - 1}}{40 \, x}\right ) - \frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {5} + 1} \log \left (\frac {\sqrt {10} \sqrt {5} x \sqrt {\sqrt {5} + 1} + 10 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {5} + 1} \log \left (-\frac {\sqrt {10} \sqrt {5} x \sqrt {\sqrt {5} + 1} - 10 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {5} - 1} \log \left (\frac {\sqrt {10} \sqrt {5} x \sqrt {\sqrt {5} - 1} + 10 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {5} - 1} \log \left (-\frac {\sqrt {10} \sqrt {5} x \sqrt {\sqrt {5} - 1} - 10 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \arctan \left (\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \log \left (\frac {x + {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \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)/(x^8-x^4-1),x, algorithm="fricas")

[Out]

1/10*sqrt(10)*sqrt(sqrt(5) + 1)*arctan(1/40*(sqrt(10)*sqrt(2)*(sqrt(5)*x - 5*x)*sqrt(sqrt(5) + 1)*sqrt((sqrt(5
)*x^2 + x^2 + 2*sqrt(x^4 - 1))/x^2) - 2*sqrt(10)*(x^4 - 1)^(1/4)*sqrt(sqrt(5) + 1)*(sqrt(5) - 5))/x) - 1/10*sq
rt(10)*sqrt(sqrt(5) - 1)*arctan(1/40*(sqrt(10)*sqrt(2)*(sqrt(5)*x + 5*x)*sqrt(sqrt(5) - 1)*sqrt((sqrt(5)*x^2 -
x^2 + 2*sqrt(x^4 - 1))/x^2) - 2*sqrt(10)*(x^4 - 1)^(1/4)*(sqrt(5) + 5)*sqrt(sqrt(5) - 1))/x) - 1/40*sqrt(10)*
sqrt(sqrt(5) + 1)*log((sqrt(10)*sqrt(5)*x*sqrt(sqrt(5) + 1) + 10*(x^4 - 1)^(1/4))/x) + 1/40*sqrt(10)*sqrt(sqrt
(5) + 1)*log(-(sqrt(10)*sqrt(5)*x*sqrt(sqrt(5) + 1) - 10*(x^4 - 1)^(1/4))/x) - 1/40*sqrt(10)*sqrt(sqrt(5) - 1)
*log((sqrt(10)*sqrt(5)*x*sqrt(sqrt(5) - 1) + 10*(x^4 - 1)^(1/4))/x) + 1/40*sqrt(10)*sqrt(sqrt(5) - 1)*log(-(sq
rt(10)*sqrt(5)*x*sqrt(sqrt(5) - 1) - 10*(x^4 - 1)^(1/4))/x) - arctan((x^4 - 1)^(1/4)/x) + 1/2*log((x + (x^4 -
1)^(1/4))/x) - 1/2*log(-(x - (x^4 - 1)^(1/4))/x)

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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)/(x^8-x^4-1),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

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maple [F]  time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {2 x^{8}-2 x^{4}-1}{\left (x^{4}-1\right )^{\frac {1}{4}} \left (x^{8}-x^{4}-1\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)/(x**8-x**4-1),x)

[Out]

Timed out

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